mathematical nequalities i pplications a - narod.hr · horst martini marko matić ... volume 19,...

Publishe

d b

y

UDC 51 ISSN 1331-4343

CONTENTS

athematical nequalities pplications

MIA&

Zagreb, Croatia Volume 19, Number 4, October 2016

Vol 19, N

o 4 (2016) 1141-1440

Ravi P. AgarwalMartin BohnerJean-Christophe BourinYuri BuragoMaria A. Hernandez CifreZeev DitzianTamás ErdélyiKálmán GyoryFrank HansenDon HintonAndrea LaforgiaLászló LosoncziLech MaligrandaHorst MartiniMarko MatićRam N. Mohapatra

M. Sal MoslehianConstantin P. NiculescuBohumir OpicDonal O’ReganZsolt PálesLars Erik PerssonLubos PickIosif PinelisSimo PuntanenSaburou SaitohJavier SoriaHari M. SrivastavaStevo StevićKenneth B. StolarskyGeorge P. H. Styan

Ma

them

atic

al Ine

qua

lities &

Ap

plic

atio

ns

1141 Zsolt Pales and Pawe PasteczkaCharacterization of the Hardy property of means and the best Hardy constants

1159 Merve Avc Ard c, Ahmet Ocak Akdemir and Erhan SetNew Ostrowski like inequalities for GG -convex and GA -convex functions

1169 Josip Pecaric and Marjan PraljakHermite interpolation and inequalities involvingweighted averages of n -convex functions

1181 M. Adil Khan, S. Ivelic Bradanovic and J. PecaricGeneralizations of Sherman’s inequality by Hermite’s interpolating polynomial

1193 Bozidar Ivankovic – One proof of the Gheorghiu inequality1203 Shoshana Abramovich – Jensen, Holder, Minkowski, Jensen-Steffensen and

Slater-Pecaric inequalities derived through N -quasiconvexity1227 Laszlo Horvath and Josip Pecaric

Recursively de ned re nements of the integral form of Jensen’s inequality1247 Ravi P. Agarwal, Saad Ihsan Butt, Josip Pecaric and Ana Vukelic

Generalization of Popoviciu type Inequalities via Green’s function and Fink’s identity1257 Saad Ihsan Butt, Ljiljanka Kvesic and Josip Pecaric

Generalization of majorization theorem via Taylor’s formula1271 J. Baric, R. Jaksic and J. Pecaric

Converses of Jessen’s inequality on time scales II1287 Miros aw Adamek – On a problem connected with strongly convex functions1295 Maja Andric, Josip Pecaric and Ivan Peric

On weighted integral and discrete Opial-type inequalities1309 Saad Ihsan Butt, Khuram Ali Khan and Josip Pecaric

Popoviciu type inequalities via Hermite’s polynomial1319 Monika Nowicka and Alfred Witkowski

Applications of the Hermite-Hadamard inequality1335 Ozge Dalmanoglu and Mediha Orkcu

On convergence properties of gamma-Stancu operators based on q -integers1349 Gorana Aras-Gazic, Josip Pecaric and Ana Vukelic

Cauchy’s error representation of Hermite interpolating polynomial and related results1363 Josip Pecaric, Dora Pokaz and Marjan Praljak

Boas-type inequality for 3 -convex functions at a point1375 Edward Neuman – Inequalities and bounds for a certain bivariate elliptic mean1387 A. Matkovic – Generalization of the Jensen-Mercer inequality by Taylor’s polynomial1399 Neven Elezovic and Lenka Mihokovic – Asymptotic behavior of power means1413 Witold Jarczyk and Kazimierz Nikodem

On some properties of strictly convex functions1417 Ludmila Nikolova and Sanja Varosanec

Chebyshev-Gruss type inequalities on time scales via two linear isotonic functionals1429 M. Emin Ozdemir and Alper Ekinci

Generalized integral inequalities for convex functions

EDITORS-IN-CHIEF

MANAGING EDITOR ASSOCIATE EDITORS

EDITORIAL BOARD

Josip Pečarić Ivan Perić Sanja Varošanec

Neven Elezović Iva Franjić Marjan Praljak

MathematicalI nequalities

& Applications

Volume 19, Number 4 (2016) 1141–1440

ZAGREB, 2016.

CONTENTS

1141 Zsolt Pales and Paweł PasteczkaCharacterization of the Hardy property of means and the best Hardy constants

1159 Merve Avcı Ardıc, Ahmet Ocak Akdemir and Erhan SetNew Ostrowski like inequalities for GG -convex and GA -convex functions

1169 Josip Pecaric and Marjan PraljakHermite interpolation and inequalities involving weighted averages of n -convex functions




Recursively defined refinements of the integral form of Jensen’s inequality1247 Ravi P. Agarwal, Saad Ihsan Butt, Josip Pecaric and Ana Vukelic



Converses of Jessen’s inequality on time scales II1287 Mirosław Adamek – On a problem connected with strongly convex functions1295 Maja Andric, Josip Pecaric and Ivan Peric










Selected papers

MATHEMATICAL INEQUALITIES AND APPLICATIONS 2015

Mostar, Bosnia and Herzegovina

November 11–15, 2015

Conference on the occasion of 60th birthdays ofProfessors Neven Elezovic, Marko Matic and Ivan Peric

Special editors

Kazimierz NikodemConstantin P. Niculescu

Sanja Varosanec

MathematicalInequalities

& Applications

Volume 19, Number 4 (2016), 1141–1158 doi:10.7153/mia-19-84

CHARACTERIZATION OF THE HARDY PROPERTY

OF MEANS AND THE BEST HARDY CONSTANTS

ZSOLT PÁLES AND PAWEŁ PASTECZKA

Dedicated to the 60th birthdays of ProfessorsNeven Elezovic, Marko Matic and Ivan Peric

(Communicated by S. Varošanec)

Abstract. The aim of this paper is to characterize in broad classes of means the so-called Hardymeans, i.e., those means M :

⋃∞n=1 R

n+ → R+ that satisfy the inequality

∞

∑n=1

M(x1 , . . . ,xn) � C∞

∑n=1

xn

for all positive sequences (xn) with some finite positive constant C . One of the main resultsoffers a characterization of Hardy means in the class of symmetric, increasing, Jensen concaveand repetition invariant means and also a formula for the best constant C satisfying the aboveinequality.

1. Introduction

Hardy’s celebrated inequality (cf. [23], [24]) states that, for p > 1,

∞

∑n=1

(x1 + · · ·+ xn

n

)p�( p

p−1

)p ∞

∑n=1

xpn , (1.1)

for all nonnegative sequences (xn) .This inequality, in integral form was stated and proved in [23] but it was also

pointed out that this discrete form follows from the integral version. Hardy’s originalmotivation was to get a simple proof of Hilbert’s celebrated inequality. About the enor-mous literature concerning the history, generalizations and extensions of this inequality,we recommend four recent books [30], [31], [44], and [46] for the interested readers.

In this paper, we follow the approach in generalizing Hardy’s inequality of thepaper [62]. The main idea is to rewrite (1.1) in terms of means.

Mathematics subject classification (2010): 26D15.Keywords and phrases: Mean, Hardy mean, Hardy constant, Hardy inequality, quasi-arithmetic mean,

Kedlaya mean, Gini mean, Gaussian product of means.The research of the first author was supported by the Hungarian Scientific Research Fund (OTKA) Grant K-111651.

c© � � , ZagrebPaper MIA-19-84

1141

http://dx.doi.org/10.7153/mia-19-84

1142 Z. PÁLES AND P. PASTECZKA

First, replacing xn by x1/pn and p by 1/p , we get that

∞

∑n=1

(xp1 + · · ·+ xp

n

n

)1/p�( 1

1− p

)1/p ∞

∑n=1

xn (1.2)

for 0 < p < 1. This inequality was also established for p < 0 by Knopp [28]. Takingthe limit p → 0, the so-called Carleman inequality (cf. [10]) can also be derived:

∞

∑n=1

n√

x1 · · ·xn � e∞

∑n=1

xn. (1.3)

It is also important to note that the constants of the right hand sides of the above in-equalities are the smallest possible. For further developments and historical remarksconcerning inequality (1.3), we refer to the paper Pecaric–Stolarsky [49].

Now define for p ∈ R the p th power (or Hölder) mean of the positive numbersx1, . . . ,xn by

Pp(x1, . . . ,xn) :=

⎧⎨⎩(xp

1 + · · ·+ xpn

n

) 1p

if p �= 0,

n√

x1 · · ·xn if p = 0.

(1.4)

The power mean P1 is the arithmetic mean which will also be denoted by A in thesequel.

Observe that all of the above inequalities are particular cases of the following one

∞

∑n=1

M(x1, . . . ,xn) � C∞

∑n=1

xn, (1.5)

where M is a mean on R+ , that is, M is a real valued function defined on the set⋃∞n=1 R

n+ such that, for all n ∈ N , x1, . . . ,xn > 0,

min(x1, . . . ,xn) � M(x1, . . . ,xn) � max(x1, . . . ,xn).

In the sequel, a mean M will be called a Hardy mean if there exists a positive realconstant C such that (1.5) holds for all positive sequences x = (xn) . The smallestpossible extended real value C such that (1.5) is valid will be called the Hardy constantof M and denoted by H∞(M) . Due to the Hardy, Carleman, and Knopp inequalities,the p th power mean is a Hardy mean if p < 1. One can easily see that the arithmeticmean is not a Hardy mean, therefore the following result holds.

THEOREM 1.1. Let p ∈ R . Then, the power mean Pp is a Hardy mean if andonly if p < 1 . In addition, for p < 1 ,

H∞(Pp) =

{(1− p

)− 1p if p �= 0,

e if p = 0.

CHARACTERIZATION OF THE HARDY PROPERTY OF MEANS 1143

The notion of power means is generalized by the concept of quasi-arithmeticmeans (cf. [24]): If I ⊆R is an interval and f : I →R is a continuous strictly monotonicfunction then the quasi-arithmetic mean M f :

⋃∞n=1 In → R is defined by

M f (x1, . . . ,xn) := f−1(

f (x1)+ · · ·+ f (xn)n

), x1, . . . ,xn ∈ I. (1.6)

By taking f as a power function or a logarithmic function on I = R+ , the resultingquasi-arithmetic mean is a power mean. It is well-known that Hölder means are theonly homogeneous quasi-arithmetic means (cf. [24], [61], [48]).

The following result which completely characterizes the Hardy means amongquasi-arithmetic means is due to Mulholland [45].

THEOREM 1.2. Let f : R+ → R be a continuous strictly monotonic function.Then, the quasi-arithmetic mean M f is a Hardy mean if and only if there exist con-stants p < 1 and C > 0 such that, for all n ∈ N and x1, . . . ,xn > 0 ,

M f (x1, . . . ,xn) � CPp(x1, . . . ,xn).

In 1938 Gini introduced another extension of power means: For p,q∈R , the Ginimean Gp,q of the variables x1, . . . ,xn > 0 is defined as follows:

Gp,q(x1, . . . ,xn) :=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(xp1 + · · ·+ xp

n

xq1 + · · ·+ xq

n

) 1p−q

if p �= q,

exp

(xp1 ln(x1)+ · · ·+ xp

n ln(xn)xp1 + · · ·+ xp

n

)if p = q.

(1.7)

Clearly, in the particular case q = 0, the mean Gp,q reduces to the p th Hölder meanPp . It is also obvious that Gp,q = Gq,p . A common generalization of quasi-arithmeticmeans and Gini means can be obtained in terms of two arbitrary real functions. Thesemeans were introduced by Bajraktarevic [2], [3] in 1958. Let I ⊆ R be an intervaland let f ,g : I → R be continuous functions such that g is positive and f/g is strictlymonotone. Define the Bajraktarevic mean B f ,g :

⋃∞n=1 In → R by

B f ,g(x1, . . . ,xn) :=( f

g

)−1(

f (x1)+ · · ·+ f (xn)g(x1)+ · · ·+g(xn)

), x1, . . . ,xn ∈ I.

One can check that B f ,g is a mean on I . In the particular case g ≡ 1, the mean B f ,g

reduces to M f , that is, the class of Bajraktarevic means is more general than that ofthe quasi-arithmetic means. By taking power functions, we can see that the Gini meansalso belong to this class. It is a remarkable result of Aczél and Daróczy [1] that thehomogeneous means among the Bajraktarevic means defined on I = R+ are exactlythe Gini means.

Finally, we recall the concept of the most general means considered in this paper,the notion of the deviation means introduced by Daróczy [12] in 1972. A functionE : I × I → R is called a deviation function on I if E(x,x) = 0 for all x ∈ I and the

function y �→ E(x,y) is continuous and strictly decreasing on I for each fixed x ∈ I .The E -deviation mean or Daróczy mean of some values x1, . . . ,xn ∈ I is now definedas the unique solution y of the equation

E(x1,y)+ · · ·+E(xn,y) = 0

and is denoted by DE(x1, . . . ,xn) . It is immediate to see that the arithmetic deviationA(x,y) = x−y generates the arithmetic mean. More generally, if E : I× I →R is of theform E(x,y) := f (x)−g(x)

( fg

)(y) for some continuous function f ,g : I →R such that

g is positive and f/g is strictly monotone then DE = B f ,g . Thus, Hölder means, quasi-arithmetic means, Gini means and Bajraktarevic means are particular Daróczy means.The class of deviation means was slightly generalized to the class of quasi-deviationmeans and this class was completely characterized by Páles in [51].

The following result, which gives necessary and also sufficient conditions for theHardy property of deviation means was established by Páles and Persson [62] in 2004.

THEOREM 1.3. Let E : R+ ×R+ → R be a deviation on R+ . If DE is a Hardymean, then there exists a positive constant C such that

DE(x1, . . . ,xn) � CA(x1, . . . ,xn)

holds for all n ∈ N and x1, . . . ,xn > 0 and there is no positive constant C∗ such that

C∗A(x1, . . . ,xn) � DE(x1, . . . ,xn)

be valid on the same domain. Conversely, if

DE(x1, . . . ,xn) � CPp(x1, . . . ,xn)

is satisfied with a parameter p < 1 and a positive constant C , then DE is a Hardymean.

As a corollary of the previous result, necessary and also sufficient conditions forthe Hardy property were established in the class of Gini means in the same paper.

THEOREM 1.4. Let p,q ∈ R . If Gp,q is a Hardy mean, then

min(p,q) � 0 and max(p,q) � 1.

Conversely, ifmin(p,q) � 0 and max(p,q) < 1,

then Gp,q is a Hardy mean.

It has been an open problem since 2004 whether the second condition was a nec-essary and sufficient condition for the Hardy property and also the best Hardy constantwas to be determined.

The necessary and sufficient condition for the Hardy property of Gini means wasfinally found by Pasteczka [47] in 2015. The key was the following general necessarycondition for the Hardy property.

LEMMA 1.5. Assume that M :⋃∞

n=1 Rn+ → R+ is a Hardy mean. Then, for all

positive non-�1 sequences (xn) ,

liminfn→∞

x−1n M(x1, . . . ,xn) < ∞.

Applying this necessary condition in the class of Gini means with the harmonicsequence xn := 1

n , Pasteczka [47] obtained the following characterization of the Hardyproperty for Gini means.

THEOREM 1.6. Let p,q ∈ R . Then Gp,q is a Hardy mean if and only if

min(p,q) � 0 and max(p,q) < 1.

There was no progress, however, in establishing the Hardy constant of the Ginimeans. There was only an upper estimate obtained by Páles and Persson in [62].

Motivated by all these preliminaries, the purpose of this paper is twofold:— To find (in terms of easy-to-check properties) a large subclass of Hardy means.— To obtain a formula for the Hardy constant in that subclass of means.

2. Means and their basic properties

For investigating the Hardy property of means, we recall several relevant notions.Let I ⊆ R be an interval and let M :

⋃∞n=1 In → I be an arbitrary mean.

We say that M is symmetric, (strictly) increasing, and Jensen convex (concave)if, for all n ∈ N , the n -variable restriction M|In is a symmetric, (strictly) increasing ineach of its variables, and Jensen convex (concave) on In , respectively. If I = R+ , wecan analogously define the notion of homogeneity of M .

The mean M is called repetition invariant if, for all n,m∈N and (x1, . . . ,xn)∈ In ,the following identity is satisfied

M(x1, . . . ,x1︸︷︷︸m-times

, . . . ,xn, . . . ,xn︸︷︷︸m-times

) = M(x1, . . . ,xn).

The mean M is strict if for any n � 2 and any non-constant vector (x1, . . . ,xn) ∈In ,

min(x1, . . . ,xn) < M(x1, . . . ,xn) < max(x1, . . . ,xn).

The mean M is said to be min -diminishing if, for any n � 2 and any non-nonstantvector (x1, . . . ,xn) ∈ In ,

M(x1, . . . ,xn,min(x1, . . . ,xn)) < M(x1, . . . ,xn).

It is easy to check that quasi-arithmetic means are symmetric, strictly increasing, repe-tition invariant, strict and min-diminishing. More generally, deviation means are sym-metric, repetition invariant, strict and min-diminishing (cf. [51]). The increasingnessof a deviation mean DE is equivalent to the increasingness of the deviation E in itsfirst variable. The Jensen concavity/convexity of quasi-arithmetic and also of deviation


means can be characterized by the concavity/convexity conditions on the generatingfunctions. All these characterizations are consequences of the general results obtainedin a series of papers by Losonczi [33, 34, 36, 35, 37, 38] (for Bajraktarevic means) andby Daróczy [14, 11, 12, 15, 16] and Páles [50, 52, 53, 54, 55, 56, 57, 58, 59, 60] (for(quasi-)deviation means).

2.1. Kedlaya means

The notion of a Kedlaya mean that we introduce below turns out to be indispens-able for the investigation of Hardy means. A mean M :

⋃∞n=1 In → I is called a Kedlaya

mean if, for all n ∈ N and (x1, . . . ,xn) ∈ In ,

M(x1)+M(x1,x2)+ · · ·+M(x1, . . . ,xn)n

� M

(x1,

x1 + x2

2, . . . ,

x1 + · · ·+ xn

n

). (2.1)

The motivation for the above terminology comes from the papers [26, 27] by Kedlaya,where he proved that the geometric mean satisfies the inequality (2.1), i.e., it is a Ked-laya mean. The next result provides a sufficient condition in order that a mean be aKedlaya mean.

THEOREM 2.1. Every symmetric, Jensen concave and repetition invariant meanis a Kedlaya mean.

Proof. Let M :⋃∞

n=1 In → I be a symmetric, Jensen concave and repetition in-variant mean. Fix n ∈ N and (x1, . . . ,xn) ∈ In . Adopting Kedlaya’s original proof, for(i, j,k) ∈ {1, . . . ,n}3 , we define

ak(i, j) : = (n−1)! ·(

n− ij− k

)(i−1k−1

)/(n−1j−1

)

=(n− i)!(n− j)!(i−1)!( j−1)!

(n− i− j + k)!(i− k)!( j− k)!(k−1)!.

To provide the corectness of this definition we assume that m! = ∞ for negative integersm (it is a natural extension of gamma function). Then, according to [26], we have thefollowing properties:

(1) ak(i, j) � 0 for all i, j, k ;

(2) ak(i, j) ∈ N∪{0} for all i, j, k ;

(3) ak(i, j) = 0 for k > min(i, j) ;

(4) ak(i, j) = ak( j, i) for all i, j, k ;

(5) ∑nk=1 ak(i, j) = (n−1)! for all i, j ;

(6) ∑ni=1 ak(i, j) =

{n!/ j for k � j,

0 for k > j.


Let us construct a matrix A of size n!×n! divided into n2 blocks (Ai, j)i, j∈{1,...,n}of size (n−1)!× (n−1)! .

The first row of each block Ai, j contains the number k exactly ak(i, j) times fork ∈ {1, . . . ,n} ; this could be done by (5). The subsequent rows are all cyclic permuta-tions of the first one. In this way each row and each column of Ai, j contains the numberk exactly ak(i, j) times.

Now, let cp(k) denote the occurrence of the number k appearing in the p th rowof A . Then, by (4), cp(k) is equal to the number of occurrences of k in the p th columnof A .

We are going to calculate cp(k) . The p th row has a nonempty intersection withthe block Ai, j if

i =⌊

p−1(n−1)!

⌋+1 =: b(p).

Whence, applying property (6), we get

cp(k) =n

∑i=1

ak(i,b(p)) =

⎧⎨⎩n!/b(p) for k � b(p),

0 for k > b(p).

Now, let us consider the matrix A′ obtained from A by replacing k �→ xk for k ∈{1, . . . ,n} . We will calculate the mean value of the elements of A′ in two differentways. First, we calculate the mean M of each column of A′ . By the Jensen concavityof M , the arithmetic mean of the results so obtained does not exceed the result of calcu-lating arithmetic mean of each row of A′ and then taking the M mean of the resultingvector of length n! . Whence, using the symmetry and the repetition invariance of M ,we obtain

1n!

((n−1)!M(x1)+ (n−1)!M(x1,x2)+ · · ·+(n−1)!M(x1,x2, . . . ,xn)

)� M

(x1,

x1 + x2

2, . . . ,

x1 + x2 + · · ·+ xn

n

)which simplifies to the inequality (2.1) to be proved. �

COROLLARY 2.2. If, in addition to the assumptions of Theorem 2.1, M is alsoincreasing and I = R+ , then

M(x1)+M(x1,x2)+ · · ·+M(x1, . . . ,xn)

� n ·M(

x1 + · · ·+ xn,x1 + · · ·+ xn

2, . . . ,

x1 + · · ·+ xn

n

).

2.2. Gaussian product

The Gaussian product of means is a broad extension of Gauss’ idea of the arith-metic-geometric mean. In 1800 (this year is due to [64]) he proposed the followingtwo-term recursion:

xn+1 =xn + yn

2, yn+1 =

√xnyn, n = 0,1, . . . ,


where x0 and y0 are positive numbers. Gauss [20, p. 370] proved that both (xn)∞n=1 and

(yn)∞n=1 converge to a common limit, which is called arithmetic-geometric mean of the

initial values x0 and y0 . J. M. Borwein and P. B. Borwein [9] extended some earlierideas [19, 32, 63] and generalized this iteration to a vector of continuous, strict means ofan arbitrary length. For several recent results about Gaussian product of means see thepapers by Baják and Páles [4, 5, 6, 7], by Daróczy and Páles [13, 17, 18], by Głazowska[21, 22], by Matkowski [39, 40, 41, 42], and by Matkowski and Páles [43].

Given N ∈ N and a vector (M1, . . . ,MN) of means defined on a common interval Iand having values in I (i.e. Mi :

⋃∞n=1 In → I for every i ∈ {1, . . . ,N} ), let us introduce

the mapping M :⋃∞

n=1 In → IN by

M(v) := (M1(v),M2(v), . . . ,MN(v)), v ∈∞⋃

n=1

In.

Whenever, for every i ∈ {1, . . . ,N} and every v ∈ ⋃∞n=1 In , the limit limk→∞[Mk(v)]i

exists and does not depend on i , then the value of this limit will be called the Gaussianproduct of (M1, . . . ,MN) evaluated at v . We will denote this limit by M⊗(v) . It iswell-known that the Gaussian product can equivalently be defined as a unique functionsatisfying the following two properties:

(i) M⊗ ◦M(v) = M⊗(v) for all v ∈⋃∞n=1 In ,

(ii) min(v) � M⊗(v) � max(v) for all v ∈⋃∞n=1 In .

Frequently, whenever each of the means Mi , i ∈ {1, . . . ,N} has a certain property,then M⊗ inherits this property. The lemma below (in view of Theorem 2.1) is its veryuseful exemplification.

LEMMA 2.3. Let I be an interval, N ∈ N , and let (M1, . . . ,MN) :⋃∞

n=1 In → IN .If, for each i ∈ {1, . . . ,N} , Mi is symmetric/homogeneous/repetition invariant/increas-ing and Jensen concave/convex, then so is their Gaussian product M⊗ .

Proof. The first four properties are naturally inherited by all of the functions[Mk]i , for k ∈ N , i ∈ {1, . . . ,N} and, finally, by their pointwise limit. The verifica-tion of the statement about the Jensen concavity is just a little bit more sophisticated.In fact, the idea presented below could also be adapted to the remaining properties.

Assume that M1, . . . ,MN are increasing and Jensen concave. We will prove thatM⊗ is Jensen concave. Let x(0) , y(0) be the equidimensional vectors and m(0) =12(x(0) + y(0)) . Let

x(k+1) = M(x(k)), y(k+1) = M(y(k)), m(k+1) = M(m(k)), k ∈ N.

We are going to prove that

[m(k)]i � 12 [x(k) + y(k)]i for any i ∈ {1, . . . ,N} and k ∈ N. (2.2)


Obviously, this holds for k = 0. Let us assume that (2.2) holds for some k ∈ N and anyi . Then, by the increasingness and Jensen concavity of Mi ,

[m(k+1)]i = Mi(m(k)) � Mi

(12(x(k) + y(k))

)� 1

2

(Mi(x(k))+Mi(y(k))

)= 1

2

([x(k+1)]i +[y(k+1)]i

)= 1

2 [x(k+1) + y(k+1)]i.

Upon taking the limit k → ∞ , one gets

M⊗

(x(0) + y(0)

2

)= M⊗(m(0)) � 1

2

(M⊗(x(0))+M⊗(y(0))

),

which proves that M⊗ is Jensen concave, indeed. �

3. Main Results

In the sequel, let I ⊆ R be a nondegenerate interval such that inf I = 0. We willdenote by �1(I) the collection of all sequences x = (xn)∞

n=1 such that, for all n ∈ N ,xn ∈ I and ‖x‖1 := ∑∞

n=1 xn is convergent, i.e., x ∈ �1 .Recall (slightly extending the definition) that, for a given mean M :

⋃∞n=1 In → I ,

the constant H∞(M) is the smallest nonnegative extended real number, called the Hardyconstant of M , such that

∞

∑n=1

M(x1, . . . ,xn) � H∞(M)∞

∑n=1

xn, (xn)∞n=1 ∈ �1(I). (3.1)

If H∞(M) is finite, then we say that M is a Hardy mean. Given also n ∈ N , we defineHn(M) to be the smallest nonnegative number such that

M(x1)+ . . .+M(x1, . . . ,xn) � Hn(M)(x1 + · · ·+ xn), (x1, . . . ,xn) ∈ In. (3.2)

Due to the mean value property of M , for n∈N , we easily obtain that 1 � Hn(M) � n .The sequence

(Hn(M)

)∞n=1 will be called the Hardy sequence of M .

Several estimates of the Hardy sequences for power means were given during theyears. For example Kaluza and Szego [25] proved Hn(Pp) � 1

n(exp(1/n)−1) ·H∞(Pp)

for p ∈ [0,1) and n∈ N . Moreover it is known [24, p.267] that Hn(P0) � (1+ 1n )n for

all n ∈ N .The basic properties of the Hardy sequence are established in the following

PROPOSITION 3.1. For every mean M :⋃∞

n=1 In → I , its Hardy sequence is non-decreasing and

limn→∞

Hn(M) = H∞(M). (3.3)

Proof. To verify the nondecreasingness of the Hardy sequence of M , let (x1, . . . ,xn)in In and ε ∈ I be arbitrary. Applying inequality (3.2) to the sequence (x1, . . . ,xn,ε) ∈In+1, we obtain

M(x1)+ · · ·+M(x1, . . . ,xn) � M(x1)+ · · ·+M(x1, . . . ,xn)+M(x1, . . . ,xn,ε)� Hn+1(M)(x1 + · · ·+ xn + ε).

Upon taking the limit ε → 0, it follows that

M(x1)+ · · ·+M(x1, . . . ,xn) � Hn+1(M)(x1 + · · ·+ xn)

for all (x1, . . . ,xn) ∈ In . Hence Hn(M) � Hn+1(M) .To prove (3.3), we will show first that Hn(M)� H∞(M) for all n∈N . If H∞(M)=

∞ then this inequality is obvious, hence we may assume that M is a Hardy mean. Fixn∈N and (x1, . . . ,xn)∈ In and choose ε ∈ I arbitrarily. Applying (3.1) to the sequence(x1, . . . ,xn,

ε2 , ε

4 , ε8 , . . .) ∈ �1(I) , one gets

M(x1)+ · · ·+M(x1, . . . ,xn)� M(x1)+ · · ·+M(x1, . . . ,xn)+M(x1, . . . ,xn,

ε2 )+M(x1, . . . ,xn,

ε2 , ε

4 )+ · · ·� H∞(M)(x1 + · · ·+ xn + ε

2 + ε4 + · · ·)

= H∞(M)(x1 + · · ·+ xn + ε).

Upon passing the limit ε → 0, we get

M(x1)+ · · ·+M(x1, . . . ,xn) � H∞(M)(x1 + · · ·+ xn),

which implies Hn(M) � H∞(M) . Using this inequality, we have also proved that in(3.3) � holds instead of equality.

To prove the reversed inequality in (3.3), let (xn)∞n=1 ∈ �1(I) be arbitrary. Then,

for all n � k , we have that

M(x1)+ · · ·+M(x1, . . . ,xn) � Hn(M)(x1 + · · ·+ xn) � Hk(M)(x1 + · · ·+ xn)

Now taking the limit as k → ∞ , we obtain that

M(x1)+ · · ·+M(x1, . . . ,xn) � limk→∞

Hk(M) · (x1 + · · ·+ xn)

holds for all n ∈ N . Finally taking the limit as n → ∞ , it follows that M satisfies

∞

∑n=1

M(x1, . . . ,xn) � limk→∞

Hk(M)∞

∑n=1

xn

which yields that the reversed inequality in (3.3) is also true. �

In what follows, we show that the inequality (3.1) is strict in a broad class ofmeans.

PROPOSITION 3.2. Let I ⊆R+ and M :⋃∞

n=1 In → I . If M is a min-diminishing,increasing and repetition invariant Hardy mean, then

∞

∑n=1

M(x1, . . . ,xn) < H∞(M)∞

∑n=1

xn, (xn)∞n=1 ∈ �1(I).

Proof. Let x = (xn)∞n=1 ∈ �1(I) be arbitrary. If xl < xk for some l < k then, for

the sequence

x′n =

⎧⎪⎨⎪⎩

xn n /∈ {k, l},xk n = l,

xl n = k,

we haveM(x1, . . . ,xn) = M(x′1, . . . ,x

′n) for n < l or n � k,

M(x1, . . . ,xn) � M(x′1, . . . ,x′n) for n ∈ {l, . . . ,k−1}.

Therefore

M(x1)+ · · ·+M(x1, . . . ,xn)+ · · · � M(x′1)+ . . .+M(x′1, . . . ,x′n)+ · · · .

Whence we may assume that x is non-increasing.Let x = (x1,x1, . . . ,xn,xn, . . .) . Then, by the repetition invariance and the min-

diminishing property of M , we get

M(x1, . . . ,xn) = M(x1, . . . , x2n),M(x1, . . . ,xn) = M(x1, . . . , x2n−1) if x1 = xn,

M(x1, . . . ,xn) < M(x1, . . . , x2n−1) if x1 �= xn.

Since xn → 0 as n → ∞ , hence x1 �= xn holds for some n . Therefore

2 ·∞

∑n=1

M(x1, . . . ,xn) <∞

∑n=1

M(x1, . . . , xn) � H∞(M)∞

∑n=1

xn = 2H∞(M)∞

∑n=1

xn.

This completes the proof of the proposition. �The next result offers a fundamental lower estimate for the Hardy constant of a

mean.

THEOREM 3.3. Let M :⋃∞

n=1 In → I be a mean. Then, for all sequences (xn)∞n=1

in I that does not belong to �1 ,

liminfn→∞

x−1n M(x1, . . . ,xn) � H∞(M). (3.4)

Proof. Assume, on the contrary, that

H∞(M) < liminfn→∞

x−1n M(x1, . . . ,xn).

Then, there exists ε > 0 and n0 such that, for all n � n0 ,

(1+ ε)H∞(M)xn < M(x1, . . . ,xn). (3.5)

Choose n1 > n0 such thatn0

∑n=1

xn � εn1

∑n=n0+1

xn (3.6)

Thus, using (3.5), Proposition 3.1, and finally (3.6), we obtain

n1

∑n=n0+1

(1+ ε)H∞(M)xn <n1

∑n=n0+1

M(x1, . . . ,xn) �n1

∑n=1

M(x1, . . . ,xn)

� Hn1(M)n1

∑n=1

xn � H∞(M)n1

∑n=1

xn � (1+ ε)H∞(M)n1

∑n=n0+1

xn.

This contradiction validates (3.4). �The main result of our paper is contained in the following theorem.

THEOREM 3.4. Let M :⋃∞

n=1 Rn+ → R+ be an increasing, symmetric, repetition

invariant, and Jensen concave mean. Then

H∞(M) = supy>0

liminfn→∞

ny·M( y

1,y2, . . . ,

yn

). (3.7)

As a trivial consequence of the above result, M is a Hardy mean if and only if thenumber H∞(M) given in (3.7) is finite.

Proof. For the proof of the theorem, denote

C := supy>0

liminfn→∞

ny·M( y

1,y2, . . . ,

yn

).

The inequality H∞(M) � C is simply a consequence of Theorem 3.3.To show the reversed inequality, we may assume that C is finite. Fix x ∈ �1(R+)

and denote y := ‖x‖1 . Then there exists a sequence (nk) , nk → ∞ such that

nk ·M( y

1,y2, . . . ,

ynk

)� (C+ 1

k )y, k ∈ N.

By the increasingness of M and by the obvious inequality x1 + · · ·+ xnk � y , the previ-ous inequality yields

nk ·M(

x1 + · · ·+ xnk ,x1 + · · ·+ xnk

2, . . . ,

x1 + · · ·+ xnk

nk

)�(C+ 1

k

)y, k ∈ N.

Therefore, in view of Corollary 2.2, we obtain

M(x1)+M(x1,x2)+ · · ·+M(x1, . . . ,xnk) � (C+ 1k )y, k ∈ N.

Upon passing the limit k → ∞ , one gets

∞

∑n=1

M(x1, . . . ,xn) � Cy = C‖x‖1 .

This completes the proof of inequality H∞(M) � C . �


COROLLARY 3.5. If, in addition to the assumptions of Theorem 3.4, M is alsohomogeneous, then

H∞(M) = limn→∞

n ·M(1, 12 , . . . , 1

n

).

Proof. In view of the previous theorem we only need to prove that the limit of thesequence (pn) exists (possible infinite), where

pn := n ·M(1, 12 , . . . , 1

n

).

For, it suffices to show that this sequence is nondecreasing. Fix n ∈ N . Let us considerthe two vectors u,v of dimension n(n+1) defined by

u := (n, . . . ,n︸︷︷︸n+1

, n2 , . . . , n

2︸︷︷︸n+1

, . . . , nn−1 , . . . , n

n−1︸︷︷︸n+1

,1, . . . ,1︸︷︷︸n+1

);

v := (n+1, . . . ,n+1︸︷︷︸n

, n+12 , . . . , n+1

2︸︷︷︸n

, . . . , n+1n , . . . , n+1

n︸︷︷︸n

,1, . . . ,1︸︷︷︸n

).

By the homogeneity and repetition invariance of M , we have that M(u) = pn andM(v) = pn+1 . Divide vectors u and v into n+1 parts of dimension n :

u(i) := ( ni , . . . ,

ni︸︷︷︸

i

, ni+1 , . . . , n

i+1︸︷︷︸n−i

), i = 0, . . . ,n;

v(i) := ( n+1i+1 , . . . , n+1

i+1︸︷︷︸n

), i = 0, . . . ,n.

For i � 1, each element ni appears (n− i+1) times in u(i−1) and i times in u(i) , that

is, (n+1) times altogether. Therefore, the arithmetic mean of u(i) , denoted by A(u(i)) ,is equal to n+1

i+1 for i = 1, . . . ,n and A(u(0)) = n .

Let u(i)k , for k = 1, . . . ,n! and i = 0, . . . ,n , denote the vectors that are obtained

from all possible permutations of the components of u(i) . Observe that

(u(0),v(1), . . . ,v(n)) =1n!

n!

∑k=1

(u(0)k , . . . ,u(n)

k ).

Then, by the increasingness, Jensen concavity and symmetry of the mean M , we obtain

pn+1 = M(v) = M(v(0),v(1), . . . ,v(n)) � M(u(0),v(1), . . . ,v(n))

� 1n!

n!

∑k=1

M(u(0)k , . . . ,u(n)

k ) = M(u(0), . . . ,u(n)) = M(u) = pn.

This proves that (pn) is non-deceasing and, therefore it has a (possibly infinite) limit. �

4. Applications

In this section we demonstrate the consequences of our results for Gini means andalso for the Gaussian product of symmetric, homogeneous, increasing, Jensen concaveand repetition invariant means, in particular, the Gaussian product of Hölder means.

4.1. Gini means

Gini means are symmetric and repetition invariant and min-diminishing (first twoproperties are simple while the third one was proved in [51]). Moreover, by the resultsof Losonczi [34, 35], the Gini mean Gp,q is increasing and Jensen concave if and onlyif pq � 0 and min(p,q) � 0 � max(p,q) � 1, respectively. In particular it implies thatHölder mean Pp is Jensen concave if and only if p � 1.

In view of Theorem 1.6, we have the characterization of pairs (p,q) such thatGp,q is a Hardy mean. In order to calculate the Hardy constant of Gini means usingCorollary 3.5, we need to establish the following result.

LEMMA 4.1. Let p,q ∈ (−∞,1) . Then

limn→∞

n ·Gp,q(1, 1

2 , . . . , 1n

)=

⎧⎪⎪⎨⎪⎪⎩( 1−q

1− p

) 1p−q

if p �= q,

e1

1−p if p = q.

Proof. For every s ∈ (−1,∞) , one has

limn→∞

1n

n

∑i=1

( in

)s=∫ 1

0xsdx =

11+ s

.

Using this equality, for p, q < 1, p �= q , we simply obtain

limn→∞

n·Gp,q(1, 1

2 , . . . , 1n

)= lim

n→∞n ·(

1+2−p +3−p + · · ·+n−p

1+2−q +3−q + · · ·+n−q

) 1p−q

= limn→∞

⎛⎝ 1

n

[(1n

)−p +(

2n

)−p +(

3n

)−p + · · ·+ (n−1n

)−p +1]

1n

[(1n

)−q +(

2n

)−q +(

3n

)−q + · · ·+ (n−1n

)−q +1]⎞⎠

1p−q

=(

1−q1− p

) 1p−q

.

The proof for the case p = q < 1 is analogous. �Using this lemma and the properties that are mentioned just before, we obtain the

following

COROLLARY 4.2. Let p,q ∈ R , min(p,q) � 0 � max(p,q) < 1 . Then

H∞(Gp,q) =

⎧⎪⎨⎪⎩(

1−q1− p

) 1p−q

p �= q,

e p = q = 0.

Proof. Due to the assumption min(p,q) � 0 � max(p,q) < 1 and in view of theresults of Losonczi [34, 35], the Gini mean Gp,q is increasing and Jensen concave.Furthermore, Gp,q is symmetric, homogeneous, and repetition invariant. Therefore, byCorollary 3.5 and Lemma 4.1, we have

H∞(Gp,q) = limn→∞

n ·Gp,q(1, 1

2 , . . . , 1n

)=

⎧⎪⎨⎪⎩(

1−q1− p

) 1p−q

p �= q,

e p = q = 0,

which was to be proved. �

4.2. Gaussian product

PROPOSITION 4.3. Let N ∈ N and let M1, . . . ,MN :⋃∞

n=1 Rn+ → R+ be symmet-

ric, homogeneous, increasing, Jensen concave and repetition invariant means. If Mi isHardy for each i ∈ {1, . . . ,N} , then so is their Gaussian product M⊗ and

H∞ (M⊗) = M⊗(H∞(M1), . . . ,H∞(MN)

). (4.1)

Proof. In view of Lemma 2.3, the Gaussian product M⊗ is a symmetric, homoge-neous, increasing, Jensen concave and repetition invariant mean. The Jensen concavityand the local boundedness by the Bernstein–Doetsch Theorem implies that M⊗ is con-cave and therefore it is also continuous (see [8], [29]). Thus, by Corollary 3.5, wehave

H∞(M⊗) = limn→∞

n ·M⊗(1, 1

2 , . . . , 1n

)= lim

n→∞n ·M⊗

(M1(1, 1

2 , . . . , 1n), . . . ,MN(1, 1

2 , . . . , 1n ))

= limn→∞

M⊗(nM1(1, 1

2 , . . . , 1n), . . . ,nMN(1, 1

2 , . . . , 1n))

= M⊗(

limn→∞

nM1(1, 12 , . . . , 1

n ), . . . , limn→∞

nMN(1, 12 , . . . , 1

n))

= M⊗ (H∞(M1), . . . ,H∞(MN)) ,

which proves formula (4.1). �

COROLLARY 4.4. Let N ∈ N and (λ1, . . . ,λN) ∈ RN then the Gaussian product

P⊗ of the Hölder means Pλ1, . . . ,PλN

is a Hardy mean if and only if max1�k�N λk < 1 .Furthermore, in this case,

H∞ (P⊗) = P⊗(H∞(Pλ1

), . . . ,H∞(PλN)). (4.2)

Proof. The first part of the statement of the above Corollary was proved in [47] byPasteczka. If λk < 1, then Pλk

is a Jensen concave mean, therefore (4.2) is a particularcase of (4.1). �

For example, for the geometric-harmonic mean P−1 ⊗P0 , i.e., for the Gaussianproduct of the harmonic mean P−1 and the geometric mean P0 , we get

H∞(P−1⊗P0) = (P−1⊗P0)(H∞(P−1),H∞(P0)) = (P−1⊗P0)(2,e) ≈ 2,318.

RE F ER EN C ES

[1] J. ACZÉL AND Z. DARÓCZY, Über verallgemeinerte quasilineare Mittelwerte, die mit Gewichtsfunk-tionen gebildet sind, Publ. Math. Debrecen, 10: 171–190, 1963.

[2] M. BAJRAKTAREVIC, Sur une équation fonctionnelle aux valeurs moyennes, Glasnik Mat.-Fiz. As-tronom. Društvo Mat. Fiz. Hrvatske Ser. II, 13: 243–248, 1958.

[3] M. BAJRAKTAREVIC, Über die Vergleichbarkeit der mit Gewichtsfunktionen gebildeten Mittelwerte,Studia Sci. Math. Hungar., 4: 3–8, 1969.

[4] SZ. BAJÁK AND ZS. PÁLES, Computer aided solution of the invariance equation for two-variableGini means, Comput. Math. Appl., 58: 334–340, 2009.

[5] SZ. BAJÁK AND ZS. PÁLES, Invariance equation for generalized quasi-arithmetic means, Aequa-tiones Math., 77: 133–145, 2009.

[6] SZ. BAJÁK AND ZS. PÁLES, Computer aided solution of the invariance equation for two-variableStolarsky means, Appl. Math. Comput., 216 (11): 3219–3227, 2010.

[7] SZ. BAJÁK AND ZS. PÁLES, Solving invariance equations involving homogeneous means with thehelp of computer, Appl. Math. Comput., 219 (11): 6297–6315, 2013.

[8] F. BERNSTEIN AND G. DOETSCH, Zur Theorie der konvexen Funktionen, Math. Ann., 76 (4): 514–526, 1915.

[9] J. M. BORWEIN AND P. B. BORWEIN, Pi and the AGM, Canadian Mathematical Society Series ofMonographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1987, A study in analyticnumber theory and computational complexity, A Wiley-Interscience Publication.

[10] T. CARLEMAN, Sur les fonctions quasi-analitiques, Conférences faites au cinquième congrès desmathématiciens scandinaves, Helsinki, page 181–196, 1932.

[11] Z. DARÓCZY, A general inequality for means, Aequationes Math., 7 (1): 16–21, 1971.[12] Z. DARÓCZY, Über eine Klasse von Mittelwerten, Publ. Math. Debrecen, 19: 211–217 (1973), 1972.[13] Z. DARÓCZY, Functional equations involving means and Gauss compositions of means, Nonlinear

Anal., 63 (5–7): e417–e425, 2005.[14] Z. DARÓCZY AND L. LOSONCZI, Über den Vergleich von Mittelwerten, Publ. Math. Debrecen, 17:

289–297 (1971), 1970.[15] Z. DARÓCZY AND ZS. PÁLES, On comparison of mean values, Publ. Math. Debrecen, 29 (1–2):

107–115, 1982.[16] Z. DARÓCZY AND ZS. PÁLES, Multiplicative mean values and entropies, In Functions, series, opera-

tors, Vol. I, II (Budapest, 1980), page 343–359, North-Holland, Amsterdam, 1983.[17] Z. DARÓCZY AND ZS. PÁLES, Gauss-composition of means and the solution of the Matkowski-Sutô

problem, Publ. Math. Debrecen, 61 (1–2): 157–218, 2002.[18] Z. DARÓCZY AND ZS. PÁLES, The Matkowski-Sutô problem for weighted quasi-arithmetic means,

Acta Math. Hungar., 100 (3): 237–243, 2003.[19] D. M. E. FOSTER AND G. M. PHILLIPS, The arithmetic-harmonic mean, Math. Comp., 42 (165):

183–191, 1984.[20] C. F. GAUSS, Nachlass: Aritmetisch-geometrisches Mittel, In Werke 3 (Göttingem 1876), page 357–

402, Königliche Gesellschaft der Wissenschaften, 1818.[21] D. GŁAZOWSKA, A solution of an open problem concerning Lagrangian mean-type mappings, Cent.

Eur. J. Math., 9 (5): 1067–1073, 2011.[22] D. GŁAZOWSKA, Some Cauchy mean-type mappings for which the geometric mean is invariant, J.

Math. Anal. Appl., 375 (2): 418–430, 2011.


[23] G. H. HARDY, Note on a theorem of Hilbert concerning series of positive terms, Proc. London Math.Soc., 23 (2), 1925.

[24] G. H. HARDY, J. E. LITTLEWOOD, AND G. PÓLYA, Inequalities, Cambridge University Press, Cam-bridge, 1934 (first edition), 1952 (second edition).

[25] T. KALUZA AND G. SZEGO, Über Reihen mit lauter positiven Gliedern. J. London Math. Soc.,2:266–272, 1927.

[26] K. S. KEDLAYA, Proof of a mixed arithmetic-mean, geometric-mean inequality, Amer. Math.Monthly, 101 (4): 355–357, 1994.

[27] K. S. KEDLAYA, Notes: A Weighted Mixed-Mean Inequality, Amer. Math. Monthly, 106 (4): 355–358,1999.

[28] K. KNOPP, Über Reihen mit positiven Gliedern, J. London Math. Soc., 3: 205–211, 1928.[29] M. KUCZMA, An Introduction to the Theory of Functional Equations and Inequalities, volume 489 of

Prace Naukowe Uniwersytetu Slaskiego w Katowicach, Panstwowe Wydawnictwo Naukowe – Uniw-ersytet Slaski, Warszawa–Kraków–Katowice, 1985. 2nd edn. (ed. by A. Gilányi), Birkhäuser, Basel,2009.

[30] A. KUFNER, L. MALIGRANDA, AND L. E. PERSSON, The Hardy Inequality: About Its History andSome Related Results, Vydavatelsky servis, 2007.

[31] A. KUFNER AND L.-E. PERSSON, Integral Inequalities with Weights, Matematický ústav AVCR,Prague, 1990.

[32] D. H. LEHMER, On the compounding of certain means, J. Math. Anal. Appl., 36: 183–200, 1971.[33] L. LOSONCZI, Über den Vergleich von Mittelwerten die mit Gewichtsfunktionen gebildet sind, Publ.

Math. Debrecen, 17: 203–208 (1971), 1970.[34] L. LOSONCZI, Subadditive Mittelwerte, Arch. Math. (Basel), 22: 168–174, 1971.[35] L. LOSONCZI, Subhomogene Mittelwerte, Acta Math. Acad. Sci. Hungar., 22: 187–195, 1971.[36] L. LOSONCZI, Über eine neue Klasse von Mittelwerten, Acta Sci. Math. (Szeged), 32: 71–81, 1971.[37] L. LOSONCZI, General inequalities for nonsymmetric means, Aequationes Math., 9: 221–235, 1973.[38] L. LOSONCZI, Inequalities for integral mean values, J. Math. Anal. Appl., 61 (3): 586–606, 1977.[39] J. MATKOWSKI, Iterations of mean-type mappings and invariant means, Ann. Math. Sil., (13): 211–

226, 1999, European Conference on Iteration Theory (Muszyna-Złockie, 1998).[40] J. MATKOWSKI, On iteration semigroups of mean-type mappings and invariant means, Aequationes

Math., 64 (3): 297–303, 2002.[41] J. MATKOWSKI,Lagrangian mean-type mappings for which the arithmetic mean is invariant, J. Math.

Anal. Appl., 309 (1): 15–24, 2005.[42] J. MATKOWSKI, Iterations of the mean-type mappings and uniqueness of invariant means, Annales

Univ. Sci. Budapest., Sect. Comp., 41: 145–158, 2013.[43] J. MATKOWSKI AND ZS. PÁLES, Characterization of generalized quasi-arithmetic means, Acta Sci.

Math. (Szeged), 81 (3–4): 447–456, 2015.[44] D. S. MITRINOVIC, J. E. PECARIC, AND A. M. FINK, Inequalities Involving Functions and Their

Integrals and Derivatives, volume 53 of Mathematics and its Applications (East European Series),Kluwer Academic Publishers Group, Dordrecht, 1991.

[45] P. MULHOLLAND, On the generalization of Hardy’s inequality, J. London Math. Soc., 7: 208–214,1932.

[46] B. OPIC AND A. KUFNER, Hardy-type Inequalities, volume 219 of Pitman Research Notes in Math-ematics, Longman Scientific & Technical, Harlow, 1990.

[47] P. PASTECZKA, On negative results concerning Hardy means, Acta Math. Hungar., 146 (1): 98–106,2015.

[48] P. PASTECZKA, Scales of quasi-arithmetic means determined by an invariance property, J. DifferenceEqu. Appl., 21 (8): 742–755, 2015.

[49] J. E. PECARIC AND K. B. STOLARSKY, Carleman’s inequality: history and new generalizations,Aequationes Math., 61 (1–2): 49–62, 2001.

[50] ZS. PÁLES, A generalization of the Minkowski inequality, J. Math. Anal. Appl., 90 (2): 456–462,1982.

[51] ZS. PÁLES, Characterization of quasideviation means, Acta Math. Acad. Sci. Hungar., 40 (3–4): 243–260, 1982.


[52] ZS. PÁLES, Inequalities for homogeneous means depending on two parameters, In E. F. Beckenbachand W. Walter, editors, General Inequalities, 3 (Oberwolfach, 1981), volume 64 of International Seriesof Numerical Mathematics, page 107–122. Birkhäuser, Basel, 1983.

[53] ZS. PÁLES, On complementary inequalities, Publ. Math. Debrecen, 30 (1–2): 75–88, 1983.[54] ZS. PÁLES, On Hölder-type inequalities, J. Math. Anal. Appl., 95 (2): 457–466, 1983.[55] ZS. PÁLES, Inequalities for comparison of means, In W. Walter, editor, General Inequalities, 4 (Ober-

wolfach, 1983), volume 71 of International Series of Numerical Mathematics, page 59–73, Birkhäuser,Basel, 1984.

[56] ZS. PÁLES, Ingham Jessen’s inequality for deviation means, Acta Sci. Math. (Szeged), 49 (1–4): 131–142, 1985.

[57] ZS. PÁLES, On the characterization of quasi-arithmetic means with weight function, AequationesMath., 32 (2–3): 171–194, 1987.

[58] ZS. PÁLES, General inequalities for quasideviation means, Aequationes Math., 36 (1): 32–56, 1988.[59] ZS. PÁLES, On a Pexider-type functional equation for quasideviation means, Acta Math. Hungar., 51

(1–2): 205–224, 1988.[60] ZS. PÁLES, On homogeneous quasideviation means, Aequationes Math., 36 (2-3): 132–152, 1988.[61] ZS. PÁLES, Nonconvex functions and separation by power means, Math. Inequal. Appl., 3 (2): 169–

176, 2000.[62] ZS. PÁLES AND L.-E. PERSSON, Hardy type inequalities for means, Bull. Austr. Math. Soc., 70 (3):

521–528, 2004.[63] I. J. SCHOENBERG, Mathematical time exposures, Mathematical Association of America, Washing-

ton, DC, 1982.[64] G. TOADER AND S. TOADER, Greek means and the arithmetic-geometric mean, RGMIA Mono-

graphs, Victoria University, 2005.

(Received January 3, 2016) Zsolt PálesInstitute of Mathematics, University of Debrecen

Egyetem tér 1, 4032 Debrecen, Hungarye-mail: [email protected]

Paweł PasteczkaInstitute of Mathematics

Pedagogical University of CracowPodchorazych str 2, 30-084 Cracow, Poland

e-mail: [email protected]

Mathematical Inequalities & [email protected]

& Applications


NEW OSTROWSKI LIKE INEQUALITIES FOR

GG–CONVEX AND GA–CONVEX FUNCTIONS

MERVE AVCI ARDIC, AHMET OCAK AKDEMIR AND ERHAN SET

(Communicated by S. Varosanec)

Abstract. In this paper, we established some Ostrowski like integral inequalities for functionswhose derivatives of absolute values are GG -convex and GA -convex functions via a new integralidentity. General results are obtained using the weighted Montgomery identity. Also, particularresults for the weight function w(t) = 1

t logb/a are given.

1. Introduction

We will start with the definiton of convexity that has utilization in all branches ofmathematics and has several applications in mathematical analysis, optimization andstatictics.

The function f : I ⊂R → R is a convex function on an interval I , if the inequality

f (tx+(1− t)y) � t f (x)+ (1− t) f (y)

holds for all x,y ∈ I and t ∈ [0,1] .Let f : I ⊂ [0,∞)→R be a differentiablemapping on I◦ , the interior of the interval

I, such that f ′ ∈ L[a,b] where a,b ∈ I with a < b. If | f ′(x)| � M , then the followinginequality holds

∣∣∣∣ f (x)− 1b−a

∫ b

af (u)du

∣∣∣∣� Mb−a

[(x−a)2 +(b− x)2

2

].

This inequality is well known in the literature as the Ostrowski inequality.In [4], Niculescu mentioned the following considerable definitions:The GG-convex functions (called also multiplicatively convex functions) are those

functions f : I → J ( I, J are subintervals of (0,∞)) such that

x,y ∈ I and λ ∈ [0,1] =⇒ f(x1−λ yλ

)� f (x)1−λ f (y)λ . (1)

Mathematics subject classification (2010): 26D15, 26A51, 26E60, 41A55.Keywords and phrases: GG -convex functions, GA -convex functions, the Holder inequality, the

weighted Montgomery identity.


1159

1160 M. AVCI ARDIC, A. OCAK AKDEMIR AND E. SET

The class of all GA-convex functions is constituted by all functions f : I → R (definedon subintervals of (0,∞)) for which

x,y ∈ I and λ ∈ [0,1] =⇒ f(x1−λ yλ

)� (1−λ ) f (x)+ λ f (y) . (2)

Besides, recall that the condition of GA-convexity is x2 f ′′ + x f ′ � 0 which implies alltwice differentiable nondecreasing convex functions are also GA-convex.

For recent results, generalizations, improvements and counterparts see the papers[3], [4], [6], [7], [8], [9] and references therein.

In [2], the authors have mentioned the weighted Montgomery identity as following(see also [1] and [5]):

f (x) =∫ b

aw(t) f (t)dt +

∫ b

aPw(x,t) f ′(t)dt, (3)

where f : [a,b] → R is differentiable on [a,b] , f ′ : [a,b] → R is integrable on [a,b]and w : [a,b] → [0,∞) is some normalized weight function, i.e. an integrable functionsatisfying

∫ ba w(t)dt = 1, W (t) =

∫ ta w(x)dx for t ∈ [0,1] . The weighted Peano kernel

is

Pw(x,t) =

{W (t), a � t � x

W (t)−1, x < t � b.

For the uniform weight function w(t) = 1b−a , t ∈ [a,b] , (3) reduces to the Montgomery

identity,

f (x) =1

b−a

∫ b

af (t)dt +

∫ b

aP(x,t) f ′(t)dt, (4)

where

P(x,t) =

⎧⎪⎨⎪⎩

t−ab−a

, a � t � x

t−bb−a

, x < t � b.

In [2], Aljinovic and Pecaric have given a discrete analogue of the weighted Mont-gomery identity for mappings of two variables and have proved some new discrete Os-trowski type inequalities.

The main aim of this paper is to prove some new integral inequalities for GG-convex and GA-convex functions by using a new integral identity.

2. New inequalities for GG- and GA-convex functions

We will give a new integral identity which is emboided in the following lemma toobtain our results. The proof of identity is based on using the weighted montgomeryidentity.

NEW OSTROWSKI LIKE INEQUALITIES FOR GG-CONVEX AND GA-CONVEX FUNCTIONS 1161

LEMMA 1. Let I ⊂ (0,∞) be an interval, a,b∈ Io , a < b. Let w be a nonnegativeintegrable function on [a,b] such that

∫ ba w(x)dx = 1 and let W (t) =

∫ ta w(x)dx. Let

f : I → R be a function differentiable on Io. Then for x ∈ [a,b]

f (x)−∫ b

aw(t) f (t)dt = log

xaI(f ′,a,x;W

)+ log

bxI(f ′,x,b;W −1

)holds, where I (F,v,u;Q) =

∫ 10 Q(u1−τvτ)F (u1−τvτ)u1−τvτdτ , provided that all inte-

grals exist.

Proof. Using the weighted Montgomery identity we get

f (x)−∫ b

aw(t) f (t)dt =

∫ b

aP(w,t) f ′(t)dt

=∫ x

aW (t) f ′(t)dt +

∫ b

x(W (t)−1) f ′(t)dt

=∫ 1

0W(x1−τaτ) f ′

(x1−τaτ)x1−τaτ log

xadτ

+∫ 1

0

(W (b1−τxτ)−1

)f ′(b1−τxτ)b1−τxτ log

bxdτ

= logxaI(f ′,a,x;W

)+ log

bxI(f ′,x,b;W −1

)where in the first integral we use substitution t = x1−τaτ , and in the second integral weuse substitution t = b1−τxτ .

REMARK 1. If we choose w(u) = 1u logb/a in Lemma 1, we get the following

equality:

logba

f (x)−∫ b

a

f (u)u

du

= log2 xa

∫ 1

0(1− τ)x1−τaτ f ′

(x1−τaτ)dτ − log2 b

x

∫ 1

0τb1−τxτ f ′

(b1−τxτ)dτ.

A new inequality for GG-convex functions is given in the following theorem.

THEOREM 1. Let I ⊂ (0,∞) be an interval, a,b ∈ Io , a < b. Let w be a nonneg-ative integrable function on [a,b] such that


∫ ta w(x)dx.

Let f : I → R be a function differentiable on Io. If | f ′|q is GG-convex function on[a,b] for some q > 1, then for all x ∈ [a,b] , following inequality holds


∣∣∣∣ f (x)−∫ b

aw(t) f (t)dt

∣∣∣∣� log

1q

xa

(∫ x

aW (t)dt

) 1p(

x | f ′(x)|qlogAa,x

[−W (x)+

∫ x

aw(t)A

logt/xloga/xa,x dt

]) 1q

+ log1q

bx

(∫ b

x(1−W(t))dt

) 1p(

b | f ′(b)|qlogAx,b

[(1−W(x))Ax,b −

∫ b

xw(t)A

log t/blogx/bx,b dt

]) 1q

where 1p = 1− 1

q and Av,u =v| f ′|q(v)u| f ′|q(u) , provided that all integrals exist.

Proof. From Lemma 1 and the Holder inequality, we get∣∣∣∣ f (x)−∫ b

aw(t) f (t)dt

∣∣∣∣ (5)

� logxaI(∣∣ f ′∣∣ ,a,x;W

)+ log

bxI(∣∣ f ′∣∣ ,x,b;1−W

)� log

xa

(∫ 1

0W(x1−τaτ)x1−τaτdτ

) 1p(∫ 1

0W(x1−τaτ)∣∣ f ′∣∣q (x1−τaτ)x1−τaτdτ

) 1q

+ logbx

(∫ 1

0

(1−W(b1−τxτ)

)b1−τxτdτ

) 1p

×(∫ 1

0

(1−W(b1−τxτ)

) ∣∣ f ′∣∣q (b1−τxτ)b1−τxτdτ) 1

q

= logxa

(I (1,a,x;W ))1p(I(| f ′|q,a,x;W

)) 1q

+ logbx

(I (1,x,b;1−W))1p(I(| f ′|q,x,b;1−W

)) 1q ,

where I(F,v,u;Q) is defined as in Lemma 1. Using the GG-convexity of | f ′|q andintegration by parts, we get

I(∣∣ f ′∣∣q ,v,u;Q

)� u | f ′|q (u)

logAv,u

[−Q(u)+Q(v)Av,u +

∫ u

vQ′(t)A

logt/ulogv/uv,u dt

]. (6)

Calculating I (1,v,u;Q) , we get

I (1,v,u;Q) =1

logu/v

∫ u

vQ(t)dt. (7)

Using (6) and (7) in (5), we get the desired result.

NEW OSTROWSKI LIKE INEQUALITIES FOR GG-CONVEX AND GA-CONVEX FUNCTIONS 1163

REMARK 2. In Theorem 1, if we choose q → 1, we get the following inequality:

∣∣∣∣ f (x)−∫ b

aw(t) f (t)dt

∣∣∣∣� log

xa

(x | f ′(x)|logAa,x

[−W (x)+

∫ x

aw(t)A

logt/xloga/xa,x dt

])

+ logbx

(b | f ′(b)|logAx,b

[(1−W(x))Ax,b −

∫ b

xw(t)A

log t/blogx/bx,b dt

]).

We can consider inequalities for different weights w, but here we give only resultfor a particular weight w(t) = 1

t logb/a .

COROLLARY 1. If assumptions of Theorem 1 are satisfied with w(t) = 1t logb/a ,

then the following inequality holds:

∣∣∣∣ f (x)− 1logb− loga

∫ b

a

f (t)t

dt

∣∣∣∣�

log1+ 1q x

a

log ba

(x−L(a,x))1p

⎡⎣x | f ′|q (x)−L(a | f ′|q (a) ,x | f ′|q (x))

log x| f ′|q(x)a| f ′|q(a)

⎤⎦

1q

+log1+ 1

q bx

log ba

(L(x,b)− x)1p

⎡⎣L(x | f ′|q (x) ,b | f ′|q (b))− x | f ′|q (x)

log b| f ′|q(b)x| f ′|q(x)

⎤⎦

1q

where L(x,y) = y−xlogy−logx .

THEOREM 2. Under the assumptions of Theorem 1, we get the following inequal-ity

∣∣∣∣ f (x)−∫ b

aw(t) f (t)dt

∣∣∣∣� x∣∣ f ′(x)∣∣ log

1q

xa

(∫ x

a

W p(t)t

dt

) 1p(

1logCa,x

[Ca,x −1]) 1

q

+b∣∣ f ′(b)

∣∣ log1q

bx

(∫ b

x

(1−W(t))p

tdt

) 1p(

1logCx,b

[Cx,b −1

]) 1q

where Cv,u =vq| f ′|q(v)uq| f ′|q(u) .


Proof. From Lemma 1, using the Holder inequality and GG-convexity of | f ′|qwe get

∣∣∣∣ f (x)−∫ b

aw(t) f (t)dt

∣∣∣∣ (8)

� logxa

(∫ 1

0W p (x1−τaτ)dτ

) 1p(∫ 1

0

∣∣ f ′∣∣q (x1−τaτ)x(1−τ)qaτqdτ) 1

q

+ logbx

(∫ 1

0

(1−W(b1−τxτ )

)pdτ) 1

p(∫ 1

0

∣∣ f ′∣∣q (b1−τxτ)b(1−τ)qxτqdτ) 1

q

� logxa

(∫ 1

0W p (x1−τaτ)dτ

) 1p(∫ 1

0

∣∣ f ′∣∣(1−τ)q (x)∣∣ f ′∣∣τq (a)x(1−τ)qaτqdτ

) 1q

+ logbx

(∫ 1

0

(1−W(b1−τxτ )

)pdτ) 1

p(∫ 1

0

∣∣ f ′∣∣(1−τ)q (b)∣∣ f ′∣∣τq (x)b(1−τ)qxτqdτ

) 1q

.

By a simple computation, we get

∫ 1

0Qp (u1−τvτ)dτ =

1log u

v

∫ u

v

Qp(t)t

dt (9)

and ∫ 1

0Cτ

v,udτ =Cv,u −1logCv,u

, (10)

where Q is a function and Cv,u is defined in theorem. Using (9) and (10) in (8) we getthe desired result.

COROLLARY 2. In Theorem 2, if we choose w(t) = 1t logb/a , then we get the fol-

lowing inequality:

∣∣∣∣ f (x)− 1logb− loga

∫ b

a

f (t)t

dt

∣∣∣∣� 1

logb− loga

(1

p+1

) 1p[log2 x

a

(L(aq∣∣ f ′∣∣q (a) ,xq

∣∣ f ′∣∣q (x))) 1

q

+ log2 bx

(L(xq∣∣ f ′∣∣q (x) ,bq

∣∣ f ′∣∣q (b))) 1

q

].

THEOREM 3. Under the assumptions of Theorem 1, we get the following inequal-ity

NEW OSTROWSKI LIKE INEQUALITIES FOR GG -CONVEX AND GA -CONVEX FUNCTIONS 1165

∣∣∣∣ f (x)−∫ b

aw(t) f (t)dt

∣∣∣∣�∣∣ f ′(x)∣∣ log

1q

xa

(∫ x

aW (t)t p−1dt

) 1p(

1logBa,x

[−W (x)+

∫ x

aB

log t/xloga/xa,x w(t)dt

]) 1q

+∣∣ f ′(b)

∣∣ log1q

bx

(∫ b

x(1−W(t))t p−1dt

) 1p

×(

1logBx,b

[(1−W(x))Bx,b−

∫ b

xB

logt/blogx/bx,b w(t)dt

]) 1q

where Bv,u = | f ′|q(v)| f ′|q(u) .

Proof. From Lemma 1, using the Holder inequality and GG-convexity of | f ′|qwe get∣∣∣∣ f (x)−∫ b

aw(t) f (t)dt

∣∣∣∣ (11)

� logxa

(∫ 1

0W(x1−τaτ)(x1−τaτ)p dτ

) 1p(∫ 1

0W(x1−τaτ) ∣∣ f ′∣∣q (x1−τaτ)dτ

) 1q

+ logbx

(∫ 1

0

(1−W(b1−τxτ )

)(b1−τxτ)p dτ

) 1p

×(∫ 1

0

(1−W(b1−τxτ )

)∣∣ f ′∣∣q (b1−τxτ)dτ) 1

q

� logxa

(∫ 1

0W(x1−τaτ)(x1−τaτ)p dτ

) 1p(∫ 1

0W(x1−τaτ)∣∣ f ′∣∣(1−τ)q (x)

∣∣ f ′∣∣τq (a)dτ) 1

q

+ logbx

(∫ 1

0

(1−W(b1−τxτ )

)(b1−τxτ)p dτ

) 1p

×(∫ 1

0

(1−W(b1−τxτ )

)∣∣ f ′∣∣(1−τ)q (b)∣∣ f ′∣∣τq (x)dτ

) 1q

.

By a simple computation, we get∫ 1

0Q(u1−τvτ)(u1−τvτ)p dτ =

1log u

v

∫ u

vQ(t)t p−1dt (12)

and ∫ 1

0Q(u1−τvτ)Bτ

v,udτ =1

logBv,u

[Q(v)Bv,u −Q(u)+

∫ u

vB

logt/ulogv/uv,u Q′(t)dt

]. (13)


Using (12) and (13) in (11) we get the desired result.

COROLLARY 3. In Theorem 3, if we choose w(t) = 1t logb/a , we get the following

inequality:∣∣∣∣ f (x)− 1logb− loga

∫ b

a

f (t)t

dt

∣∣∣∣� 1

p1p logb/a

⎧⎪⎨⎪⎩log1+ 1

qxa

(xp−L(ap,xp))1p

⎛⎝L(| f ′|q (a) , | f ′|q (x))−| f ′|q (x)

log | f ′ |q(a)| f ′|q(x)

⎞⎠

1q

+ log1+ 1q

xb

(L(xp,bp)− xp)1p

⎛⎝ | f ′|q (x)−L(| f ′|q (x) , | f ′|q (b))

log | f ′ |q(x)| f ′|q(b)

⎞⎠

1q

⎫⎪⎬⎪⎭ .

THEOREM 4. Under the assumptions of Theorem 1, we get the following inequal-ity ∣∣∣∣ f (x)−∫ b

aw(t) f (t)dt

∣∣∣∣� log

1q

xa

(∫ x

aW p(t)dt

) 1p(

x | f ′|q (x)logAa,x

(Aa,x −1)) 1

q

+ log1q

bx

(∫ b

x(1−W(t))p dt

) 1p(

b | f ′|q (b)logAx,b

(Ax,b −1

)) 1q

,

where Av,u defined as in Theorem 1.

Proof. By a similar argument to the proof of previous theorems, since | f ′|q isGG-convex function on [a,b], from Lemma 1 and the Holder integral inequality, weget ∣∣∣∣ f (x)−∫ b

aw(t) f (t)dt

∣∣∣∣ (14)

� logxa

(∫ 1

0W p (x1−τaτ)x1−τaτdτ

) 1p(∫ 1

0

∣∣ f ′∣∣q (x1−τaτ)x1−τaτdτ) 1

q

+ logbx

(∫ 1

0

(1−W(b1−τxτ )

)pb1−τxτdτ

) 1p(∫ 1

0

∣∣ f ′∣∣q (b1−τxτ)b1−τxτdτ) 1

q

� logxa

(∫ 1

0W p(x1−τaτ)x1−τaτdτ

) 1p(∫ 1

0

∣∣ f ′∣∣(1−τ)q (x)∣∣ f ′∣∣τq (a)x1−τaτdτ

) 1q

+ logbx

(∫ 1

0(1−W (b1−τxτ))pb1−τxτdτ

) 1p(∫ 1

0

∣∣ f ′∣∣(1−τ)q (b)∣∣ f ′∣∣τq (x)b1−τxτdτ

) 1q

.

NEW OSTROWSKI LIKE INEQUALITIES FOR GG -CONVEX AND GA -CONVEX FUNCTIONS 1167

By a simple computation, we get∫ 1

0Qp (u1−τvτ)u1−τvτdτ =

1log u

v

∫ u

vQp(t)dt (15)

and ∫ 1

0Aτ

v,udτ =1

logAv,u[Av,u −1] . (16)

If we use (15) and (16) in (14) we get the desired result.

Lastly, we will give a new result for GA-convex functions as following:

THEOREM 5. Let I ⊂ (0,∞) be an interval, a,b ∈ Io , a < b. Let w be a nonneg-ative integrable function on [a,b] such that


∫ ta w(x)dx.

Let f : I →R be a function differentiable on Io. If | f ′|q is GA-convex function on [a,b]for some q > 1, then for x ∈ [a,b] , following inequality holds∣∣∣∣ f (x)−∫ b

aw(t) f (t)dt

∣∣∣∣� log

1q

xa

(∫ x

aW (t)t p−1dt

) 1p (∣∣ f ′(x)∣∣q κ1 +

∣∣ f ′(a)∣∣q κ2

) 1q

+ log1q

bx

(∫ b

x(1−W(t))t p−1dt

) 1p (∣∣ f ′(b)

∣∣q κ3 +∣∣ f ′(x)∣∣q κ4

) 1q ,

where 1p = 1− 1

q and

κ1 =∫ x

a

[log t

x

log ax

(1− log t

x

2log ax

)]w(t)dt

κ2 =12

∫ x

a

(log t

x

log ax

)2

w(t)dt

κ3 = (1−W(x))−∫ b

x

[log t

b

log xb

(1− log t

b

2log xb

)]w(t)dt

κ4 = (1−W(x))− 12

∫ b

x

(log t

b

log xb

)2

w(t)dt,

provided that all integrals exist.

Proof. Similar to the proof of the previous theorems, by using Lemma 1, Holderintegral inequality and GA-convexity of | f ′|q , one can immediately get the result. Weomit the details.

REMARK 3. Results similar to Theorems 2, 3, 4 can also be obtained for GA-convex functions and some applications for special means can be given. It is left to theinterested readers.


RE F ER EN C ES

[1] A. AGLIC ALJINOVIC, J. PECARIC, I. PERIC, Estimates of the difference between two weightedintegral means via weighted Montgomery identity, Math. Inequal. Appl. 7 (2004), 315–336.

[2] A. AGLIC ALJINOVIC AND J. PECARIC, A discrete weighted Montgomery identity and Ostrowskitype inequalities for functions of two variables, Sarajevo Journal of Mathematics, Vol. 9 (21) (2013),47–56.

[3] S. S. DRAGOMIR, Inequalities of Jensen Type for GA-Convex Functions, RGMIA Research ReportCollection, 18 (2015), Article 35.

[4] C. P. NICULESCU, Convexity according to the geometric mean, Math. Inequal. Appl., 3 (2) (2000),155–167.

[5] J. PECARIC, On the Cebysev inequality, Bul. Inst. Politehn. Timisoara, 25 (39) (1980), 5–9.[6] R. A. SATNOIANU, Improved GA-convexity inequalities, Journal of Inequalities in Pure and Applied

Mathematics, Volume 3, Issue 5, Article 82, 2002.[7] X.-M. ZHANG, Y.-M. CHU, AND X.-H. ZHANG, The Hermite-Hadamard type inequality of GA-

convex functions and its application, Journal of Inequalities and Applications, Volume 2010, ArticleID 507560, 11 pages.

[8] T.-Y. ZHANG, A.-P. JI AND F. QI, Some inequalities of Hermite-Hadamard type for GA-convexfunctions with applications to means, Le Matematiche, Vol. LXVIII (2013) – Fasc. I, pp. 229–239.

[9] Z. ZHANG, W. WEI AND J.-R. WANG, Generalization of Hermite-Hadamard Inequalities InvolvingHadamard Fractional Integral Inequalities, Filomat 29: 7 (2015), 1515–1524.

(Received January 6, 2016) Merve Avcı ArdıcAdıyaman University, Faculty of Science and Arts

Department of MathematicsAdıyaman, Turkey


Ahmet Ocak AkdemirAgrı Ibrahim Cecen University

Faculty of Science and Letters, Department of MathematicsAgrı, Turkey


Erhan SetOrdu University, Faculty of Science and Letters

Department of MathematicsOrdu, Turkey




& Applications


HERMITE INTERPOLATION AND INEQUALITIES INVOLVING

WEIGHTED AVERAGES OF n–CONVEX FUNCTIONS

JOSIP PECARIC AND MARJAN PRALJAK

(Communicated by C. P. Niculescu)

Abstract. By using Hermite interpolation we obtain Popoviciu-type inequalities containing sums∑m

i=1 pi f (xi) , where f is an n -convex function. We also give integral analogues of the results,as well as bounds for integral remainders of identities associated with the obtained inequalities.

1. Introduction

Pecaric [5] proved the following result (see also [6, p. 262]):

PROPOSITION 1.1. The inequality

m

∑i=1

pi f (xi) � 0 (1)

holds for all convex functions f if and only if the m-tuples x = (x1, . . . ,xm), p =(p1, . . . , pm) ∈ R

m satisfy

m

∑i=1

pi = 0 andm

∑i=1

pi|xi − xk| � 0 for k ∈ {1, . . . ,m}. (2)

Sincem

∑i=1

pi|xi− xk| = 2m

∑i=1

pi(xi − xk)+ −m

∑i=1

pi(xi − xk),

where y+ = max(y,0) , it is easy to see that condition (2) is equivalent to

m

∑i=1

pi = 0,m

∑i=1

pixi = 0 andm

∑i=1

pi(xi − xk)+ � 0 for k ∈ {1, . . . ,m−1}. (3)

Mathematics subject classification (2010): 26D15, 26D10.Keywords and phrases: n -convex functions, Hermite interpolation, Cebysev functional.This work has been fully supported by Croatian Science Foundation under the project 5435.


1169


1170 J. PECARIC AND M. PRALJAK

Let A denote the linear operator A( f ) = ∑mi=1 pi f (xi) , let w(x,t) = (x− t)+ and

x(1) � x(2) � . . . � x(m) be the sequence x in ascending order. Notice that A(w(·,xk)) =∑m

i=1 pi(xi − xk)+ . For t ∈ [x(k),x(k+1)] we have

A(w(·,t)) = A(w(·,x(k)))+ (x(k)− t) ∑{i:xi>x(k)}

pi,

so the mapping t �→ A(w(·,t)) is linear on [x(k),x(k+1)] . Furthermore, A(w(·,x(m)) = 0,so condition (3) is equivalent to

m

∑i=1

pi = 0,m

∑i=1

pixi = 0 andm

∑i=1

pi(xi − t)+ � 0 for every t ∈ [x(1),x(m−1)]. (4)

It turns out that condition (4) is appropriate for extension of Proposition 1.1 to theintegral case and the more general class of n -convex functions.

DEFINITION 1.2. The n -th order divided difference of a function f : I → R ,where I is an interval in R , at distinct points x0, ...,xn ∈ I is defined recursively (see[6]) by

f [xi] = f (xi), (i = 0, . . . ,n)

and

f [x0, . . . ,xn] =f [x1, . . . ,xn]− f [x0, . . . ,xn−1]

xn− x0.

The function f is said to be n -convex on I , n � 0, if for all choices of (n+1) distinctpoints in I, the n -th order divided difference of f satisfies

f [x0, ...,xn] � 0.

The value f [x0, . . . ,xn] is independent of the order of the points x0, . . . ,xn . If f (n)

exists, then f is n -convex if and only if f (n) � 0. For 1 � k � n−2, a function f isn -convex if and only if f (k) exists and is (n− k)-convex.

The following result is due to Popoviciu [7, 8] (see [10, 6] also).

PROPOSITION 1.3. Let n � 2 . Inequality (1) holds for all n-convex functionsf : [a,b] → R if and only if the m-tuples x ∈ [a,b]m , p ∈ R

m satisfy

m

∑i=1

pixki = 0, for all k = 0,1, . . . ,n−1 (5)

m

∑i=1

pi(xi − t)n−1+ � 0, for every t ∈ [a,b]. (6)

In fact, Popoviciu proved a stronger result – it is enough to assume that (6) holdsfor every t ∈ [x(1),x(m−n+1)] and then, due to (5), it is automatically satisfied for everyt ∈ [a,b] . The integral analogue (see [9, 6]) is given in the next proposition.

HERMITE INTERPOLATION AND WEIGHTED AVERAGES OF n -CONVEX FUNCTIONS 1171

PROPOSITION 1.4. Let n � 2 , p : [α,β ] → R and g : [α,β ] → [a,b] . Then, theinequality ∫ β

αp(x) f (g(x))dx � 0 (7)

holds for all n-convex functions f : [a,b]→ R if and only if

∫ β

αp(x)g(x)k dx = 0, for all k = 0,1, . . . ,n−1∫ β

αp(x)(g(x)− t)n−1

+ dx � 0, for every t ∈ [a,b].(8)

In this paper we will derive inequalities of type (1) and (7) for n -convex functionsby making use of the Hermite interpolation. Let −∞ < a � a1 < a2 < · · ·< ar � b < ∞ ,r � 2. The Hermite interpolation of a function f ∈Cn[a,b] is of the form

f (x) = PH(x)+ eH(x)

where PH is the unique polynomial of degree n− 1, called the Hermite interpolatingpolynomial of f , satisfying

P(i)H (a j) = f (i)(a j), 0 � i � k j, 1 � j � r,

r

∑j=1

k j + r = n.

The associated error eH(x) can be represented in terms of the Green’s functionGH,n(x,s) for the multipoint boundary value problem

z(n)(x) = 0, z(i)(a j) = 0, 0 � i � k j, 1 � j � r,

that is, the following result holds (see [2]):

THEOREM 1.5. Let f ∈Cn[a,b] , and let PH be its Hermite interpolating polyno-mial. Then

f (x) = PH(x)+ eH(x)

=r

∑j=1

k j

∑i=0

Hi j(x) f (i)(a j)+b∫

a

GH,n(x,s) f (n)(s)ds, (9)

where Hi j are the fundamental polynomials of the Hermite basis defined by

Hi j(x) =1i!

w(x)

(x−a j)k j+1−i

k j−i

∑k=0

1k!

dk

dxk

((x−a j)k j+1

w(x)

)∣∣∣x=a j

(x−a j)k, (10)

where

w(x) =r

∏j=1

(x−a j)k j+1 (11)


and GH,n is the Green’s function defined by

GH,n(x,s) =

⎧⎨⎩∑l

j=1 ∑k ji=0

(a j−s)n−i−1

(n−i−1)! Hi j(x), s � x,

−∑rj=l+1 ∑

k ji=0

(a j−s)n−i−1

(n−i−1)! Hi j(x), s � x(12)

for all al � s � al+1 , l = 0,1, . . . ,r (a0 = a,ar+1 = b).

The following are some special cases of the Hermite interpolation of functions:(i) (m,n−m) conditions: r = 2, a1 = a , a2 = b , 1 � m � n−1, k1 = m−1 and

k2 = n−m−1. In this case

f (x) =m−1

∑i=0

τi(x) f (i)(a)+n−m−1

∑i=0

ηi(x) f (i)(b)+∫ b

aGm,n(x,s) f (n)(s)ds,

where

τi(x) =1i!

(x−a)i( x−b

a−b

)n−m m−1−i

∑k=0

(n−m+ k−1

k

)( x−ab−a

)k, (13)

ηi(x) =1i!

(x−b)i( x−a

b−a

)m n−m−1−i

∑k=0

(m+ k−1

k

)( x−ba−b

)k

(14)

and the Green’s function Gm,n is of the form

Gm,n(x,s) =

⎧⎪⎨⎪⎩

∑m−1j=0

[∑m−1− j

p=0

(n−m+p−1p

)(x−ab−a

)p] (x−a) j(a−s)n− j−1

j!(n− j−1)!

(b−xb−a

)n−m, s � x,

−∑n−m−1i=0

[∑n−m−1−i

q=0

(m+q−1q

)(b−xb−a

)q] (x−b)i(b−s)n−i−1

i!(n−i−1)!

(x−ab−a

)m, s � x

(15)(ii) Taylor’s two-point condition: m ∈ N , n = 2m , r = 2, a1 = a , a2 = b and

k1 = k2 = m−1. In this case

f (x) =m−1

∑i=0

m−i−1

∑k=0

(m+ k−1

k

)[ (x−a)i

i!

( x−ba−b

)m( x−ab−a

)kf (i)(a)

+(x−b)i

i!

( x−ab−a

)m( x−ba−b

)k

f (i)(b)]+∫ b

aG2T,m(x,s) f (2m)(s)ds,

where the Green’s function G2T,m is of the form

G2T,m(x,s) =(−1)m

(2m−1)!

⎧⎨⎩

pm(x,s)∑m−1k=0

(m+k−1k

)(x− s)m−1−kqk(x,s), s � x,

qm(x,s)∑m−1k=0

(m+k−1k

)(s− x)m−1−k pk(x,s), x � s,

where p(x,s) = (s−a)(b−x)(b−a) and q(x,s) = p(s,x).

The following lemma yields the sign of the Green’s function (12) on certain inter-vals (see Lemma 2.3.3, page 75, in [2]).

HERMITE INTERPOLATION AND WEIGHTED AVERAGES OF n-CONVEX FUNCTIONS 1173

LEMMA 1.6. The Green’s function GH,n given by (12) and w given by (11) satisfy

GH,n(x,s)w(x)

> 0, for a1 � x � ar, a1 < s < ar.

Integration by parts easily yields that for any function f ∈C2[a,b] the followingholds

f (x) =b− xb−a

f (a)+x−ab−a

f (b)+∫ b

aG(x,s) f ′′(s)ds, (16)

where the function G : [a,b]× [a,b]→ R is the Green’s function of the boundary valueproblem

z′′(x) = 0, z(a) = z(b) = 0

and is given by

G(x,s) =

⎧⎨⎩

(x−b)(s−a)b−a , for a � s � x,

(s−b)(x−a)b−a , for x � s � b.

(17)

The function G is continuous, symmetric and convexwith respect to both variablesx and s .

2. Main results

We will start this section with several identities.

THEOREM 2.1. Let −∞ < a � a1 < a2 < · · · < ar � b < ∞ , r � 2 , ∑rj=1 k j + r =

n, f ∈Cn[a,b] , x ∈ [a,b]m , p ∈ Rm and let Hi j and GH,n be given by (10) and (12).

Then

m

∑k=1

pk f (xk) =r

∑j=1

k j

∑i=0

m

∑k=1

pkHi j(xk) f (i)(a j) +∫ b

a

m

∑k=1

pkGH,n(xk,s) f (n)(s)ds. (18)

Proof. By applying identity (9) at xk , multiplying it by pk and summing up weobtained the required identity. �

The integral version of the previous theorem is the following:


n, f ∈ Cn[a,b] , g : [α,β ] → [a,b] , p : [α,β ] → R and let Hi j and GH,n be given by(10) and (12). Then

∫ β

αp(x) f (g(x))dx =

r

∑j=1

k j

∑i=0

f (i)(a j)∫ β

αp(x)Hi j(x)dx

+∫ b

a

(∫ β

αp(x)GH,n(g(x),s)dx

)f (n)(s)ds.



n−2 , f ∈Cn[a,b] , x ∈ [a,b]m , p ∈ Rm and let Hi j and GH,n−2 be given by (10) and

(12). Then

m

∑k=1

pk f (xk) =f (b)− f (a)

b−a

m

∑k=1

pkxk +b f (a)−a f (b)

b−a

m

∑k=1

pk

+r

∑j=1

k j

∑i=0

f (i+2)(a j)∫ b

a

m

∑k=1

pkG(xk,s)Hi j(s)ds

+∫ b

a

∫ b

a

m

∑k=1

pkG(xk,s)GH,n−2(s,t) f (n)(t)dt ds. (19)

Proof. Applying identity (16) at xk , multiplying it by pk and summing up weobtainm

∑k=1

pk f (xk)=f (b)− f (a)

b−a

m

∑k=1

pkxk+b f (a)−a f (b)

b−a

m

∑k=1

pk +∫ b

a

m

∑k=1

pkG(xk,s) f ′′(s)ds.

(20)By Theorem 1.5, f ′′(s) can be expressed as

f ′′(s) =r

∑j=1

k j

∑i=0

Hi j(s) f (i+2)(a j)+b∫

a

GH,n−2(s,t) f (n)(t)dt. (21)

Inserting (21) in (20) we get (19). �We also state the integral version of the previous theorem.


n−2 , f ∈Cn[a,b] , g : [α,β ]→ [a,b] , p : [α,β ]→ R and let Hi j and GH,n−2 be givenby (10) and (12). Then∫ β

αp(x) f (g(x))dx =

f (b)− f (a)b−a

∫ β

αp(x)g(x)dx+

b f (a)−a f (b)b−a

∫ β

αp(x)dx

+r

∑j=1

k j

∑i=0

f (i+2)(a j)∫ b

a

(∫ β

αp(x)G(g(x),s)dx

)Hi j(s)ds

+∫ b

a

∫ b

a

(∫ β

αp(x)G(g(x),s)dx

)GH,n−2(s,t) f (n)(t)dt ds.

Next we will use the identities proven above to derive inequalities.


n, x∈ [a,b]m , p∈Rm and let Hi j and GH,n be given by (10) and (12). If f : [a,b]→R

is n-convex andm

∑k=1

pkGH,n(xk,s) � 0 for all s ∈ [a,b], (22)

thenm

∑k=1

pk f (xk) �r

∑j=1

k j

∑i=0

m

∑k=1

pkHi j(xk) f (i)(a j). (23)

If the inequality in (22) is reversed, then the inequality in (23) is reversed also.

Proof. If (22) holds, then the second term on the right hand side (18) is nonnega-tive. �


n, x ∈ [a,b]m , p : [α,β ] → R and let Hi j and GH,n be given by (10) and (12). Iff : [a,b] → R is n-convex and∫ β

αp(x)GH,n(g(x),s)dx � 0 for all s ∈ [a,b], (24)

then ∫ β

αp(x) f (g(x))dx �

r

∑j=1

k j

∑i=0

f (i)(a j)∫ β

αp(x)Hi j(x)dx. (25)

If the inequality in (24) is reversed, then the inequality in (25) is reversed also.

THEOREM 2.7. Let −∞ < a = a1 < a2 < · · · < ar = b < ∞ , r � 2 , ∑rj=1 k j + r =

n− 2 , x ∈ [a,b]m , p ∈ Rm and let Hi j and GH,n−2 be given by (10) and (12). Let

f : [a,b] → R be n-convex and

m

∑k=1

pkG(xk,s) � 0 for all s ∈ [a,b], (26)

and consider the inequality

m

∑k=1

pk f (xk) � f (b)− f (a)b−a

m

∑k=1


b−a

m

∑k=1

pk

+r

∑j=1

k j

∑i=0

f (i+2)(a j)∫ b

a

m

∑k=1

pkG(xk,s)Hi j(s)ds. (27)

(i) If k j for j = 2, . . . ,r are odd, then (27) holds.

(ii) If k j for j = 2, . . . ,r−1 are odd and kr is even, then the reverse of (27) holds.

Proof. (i) Assume first that f ∈ Cn[a,b] . Due to the assumptions w given by(11) satisfies w(x) � 0 for all x and, hence, by Lemma 1.6, GH,n−2(s, t) � 0 for alls,t ∈ [a,b] . Therefore, the last term on the right hand side of (19) is nonnegative,so inequality (27) holds. The inequality for general f follows since every n -convexfunction can be obtained, by making use of the Bernstein polynomials, as a uniformlimit of n -convex functions with a continuous n -th derivative (see [6]).

(ii) Under these assumptions w(x) � 0, so GH,n−2(s,t) � 0. The rest of the proofis the same as in (i) . �

THEOREM 2.8. Let −∞ < a = a1 < a2 < · · · < ar = b < ∞ , r � 2 , ∑rj=1 k j + r =

n−2 , g : [α,β ] → R , p : [α,β ] → R and let Hi j be given by (10). Let f : [a,b] → R

be n-convex and ∫ β

αp(x)G(g(x),s)dx � 0 for all s ∈ [a,b],

and consider the inequality

∫ β

αp(x) f (g(x))dx � f (b)− f (a)

b−a

∫ β

αp(x)g(x)dx+

b f (a)−a f (b)b−a

∫ β

αp(x)dx

+r

∑j=1

k j

∑i=0

f (i+2)(a j)∫ b

a

(∫ β

αp(x)G(g(x),s)dx

)Hi j(s)ds. (28)

(i) If k j for j = 2, . . . ,r are odd, then (28) holds.

(ii) If k j for j = 2, . . . ,r−1 are odd and kr is even, then the reverse of (28) holds.

In the case of the (m,n−m) conditions we have the following corollary.

COROLLARY 2.9. Let τi and ηi be given by (13) and (14) and let x ∈ [a,b]m

and p ∈ Rm be such that (26) holds. Let f : [a,b] → R be n-convex and consider the

inequality

m

∑k=1


m

∑k=1


b−a

m

∑k=1

pk

+∫ b

a

(m

∑k=1

pkG(xk,s)

)(l−1

∑i=0

τi(s) f (i+2)(a)+n−l−1

∑i=0

ηi(s) f (i+2)(b)

)ds. (29)

(i) If n− l is even, then (29) holds.

(ii) If n− l is odd, then the reverse of (29) holds.

In the case of Taylor’s two point conditions we have the following corollary.

COROLLARY 2.10. Let x ∈ [a,b]m and p ∈ Rm be such that (26) holds. Let f :

[a,b]→ R be n-convex and consider the inequality

m

∑k=1


m

∑k=1


b−a

m

∑k=1

pk +∫ b

a

(m

∑k=1

pkG(xk,s)

)

×(

l−1

∑i=0

l−i−1

∑k=0

(l + k−1

k

)[ (s−a)i

i!

( s−ba−b

)l( s−ab−a

)kf (i+2)(a)

+(s−b)i

i!

( s−ab−a

)l( s−ba−b

)k

f (i+2)(b)])

ds. (30)

(i) If l is even, then (30) holds.

(ii) If l is odd, then the reverse of (30) holds.

THEOREM 2.11. Let −∞ < a = a1 < a2 < · · ·< ar = b < ∞ , r � 2 , ∑rj=1 k j +r =

n−2 , let x ∈ [a,b]m and p ∈ Rm satisfy (2), and let Hi j and GH,n−2 be given by (10)

and (12). Let f : [a,b]→ R be n-convex and consider the inequality

m

∑k=1

pk f (xk) �r

∑j=1

k j

∑i=0

f (i+2)(a j)∫ b

a

m

∑k=1

pkG(xk,s)Hi j(s)ds (31)

and the function

F(x) =r

∑j=1

k j

∑i=0

f (i+2)(a j)∫ b

aG(x,s)Hi j(s)ds. (32)

(i) If k j for j = 2, . . . ,r are odd, then (31) holds. Furthermore, if the function F isconvex, then inequality (1) holds.

(ii) If k j for j = 2, . . . ,r− 1 are odd and kr is even, then the reverse of (31) holds.Furthermore, if the function F is concave, then the reverse of inequality (1)holds.

Proof. The function G(x,s) is convex in the first variable, so assumption (26) issatisfied by Proposition 1.1. Now, the claims of the theorem follow from Theorem 2.7and Proposition 1.1. �

THEOREM 2.12. Let −∞ < a = a1 < a2 < · · ·< ar = b < ∞ , r � 2 , ∑rj=1 k j +r =

n−2 , let g : [α,β ]→R and p : [α,β ]→R satisfy (8), and let Hi j and GH,n−2 be givenby (10) and (12). Let f : [a,b] → R be n-convex and consider the inequality

∫ β

αp(x) f (x)dx �

r

∑j=1

k j

∑i=0

f (i+2)(a j)∫ b

a

(∫ β

αp(x)G(g(x),s)dx

)Hi j(s)ds (33)

and the function F given by (32).

(i) If k j for j = 2, . . . ,r are odd, then (33) holds. Furthermore, if the function F isconvex, then inequality (7) holds.

(ii) If k j for j = 2, . . . ,r− 1 are odd and kr is even, then the reverse of (33) holds.Furthermore, if the function F is concave, then the reverse of inequality (7)holds.

3. Bounds for identities related to the Popoviciu-type inequalities

Let f ,h : [a,b] → R be two Lebesgue integrable functions. We consider theCebysev functional

T ( f ,h) =1

b−a

∫ b

af (x)h(x)dx−

(1

b−a

∫ b

af (x)dx

)(1

b−a

∫ b

ah(x)dx

).

The following results can be found in [4].

PROPOSITION 3.1. Let f : [a,b] → R be a Lebesgue integrable function and h :[a,b]→ R be an absolutely continuous function with (·−a)(b−·)[h′]2 ∈ L[a,b] . Thenwe have the inequality

|T ( f ,h)| � 1√2

(1

b−a|T ( f , f )|

∫ b

a(x−a)(b− x)[h′(x)]2 dx

) 12

. (34)

The constant 1√2

in (34) is the best possible.

PROPOSITION 3.2. Let h : [a,b]→R be a monotonic nondecreasing function andlet f : [a,b] → R be an absolutely continuous function such that f ′ ∈ L∞[a,b] . Thenwe have the inequality

|T ( f ,h)| � 12(b−a)

‖ f ′‖∞

∫ b

a(x−a)(b− x)dh(x). (35)

The constant 12 in (35) is the best possible.

For m-tuples p = (p1, . . . , pm) ∈ Rm , x = (x1, . . . ,xm) ∈ [a,b]m and the functions

G and GH,n given by (17) and (12) denote

δ1(t) =m

∑k=1

pkGH,n(xk,t), for t ∈ [a,b]. (36)

δ2(t) =∫ b

a

m

∑k=1

pkG(xk,s)GH,n−2(s,t)ds, for t ∈ [a,b]. (37)

Now, we are ready to state the main results of this section.

THEOREM 3.3. Let −∞ < a � a1 < a2 < · · ·< ar � b < ∞ , r � 2 , let f : [a,b]→R be such that f (n) is an absolutely continuous function with (·−a)(b−·)[ f (n+1)]2 ∈L[a,b] , x ∈ [a,b]m , p ∈ R

m and let Hi j , δ1 and δ2 be given by (10), (36) and (37).(i) If ∑r

j=1 k j + r = n, then

m

∑k=1

pk f (xk) =r

∑j=1

k j

∑i=0

m

∑k=1

pkHi j(xk) f (i)(a j)

+f (n−1)(b)− f (n−1)(a)

b−a

∫ b

aδ1(s)ds+R1

n( f ;a,b), (38)

HERMITE INTERPOLATION AND WEIGHTED AVERAGES OF n-CONVEX FUNCTIONS 1179

where the remainder R1n( f ;a,b) satisfies the estimation

|R1n( f ;a,b)| �

(b−a

2|T (δ1,δ1)|

∫ b

a(s−a)(b− s)[ f (n+1)(s)]2 ds

) 12

. (39)

(ii) If ∑rj=1 k j + r = n−2 , then

m

∑k=1

pk f (xk) =f (b)− f (a)

b−a

m

∑k=1


b−a

m

∑k=1

pk

+r

∑j=1

k j

∑i=0

f (i+2)(a j)∫ b

a

m

∑k=1

pkG(xk,s)Hi j(s)ds

+f (n−1)(b)− f (n−1)(a)

b−a

∫ b

aδ2(s)ds+R2

n( f ;a,b), (40)

where the remainder R2n( f ;a,b) satisfies the estimation

|R2n( f ;a,b)| �

(b−a

2|T (δ2,δ2)|

∫ b

a(s−a)(b− s)[ f (n+1)(s)]2 ds

) 12

.

Proof. (i) Applying Proposition 3.1 with f → δ1 and h → f (n) we get

∣∣∣∣∫ b

aδ1(s) f (n)(s)ds− 1

b−a

∫ b

aδ1(s)ds

∫ b

af (n)(s)ds

∣∣∣∣�(

b−a2

|T (δ1,δ1)|∫ b

a(s−a)(b− s)[ f (n+1)(s)]2 ds

) 12

. (41)

From identities (18) and (38) we obtain

∫ b

aδ1(s) f (n)(s)ds =

f (n−1)(b)− f (n−1)(a)b−a

∫ b

aδ1(s)ds+R1

n( f ;a,b),

where the estimate (39) follows from (41).(ii) Analogous as in (i) . �By using Proposition 3.2 we obtain the following Gruss type inequality.

THEOREM 3.4. Let −∞ < a � a1 < a2 < · · · < ar � b < ∞ , r � 2 , let x , p ,Hi j , δ1 , δ2 and n be as in Theorem 3.3 and let f : [a,b] → R be such that f (n) isan absolutely continuous function with f (n+1) � 0 . Then representations (38) and (40)hold with the remainders Ri

n( f ;a,b) , i = 1,2 , satisfying the bounds

|Rin( f ;a,b)|�‖δ ′

i ‖∞

[b−a2

(f (n−1)(b)+ f (n−1)(a)

)− f (2n−2)(b)+ f (2n−2)(a)

]. (42)


Proof. If we apply Proposition 3.2 with f → δi and h → f (n) we obtain∣∣∣∣∫ b

aδi(s) f (n)(s)ds− 1

b−a

∫ b

aδi(s)ds

∫ b

af (n)(s)ds

∣∣∣∣�12‖δ ′

i ‖∞

∫ b

a(s−a)(b−s) f (n+1)(s)ds.

Since∫ b

a(s−a)(b− s) f (n+1)(s)ds =

∫ b

a(2s−a−b) f (n)(s)ds

= (b−a)[f (n−1)(b)+ f (n−1)(a)

]−2[f (n−2)(b)− f (n−2)(a)

], (43)

using (43) and identities (18) or (19) we deduce (42). �

REMARK 3.5. We can construct linear functionals by taking differences of theleft and right hand sides of the inequalities from Theorems 2.5, 2.6, 2.7 and 2.8. Byusing similar methods as in [1, 3] we can prove mean value results for these functionals,as well as construct new families of exponentially convex functions and Cauchy-typemeans. Then, by using some known properties of exponentially convex functions, wecan derive new inequalities and prove monotonicity of the obtained Cauchy-type meansanalogously as in [1, 3].

RE F ER EN C ES

[1] M. ADIL KHAN, N. LATIF, J. PECARIC, Generalization of majorization theorem, J. Math. Inequal. 9(2015), 847–872

[2] R. P. AGARWAL, P. J. Y. WONG, Error Inequalities in Polynomial Interpolation and Their Applica-tions, Kluwer Academic Publishers, Dordrecht/Boston/London, 1993.

[3] S. I. BUTT, K. A. KHAN, J. PECARIC,Popoviciu type inequalities via Green function and generalizedMontgomery identity, Math. Ineq. Appl. 18 (2015), 1519–1538.

[4] P. CERONE AND S. S. DRAGOMIR, Some new Owstrowski-type bounds for the Cebysev functionaland applications, J. Math. Inequal., 8 (1) (2014), 159–170.

[5] J. PECARIC, On Jessen’s Inequality for Convex Functions, III, J. Math. Anal. Appl., 156 (1991), 231–239.

[6] J. E. PECARIC, F. PROSCHAN AND Y. L. TONG, Convex functions, partial orderings and statisticalapplications, Academic Press, New York, 1992.

[7] T. POPOVICIU, Notes sur les fonctions convexes d’orde superieur III, Mathematica (Cluj) 16, (1940),74–86.

[8] T. POPOVICIU, Notes sur les fonctions convexes d’orde superieur IV, Disqusitiones Math. 1, (1940),163–171.

[9] T. POPOVICIU, Notes sur les fonctions convexes d’orde superieur IX, Bull. Math. Soc. Roumaine Sci.43, (1941), 85–141.

[10] T. POPOVICIU, Les fonctions convexes, Herman and Cie, Editeurs, Paris 1944.

(Received January 25, 2016) Josip PecaricFaculty Of Textile Technology, University Of Zagreb

Prilaz Baruna Filipovica 28a, 10000 Zagreb, Croatiae-mail: [email protected]

Marjan PraljakFaculty of Food Technology and Biotechnology

University Of ZagrebPierottijeva 6, 10000 Zagreb, Croatia




& Applications


GENERALIZATIONS OF SHERMAN’S INEQUALITY

BY HERMITE’S INTERPOLATING POLYNOMIAL

M. ADIL KHAN, S. IVELIC BRADANOVIC AND J. PECARIC

(Communicated by K. Nikodem)

Abstract. Generalizations of Sherman’s inequality for convex functions of higher order are ob-tained by applying Hermite’s interpolating polynomials. The results for particular cases, namely,Lagrange, (m,n−m) and two-point Taylor interpolating polynomials are also cosidered. TheGruss and Ostrowski type inequalities related to these generalizations are given.

1. Introduction

We start with the concept of majorization which is exactly a partial ordering ofvectors and determines the degree of similarity between the vector elements.

For fixed m � 2, let x = (x1, ...,xm) and y = (y1, ...,ym) denote two m-tuples. Letx[1] � x[2] � ... � x[m] and y[1] � y[2] � ... � y[m] be their ordered components. We saythat x majorizes y or y is majorized by x and write y ≺ x if

k

∑i=1

y[i] �k

∑i=1

x[i], k = 1, ....,m−1, andm

∑i=1

yi =m

∑i=1

xi. (1)

A notation from real vectors may be extended to real matrices. Let Mml(R) de-notes the space of m× l real matrices . A matrix A = (ai j) ∈ Mml(R) is called rowstochastic if all of its entries are greater than or equal to zero and the sum of the en-tries in each row is equal to 1. A square matrix A = (ai j) ∈ Mll(R) is called doublestochastic if all of its entries are greater than or equal to zero and the sum of the entriesin each column and each row is equal to 1.

The majorization theorem, due to Hardy et al (1929 [6]), gives connections withmatrix theory (see also [8, p. 333]).

THEOREM 1. Let x, y ∈ Rm. Then the following statements are equivalent:

(i) y ≺ x ;

Mathematics subject classification (2010): 26D15.Keywords and phrases: Majorization, n -convexity, Schur-convexity, Sherman’s theorem, Hermite’s

interpolating polynomial, Chebyshev functional, Gruss type inequalities, Ostrowski type inequalities, expo-nentially convex functions.

The research of the second and third author has been fully supported by Croatian Science Foundation under the project5435.


1181


1182 M. ADIL KHAN, S. IVELIC BRADANOVIC AND J. PECARIC

(ii) There is a doubly stochastic matrix A such that y = xA;

(iii) The inequality ∑mi=1 φ(yi) � ∑m

i=1 φ(xi) holds for each convex function φ : R →R .

S. Sherman [10] obtained the following general result.

THEOREM 2. Let [α,β ] ⊂ R and for fixed l,m ∈ N, l,m � 2, let x ∈ [α,β ]l ,y ∈ [α,β ]m, u ∈ [0,∞)l, v ∈ [0,∞)m and

y = xAT and u = vA (2)

for some row stochastic matrix A ∈ Mml(R). Then for every convex function φ :[α,β ] → R we have

m

∑q=1

vqφ(yq) �l

∑p=1

upφ(xp). (3)

Sherman obtained this useful generalization replacing the classical concept of ma-jorization y ≺ x by the notion of weighted majorization (2) for two pairs (x,u) and(y,v), where x = (x1, ...,xl) and y = (y1, ...,ym) are real vectors and u = (u1, ...,ul)and v = (v1, ...,vm) are corresponding nonnegative weights. Here AT denotes thetranspose of a matrix A. In particular, if m = l and up = vq for all p,q = 1, ...,m ,the condition u = vA assures the stochasticity on columns, so in that case we deal withdoubly stochastic matrices. Then, as a special case of Sherman’s inequality, we get theweighted version of majorization’s inequality:

m

∑p=1

upφ(yp) �m

∑p=1

upφ(xp).

Denoting Um = ∑mp=1 up and putting y1 = y2 = ... = ym = 1

Um∑m

p=1 upxp, we obtainJensen’s inequality in the form

φ

(1

Um

m

∑p=1

upxp

)� 1

Um

m

∑p=1

upφ(xp).

In this paper, we recall generalizations of Sherman’s result for convex functionsof the higher order. Moreover, we obtain extension to real, not necessary nonnegativeweights u , v and matrix A . For some related results see also [1], [2], [7].

In sequel, we always assume that [α,β ] ⊂ R without having to be emphasized.The notion of n -convexity was defined in terms of divided differences by Popovi-

ciu [9]. A function φ : [α,β ] → R is n -convex, n � 0, if its n th order divided differ-ences [x0, ...,xn;φ ] are nonnegative for all choices of (n+1) distinct points xi ∈ [α,β ],i = 0, ...,n. Thus, a 0-convex function is nonnegative, a 1-convex function is nonde-creasing and a 2-convex function is convex in the usual sense. If φ (n) exists then φ isn -convex iff φ (n) � 0 (see [8]).

GENERALIZATION OF SHERMAN’S INEQUALITY BY HERMITE’S POLYNOMIAL 1183

2. Preliminaries

Let α � a1 < a2 < .. . < ar � β , (r � 2) be the given points. For φ ∈Cn([α,β ])(n � r ) a unique polynomial ρH(s) of degree (n−1) exists, such that Hermite condi-tions hold:

ρ (i)H (a j) = φ (i)(a j), 0 � i � k j, 1 � j � r, (H)

wherer∑j=1

k j + r = n.

In particular, for r = n, k j = 0 for all j, we have Lagrange conditions:

ρL(a j) = φ(a j), 1 � j � n.

For r = 2, 1 � m � n− 1, k1 = m− 1, k2 = n−m− 1, we have Type (m,n−m)conditions:

ρ (i)(m,n)(α) = φ (i)(α), 0 � i � m−1,

ρ (i)(m,n)(β ) = φ (i)(β ), 0 � i � n−m−1.

For n = 2m, r = 2 and k1 = k2 = m−1, we have Two-point Taylor conditions:

ρ (i)2T (α) = φ (i)(α), ρ (i)

2T (β ) = φ (i)(β ), 0 � i � m−1.

The following theorem and remark can be found in [3].

THEOREM 3. Let α � a1 < a2 < .. . < ar � β , (r � 2 ), be the given points andφ ∈Cn([α,β ]), (n � r ) . Let ρH(s) be the Hermite inrepolating polynomial. Then

φ(t) = ρH(t)+RH,n(φ ,t) (4)

=r

∑j=1

k j

∑i=0

Hi j(t)φ (i)(a j)+∫ β

αGH,n(t,s)φ (n)(s)ds,

where Hi j are fundamental polynomials of the Hermite basis defined by

Hi j(t) =1i!

ω(t)

(t−a j)k j+1−i

k j−i

∑k=0

1k!

dk

dtk

((t −a j)

k j+1

ω(t)

)∣∣∣∣∣t=a j

(t −a j)k, (5)

where

ω(t) =r

∏j=1

(t−a j)k j+1,

and GH,n(t,s) is defined by

GH,n(t,s) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

l∑j=1

k j

∑i=0

(a j−s)n−i−1

(n−i−1)! Hi j(t); s � t,

− r∑

j=l+1

k j

∑i=0

(a j−s)n−i−1

(n−i−1)! Hi j(t); s � t,

(6)

for all al � s � al+1 ; l = 0, . . . ,r with a0 = α and ar+1 = β .


REMARK 1. For Lagrange conditions, from Theorem 3 we have

φ(t) = ρL(t)+RL(φ ,t)

where ρL(t) is the Lagrange interpolating polynomial i.e.

ρL(t) =n

∑j=1

n

∏k=1k �= j

(t−ak

a j −ak

)φ(a j)

and the remainder RL(φ ,t) is given by

RL(φ ,t) =∫ β

αGL(t,s)φ (n)(s)ds

with

GL(t,s) =1

(n−1)!

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

l∑j=1

(a j − s)n−1n∏k=1k �= j

(t−aka j−ak

), s � t

−n∑

j=l+1(a j − s)n−1

n∏k=1k �= j

(t−aka j−ak

), s � t

(7)

al � s � al+1, l = 1,2, ...,n−1 with a1 = α and an = β .For type (m,n−m) conditions, from Theorem 3 we have

φ(t) = ρ(m,n)(t)+R(m,n)(φ , t)

where ρ(m,n)(t) is (m,n−m) interpolating polynomial, i.e.

ρ(m,n)(t) =m−1

∑i=0

τi(t)φ (i)(α)+n−m−1

∑i=0

ηi(t)φ (i)(β ),

with

τi(t) =1i!

(t−α)i(

t−βα −β

)n−m m−1−i

∑k=0

(n−m+ k−1

k

)(t−αβ −α

)k

(8)

and

ηi(t) =1i!

(t−β )i(

t −αβ −α

)m n−m−1−i

∑k=0

(m+ k−1

k

)(t−βα −β

)k

. (9)

and the remainder R(m,n)(φ ,t) is given by

R(m,n)(φ ,t) =∫ β

αG(m,n)(t,s)φ (n)(s)ds

with

G(m,n)(t,s) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

m−1∑j=0

[m−1− j

∑p=0

(n−m+p−1p

)(t−αβ−α

)p]

(t−α) j(α−s)n− j−1

j!(n− j−1)!

(β−tβ−α

)n−m, s � t

-n−m−1

∑i=0

[n−m−i−1

∑q=0

(m+q−1q

)( β−tβ−α

)q (t−β )i(β−s)n−i−1

i!(n−i−1)!

](t−αβ−α

)m, t � s.

(10)

For Type Two-point Taylor conditions, from Theorem 3 we have

φ(t) = ρ2T (t)+R2T (φ ,t)

where ρ2T (t) is the two-point Taylor interpolating polynomial i.e,

ρ2T (t) =m−1

∑i=0

m−1−i

∑k=0

(m+k−1k

)[φ (i)(α) (t−α)i

i!

(t−βα−β

)m(t−αβ−α

)k(11)

+φ (i)(β ) (t−β )ii!

(t−αβ−α

)m( t−βα−β

)k]

and the remainder R2T (φ ,t) is given by

R2T (φ ,t) =∫ β

αG2T (t,s)φ (n)(s)ds

with

G2T (t,s) =

⎧⎪⎪⎨⎪⎪⎩

(−1)m

(2m−1)! pm(t,s)

m−1∑j=0

(m−1+ jj

)(t − s)m−1− jq j(t,s), s � t;

(−1)m

(2m−1)!qm(t,s)

m−1∑j=0

(m−1+ jj

)(s− t)m−1− j p j(t,s), s � t;

(12)

where p(t,s) = (s−α)(β−t)β−α , q(t,s) = p(s,t),∀t,s ∈ [α,β ].

3. Generalizations of Sherman’s inequality

Applying Hermite’s interpolating polynomial we obtain a generalization of Sher-man’s theorem which holds for real, not necessary nonnegative weights u , v and amatrix A and without assumption (2).

THEOREM 4. Let α � a1 < a2 < .. . < ar � β (r � 2 ) be the given points, k j �0, j = 1, ...,r, with

r∑j=1

k j + r = n. Let φ ∈ Cn([α,β ]) be n-convex and x ∈ [α,β ]l ,

y ∈ [α,β ]m, u ∈ Rl and v ∈ R

m. If

l

∑p=1

upGH,n(xp,s)−m

∑q=1

vqGH,n(yq,s) � 0, s ∈ [α,β ], (13)

then

l

∑p=1

upφ(xp)−m

∑q=1

vqφ(yq) (14)

�l

∑p=1

up

r

∑j=1

k j

∑i=0

φ (i)(a j)Hi j(xp)−m

∑q=1

vq

r

∑j=1

k j

∑i=0

φ (i)(a j)Hi j(yq),

where GH,n and Hi j are defined as in (6) and (5), respectively.

Proof. Since φ ∈Cn([α,β ]), applying Theorem 3 onl∑

p=1upφ(xp)−

m∑

q=1vqφ(yq),

we get the identity

l

∑p=1

upφ(xp)−m

∑q=1

vqφ(yq) =r

∑j=1

k j

∑i=0

φ (i)(a j)

[l

∑p=1

upHi j(xp)−m

∑q=1

vqHi j(yq)

](15)

+∫ β

α

[l

∑p=1

upGH,n(xp,s)−m

∑q=1

vqGH,n(yq,s)

]φ (n)(s)ds.

Since φ is n -convex on [α,β ], then we have φ (n) � 0 on [α,β ] . Moreover, the in-equality (14) holds. �

Under Sherman’s assumptions the following generalizations hold.

THEOREM 5. Let all the assumptions of Theorem 4 be satisfied. Additionally, letvectors u, v be nonnegative and let (2) holds for some row stochastic matrix A ∈Mml(R). If (14) holds and the function

F(·) =r

∑j=1

k j

∑i=0

φ (i)(a j)Hi j(·) (16)

is convex on [α,β ] then the inequality (3) holds.

Proof. If (14) holds, the right hand side of (14) can be written in the form

l

∑p=1

upF(xp)−m

∑q=1

vqF(yq),

where F is defined by (16). If F is convex, then by Sherman’s theorem we have

l

∑p=1

upF(xp)−m

∑q=1

vqF(yq) � 0,

i.e. the right-hand side of (14) is nonnegative, so (3) immediately follows. �By using Lagrange conditions we get the following generalization of Sherman’s

theorem.

COROLLARY 1. Let α � a1 < a2 < .. . < an � β (n � 2 ) be the given points andφ ∈Cn([α,β ]) be n-convex. Let x ∈ [α,β ]l, y ∈ [α,β ]w, u ∈ [0,∞)l and v ∈ [0,∞)w

be such that (2) holds for some row stochastic matrix A ∈ Mwl(R).

(i) Ifl

∑p=1

upGL(xp,s)−w

∑q=1

vqGL(yq,s) � 0, s ∈ [α,β ],


then

l

∑p=1

upφ(xp)−w

∑q=1

vqφ(yq) (17)

�l

∑p=1

up

n

∑j=1

φ(a j)n

∏u=1u �= j

(xp−au

a j −au

)−

w

∑q=1

vq

n

∑j=1

φ(a j)n

∏u=1u �= j

(yq −au

a j −au

),

where GL is defined as in (7).

(ii) If (17) holds and the function

F(·) =n

∑j=1

φ(a j)n

∏u=1u �= j

( ·−au

a j −au

)

is convex on [α,β ] then

w

∑q=1

vqφ(yq) �l

∑p=1

upφ(xp).

By using type (m,n−m) conditions we can give the following result.

COROLLARY 2. Let n � 2, 1 � m � n−1 and φ ∈Cn([α,β ]) be n-convex. Letx∈ [α,β ]l, y∈ [α,β ]w, u∈ [0,∞)l and v∈ [0,∞)w be such that (2) holds for some rowstochastic matrix A ∈ Mwl(R).

(i) Ifl

∑p=1

upG(m,n)(xp,s)−w

∑q=1

vqG(m,n)(yq,s) � 0, s ∈ [α,β ],

then

l

∑p=1

upφ(xp)−w

∑q=1

vqφ(yq) �l

∑p=1

up

(m−1

∑i=0

τi(xp)φ (i)(α)+n−m−1

∑i=0

ηi(xp)φ (i)(β )

)

−w

∑q=1

vq

(m−1

∑i=0

τi(yq)φ i(α)+n−m−1

∑i=0

ηi(yq)φ (i)(β )

),

(18)

where τi, ηi and G(m,n) are defined as in (8), (9) and (10), respectively.

(ii) If (18) holds and the function

F(·) =m−1

∑i=0

τi(·)φ (i)(α)+n−m−1

∑i=0

ηi(·)φ (i)(β )


is convex on [α,β ] then

u

∑q=1

vqφ(yq) �w

∑p=1

upφ(xp).

By using Two-point Taylor conditions we can give the following result.

COROLLARY 3. Let m � 1 and φ ∈C2m([α,β ]) be 2m-convex. Let x ∈ [α,β ]l ,y ∈ [α,β ]w, u ∈ [0,∞)l and v ∈ [0,∞)w be such that (2) holds for some row stochasticmatrix A ∈ Mwl(R).

(i) Ifl

∑p=1

upG2T (xp,s)−w

∑q=1

vqG2T (yq,s) � 0, s ∈ [α,β ],

thenl

∑p=1

upφ(xp)−w

∑q=1

vqφ(yq) �l

∑p=1

upρ2T (xp)−w

∑q=1

vqρ2T (yq), (19)

where ρ2T and G2T are defined as in (11) and (12), respectively.

(ii) Moreover, if the function ρ2T is convex on [α,β ] , then

w

∑q=1

vqφ(yq) �l

∑p=1

upφ(xp).

REMARK 2. Motivated by the inequality (14), under the assumptions of Theorem4, we define the linear functional A : Cn([α,β ]) → R by

A(φ) =l

∑p=1

upφ(xp)−m

∑q=1

vqφ(yq) (20)

−r

∑j=1

k j

∑i=0

φ (i)(a j)

[l

∑p=1

upHi j(xp)−m

∑q=1

vqHi j(yq)

].

Then for every n -convex functions φ ∈Cn([α,β ]) we have A(φ) � 0. Using the linear-ity and positivity of this functional we may derive corresponding mean-value theoremsapplying the same method as given in [2]. Moreover, we could produce new classes ofexponentially convex functions and as outcome we get new means of the Cauchy type.Here we also refer to [7] with related results.

4. Gruss and Ostrowski type inequalities

P. L. Chebyshev [5] obtained the following inequality

|T ( f ,g)| � 112

(b−a)2∥∥ f ′∥∥

∞

∥∥g′∥∥∞

where f ,g : [α,β ] → R are absolutely continuous functions whose derivatives f ′ andg′ are bounded and T ( f ,g) is so-called Chebyshev functional defined as

T ( f ,g) :=1

β −α

∫ β

αf (t)g(t)dt− 1

β −α

∫ β

αf (t)dt · 1

β −α

∫ β

αg(t)dt. (21)

Here ‖·‖∞ denotes the norm in L∞[α,β ], the space of essentially bounded functionson [α,β ] , defined by ‖ f‖∞ = esssup

t∈[α ,β ]| f (t)|. We also use notation ‖·‖p , p � 1, for Lp

norm.P. Cerone and S. S. Dragomir [4], considering the Chebyshev functional (21), ob-

tained the following two related results.

THEOREM 6. Let f : [α,β ] → R be Lebesgue integrable and g : [α,β ] → R beabsolutely continuous with (·−α)(β −·)(g′)2 ∈ L1[α,β ]. Then

|T ( f ,g)| � 1√2[T ( f , f )]

12

1√β −α

(∫ β

α(x−α)(β − x)[g′(x)]2dx

) 12

. (22)

The constant 1√2


THEOREM 7. Let g : [α,β ]→ R be monotonic nondecreasing and f : [α,β ]→R

be absolutely continuous with f ′ ∈ L∞[α,β ]. Then

|T ( f ,g)| � 12(β −α)

∥∥ f ′∥∥

∞

∫ β

α(x−α)(β − x)dg(x). (23)


In following results we consider the function B : [α,β ] → R , defined under as-sumptions of Theorem 4, by

B(s) =l

∑p=1

upGH,n(xp,s)−m

∑q=1

vqGH,n(yq,s), (24)

where x ∈ [α,β ]l , y ∈ [α,β ]m, u ∈ Rl, v ∈ R

m and GH,n is defined as in (6).

THEOREM 8. Let α � a1 < a2 < .. . < ar � β (r � 2 ) be the given points, k j � 0,

j = 1, ...,r, withr∑j=1

k j + r = n. Let φ : [α,β ] → R be such that φ (n) is an absolutely

continuous on [α,β ] with (· −α)(β − ·)(φ (n+1))2 ∈ L1[α,β ]. Let x ∈ [α,β ]l , y ∈[α,β ]m, u ∈ R

l , v ∈ Rm and Hi j and B be defined as in (5) and (24), respectively.

Then the remainder R(φ ;α,β ) defined by

R(φ ;α,β ) =l

∑p=1

upφ(xp)−m

∑q=1

vqφ(yq)

−r

∑j=1

k j

∑i=0

φ (i)(a j)

[l

∑p=1

upHi j(xp)−m

∑q=1

vqHi j(yq)

]

− φ (n−1)(β )−φ (n−1)(α)β −α

∫ β

αB(s)ds (25)

satisfies the estimation

|R(φ ;α,β )| �√

β −α√2

[T (B,B)]12

(∫ β

α(s−α)(β − s)[φ (n+1)(s)]2ds

) 12

. (26)

Proof. Comparing (15) and (25) we have

R(φ ;α,β ) =∫ β

αB(s)φ (n)(s)ds− φ (n−1)(β )−φ (n−1)(α)

β −α

∫ β

αB(s)ds

=∫ β

αB(s)φ (n)(s)ds− 1

β −α

∫ β

αφ (n)ds

∫ β

αB(s)ds = (β −α)T (B,φ (n)).

Applying Theorem 6 on the functions B and φ (n) we obtain (26). �

Using Theorem 7 we obtain the Gruss type inequality.

THEOREM 9. Let α � a1 < a2 < .. . < ar � β (r � 2 ) be the given points, k j � 0,

j = 1, ...,r, withr∑j=1

k j + r = n. Let φ ∈Cn([α,β ]) be such that φ (n+1) � 0 on [α,β ]

and x ∈ [α,β ]l , y ∈ [α,β ]m, u ∈ Rl, v ∈ R

m and Hi j and B be defined as in (5)and (24), respectively. Then the remainder R(φ ;α,β ) defined by (25) satisfies theestimation

|R(φ ;α,β )| � ‖B′‖∞

[φ (n−1)(β )+ φ (n−1)(α)

2− φ (n−2)(β )−φ (n−2)(α)

β −α

]. (27)

Proof. Since R(φ ;α,β ) = (β −α)T (B,φ (n)) , applying Theorem 7 on the func-tions B and φ (n) we obtain (27). �

We present the Ostrowski type inequality related to generalizations of Sherman’sinequality.

THEOREM 10. Let α � a1 < a2 < .. . < ar � β (r � 2 ) be the given points, k j �0, j = 1, ...,r, with

r∑j=1

k j + r = n. Let φ ∈ Cn([α,β ]) and x ∈ [α,β ]l, y ∈ [α,β ]m,

u ∈ Rl and v ∈ R

m. Let 1 � p,q � ∞ , 1/p+1/q = 1 and∣∣∣φ (n)

∣∣∣p ∈ Lp [α,β ] . Then

∣∣∣∣∣l

∑p=1

upφ(xp)−m

∑q=1

vqφ(yq)−r

∑j=1

k j

∑i=0

φ (i)(a j)

[l

∑p=1

upHi j(xp)−m

∑q=1

vqHi j(yq)

]∣∣∣∣∣�∥∥∥φ (n)

∥∥∥p‖B‖q , (28)

where Hi j and B are defined as in (5) and (24), respectively.The constant ‖B‖q is sharp for 1 < p � ∞ and the best possible for p = 1 .

Proof. Under ussumption of theorem the identity (15) holds. Applying the well-known Holder inequality to (15), we have∣∣∣∣∣

l

∑p=1

upφ(xp)−m

∑q=1

vqφ(yq)−r

∑j=1

k j

∑i=0

φ (i)(a j)

[l

∑p=1

upHi j(xp)−m

∑q=1

vqHi j(yq)

]∣∣∣∣∣=

∣∣∣∣∣∫ β

α

[l

∑p=1

upGH,n(xp,s)−m

∑q=1

vqGH,n(yq,s)

]φ (n)(s)ds

∣∣∣∣∣=∣∣∣∣∫ β

αB(s)φ (n)(s)ds

∣∣∣∣� ∥∥∥φ (n)∥∥∥

p

(∫ β

α|B(s)|qds

) 1q

The proof of the sharpness is analog to one in proof of Theorem 11 in [2]. �

Acknowledgement. The authors would like to thank the editor and referees for thevaluable comments and suggestions on the manuscript.

RE F ER EN C ES

[1] M. ADIL KHAN, N. LATIF AND J. PECARIC, Generalization of majorization theorem, J. Math. In-equal., 9, 3 (2015), 847–872.

[2] RAVI P. AGARWAL, S. IVELIC BRADANOVIC, J. PECARIC, Generalizations of Sherman’s inequalityby Lidstone’s interpolating polynomial, J. Inequal. Appl., 6, 2016 (2016).

[3] RAVI P. AGARWAL, P. J. Y. WONG, Error Inequalities in Polynomial Interpolation and their Appli-cations, Kluwer Academic Publisher, Dordrecht, 1993.

[4] P. CERONE, S. S. DRAGOMIR, Some new Ostrowski-type bounds for the Cebysev functional andapplications, J. Math. Inequal., 8, 1 (2014), 159–170.

[5] P. L. CHEBYSHEV, Sur les expressions approximatives des integrales definies par les autres prisesentre les memes limites, Proc. Math. Soc. Charkov, 2, (1882) 93–98.

[6] G. H. HARDY, J. E. LITTLEWOOD, G. POLYA, Inequalities, Cambridge University Press, 2nd ed.,Cambridge 1952.

[7] S. IVELIC BRADANOVIC, J. PECARIC, Generalizations of Sherman’s inequality, Per. Math. Hung. toappear.

[8] J. E. PECARIC, F. PROSCHAN, Y. L. TONG, Convex Functions, Partial Orderings, and StatisticalApplications, Academic Press, New York, 1992.


[9] T. POPOVICIU, Sur l’approximation des fonctions convexes d’ordre superier, Mathematica, 10,(1934), 49–54.

[10] S. SHERMAN, On a theorem of Hardy, Littlewood, Polya and Blackwell, Proc. Nat. Acad. Sci. USA,37, 1 (1957), 826–831.

(Received January 26, 2016) Muhammad Adil KhanDepartment of Mathematics, University of Peshawar

Peshawar 25000, Pakistane-mail: [email protected]

S. Ivelic BradanovicFaculty of Civil Engineering, Architecture and Geodesy

University of SplitMatice hrvatske 15, 21000 Split, Croatia


Josip PecaricFaculty of Textile Technology, University of Zagreb

Prilaz Baruna Filipovica 30, 10000 Zagreb, Croatiae-mail: [email protected]



& Applications


ONE PROOF OF THE GHEORGHIU INEQUALITY

BOZIDAR IVANKOVIC


Abstract. The Gheorghiu inequality is a reverse Holder’s inequality. In this article, the Gheo-rghiu inequality is proven by using a property of a two – variable function. Original Gheorghiu’sresult is presented and compared with obtained result.

1. Introduction and preliminaries

A class of real function defined on a set Ω is noted with L if for any pair f ,g∈Ltheir linear combination α f + βg ∈ L for all α,β ∈ R and if L consists constantfunctions.

A linear mean E is a linear functional defined on L with property that if f (t) � 0on Ω , then E( f ) � 0 and with property that E(1) = 1. The function noted by 1presents the basic constant function with 1(t) = 1 for every t ∈ Ω .

Jensen’s inequality and McShane’s extension are given according the [8].

THEOREM 1. (Jensen) Let g1 ∈ L , such that g1(t) ∈ [a,A] ⊂ R for all t ∈ Ω .Let E be a linear mean on L . If ϕ : [a,A]→ R is a continuous concave function, thenϕ(g1) ∈ L , E(ϕ(g1)) ∈ [a,A] and

E(ϕ(g1)) � ϕ(E(g1)).

THEOREM 2. (McShane, case on rectangular) Let g1,g2 ∈ L such that(g1(t),g2(t)) ∈ D = [a,A]× [b,B] for all t ∈ Ω . Let E be a linear mean on L . Ifϕ : D → R is a continuous concave function, then ϕ(g1,g2) ∈ L , (E(g1),E(g2)) ∈ Dand

E(ϕ(g1,g2) � ϕ(E(g1),E(g2)). (1)

In [6] author proved Theorem 3 that characterized the right hand side in the Mc-Shane inequality for a measure space.

Mathematics subject classification (2010): 26D15, 26D99.Keywords and phrases: Convex functions, linear functionals, Jensen’s inequality, reverse Jessen’s in-

equality, reverse Holder’s inequality, Gheorghiu inequality.This work has been fully supported by Croatian Science Foundation under the project 5435.


1193


1194 B. IVANKOVIC

THEOREM 3. Let (Ω,Σ,μ) be a measure space such that 0 < μ(A) < 1 < μ(B) <∞ for some A,B ∈ Σ and let bijections ϕ1,ϕ2,ψ1,ψ2 : (0,∞) → (0,∞) be such thatψ1 ◦ϕ1(t)

t� c � t

ψ2 ◦ϕ2(t). If

∫Ω

xy dμ � ψ1

(∫Ω(x)ϕ1 ◦ |x| dμ

)ψ2

(∫Ω(y)ϕ2 ◦ |y| dμ

)

for all nonnegative μ -integrable simple functions x,y : Ω →R (where Ω(x) stands forthe support of x), then there exists a real p > 1 such that

ϕ1(t)ϕ1(1)

= t p,ψ1(t)ψ1(1)

= t1/p,ϕ2(t)ϕ2(1)

= tq,ψ2(t)ψ2(1)

= t1/q, t > 0,

where1p

+1q

= 1 .

A part of Csiszar and Mori conversion for (1) is given below. Complete conversionis given in [1]

THEOREM 4. Let ϕ : D → R be a concave function and suppose that ϕ(a,b)+ϕ(A,B)− ϕ(A,b)− ϕ(a,B) � 0 . If (B− b)E(g1) + (A− a)E(g2) � AB− ab, thenλE(g1)+ μE(g2)+ ν � E(ϕ(g1,g2)) � ϕ(E(g1),E(g2)), with

λ =ϕ(A,b)−ϕ(a,b)

A−a, μ =

ϕ(a,B)−ϕ(a,b)B−b

,

and

ν =AB−ab

(A−a)(B−b)ϕ(a,b)− b

B−bϕ(a,B)− a

A−aϕ(A,b).

More general conversion and refinement in Theorem 5 is proven in [2] by consid-ering the functions

Mij(t,s) =(λi + λ j)t +(μi + μ j)s+ νi + ν j

2+

|(λi −λ j)t +(μi− μ j)s+ νi−ν j|2

and

mi j(t,s) =(λi + λ j)t +(μi + μ j)s+ νi + ν j

2− |(λi −λ j)t +(μi− μ j)s+ νi −ν j|

2.

Given coefficients are λ1 = λ4 =ϕ(A,b)−ϕ(a,b)

A−a, μ1 = μ3 =

ϕ(a,B)−ϕ(a,b)B−b

;

λ2 = λ3 =ϕ(A,B)−ϕ(a,B)

A−a; μ2 = μ4 =

ϕ(A,B)−ϕ(A,b)B−b

; ν1 = ϕ(a,b)−λ1a−μ1b ;

ν2 = ϕ(A,B)−λ2A− μ2B ; ν3 = ϕ(a,B)−λ3a− μ3B and ν4 = ϕ(A,b)−λ4A− μ4b .

ONE PROOF OF THE GHEORGHIU INEQUALITY 1195

),( ba�

),( Ba�),( BA�

),( bA�

Aa

B

b ))(),(( 21 gEgE

�� )()( 21 gEgE

))(),(( 21 gEgE�

Figure 1: Conversion by Csiszar and Mori

THEOREM 5. Suppose ϕ : D → R is a continuous and concave function.If ϕ(a,b)+ ϕ(A,B)−ϕ(A,b)−ϕ(a,B)� 0 , then

M12(E(g1),E(g2)) � E(m34(g1,g2)) � E(ϕ(g1,g2)) � ϕ(E(g1),E(g2)). (2)

If ϕ(a,b)+ ϕ(A,B)−ϕ(A,b)−ϕ(a,B)� 0 , then

M34(E(g1),E(g2)) � E(m12(g1,g2)) � E(ϕ(g1,g2)) � ϕ(E(g1),E(g2)).

2. Main result and applications

Conversion of (1) by a two-variable function is given. Under the special conditionsthe Georhiu-type inequality is proven. The following general result is given in [4].

THEOREM 6. (General result) Let ϕ ,ψ : D→ R be continuous, let ϕ be concaveand for g1,g2 ∈ L let us assume that (g1(t),g2(t)) ∈ D for all t ∈ Ω . Let E be alinear mean on L . Suppose that ϕ(D) ⊆ U and ψ(D) ⊆ V and suppose that F :U ×V ⊆ R

2 → R is increasing in the first variable.If ϕ(a,b)+ ϕ(A,B)−ϕ(A,b)−ϕ(a,B)� 0 , then

min(t,s)∈D

F (M12(t,s),ψ(t,s)) � F (E(ϕ(g1,g2)),ψ(E(g1),E(g2))) .


min(t,s)∈D

F (M34(t,s),ψ(t,s)) � min(t,s)∈D

F (E(ϕ(g1,g2)),ψ(E(g1),E(g2))) .

1196 B. IVANKOVIC

),( ba�

),( Ba�),( BA�

),( bA�

Aa

B

b),( st

),(12 stM

),(34 stm

Figure 2: Conversions in Theorem 5

Proof. If ϕ(a,b)+ϕ(A,B)−ϕ(A,b)−ϕ(a,B)� 0, then by Theorem 5, inequality(2) holds. Since (E(g1),E(g2)) ∈ D , then

min(t,s)∈D

F (M12(t,s),ψ(t,s)) � F ((M12(E(g1),E(g2)),ψ(E(g1),E(g2))) .

By Theorem 5, we have that M12(E(g1),E(g2)) � E(ϕ(g1,g2)) . Since F is increasingin the first variable, we get

min(t,s)∈D

F (M12(t,s),ψ(t,s)) � F (E(ϕ(g1,g2)),ψ(E(g1),E(g2)))

and obtain the desired inequality. �

A multiplicative conversion is made by taking F (x,y) =xy

.

COROLLARY 1. Suppose that assumptions of Theorem 6 hold with ϕ(D) > 0 ad-ditionally.


min(t,s)∈D

M12(t,s)ϕ(t,s)

·ϕ(E(g1),E(g2)) � E(ϕ(g1,g2)) � ϕ(E(g1),E(g2)).

In opposite, if ϕ(a,b)+ ϕ(A,B)−ϕ(A,b)−ϕ(a,B)� 0 , then

min(t,s)∈D

M34(t,s)ϕ(t,s)

·ϕ(E(g1),E(g2)) � E(ϕ(g1,g2)) � ϕ(E(g1),E(g2)).

A conversion by medium value is given by the next lemma.

LEMMA 1. Assume that ϕ ,g1,g2 and E are as in Theorem 6. Let α,β � 0 suchthat α + β = 1.

If ϕ(a,b)+ ϕ(A,B)−ϕ(A,b)−ϕ(a,B)� 0 then

(αλ1 + β λ2)E(g1)+ (αμ1 + β μ2)E(g2)+ αν1 + β ν2 � E(ϕ(g1,g2)).

If ϕ(a,b)+ ϕ(A,B)−ϕ(A,b)−ϕ(a,B)� 0, then

(αλ3 + β λ4)E(g1)+ (αμ3 + β μ4)E(g2)+ αν3 + β ν4 � E(ϕ(g1,g2)).

Proof. Considering that

M12(E(g1),E(g2) = max{λ1E(g1)+ μ1E(g1)+ ν1,λ2E(g1)+ μ2E(g2)+ ν2},we obtain the first inequality if ϕ(a,b)+ ϕ(A,B)−ϕ(A,b)−ϕ(a,B)� 0. �

),( ba�

),( Ba�),( BA�

),( bA�

Aa

B

b))(),(( 21 gEgE

))(),(( 2112 gEgEM

212)21121 )(()()( �� gEgE

Figure 3: Conversion by a medium value

A conversion with very special condition is given bellow.

PROPOSITION 1. Suppose that assumptions of Lemma 1 hold.

(i) If ϕ(a,b)+ ϕ(A,B)−ϕ(A,b)−ϕ(a,B)� 0 and ν1 ·ν2 < 0, then

U12E(g1)+V12E(g2) � E(ϕ(g1,g2)) � ϕ(E(g1),E(g2)),

where:

U12 =ν2λ1−ν1λ2

ν2 −ν1, V12 =

ν2μ1−ν1μ2

ν2 −ν1.

1198 B. IVANKOVIC

(ii) If ϕ(a,b)+ ϕ(A,B)−ϕ(A,b)−ϕ(a,B)� 0 and ν3 ·ν4 < 0, then

U34 =ν4λ3−ν3λ4

ν4 −ν3, V34 =

ν4μ3−ν3μ4

ν4 −ν3.

Proof. Solving the system

{α + β = 1

αν1 + β ν2 = 0by α,β we obtain that αλ1+β λ2 =

U1,2 and αμ1 + β μ2 = V1,2 . �Considering the Corollary 1, the next Proposition is given.

PROPOSITION 2. Let ϕ : D→R be a continuous concave positive function, g1,g2

∈ L and a linear mean E on L .

(i) If ϕ(a,b)+ ϕ(A,B)−ϕ(A,b)−ϕ(a,B)� 0 and ν1 ·ν2 < 0, then:

min(t,s)∈D

U12t +V12sϕ(t,s)

·ϕ(E(g1),E(g2)) � E(ϕ(g1,g2)) � ϕ(E(g1),E(g2)). (3)

(ii) If ϕ(a,b)+ ϕ(A,B)−ϕ(A,b)−ϕ(a,B)� 0 and ν3 ·ν4 < 0, then:

min(t,s)∈D

U34t +V34sϕ(t,s)

·ϕ(E(g1),E(g2)) � E(ϕ(g1,g2)) � ϕ(E(g1),E(g2)),

where values U12 V12 , U34 and V34 are given in Proposition 2.

Gheorghiu’s type inequality is a converse of Holder’s type inequality. The originalGheorghiu inequality from [9] will be presented in the next section. Here the proof fora refinement is presented.

THEOREM 7. Let E be a linear mean on L and for g1,g2 ∈ L let us assumethat g1(Ω) ⊂ [a,A] and g2(Ω) ⊂ [b,B] for positive real numbers a,b. Let p,q be

positive real numbers such that1p

+1q

= 1 holds. Then

p1p q

1q (abAB)

1pq

((AB)

1p − (ab)

1p

) 1p((AB)

1q − (ab)

1q

) 1q

AB−ab· (E(g1))

1p (E(g2))

1q

� E(

g1p1 ·g

1q2

)� (E(g1))

1p (E(g2))

1q . (4)

Proof. The function ϕ(x,y) = x1p y

1q is continuous, concave and positive for all

(x,y) ∈ [a,A]× [b,B] . Because(A

1p −a

1p

)(B

1q −b

1q

)> 0, for application of Propo-

sition 2 it is enough to prove that

ν1 = a1p b

1q −ab

1qA

1p −a

1p

A−a−a

1p b

B1q −b

1q

B−b� 0


and

ν2 = A1p B

1q −AB

1qA

1p −a

1p

A−a−A

1p B

B1q −b

1q

B−b� 0.

Since the function f (x) = x1p is concave for p > 1, f ′(x) is continuous and de-

creasing. So there exists c ∈ [a,A] such that f ′(c) =A

1p −a

1p

A−aand f ′(a) � f ′(c) �

f ′(A) which gives

1pa

1p−1 � A

1p −a

1p

A−a� 1

pA

1p−1

Multiplying with ab1q and AB

1q we get

a1p b

1q

p� ab

1qA

1p −a

1p

A−aand AB

1qA

1p −a

1p

A−a� A

1p B

1q

p.

Similar consideration on f (x) = x1q gives

1qb

1q−1 � B

1q −b

1q

B−b� 1

pA

1p−1.

Multiplying with a1p b and A

1p B we get

a1p b

1q

q� a

1p b

B1q −b

1q

B−band A

1p B

B1q −b

1q

B−b� A

1p B

1q

q.

Now we have

ν1 = a1p b

1q −ab

1qA

1p −a

1p

A−a−a

1p b

B1q −b

1q

B−b

ν1 � a1p b

1q − a

1p b

1q

p− a

1p b

1q

q= a

1p b

1q

(1− 1

p− 1

q

)= 0.

ν2 = A1p B

1q −AB

1qA

1p −a

1p

A−a−A

1p B

B1q −b

1q

B−b

ν2 � A1p B

1q − A

1p B

1q

p− A

1p B

1q

q= A

1p B

1q

(1− 1

p− 1

q

)= 0.

Note that (4) is equal to (3) by coefficients

U12 =B

1q b

1q

((AB)

1p − (ab)

1p

)AB−ab

and V12 =A

1p a

1p

((AB)

1q − (ab)

1q

)AB−ab

.

1200 B. IVANKOVIC

It is necessary to minimize min(t,s)∈D

U12t +V12s

t1p s

1q

= min(t,s)∈D

(U12 ·

( ts

) 1q+V12 ·

(st

) 1p

).

Supstitution z =ts

and easy calculus gives min(t,s)∈D

U12t +V12sϕ(t,s)

= U1p

12V1q

12p1p q

1q . Sub-

stituting the all above in (3) we get (4) and the proof is done. �

REMARK 1. Using substitutions g1 � gp1 and g1 � gp

1 in the previous Theoramwe get the following

p1p q

1q (AbBq−abqB)

1p (ApaB−apbA)

1q

ApBq−apbq · (E(gp1))

1p (E(gq

2))1q

� E(g1 ·g2) � (E(gp1))

1p (E(gq

2))1q . (5)

Normalized Gheorghiu inequality in the case that (Ω, p,F ) is a probability spaceis given in [5]. Functions g1 = X and g2 = Y are random variables and E(g1) = E[X ]is the mathematical expectation of random variable X .

COROLLARY 2. Suppose that random variables X and Y capture their values

0 < α � X � 1 and 0 < β � Y � 1 . Equality1p

+1q

= 1 implies

p1p q

1q (β −αβ q)

1p (α −α pβ )

1q

1−α pβ q (E[X p])1p (E[Yq])

1q � E[XY ] � (E[X p])

1p (E[Yq])

1q

in the case of positive p and q.

3. Original Gheorghiu’s inequality

In the article [9], the converse of Holder’s inequality was obtained. Here it ispresented in the next theorem.

THEOREM 8. Suppose that a1,a2, . . .an,b1,b2, . . .bn are given 2n positive realnumbers. Let pair (a,A) represents the minimal and maximal number among a1,a2, . . . ,an and in the same manner let (b,B) be the pair of those among the b1,b2, . . . ,bn . As-sume that p is a real number greater than 1. Then we have

1 �

(∑n

k=1 apk

)(∑n

k=1 bp

p−1k

)p−1

(∑nk=1 akbk)

p � μ , (6)

where

μ =(p−1)p−1

pp · Ap−1

ap−1 · Bb·

(1− apb

pp−1

ApBp

p−1

)p

(1− ab

1p−1

AB1

p−1

)(1− ap−1b

Ap−1B

)p−1. (7)


The left inequality has been demonstrated by Holder and Jensen. Theorem 8 couldbe modulated in the terms that are given in the introduction of this paper.

THEOREM 7’. Suppose that Ω = {1,2,3, . . . ,n} and g1,g2 : Ω → R are givenreal functions. Let a = min{g1(k),k ∈Ω} , A = max{g2(k),k ∈Ω} , b = min{g2(k),k ∈Ω} and B = max{g2(k),k ∈ Ω} . Let E(g) =

1n

n

∑k=1

g(k) . If p,q > 1 ,1p

+1q

= 1 , then

1 � E(gp1)

1p ·E(gq

2)1q

E(g1 ·g2)� μ

1p , (8)

where μ is given by (7).

Inequality (8) can be expressed as the chain of inequalities alike the (5):

μ− 1p ·E(gp

1)1p ·E(gq

2)1q � E(g1 ·g2) � E(gp

1)1p ·E(gq

2)1q (9)

The next proposition shows that left inequality in (5) is better than the left inequality in(9).

PROPOSITION 3. Under the assumptions of Theorem 7’, the next is valid:

p1p q

1q (AbBq−abqB)

1p (ApaB−apbA)

1q

ApBq−apbq = p ·μ− 1p . (10)

Proof. Using elementary algebra we get

μ− 1p =

q1q a

1q b

1p

(AB

qp −ab

qp

) 1p(A

pq B−a

pq b) 1

q

p1q A

pq2 +1−p

Bqp2 +1−q

(ApBq−apbq).

Separately, using relation p−1 =pq

, we havepq2 +1− p =−1

qand

qp2 +1−q =− 1

p.

The proof is prolonging with

μ− 1p =

q1q A

1q a

1q B

1p b

1p

(AB

qp −ab

qp

) 1p(A

pq B−a

pq b) 1

q

p1q (ApBq−apbq)

.

By selective multiplying factors and brackets with the same exponent we have

μ− 1p =

q1q

(AbB

qp+1 −ab

qp+1B

) 1p(aA

pq +1B−a

pq +1bA

) 1q

p1q (ApBq−apbq)

.

Considering thatqp

+1 = q andpq

+1 = q we finally obtain that

μ− 1p =

p1p q

1q (AbBq−abqB)

1p (aApB−apbA)

1q

p(ApBq−apbq).

The last equation is the same as (10) and the proof is finished. �

1202 B. IVANKOVIC

RE F ER EN C ES

[1] V. CSISZAR AND T. F. MORI, The convexity method of proving moment-type inequalities, Statisticsand Probability Letters 66, 1 (2004), 303–313.

[2] V. CULJAK, B. IVANKOVIC AND J. E. PECARIC, On Jensen-McShane’s inequality, Periodica Math-ematica Hungarica 58, 2 (2009), 139–154.

[3] B. IVANKOVIC, S. IZUMINO, J. E. PECARIC AND M. TOMINAGA, On Jensen-McShane’s inequality,Journal of Inequalities in Pure and Applied Mathematics 8, 3 (2007), 88–97.

[4] B. IVANKOVIC, Konverzije i poboljsanja Jensenove nejednakosti za funkcije vise varijabli, Doctoralthesis Zagreb, 2010 (croatian).

[5] S. IZUMINO AND M. TOMINAGA, Estimations in Holder’s type inequalities, Mathematical Inequali-ties and Applications 4, 1 (2001), 163–187.

[6] J. MATKOWSKI, A Converse of the Holder Inequality Theorem Mathematical Inequalities and Appli-cations, 12, 1 (2009), 21–32.

[7] D. MITRINOVIC, J. E. PECARIC AND A. M. FINK, Classical and New Inequalities in Analysis,Kluwer. Acad. Pub., Boston Dordrecht, 1993.

[8] J. E. PECARIC, F. PROSCHAN, AND Y. L. TONG, Convex Functions, Partial Orderings, and Statisti-cal Applications, Academic Press, Inc. San Diego, 1992.

[9] M. SERBAN AND A. GHEORGHIU, Note sur une Inegalite de Cauchy, Bull. Math. Soc. Rommainedes Sciences 35, 1 (1933), 117–119.

(Received January 28, 2016) Bozidar IvankovicFaculty of Transport and Traffic Sciences

University of ZagrebZagreb, Croatia




& Applications


JENSEN, HOLDER, MINKOWSKI, JENSEN–STEFFENSEN

AND SLATER–PECARIC INEQUALITIES

DERIVED THROUGH N –QUASICONVEXITY

SHOSHANA ABRAMOVICH

In honour of ProfessorsNeven Elezovic, Marko Matic and Ivan Peric

on the occasion of their 60th birthdays


Abstract. Jensen, Holder, Minkowski, Jensen-Steffensen and Slater-Pecaric type inequalities de-rived by the properties of γ -quasiconvex functions that we deal with here, can be seen as analogto these for superquadratic functions and refinements of these for convex functions.

1. Introduction

We deal here with inequalities satisfied by one of the many variants of convexfunctions. These functions are called γ -quasiconvex functions and have already beendealt with by S. Abramovich, L.-E. Persson and N. Samko. The basic facts on γ -quasiconvexity and superquadracity on which this paper is built, can be found in [4],[6], and [7].

The importance of convex functions is obvious and widely acknowledged. Nu-merous publications deal with convex functions, their properties and applications. Inparticular we refer to the classical 1964 book “Inequalities” by Hardy, Littlewood andPolya [9], the 1992 book “Convex functions, partial ordering and statistical applica-tions” by Pecaric, Proschan and Tong [15] and to the 2006 book “Convex functions andtheir applications – a contemporary approach” by Niculescu and Persson [13]. Out ofdealing with the classical convex functions evolved many generalizations and refine-ments of this notion, see in particular Capter 2 in [13].

Among the many types of refinements and generalizations of convex functions arethe usual quasiconvexity,Morrey-convexity,Reitz-convexity, h-convexity, superquadrac-ity and many others.

The subject of variants of convex functions and the comparison between them de-serves at least every decade a large updated suvey which is out of the scope of this

Mathematics subject classification (2010): 26D15.Keywords and phrases: Jensen’s inequality, Holder’s inequality, Minkowski inequality, convex func-

tion, γ -quasiconvex functions, N -quasiconvex functions, superquadratic functions.


1203


1204 S. ABRAMOVICH

paper. We compare here results obtained through the use of γ -quasiconvexity and su-perquadracity as these notions are the subjects of this paper, see Remark 1 and Theo-rem 4.

The notion of γ -quasiconvexity with which we deal here is related but is not aspecial case of any of the cases mentioned above. Therefore it is reasonable to assumethat our new and natural notion of γ -quasiconvexity will bring about new results andapplications.

Currently the following is already known: The original Hardy’s inequality has a“turning point” (the point where the inequality is reversed) at p = 1. This inequalitycan be proved directly by the properties of convex functions. (The proof can be foundin [16] and its references.) But by using the γ -quasiconvexity we get a refined variantof the original Hardy’s inequality where the turning point is any p > 1 (see [6] and [7]).

It is known that most of the classical inequalities can be obtained by the propertiesof convex functions, therefore it is reasonable to assume that using the properties ofγ -quasiconvexity will bring about proofs of generalizationd and refinements of moreclassical inequalities.

We know that if f := R+ → R

+ is a concave function then f (x)x is not increasing

(see for instance [11, page 142] and [17]). This is one of the reasons it is natural todeal with quasi-monotone functions, that is with γ -quasidecreasing functions. Aboutthe importance of this notion especially to theories related to approximation and inter-polations see [11].

The notion of a γ -quasiconvex function is analog to a quasimonotone function(that is to γ -quasiincreasing functions). Therefore we hope to get in future publicationsanalog results to those we know about quasimonotone functions, in addition to thosementioned above related to Hardy’s inequalities and to those dealt with in this paperwhich are related in particular to Jensen, Holder and Slater Pecaric inequalities.

γ -quasiconvex functions and superquadratic functions are closely related and there-fore it is interesting to show side by side results related to these two sets.

We start with a definition of and lemmas about γ -quasiconvexity.

DEFINITION 1. Let γ be a real number. A real-valued function f defined on aninterval [0,b) with 0 < b � ∞ is called γ -quasiconvex if it can be represented as theproduct of a comvex function and the power function xγ .

A convex function ϕ on [0,b) , 0 < b � ∞ is characterized by the inequality

ϕ(y)−ϕ(x) � Cϕ (x) (y− x), ∀x,y ∈ (0,b], Cϕ ∈ R, (1.1)

from which we establish easily the following lemmas:

LEMMA 1. [6, Lemma1] Let ψγ (x) = xγ ϕ (x) , γ ∈ R, where ϕ is convex on[0,b) , that is, ψγ is a γ -quasiconvex function. Then

ψγ (y)−ψγ (x) � ϕ (x) (yγ − xγ)+Cϕ (x)yγ (y− x) , (1.2)

holds for all x ∈ [0,b) , y ∈ [0,b) , where Cϕ (x) is defined by (1.1).

INEQUALITIES DERIVED FROM N -QUASICONVEXITY 1205

The following is derived by some computation on the right handside of (1.2), seealso [7, Lemma 2]:

LEMMA 2. [7] Let ϕ be convex differentiable function and let ψk (x) = xkϕ (x) ,k = 0,1, ...,N, then the function ψN (x) = xNϕ (x) , satisfies for x,y ∈ [a,b) , a � 0

ψN (y)−ψN (x) (1.3)

� (ψN (x))′ (y− x)+ (y− x)2N

∑k=1

yk−1 (ψN−k (x))′

= (ψN (x))′ (y− x)+ (y− x)2∂∂x

(xN − yN

x− yϕ (x)

).

Now we quote a definition and some basic properties of superquadratic functions.

DEFINITION 2. [4, Definition 2.1] A function ϕ : [0,∞) → R is superquadraticprovided that for all x � 0 there exists a constant C(x) ∈ R such that

ϕ (y)−ϕ (x)−ϕ (|y− x|) � C (x)(y− x) (1.4)

for all y � 0.

From this definition we get that when ϕ is a superquadratic function, if ϕ � 0,then ϕ is convex and ϕ(0) = ϕ ′(0) = 0, see [4].

When ϕ : [0,b)→R is differentiable non-negative, incresing convex and ϕ(0) = 0the function ψN (x) = xNϕ (x) is not only N -quasiconvex where N is a non-negativeinteger but also superquadratic. In particular the power functions f (x) = xp, p � 2,x � 0 are superquadratic functions as well as 1-quasiconvex functions. The powerfunctions f (x) = xp, p � N +1, N � 1, x � 0, are also N -quasiconvex functions.

In Section 2 we deal with Jensen’s type and Slater-Pecaric type inequalities whenthe coefficients αi � 0, i = 1, ...,n. In Section 5 we deal with inequalities for whichthe coefficients are not always non-negative. We call these coefficients Steffensen’scoefficients. For such coefficients and for a function ϕ we get:

LEMMA 3. Let ϕ : [0,∞) → R be a given function, let xxxxx be a nonnegative mono-tonic n-tuple in R

n, and ρρρρρ a real n-tuple satisfying Steffensen’s coefficients, that is

0 � Pj � Pn, j = 1, ...,n, Pn > 0, (1.5)

Pj =j

∑i=1

ρi, Pj =n

∑i= j

ρi, j = 1, ...,n

Then

n

∑i=1

ρiϕ (xi) =k−1

∑j=1

Pj(ϕ (x j)−ϕ

(x j+1

))+Pkϕ (xk) (1.6)

+Pk+1ϕ (xk+1)+n

∑j=k+2

Pj(ϕ (x j)−ϕ

(x j−1

)).

1206 S. ABRAMOVICH

Identity (1.6) is used in the proofs in Section 5 related to N -quasiconvex functionsin a similar way as they are used in [15] and [1] for convex functions, in [2] for su-perquadratic functions and in [7] for 1-quasiconvex functions.

By using the results stated in Section 2 we get in Section 3 Holder’s type inequal-ities which are of the type

∫f gdν ≶

(∫gqdν

)1/q(∫f pdν

)1/p

H ( f ,g)

that lately are widely discussed (see for instance [10], [12], [14], [19] and their refer-ences).

In Section 4 we prove Minkowski type inequalities by using again the results statedin Section 2.

In Section 5 we get more inequalities which are derived from the results fromSection 2.

In Section 6 we get inequalities related to differences of “Jensen’s gap” motivatedby the work of Dragomir in [8]. The results in this section are analog to the results in[3].

2. Jensen and Slater-Pecaric type inequalities for N -quasiconvex functions

We quote first extensions of Jensen and of Slater-Pecaric inequalities for super-quadratic functions which are proved in [4] and stated in Lemma A and in TheoremB.

LEMMA A. [4, Lemma 2.3] Supppose that ψ is superquadratic on [0,b) then

∫Ω

ψ ( f (s))dμ (s)−ψ(∫

Ωf (s)dμ (s)

)�∫

Ωψ(∣∣∣∣ f (s)−

∫Ω

f (σ)dμ (σ)∣∣∣∣)

dμ (s) ,

(2.1)where f is any non-negative μ -integrable function on a probability measure space(Ω,μ) and

∫Ω f (s)dμ (s) > 0 .

The discrete version of (2.1) is:Suppose that ψ is superquadratic on [0,b) . Let 0 � xi < b, i = 1, ...,n and let

x = ∑ni=1 αixi where αi � 0 and ∑n

i=1 αi = 1. Then

n

∑i=1

αiψ (xi)−ψ (x) �n

∑i=1

αiψ (|xi − x|) . (2.2)

LEMMA B. [4, Theorem 2.4] Suppose that ψ is superquadratic and that C ( f (s))is given as in Definition 2. If μ is a probability measure, f is any non-negative μ -measurable function,

∫C ( f (s))dμ (s) �= 0, and m and M as defined by

m =∫

f (s)dμ (s) and M =∫

f (s)C ( f (s))dμ (s)∫C ( f (s))dμ (s)

.

INEQUALITIES DERIVED FROM N-QUASICONVEXITY 1207

then

ψ (m)+∫

ψ (| f (s)−m|)dμ (s)

�∫

ψ ( f (s))dμ (s)

� ψ (M)−∫

ψ (| f (s)−M|)dμ (s) .

The discrete version is: Suppose that ψ is superquadratic and C is as in Definition2. Let xi � 0, i = 1, ...,n and let αi � 0 , ∑n

i=1 αi = 1. If ∑ni=1 αiC (xi) �= 0 we define

M = ∑ni=1 αixiC(xi)

∑ni=1 αiC(xi)

. Then

n

∑i=1

αiψ (xi) � ψ (M)−n

∑i=1

αiψ (|xi−M|) ,

The following Theorem 1 may be considered an analog of lemmas A and B. Init we get refinements of Jensen’s inequality and Slater-Pecaric inequality (see [1] and[15]). The refinements are obtained just by using (1.3) in Lemma 2 for each i and thensumming up for i = 1, ...,n .

THEOREM 1. Let ϕ : [a,b) → R, a � 0 be convex differentiable function, andlet ψk (x) be ψk (x) = xkϕ (x) , k = 0,1, ...,N, where ψ0 = ϕ . Let αi � 0, xi ∈ [a,b) ,i = 1, ...,n, ∑n

i=1 αi = 1 . Then:1) A Jensen’s type inequality holds where x = ∑n

i=1 αixi :

n

∑i=1

αiψN (xi)−ψN (x) (2.3)

�n

∑i=1

αiϕ (x)(xNi − xN)+ n

∑i=1

αiϕ′(x)xN

i (xi − x)

=n

∑i=1

N

∑k=1

αi (xi − x)2 xk−1i (ψN−k (x))′

=n

∑i=1

αi (xi − x)2 ∂∂x

(xN − xN

i

x− xiϕ (x)

).

If ϕ is also non-negative and increasing then for N = 2, ..., the above inequality refinesJensen’s inequality. For N = 1 we get for ψ1 (x) = xϕ (x)

n

∑i=1

αiψ1 (xi)−ψ1 (x) �n

∑i=1

αiϕ′(x)xi (xi − x) =

n

∑i=1

αiϕ′(x) (xi − x)2 . (2.4)

If ϕ is increasing and convex (and not necessarily non-negative) then again (2.4) is arefinement of Jensen’s inequality.

1208 S. ABRAMOVICH

2) For a fixed C ∈ [a,b) we get when αi � 0, i = 1, ...,n, ∑ni=1 αi = 1 that

CNϕ (C)−n

∑i=1

αixNi ϕ (xi)

= ψN (C)−n

∑i=1

αiψN (xi)

�n

∑i=1

αi(xNi ϕ (xi)

)′(C− xi)+

n

∑i=1

αi (C− xi)2

N

∑k=1

Ck−1 (ψN−k (xi))′

=n

∑i=1

αi(xNi ϕ (xi)

)′(C− xi)+

n

∑i=1

αi (C− xi)2 ∂

∂xi

(xNi −CN

xi−Cϕ (xi)

).

3) Especially if ∑ni=1 αiψ ′

N (xi) > 0, and if C = MψN = ∑ni=1 αixiψ ′

N (xi)∑n

i=1 αiψ ′N(xi)

∈ [a,b) , then

by using ∑ni=1 αiψ ′

N (xi)(MψN − xi

)= 0 we get a Slater-Pecaric type inequality

ψN(MψN

)− n

∑i=1

αiψN (xi)

�n

∑i=1

N

∑k=1

αi(MψN − xi

)2Mk−1

ψN(ψN−k (xi))

′

=n

∑i=1

αi(MψN − xi

)2 ∂∂xi

(MN

ψN− xN

i

MψN − xiϕ (xi)

).

If ϕ is also non-negative and increasing then for N = 1, ... the above inequality is arefinement of Slater Pecaric inequality.

Theorem 1 Case 1 appears in [5, Corollary 1].We get in [7, Theorem 1] the integral form of Jensen’s type inequality for γ -

quasiconvex functions and the special case when γ = 1 is:

LEMMA C. [7] Let f be a non-negative function. Let f and ϕ ◦ f be μ -integrable functions on the probability measure space (Ω,μ) and

∫Ω f (s)dμ (s) > 0.

Let also ψ (x) = xϕ (x) . If ϕ is a differentiable convex on [0,b) , 0 < b � ∞∫Ω

ψ ( f (s))dμ (s)−ψ(∫

Ωf (s)dμ (s)

)

�∫

Ωϕ ′(∫

Ωf (σ)dμ (σ)

)(f (s)−

∫Ω

f (σ)dμ (σ))2

dμ (s) .

hold. If ϕ is also increasing we get a refinement of Jensen’s inequality.

EXAMPLE 1. Let ϕ (x) = ex3, ψ (x) = xex3

then from the convexity of ψ we get

that∫ 10 ψ (x)dx � e

182 and from the 1-quasiconvexitywe get the better result

∫ 10 ψ (x)dx

� 5e18

8 .

REMARK 1. In [7, Proposition 5] it is proved that: Let f be a non-negative func-tion. Let f and ϕ ◦ f be μ -integrable functions on the probability measure space(Ω,μ) and

∫Ω f (s)dμ (s) > 0. Let also ψ (x) = xϕ (x) . If ϕ is a differentiable non-

negative convex increasing on [0,b) , 0 < b � ∞ and ϕ (0) = limz→0+

zϕ ′(z) = 0 then ψ

is also superquadratic and the inequalities∫Ω

ψ ( f (s))dμ (s)−ψ(∫

Ωf (s)dμ (s)

)

�∫

Ωϕ ′(∫

Ωf (σ)dμ (σ)

)(f (s)−

∫Ω

f (σ)dμ (σ))2

dμ (s)

�∫

Ωψ(∣∣∣∣ f (s)−

∫Ω

f (σ)dμ (σ)∣∣∣∣)

dμ (s) ,

hold when 0 < f (s) � 2∫

Ω f (σ)dμ (σ) for every s ∈ Ω, in particular when 0 < a �f (s) � 2a, s ∈ Ω.

The discrete form says there that: when 0< xi � 2x, i = 1, ...,n and ψ (x) = xϕ (x)where ϕ (x) is non-negative increasing differentiable and convex then Inequality (2.4)is better than (2.2) when ϕ (0) = lim

z→0+zϕ ′

(z) = 0.

3. Holder type inequalities derived from γ -quasiconvexity and superquadracity

In this section we use Jensen’s type inequalities to prove new Holder type inequal-ities and reversed Holder type inequalities. We use in particular Lemma A, Lemma Cand the following lemmas D and E to get refinements for p � 2 of Holder inequality,lower bounds for 1 0.

Then

−I1 +(∫

Ωf (s)dμ (s)

)p

�∫

Ω( f (s))p dμ (s) �

(∫Ω

f (s)dμ (s))p

,

where

I1 = p

(∫Ω

f (s)dμ (s))p(

1−∫

Ωf (s)dμ (s)

∫Ω

( f (s))−1 dμ (s))

> 0.

LEMMA E. [7, Corollary 2] Let 0 0. Then

− I2 +(∫

Ωf (s)dμ (s)

)p

�∫

Ω( f (s))p dμ (s) �

(∫Ω

f (s)dμ (s))p

, (3.1)

where

I2 = p

(∫Ω

f (s)dμ (s))p−1 ∫

Ω

( f (s)− x)2

f (s)dμ (s) . (3.2)

1210 S. ABRAMOVICH

As the power functions ϕ (x) = xp, x � 0 are superquadratic when p � 2 andsubquadratic when 1 � p � 2, we get from Lemma A that for p � 2

∫Ω

( f (s))p dμ (s)−(∫

Ωf (s)dμ (s)

)p

�∫

Ω

(∣∣∣∣ f (s)−∫

Ωf (σ)dμ (σ)

∣∣∣∣)p

dμ (s) ,

(3.3)holds, where f is any non-negative μ -integrable function on a probability measurespace (Ω,μ) and

∫Ω f (s)dμ (s) > 0.

In [18, Theorem 1.4] a refinement of Holder’s inequality is proved:

THEOREM 2. For p � 2 and for any two non-negative ν -measurable functions fand g and for 1

p + 1q = 1 we get a refinement of Holder inequality

∫Ω

f gdν �(∫

Ωf pdν −

∫Ω

∣∣∣∣ f −gq−1∫

Ω f gdν∫Ω gqdν

∣∣∣∣p dν) 1

p(∫

Ωgqdν

) 1q

=(∫

Ωf pdν −

∫Ω

∣∣∣∣ f g1−q−∫


∣∣∣∣p gqdν) 1

p(∫

Ωgqdν

) 1q

.

In the case 1 < p � 2 we get for any two non-negative ν -measurable functions f and

g when∫

Ω f pdν �∫

Ω

∣∣∣ f −gq−1∫


∣∣∣p dν, that

(∫Ω

f pdν) 1

p(∫

Ωgqdν

) 1q

�∫

Ωf gdν �

(∫Ω

f pdν −∫

Ω

∣∣∣∣ f −gq−1∫


∣∣∣∣p dν) 1

p(∫

Ωgqdν

) 1q

.

From Lemma C we get that for the 1-quasiconvex functions ϕ (x) = xp, x � 0,p � 2 the inequality

∫Ω

( f (s))p dμ (s)−(∫

Ωf (s)dμ (s)

)p

(3.4)

� (p−1)(∫

Ωf (s)dμ (s)

)p−2∫Ω

(f (s)−

∫Ω

f (s)dμ (s))2

dμ (s)

holds.Theorem 3, which is another refinement of Holder inequality, follows in the same

way that Holder’s inequality follows from Jensen’s inequality by fixing a non-negativeν -measurable functions f and g and applying (3.4) with f g1−q in place of f and

gqdν∫Ω gqdν in place of dμ where 1

p + 1q = 1:

THEOREM 3. Let p � 2 and define q by 1p + 1

q = 1 . Then for any two nonnegativeν -measurable functions f and g∫

Ωf gdν (3.5)

�(∫

Ωf pdν − (p−1)

(∫Ω f gdν∫Ω gqdν

)p−2∫Ω

(f g(1−q)−

∫Ω f gdν∫Ω gqdν

)2

gqdν

) 1p

×(∫

Ωgqdν

) 1q

.

If 1 < p � 2 we get when∫

Ω f pdν � (p−1)(∫


)p−2 ∫Ω

(f g(1−q)−


)2gqdν,

that (∫Ω

f pdν) 1

p(∫

Ωgqdν

) 1q

(3.6)

�∫

Ωf gdν

�(∫

Ωf pdν − (p−1)


)p−2∫Ω

(f g(1−q)−


)2

gqdν

) 1p

×(∫

Ωgqdν

) 1q

.

The last inequalities emphasize that through the 1 -quasiconvexity and 1 -quasiconcavitynotions we get refined Holder inequality for p � 2 in (3.5) and a lower bound in (3.6)for 1 < p � 2.

From Remark 1 it follows that:

THEOREM 4. Under the same conditions as in Theorems 2 and 3 we get that therefinement of Holder inequality derived from the 1-quasiconvexity of xp , x � 0 , p � 2 is

better that the refinement derived from its superquadracity when 0 � f g(1−q) � 2∫


that is we get that

∫Ω

f gdν �(∫

Ωf pdν −Δ1

) 1p(∫

Ωgqdν

) 1q

�(∫

Ωf pdν −Δ2

) 1p(∫

Ωgqdν

) 1q

,

where

Δ1 = (p−1)(∫


)p−2∫Ω

(f g(1−q)−


)2

gqdν

1212 S. ABRAMOVICH

and

Δ2 =∫

Ω

∣∣∣∣ f g(1−q)−∫


∣∣∣∣p gqdν.

From Lemma E we get a two sided Holder type inequality:

THEOREM 5. Let 0 < p � 1 , f and g be non-negative μ -measurable functionson the probability measure space (Ω,ν) then

(∫Ω

f pdν) 1

p(∫

Ωgqdν

) 1q

(3.7)

�∫

Ωf gdν

�(∫

Ωf pdν + p


)p−1∫Ω

(f g(1−q)−


)2 g2q−1

fdν

) 1p

×(∫

Ωgqdν

) 1q

.

Proof. To get a refinement of Holder inequality, from Lemma E we fix as before anon-negative ν -measurable functions f and g and apply (3.1) and (3.2) with f g1−q inplace of f and gqdν∫

Ω gqdν in place of dμ where 1p + 1

q = 1 and get the right side of (3.7)by a simple computation, and together with Holder inequality for 0 < p � 1 which saysthat (∫

Ωf pdν

) 1p(∫

Ωgqdν

) 1q

�∫

Ωf gdν

(3.7) is obtained. �Similarly we get from Lemma D that

THEOREM 6. Let 0 < p � 1, f and g be non-negative μ -measurable functionson the probability measure space (Ω,ν) , then∫

Ωf gdν

�(∫

Ωf pdν + p


)p(∫Ω

gqdν −∫


∫Ω

g2q−1

fdν)) 1

p

×(∫

Ωgqdν

) 1q

.

Holder type inequality for 0 < p � 12 and for 1

2 � p < 1 which we get now arederived again from the theorems related to 1-quasiconvex functions but are obtainedby different substitutions that those employed up to now.

THEOREM 7. Let 0 < p � 12 and define 1

p + 1q = 1. then for any positive ν -

measurable function f and g

∫Ω

f gdν �(∫

Ωf pdν

) 1p(∫

Ωgqdν

) 1q

(3.8)

×[1+(

1p−1

)∫Ω

(f p ∫

Ω gqdν −gq ∫Ω f pdν∫

Ω f pdν

)2 g−q∫Ω gqdν

dν

]

is derived, which is a refinement of Holder inequality.For 1

2 � p < 1, we get the reverse of inequality (3.8) and together with Holderinequality for 0 < p < 1(∫

Ωgqdν

) 1q(∫

Ωf pdν

) 1p

(3.9)

�∫

Ωf gdν

�(∫

Ωgqdν

) 1q(∫

Ωf pdν

) 1p

×[1+(

1p−1

)∫Ω

(f p ∫

Ω gqdν −gq ∫Ω f pdν∫

Ω f pdν

)2 g−q∫Ω gqdν

dν

]

is derived.

Proof. For simplicity we denote∫

Ω as∫

. For 1p � 2 we use the inequality for the

1-quasiconvex function x1p

∫f

1p dμ �

(∫f dμ

) 1p(

1+(

1p−1

)∫ (f − ∫ f dμ∫

f dμ

)2

dμ

). (3.10)

We fix now non-negative ν measurable functions f and g and apply (3.10) with f pg−q

in place of f and dμ = gqdν∫gqdν . Therefore

∫f dμ is replaced by

∫f pdν∫gqdν ,

∫f

1p dμ is

replaced by∫

f gdν∫gqdν , and apply (3.10) and get

∫f gdν∫gqdν

�(∫

f pdν∫gqdν

) 1p

⎡⎢⎣1+

(1p−1

)∫ ⎛⎝ f pg−q−∫

f pdν∫gqdν∫

f pdν∫gqdν

⎞⎠2

gq∫gqdν

dν

⎤⎥⎦ . (3.11)

from which (3.8) is obtained.

The proof of (3.9) is similar using the 1-quasiconcavity of x1p , 1

2 � p < 1. �

Similarly, using the superquadracity of x1p , x � 0, 1

p � 2 and the subquadracity

of x1p , 1 � 1

p � 2 and under the same condition as in 7 we get in a similar way when

1214 S. ABRAMOVICH

0 < p � 12 the inequality

∫f gdν �

(∫f pdν

) 1p(∫

gpdν) 1

q

×[1+

∫ ∣∣∣∣ f p ∫ gqdν −gq ∫ f pdν∫f pdν

∣∣∣∣1p gdv∫

gpdν

]

and the reverse inequality holds when 12 � p < 1, and together with Holder inequality

for 0 < p < 1 we get

(∫f pdν

) 1p(∫

gpdν) 1

q

�∫

f gdν

�(∫

f pdν) 1

p(∫

gpdν) 1

q

×[1+

∫ ∣∣∣∣ f p ∫ gqdν −gq ∫ f pdν∫f pdν

∣∣∣∣1p gdv∫

gpdν

]

4. Minkowski type inequalities using 1 -quasiconvexity

By using Theorem 3 we get Minkowski type inequalities:

THEOREM 8. Let p � 2 and let 1q = 1− 1

p . Then for any two non-negative ν -measurable functions f and g

(∫( f +g)p dν

) 1p

�(∫

f pdν −D

(∫f ( f +g)p−1 dν

)p−2) 1

p

(4.1)

+

(∫gpdν −D

(∫g( f +g)p−1 dν

)p−2) 1

p

where

D = (p−1)

⎛⎜⎝∫

⎛⎜⎝(g∫

f ( f +g)p−1 dν − f∫

g( f +g)p−1 dν)2

( f +g)p−2

(∫

( f +g)p dν)p

⎞⎟⎠dν

⎞⎟⎠ .

(4.2)

Proof. Inequality (4.1) follows from inequality (3.5) in the same way that Min-kowski’s inequality follows from Holder’s and as Minkowski’s inequality for super-quadratic functions xp, p � 2, x � 0 follows from Holder’s inequality for superquad-ratic functions (see [18]).

Let p � 2 and apply (3.5) with g replaced by ( f +g)p−1 and we get

∫f ( f +g)p−1 dν �

(∫( f +g)p dν

) 1q

×⎡⎣∫ f pdν − (p−1)

(∫f ( f +g)p−1 dν∫

( f +g)p dν

)p−2

×∫ (

ff +g

−∫

f ( f +g)p−1 dν∫( f +g)p dν

)2

( f +g)p dν

⎤⎦

1p

.

Interchanging the roles of f and g yields

∫g( f +g)p−1 dν �

(∫( f +g)p dν

) 1q

⎡⎣∫ gpdν − (p−1)

(∫g( f +g)p−1 dν∫

( f +g)p dν

)p−2

×∫ (

gf +g

−∫

g( f +g)p−1 dν∫( f +g)p dν

)2

( f +g)p dν

⎤⎦

1p

.

Adding the last two inequalities gives after simple computation Inequality (4.1). �The following Theorem 9 follows from inequality (3.6) by a similar argument as

Theorem 8 follows from inequality (3.5).

THEOREM 9. Let 1 < p � 2 and let 1q = 1− 1

p . Then for any two non-negativeν -measurable functions f and g

(∫f pdν

) 1p

+(∫

gpdν) 1

p

�(∫

( f +g)p dν) 1

p

(4.3)

�(∫

f pdν −D

(∫f ( f +g)p−1 dν

)p−2) 1

p

+

(∫gpdν −D

(∫g( f +g)p−1 dν

)p−2) 1

p

where

D = (p−1)

⎛⎜⎝∫

⎛⎜⎝(g∫

f ( f +g)p−1 dν − f∫

g( f +g)p−1 dν)2

( f +g)p−2

(∫

( f +g)p dν)p

⎞⎟⎠dν

⎞⎟⎠ .

(4.4)

and∫

f pdν � D(∫

f ( f +g)p−1 dν)p−2

,∫

gpdν � D(∫

g( f +g)p−1 dν)p−2

.

1216 S. ABRAMOVICH

Now we get Minkowski’s type inequalities when 0 < p � 12 and when 1

2 � p < 1.

THEOREM 10. Let 0 < p � 12 and define 1

p + 1q = 1. Then for any two non-

negative ν -measurable functions f and g

(∫( f +g)p dν

) 1p

(4.5)

�(∫

f pdν) 1

p

×[1+(

1p−1

)∫ ( ( f +g)p ∫ f pdν − f p ∫ ( f +g)p dν∫f pdν

)2 ( f +g)−p∫( f +g)p dν

dν

]

+(∫

gpdν) 1

p

×[1+(

1p−1

)∫ ( ( f +g)p ∫ gpdν −gp ∫ ( f +g)p dν∫gpdν

)2 ( f +g)−p∫( f +g)p dν

dν

].

When 12 � p < 1 we get

(∫f pdν

) 1p

+(∫

gpdν) 1

p

(4.6)

�(∫

( f +g)p dν) 1

p

�(∫

f pdν) 1

p

×[1+(

1p−1

)∫ ( ( f +g)p ∫ f pdν − f p ∫ ( f +g)p dν∫f pdν

)2 ( f +g)−p∫( f +g)p dν

dν

]

+(∫

gpdν) 1

p

×[1+(

1p−1

)∫ ( ( f +g)p ∫ gpdν −gp ∫ ( f +g)p dν∫gpdν

)2 ( f +g)−p∫( f +g)p dν

dν

].

Proof. We use inequality (3.10) for 1p > 2 to get (4.5). We fix non-negative ν -

measurable functions f and g and apply (3.10) with(

ff+g

)pin place of f and

dμ = ( f+g)pdν∫( f+g)pdν . Therefore

∫f pdν∫

( f+g)pdν is in place of∫

f dμ , ff+g is in place of f

1p ,

∫f ( f+g)p−1dν∫( f+g)pdν is in place of

∫f

1p dμ and get

∫f ( f +g)p−1 dν∫

( f +g)p dν(4.7)

�( ∫

f pdν∫( f +g)p dν

) 1p

×⎡⎣1+

(1p−1

)∫ ( ( f +g)−p f p − ∫ f p (∫

( f +g)p dν)−1∫f pdν (

∫( f +g)p dν)−1

)2( f +g)p∫( f +g)p dν

dν

⎤⎦ .

Interchanging the roles of f and g yields

∫g( f +g)p−1 dν∫

( f +g)p dν(4.8)

�( ∫

gpdν∫( f +g)p dν

) 1p

×⎡⎣1+

(1p−1

)∫ ⎛⎝( ( f+g)−p gp−∫ gp (∫

( f+g)p dν)−1∫gpdν (

∫( f+g)p dν)−1

)2( f+g)p∫( f+g)p dν

⎞⎠dν

⎤⎦ .

Adding the last two inequalities gives (4.5). Similarly together with Minkowski’s in-equality for 0 < p < 1 we get (4.6) for 1

2 � p < 1. �

5. Jensen and Slater-Pecaric type inequalities for Steffensen’s coefficients

In Section 2 we dealt with Jensen’s type and Slater-Pecaric type inequalities whenthe coefficients αi � 0, i = 1, ...,n.

We prove now a Jensen-Steffensen type inequality and a Slater-Pecaric type in-equality for N -quasiconvex functions, when N is an integer, and the coefficients arenot necessarily non-negative.

An extension of Jensen Steffensen inequality is proved in [2] for a non-negativesuperquadratic function which is therefore also increasing and convex:

THEOREM 11. Let ψ : [0,∞)→R be differentiable superquadratic and nonnega-tive. Let xxxxx be a nonnegative monotonic n-tuple in R

n, and ρρρρρ a real n-tuple satisfyingSteffensen’s coefficients. Let x be defined by x = 1

Pn∑n

i=1 ρixi. Then

n

∑i=1

ρiψ (xi)−Pnψ (x) �k−1

∑j=1

Pjψ(∣∣x j − x j+1

∣∣)+Pkψ (|xk − x|)

+Pk+1ψ (|xk+1− x|)+n

∑j=k+2

Pjψ(∣∣x j − x j−1

∣∣)

1218 S. ABRAMOVICH

�(

k

∑i=1

Pi +n

∑i=k+1

Pi

)ψ

(∑n

i=1 ρi (|xi − x|)∑k

i=1 Pi + ∑ni=k+1 Pi

)

� ((n−1)Pn)ψ(

∑ni=1 ρi (|xi − x|)(n−1)Pn

)holds where k ∈ {1, ...,n−1} satisfies xk � x � xk+1 , unless one of the following twocases occurs:

(1) either x = x1 or x = xn ,

(2) there exists k ∈ {3, ...,n−2} such that x = xk and Pj(x j − x j+1

)= 0, j =

1, ...,k−1 , P j(x j − x j−1

)= 0, j = k+1, ...,n.

In these two cases ∑ni=1 ρiψ (xi)−Pnψ (x) = 0.

An extension of Slater-Pecaric inequality is proved in [2], for a non-negative su-perquadratic function which is therefore also increasing and convex:

THEOREM 12. [2] Let ψ : [0,∞)→R be a differentiable nonnegative superquad-ratic function. Let ρρρρρ = (ρ1, ...,ρn) be Jensen-Steffensen coefficients and xxxxx=(x1, ...,xn)be a non-negative increasing n-tuple. If ∑n

i=1 ρiψ ′ (xi) �= 0 we define M = ∑ni=1 ρixiψ ′(xi)

∑ni=1 ρiψ ′(xi)

.

Then:Case A: for s satisfying xs � M � xs+1, s+1 � n,

n

∑i=1

ρiψ (xi)

� Pnψ(M)−(

s−1

∑j=1

Pjψ(x j+1−x j)+Psψ(M−xs)+Ps+1ψ(xs+1−M)+n

∑j=s+2

Pjψ(x j−x j−1)

)

� Pnψ (M)−(

s

∑j=1

Pj +n

∑j=s+1

Pj

)ψ

(∑n

i=1 ρi |xi −M|∑s

j=1 Pj + ∑nj=s+1 Pj

)

� Pnψ (M)− ((n−1)Pn)ψ(

∑ni=1 ρi |xi −M|(n−1)Pn

)holds, unless one of the following two cases occurs:



)= 0, j =

1, ...,s−1 ,Pj(x j − x j−1

)= 0, j = s+1, ...,n. In these two cases ∑n

i=1 ρiψ (xi)−Pnψ

(MψN

)= 0.

Case B: for M > xn : ∑ni=1 ρiψ (xi) � Pnψ (M)− (nPn)ψ

(∑n

i=1 ρi|xi−M|nPn

).

For 1-quasiconvex function ψ we present a Jensen’s type inequality obtained in[7, Theorem 3]:

THEOREM 13. Let ρ1, ...,ρn be Jensen-Steffensen coefficients, that is, 0 � Pk =∑k

i=1 ρi � Pn, Pk = ∑ni=k ρi � 0, Pn > 0, k = 1, ...,n, and let x = (x1, ...,xn) > 0 satisfy

0 < x1 � ... � xn . Let ϕ be non-negative, increasing differentiable convex functiondefined on x � 0, and let ψ (x) = xϕ (x) . Let x = ∑n

i=1ρixiPn

. Let s be the integer thatsatisfies 0 < xs � x � xs+1 � xn . Then we get

n

∑i=1

ρiψ (xi)−Pnψ (x)

� ϕ ′ (x1)

(s

∑j=1

Pj + ∑j=s+1

Pj

)(∑n

i=1 ρi |xi − x|∑s

j=1 Pi + ∑nj=s+1 Pj

)2

� ϕ ′ (x1)Pn max{s,n− s}(

∑ni=1 ρi |xi − x|

Pn max{s,n− s})2

� ϕ ′ (x1)(n−1)Pn

(∑n

i=1 ρi |xi − x|(n−1)Pn

)2

� 0.

We state now a Jensen-Steffensen type inequality and Slater Pecaric type inequal-ity for N -quasiconvex functions, when N is an integer. The proof of this theorem uses(1.3) and some of the techniques used in [2]. This is done using identity (1.6) for theconvex function ψN .

THEOREM 14. Let ρ1, ...,ρn be Jensen-Steffensen coefficients, and let x= (x1, ...,xn) satisfy 0 < x1 � ... � xn . Let ϕ be non-negative, increasing differentiable convexfunction defined on x � 0, and let ψN (x) = xNϕ (x) where N is an integer . Let x =∑n

i=1ρixiPn

. Let s be the integer that satisfies 0 < xs � x � xs+1 � xn . Then

n

∑i=1

ρiψN (xi)−PnψN (x) (5.1)

�N

∑k=1

xk−11 ψ ′

N−k (x1)

(s

∑j=1

Pj + ∑j=s+1

Pj

)(∑n

j=1 ρ j∣∣x j − x

∣∣∑s


)2

=

(s

∑j=1

Pj + ∑j=s+1

Pj

)(∑n

j=1 ρ j∣∣x j − x

∣∣∑s


)2∂∂x

(xN − xN

1

x− x1ϕ (x)

)/x=x1

� (Pn max{s,n− s})−1

(n

∑i=1

ρi |xi − x|)2

∂∂x

(xN − xN

1

x− x1ϕ (x)

)/x=x1

� ((n−1)Pn)−1

(n

∑i=1

ρi |xi − x|)2

∂∂x

(xN − xN

1

x− x1ϕ (x)

)/x=x1 � 0

holds, unless one of the following two cases occurs:

1220 S. ABRAMOVICH


)= 0, j =

1, ...,k−1 , P j(x j − x j−1

)= 0, j = k+1, ...,n.

In these two cases ∑ni=1 ρiψ (xi)−Pnψ (x) = 0.

A refinement of Slater-Pecaric inequality in case that ψN is N -quasiconvex func-tions uses the same techniques as in Theorem 12 and in the proof of Theorem 13 is asfollows:

THEOREM 15. Under the same conditions as in Theorem 14 on (ρ1, ...,ρn) , on(x1, ...,xn) and on ψk (x) = xkϕ (x) , k = 0,1, ...,N, if ∑n

i=1 ρiψ ′N (xi) �= 0, we define

MψN = ∑ni=1 ρixiψ ′

N (xi)∑n

i=1 ρiψ ′N (xi)

. Then,

Case A: for s satisfying xs � MψN � xs+1, s+1 � n,

n

∑i=1

ρiψN (xi)−PnψN(MψN

)(5.2)

� −N

∑k=1

xk−11 ψ ′

N−k (x1)

(s

∑j=1

Pj + ∑j=s+1

Pj

)(∑n

j=1 ρ j∣∣x j − x

∣∣∑s


)2

= −(

s

∑j=1

Pj + ∑j=s+1

Pj

)−1( n

∑j=1

ρ j∣∣x j − x

∣∣)2∂∂x

(xN − xN

1

x− x1ϕ (x)

)/x=x1

� −(Pn max{s,n− s})−1

(n

∑i=1

ρi |xi − x|)2

∂∂x

(xN − xN

1

x− x1ϕ (x)

)/x=x1

� −((n−1)Pn)−1

(n

∑i=1

ρi |xi − x|)2

∂∂x

(xN − xN

1

x− x1ϕ (x)

)/x=x1 � 0

holds, unless one of the following two cases occurs:



)= 0, j =

1, ...,s−1 , P j(x j − x j−1

)= 0, j = s+1, ...,n.

In these two cases ∑ni=1 ρiψ (xi)−Pnψ

(MψN

)= 0.

Case B: for MψN > xn ,

n

∑i=1

ρiψN (xi)−PnψN(MψN

)

� −(nPn)−1

(n

∑i=1

ρi∣∣xi−MψN

∣∣)2∂∂x

(xN − xN

1

x− x1ϕ (x)

)/x = x1.


Proof (of Theorem 14). The proof follows step by step the proof of [7, Theorem 3].

Here we only replace ϕ ′(x1) with ∂

∂x

(xN−xN

1x−x1

ϕ (x))

/x=x1 which is non-negative when

ϕ is non-negative increasing and convex. Therefore the detailed proof is omitted. �

Proof (of Theorem 15). It was proved in [1] that when ρρρρρ is satisfying (1.5), xxxxx isincreasing, and ψN is non-negative increasing and convex and that ∑n

i=1 ρiψ ′N (xi) > 0,

∑ni=1 ρixiψ ′

N (xi) � 0, we get that x1 � ∑ni=1 ρixiψ ′

N(xi)∑n

i=1 ρiψ ′N (xi)

= MψN holds.

Case A: For x1 � xs � MψN � xs+1 � xn, we use identity (1.6) for s∈{1, ...,n−1},and as Pj � 0, Pj � 0, j = 1, ...,n , and ϕ is non-negative increasing and convex func-tion, we get that the N -quasiconvex function ψN satisfies

PnψN(MψN

)− n

∑i=1

ρiψN (xi) (5.3)

=s−1

∑j=1

Pj(ψN(x j+1

)−ψN (x j))+Ps

(ψN(MψN

)−ψN (xs))

+Ps+1(ψN(MψN

)−ψN (xs+1))+

n

∑j=s+2

Pj(ψN(x j−1

)−ψN (x j))

�[

s−1

∑j=1

Pjψ′N (x j)

(x j+1− x j

)+Psψ

′N (xs)

(MψN − xs

)

+Ps+1ψ′N (xs+1)

(MψN − xs+1

)+

n

∑j=s+2

Pjψ′N (x j)

(x j−1− x j

)]

+

[s−1

∑j=1

Pj(x j+1− x j

)2 ∂∂x j

(xN

j − xNj+1

x j − x j+1ϕ (x j)

)

+Ps(MψN − xs

)2 ∂∂xs

(xNs −MN

ψN

xs−MψN

ϕ (xs)

)

+Ps+1(xs+1−MψN

)2 ∂∂xs+1

(xNs+1−MN

ψN

xs+1−MψN

ϕ (xs+1)

)

+n

∑j=s+2

Pj(x j − x j−1

)2 ∂∂x j

(xN

j − xNj−1

x j − x j−1ϕ (x j)

)].

It is shown in [2] that under our conditions on ρ , the first parenthesis in the righthandside of (5.3) for the convex functions ψN is non-negative. Then from the N -quasiconvexity of ψN , the convexity of f (x) = x2 we get from (5.3) that

PnψN(MψN

)− n

∑i=1

ρiψN (xi)

� [0]+s−1

∑j=1

Pj(x j+1− x j

)2 ∂∂x j

(xN

j − xNj+1

x j − x j+1ϕ (x j)

)

1222 S. ABRAMOVICH

+Ps(MψN − xs

)2 ∂∂xs

(xNs −MN

ψN

xs−MψN

ϕ (xs)

)

+Ps+1(xs+1−MψN

)2 ∂∂xs+1

(xNs+1−MN

ψN

xs+1−MψN

ϕ (xs+1)

)

+n

∑j=s+2

Pj(x j − x j−1

)2 ∂∂x j

(xN

j − xNj−1

x j − x j−1ϕ (x j)

)

�(

s

∑j=1

Pj +n

∑j=s+1

Pj

)(∑s−1

j=1 Pj(x j+1− x j

)+Ps

(MψN − xs

)∑s


+Ps+1

(xs+1−MψN

)+ ∑n

j=s+2 Pj(x j − x j−1

)∑s


)2∂∂x

(xN1 − xN

x1− xϕ (x)

)/x = x1

=

(s

∑j=1

Pj +n

∑j=s+1

Pj

)−1( n

∑i=1

ρi∣∣xi −MψN

∣∣)2∂∂x

(xN1 − xN

x1− xϕ (x)

)/x = x1

In the same way as in the proof of the Theorem 13, we conclude that(s

∑j=1

Pj +n

∑j=s+1

Pj

)−1( n

∑j=1

ρi (|xi −M|))2

�((n−1)Pn)−1

(n

∑i=1

ρi(∣∣xi −MψN

∣∣))2

holds and hence we get (5.2).The proof of the special cases (1) and (2) in the theorem are the same as the proofs

of the equality cases in Theorem 2 and as proved in [1] and in [2].Case B for M > xn is proved similarly to Case A.Hence the proof of Theorem 15 is complete. �

Theorem 15, besides being a refinement of Slater-Pecaric inequality is also ananalog of Theorem 12 which deals with non-negative superquadratic functions.

6. Bounds for differences of “Jensen’s gap” for N -quasiconvex functions

In this section we state one of many results that can be derived from the pre-vious theorems. First we quote a result from [3] about the difference between two“Jensen’s gaps” ∑n

i=1 piψ (xi)−ψ (xp) and ∑ni=1 qiψ (xi)−ψ (xq) . Then we present

a new theorem with results when ψ is a N -quasiconvex function. In particular for a1-quasiconvex function ψ the result is interesting.

The proofs in this section like the proofs in [3] employ some of the techniquesused in [8].

In [3, Theorem 2] the following is proved:


THEOREM 16. Suppose that ψ : I → R, where I is [0,a] or [o,∞) is superquad-ratic. Let xi ∈ I, i = 1, ...,n, xp = ∑n

i=1 pixi, pi � 0, i = 1, ...,n, ∑ni=1 pi = 1 and

xq = ∑ni=1 qixi, qi � 0, i = 1, ...,n, ∑n

i=1 qi = 1. Then, for m = min1�i�n

(piqi

)(

n

∑i=1

piψ (xi)−ψ (xp)

)−m

(n

∑i=1

qiψ (xi)−ψ (xq)

)(6.1)

� mψ

(∣∣∣∣∣n

∑i=1

(qi − pi)xi

∣∣∣∣∣)

+n

∑i=1

(pi −mqi)ψ

(∣∣∣∣∣xi −n

∑j=1

p jx j

∣∣∣∣∣)

and for M = max1�i�n

(piqi

)(

n

∑i=1

piψ (xi)−ψ (xp)

)−M

(n

∑i=1

qiψ (xi)−ψ (xq)

)(6.2)

� −n

∑i=1

(Mqi− pi)ψ

(∣∣∣∣∣xi −n

∑j=1

q jx j

∣∣∣∣∣)−ψ

(∣∣∣∣∣n

∑i=1

(pi−qi)xi

∣∣∣∣∣)

.

If the superquadratic function is also nonnegative and therefore is also convex,then (6.1) and (6.2) refine the following theorem by Dragomir in [8]:

THEOREM 17. Under the same conditions on p , q , x, xp, xq, m and M, as inTheorem 16, if ψ is convex then

M

(n

∑i=1

qiψ (xi)−ψ (xq)

)�

n

∑i=1

piψ (xi)−ψ (xp) (6.3)

� m

(n

∑i=1

qiψ (xi)−ψ (xq)

).

Now we show another refinement of Theorem 17 this time for N -quasiconvexfunction ψN .

THEOREM 18. Suppose that ψN : I → R where I is [a,b) , 0 � a, b � ∞, is N -quasiconvex function, that is ψN = xNϕ (x) , N = 1,2, ... where ϕ is convex on [a,b) .Let p , q , x , xp, xq, m and M be as in Theorem 16, then(

n

∑i=1

piψN (xi)−ψN (xp)

)−m

(n

∑i=1

qiψN (xi)−ψN (xq)

)(6.4)

�n

∑i=1

(pi −mqi)(xi − xp)2 ∂∂xp

(xNi − xN

p

xi − xpϕ (xp)

)

+m(xq − xp)2 ∂∂xp

(xNq − xN

p

xq − xpϕ (xp)

),

1224 S. ABRAMOVICH

and (n

∑i=1


)−M

(n

∑i=1


)(6.5)

�n

∑i=1

(pi −Mqi)(xi − xq)2 ∂∂xq

(xNi − xN

q

xi − xqϕ (xq)

)

−M (xq− xp)2 ∂

∂xq

(xNq − xN

p

xq− xpϕ (xq)

).

For N = 1 we get that(n

∑i=1

piψ1 (xi)−ψ1 (xp)

)−m

(n

∑i=1

qiψ1 (xi)−ψ1 (xq)

)(6.6)

� ϕ′(xp)

((n

∑i=1

pix2i − (xp)

2

)−m

(n

∑i=1

qix2i − (xq)

2

)),

and (n

∑i=1

piψ1 (xi)−ψ1 (xp)

)−M

(n

∑i=1

qiψ1 (xi)−ψ1 (xq)

)(6.7)

� ϕ′(xq)

((n

∑i=1

pix2i − (xp)

2

)−M

(n

∑i=1

qix2i − (xq)

2

)).

In particular if ϕ is also non-negative increasing then (6.4)–(6.7) are refinements of(6.3).

Proof. To prove (6.4) we define y and d as

yi ={

xi, i = 1, ...,nxq, i = n+1

, di ={

pi −mqi, i = 1, ...,nm, i = n+1

. (6.8)

From (6.8) we get that y = ∑n+1i=1 diyi = ∑n

i=1 pixi = xp Then (2.3) for y and d is(n

∑i=1


)−m

(n

∑i=1


)

=n

∑i=1

(pi −mqi)ψN (xi)+mψN (xq)−ψN (xp)

=n+1

∑i=1

diψN (yi)−ψN (xp) �n+1

∑i=1

di (yi− y)2 ∂∂y

(yN − yN

i

y− yiϕ (y)

).

which after using again (6.8) is (6.4).


To get (6.5), we choose z and r as

zi ={

xi, i = 1, ...,nxp, i = n+1

, ri ={

qi− piM , i = 1, ...,n

1M , i = n+1

.

Then, as ∑n+1i=1 ri = 1, ri � 0, i = 1, ...,n+ 1 and ∑n+1

i=1 rizi = ∑ni=1 qixi = xq , we

get that (n

∑i=1


)− 1

M

(n

∑i=1


)

=n

∑i=1

(qi − pi

M

)ψN (xi)+

1M

ψN (xp)−ψN (xq)

=n+1

∑i=1

riψN (zi)−ψN

(n+1

∑i=1

rizi

)

�n

∑i=1

(qi − pi

M

)(xi − xq)

2 ∂∂xq

(xNq − xN

i

xq − xiϕ (xq)

)

+1M

(xp− xq)2 ∂

∂xq

(xNq − xN

p

xq− xpϕ (xq)

),

which is equivalent to (6.5). �

Acknowledgements. The author wish to thank L.-E. Persson whose many advicesare as usual very useful and to the referee for the illuminating remarks.

RE F ER EN C ES

[1] S. ABRAMOVICH, M. KLARICIC BACULA, M. MATIC, AND J. PECARIC, A Variant of Jensen-Steffensen’s Inequality and Quazi Arithmetic Means, Journal of Mathematical Analysis and Applica-tions, 307 (2005) 370–386.

[2] S. ABRAMOVICH, S. BANIC, M. MATIC AND J. PECARIC, Jensen Steffensen’s and Related Inequali-ties, for Superquadratic Functions, Mathematical Inequalities and Applications, 11, (2008), pp. 23–41.

[3] S. ABRAMOVICH AND S. S. DRAGOMIR, Normalized Jensen Functional, Superquadracity and Re-lated Inequalities, International Series of Numerical Mathematics, Birkhauser Verlag, 157, (2008)217–228.

[4] S. ABRAMOVICH, G. JAMESON AND G. SINNAMON, Refining Jensen’s Inequality, Bulletin Math-ematique de la Societe des Sciences Mathematiques de Roumanie, (Novel Series) 47 (95), (2004),3–14.

[5] S. ABRAMOVICH, L.-E. PERSSON, Some new estimates of the “Jensen Gap”, J. Inequal. Appl.(2016), 2016:39, 9 pp.

[6] S. ABRAMOVICH, L.-E. PERSSON, AND N. SAMKO, Some new scales of refined Jensen and Hardytype inequalities, Math. Inequal. Appl., 17, (2014), 1105–1114.

[7] S. ABRAMOVICH, L.-E. PERSSON, AND N. SAMKO, On γ -quasiconvexity, superquadracity andtwo-sided reversed Jensen type inequalities, Math. Inequal. Appl. 18 (2), (2015), 615–627.

[8] S. S. DRAGOMIR, Bounds for the normalised Jensen functional, Bull. Austral. Math. Soc. 74 (2006),471–478.

[9] G. H. HARDY, J. E. LITTLEWOOD AND G. POLYA, Inequalities, Cambridge University press, 1964.

1226 S. ABRAMOVICH

[10] E. G. KWON AND J. E. BAE, On a refined Holder’s inequality, J. Math. Inequal., 10, no. 1, (2016),261–268.

[11] L. LARSSON, L. MALIGRANDA, J. PECARIC AND L.-E. PERSSON, Multiplicative inequalities ofCarlson type and interpolation, World Scientific 2006.

[12] J. MATKOWSKI, Aconverse of Holder inequality theorem, Math. Inequal. Appl. 12, no. 1, (2009),21–32.

[13] C. NICULESCU AND L.-E. PERSSON, Convex functions and their applications, a contemporary ap-proach, CMS books in mathematics, 23, Springer New York, 2006.

[14] L. NIKOLOVA AND S. VAROSANEC, Refinement of Holder’s inequality derived from functions ψp,q,λand φp,q,λ , Ann. Funct. Anal, 2, no. 1 (2011), 72–83.

[15] J. PECARIC, F. PROSCHAN, Y. L. TONG, Convex Functions, Partial Orderings, and Statistical Appli-cations, Academic Press, New York, 1992.

[16] L.-E. PERSSON AND N. SAMKO, What should have happen if Hardy had discovered this?, J. Inequal.Appl. (2012), 2012:29.

[17] L.-E. PERSSON, N. SAMKO AND P. WALL, Quasi-monotone weight functions and their characteris-tics and applications, Math. Inequal. Appl. 15 (2012) no. 3, 685–705.

[18] G. SINNAMON, Refining the Holder and Minkowski inequalities, J. Inequal. and Appl. 6 (2001), 633–640.

[19] J.-F. TIAN, Property of Holder-type inequality and its application, Math. Inequal. Appl. 16, no. 3,(2013), 831–841.

(Received February 1, 2016) Shoshana AbramovichDepartment of Mathematics, University of Haifa

Israele-mail: [email protected]



& Applications


RECURSIVELY DEFINED REFINEMENTS OF

THE INTEGRAL FORM OF JENSEN’S INEQUALITY

LASZLO HORVATH AND JOSIP PECARIC


Abstract. In this paper we establish infinite chains of integral inequalities related to the classicalJensen’s inequality by using special refinements of the discrete Jensen’s inequality. As applica-tions, we introduce and study new integral means (generalized quasi-arithmetic means), and giverefinements of the left hand side of Hermite-Hadamard inequality.

1. Introduction

The integral form and the discrete version of Jensen’s inequality provide the start-ing point for much of the discussion in this paper. They can be stated as follows:

THEOREM A. (classical Jensen’s inequality, see [7]) Let g be an integrable func-

tion on a probability space (X ,A ,μ) taking values in an interval I ⊂ R . Then∫X

gdμ

lies in I . If f is a convex function on I such that f ◦ g is integrable, then

f

⎛⎝∫

X

gdμ

⎞⎠�

∫X

f ◦ gdμ .

THEOREM B. (discrete Jensen’s inequality, see [7]) Let C be a convex subsetof a real vector space V , and let f : C → R be a convex function. If p1, . . . , pn are

nonnegative numbers withn

∑i=1

pi = 1 , and v1, . . . ,vn ∈C, then

f

(n

∑i=1

pivi

)�

n

∑i=1

pi f (vi) .

Jensen obtained his famous inequality in [16]. There is an extensive theory forthe study of refinements of the discrete Jensen’s inequality, see [15], but there are only

Mathematics subject classification (2010): 26A51 (26D15).Keywords and phrases: Classical and discrete Jensen’s inequality, convex function, generalized quasi-

arithmetic means, Hermite-Hadamard inequality.The research of the 2nd author has been fully supported by Croatian Science Foundation under the project 5435.


1227


1228 L. HORVATH AND J. PECARIC

few papers dealing with refinements of the classical Jensen’s inequality, see Rooin [18],Horvath [9] and Horvath and Pecaric [14]. In this paper we establish infinite chains ofintegral inequalities related to the classical Jensen’s inequality. The key of our treatmentis special refinements of the discrete Jensen’s inequality which have been developed inHorvath [13]. As an immediate application, new infinite refinements of the classicalJensen’s inequality are derived. We essentially follow the approach of Brnetic, Pearceand Pecaric [3], but our treatment is applicable in a more general environment. In Sec-tion 3 we consider our results in some interesting special cases. In Section 4 some newintegral means (generalized quasi-arithmetic means) are introduced, and their proper-ties are studied. Section 5 is devoted to refinements of the left hand side of Hermite-Hadamard inequality.

2. Preliminaries and the main inequalities

N and N+ denote the set of nonnegative and positive integers, respectively.Before proceeding to the results we present some hypotheses, and an inequality

from [13] which will be needed.(H1 ) Let n ∈ N+ be fixed, and denote

Sk :=

{(i1, . . . , in) ∈ N

n+ |

n

∑j=1

i j = n+ k−1

}, k ∈ N+. (1)

(H2 ) Let(a j (m))m∈N+

, 1 � j � n

be strictly increasing sequences such that

α := a1 (1) = . . . = an (1) > 0. (2)

(H3 ) Let p1, . . . , pn be nonnegative numbers withn

∑j=1

p j = 1.

Under the hypotheses (H1 ) and (H2 ), define the finite sequences

(uk (i1, . . . , in))(i1,...,in)∈Sk, k ∈ N+

recursively by

u1 (1, . . . ,1) :=1α

, (3)

and for every (i1, . . . , in) ∈ Sk+1 (see (1))

uk+1 (i1, . . . , in) := ∑{l∈{1,...,n}|il �=1}

1

1+ al(il−1)al(il)−al(il−1) +

n

∑j=1j �=l

a j(i j)a j(i j+1)−a j(i j)

× al (il −1)al (il)−al (il −1)

uk (i1, . . . , il−1, il −1, il+1, . . . , in) .

Now we state one of the main results in [13].

REFINEMENTS OF INTEGRAL FORM OF JENSEN’S INEQUALITY 1229

THEOREM 1. Assume (H1 –H3 ). Let C be a convex subset of a real vector spaceX , and {x1, . . . ,xn} be a finite subset of C . If f : C → R is a convex function, then

f

(n

∑j=1

p jx j

)= T1 � . . . � Tk � Tk+1 � . . . �

n

∑j=1

p j f (x j),

where for each k ∈ N+

Tk = Tk,n(x1, . . . ,xn; p1, . . . , pn;a1, . . . ,an)

:= ∑(i1,...,in)∈Sk

uk (i1, . . . , in)

(n

∑j=1

a j (i j) p j

)f

⎛⎜⎜⎜⎝

n

∑j=1

a j (i j) p jx j

n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎠ .

We follow this section by introducing some notations. Let (Xi,Bi,μi) (i ∈ Tl :={1, . . . , l}) be probability spaces for some l ∈ N+ , l � 2. The σ -algebra in XTl :=X1× . . .×Xl generated by the projection mappings

pri : X1× . . .×Xl → Xi, pri (x1, . . . ,xl) = xi (i = 1, . . . , l)

is denoted by BTl . μTl means the product measure on BTl : this is the only measureon BTl (the measures are σ -finite) which satisfies

μTl (B1× . . .×Bl) = μ1 (B1) . . .μl (Bl) , Bi ∈ Bi, (i = 1, . . . , l) .

The l -fold product of the probability space (X ,B,μ) is denoted by(Xl,Bl ,μ l

).

The following abbreviationswill be used: dμTl (x) := dμTl (x1, . . . ,xl) and dμ l (x):= dμ l (x1, . . . ,xl) .

λ l is always means the Lebesgue measure on the Borel sets of Rl .

Our first purpose is to obtain an extended and refined version of the classicalJensen’s inequality.

THEOREM 2. Assume (H1 –H3 ). Suppose the following hypotheses are also hold(H4 ) Let (Xi,Bi,μi) (i = 1, . . . ,n) be probability spaces.(H5 ) For each i = 1, . . . ,n, let gi be a μi -integrable function on Xi taking values

in an interval I ⊂ R .(H6 ) Let f be a convex function on I such that f ◦ gi is μi -integrable on Xi

(i = 1, . . . ,n) .Then(a)

f

⎛⎝ n

∑i=1

pi

∫Xi

gidμi

⎞⎠� T1 � . . . � Tk � Tk+1 � . . . �

n

∑i=1

pi

∫Xi

f ◦ gidμi, (4)


where

Tk = Tk,n( f ;gi;μi; pi;ai)

:= ∑(i1,...,in)∈Sk

uk (i1, . . . , in)

(n

∑j=1

a j (i j) p j

)

×∫

XTn

f

⎛⎜⎜⎜⎝

n

∑j=1

a j (i j) p jg j (x j)

n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎠dμTn (x) , k ∈ N+. (5)

(b) For each k ∈ N+ and all t ∈ [0,1]

f

⎛⎝ n

∑i=1

pi

∫Xi

gidμi

⎞⎠� Hk (0) � Hk (t) � Tk, (6)

where

Hk (t) = Hk,n (t; f ;gi;μi; pi;ai)

:= ∑(i1,...,in)∈Sk

uk (i1, . . . , in)

(n

∑j=1

a j (i j) p j

)

×∫

XTn

f

⎛⎜⎜⎜⎜⎜⎝t

n

∑j=1


n

∑j=1

a j (i j) p j

+(1− t)

n

∑j=1

a j (i j) p j

∫Xj

g jdμ j

n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎟⎟⎠dμTn (x) . (7)

REMARK 1. By (H3 –H6 ), the hypotheses of Lemma 2.1 in [8] are all satisfied,and so it yields that all the integrals in (5) and (7) exist and finite.

Assume (H1 –H6 ).(a) If n = 1, then Sk = {k} and uk (k) = 1

a1(k)for all k ∈ N+ , and therefore

Tk = T1 =∫X1

f ◦ g1dμ1, k ∈ N+,

and

Hk (t) = H1 (t) =∫X1

f

⎛⎝tg1 (x1)+ (1− t)

∫X1

g1dμ1

⎞⎠dμ1 (x1) , t ∈ [0,1] , k ∈ N+.


We can see that for n = 1 (4) is trivial (the classical Jensen’s inequality), while (6)gives

f

⎛⎝∫

X1

g1dμ1

⎞⎠= H1 (0) � H1 (t) � H1 (1) =

∫X1

f ◦ g1dμ1, t ∈ [0,1] .

(b) Suppose n � 2. Then S1 = {(1, . . . ,1)} , and hence

T1 =∫

XTn

f

(n

∑j=1

p jg j (x j)

)dμTn (x) ,

and

H1 (t) =∫

XTn

f

⎛⎜⎝t

n

∑j=1

p jg j (x j)+ (1− t)n

∑j=1

p j

∫Xj

g jdμ j

⎞⎟⎠dμTn (x) , t ∈ [0,1] .

Now we summarize the essential properties of the function Hk defined in (7).

THEOREM 3. Assume (H1 –H6 ). Then for each k ∈ N+(a) Hk is convex and increasing.(b)

Hk (0) � f

⎛⎝ n

∑i=1

pi

∫Xi

gidμi

⎞⎠ , Hk (1) = Tk.

(c) Hk is continuous on [0,1[ .(d) If f is continuous, then Hk is continuous on [0,1] .

It is easy to construct examples which show that Hk is not continuous at 1 ingeneral.

The following refinements of the classical Jensen’s inequality are immediate con-sequences of Theorem 2.

THEOREM 4. Assume (H1 –H3 ). The hypotheses (H4 –H6 ) are replaced by(H4 ) Let (X ,B,μ) be a probability space.(H5 ) Let g be a μ -integrable function on X taking values in an interval I ⊂ R .(H6 ) Let f be a convex function on I such that f ◦ g is μ -integrable on X .Then(a)

f

⎛⎝∫

X

gdμ

⎞⎠� T1 � . . . � Tk � Tk+1 � . . . �

∫X

f ◦ gdμ ,


where

Tk = Tk,n( f ;g;μ ; pi;ai)

= ∑(i1,...,in)∈Sk

uk (i1, . . . , in)

(n

∑j=1

a j (i j) p j

)

×∫Xn

f

⎛⎜⎜⎜⎝

n

∑j=1

a j (i j) p jg(x j)

n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎠dμn (x) , k ∈ N+.


f

⎛⎝∫

X

gdμ

⎞⎠= Hk (0) � Hk (t) � Tk,

where

Hk (t) = Hk,n (t; f ;g;μ ; pi;ai)

= ∑(i1,...,in)∈Sk

uk (i1, . . . , in)

(n

∑j=1

a j (i j) p j

)

×∫Xn

f

⎛⎜⎜⎜⎝t

n

∑j=1

a j (i j) p jg(x j)

n

∑j=1

a j (i j) p j

+(1− t)∫X

gdμ

⎞⎟⎟⎟⎠dμn (x) .

3. Results when recursion is explicitly represented

We first recall the following example from [13].

EXAMPLE 1. Assume (H1 ) and (H3 ). Let α > 0, a � 0 and b j ∈ R (1 � j � n)such that the numbers a+b j are all positive. Define the sequences (a j (m))m∈N+

by

a j (m) := αm−1

∏i=1

(1+

1ai+b j

), m ∈ N+, 1 � j � n. (8)

Then these sequences are strictly increasing and

α = a1 (1) = . . . = an (1) > 0,

thus they satisfy (H2 ).


In this case it can be proved that for every k ∈ N+

uk (i1, . . . , in) =1α

k−1

∏j=1

1

1+a(n+ j−1)+n

∑l=1

bl

n

∏j=1

(i j−1

∏m=1

(am+b j)

)

× (k−1)!(i1−1)! . . .(in−1)!

, (i1, . . . , in) ∈ Sk. (9)

As illustrations, we just consider Theorem 4 in two special cases of the previousexample.

The first part of the next result can be considered as the integral version of Theorem1 (a) in [12].

COROLLARY 1. Assume (H1 ), (H3 ), and (H4 –H6 ). By choosing α = a = 1 andb j = 0 (1 � j � n) in (8), we have

(a)

f

⎛⎝∫

X

gdμ

⎞⎠� T1 � . . . � Tk � Tk+1 � . . . �

∫X

f ◦ gdμ ,

where

Tk = Tk,n( f ;g;μ ; pi)

=1(n+k−1

k−1

) ∑(i1,...,in)∈Sk

(n

∑j=1

i j p j

)∫Xn

f

⎛⎜⎜⎜⎝

n

∑j=1

i j p jg(x j)

n

∑j=1

i j p j

⎞⎟⎟⎟⎠dμn (x) , k ∈ N+.


f

⎛⎝∫

X

gdμ

⎞⎠= Hk (0) � Hk (t) � Tk,

where

Hk (t) = Hk,n (t; f ;g;μ ; pi)

=1(n+k−1

k−1

) ∑(i1,...,in)∈Sk

(n

∑j=1

i j p j

)∫Xn

f

⎛⎜⎜⎜⎝t

n

∑j=1

i j p jg(x j)

n

∑j=1

i j p j

+(1− t)∫X

gdμ

⎞⎟⎟⎟⎠dμn (x) .

(c) For each k ∈ N+

Tk,n

(f ;g;μ ;

1n

)� Tk,n( f ;g;μ ; pi),


where

Tk,n

(f ;g;μ ;

1n

)=

1(n+k−2k−1

) ∑(i1,...,in)∈Sk

∫Xn

f

(1

n+ k−1

n

∑j=1

i jg(x j)

)dμn (x)

(d) If pi > 0 (1 � i � n) , then for each k ∈ N+ and for each l ∈ N+

f

⎛⎝∫

X

gdμ

⎞⎠� Tk � Al �

∫X

f ◦ gdμ ,

where

Al = Al,n ( f ;g;μ ; pi)

:=1(n+l−1

l−1

) ∑i1+...+in=l

i j∈N; 1� j�n

(l

∑j=1

i j p j

)∫Xn

f

⎛⎜⎜⎜⎜⎝

l

∑j=1

i j p jg(x j)

l

∑j=1

i j p j

⎞⎟⎟⎟⎟⎠dμn (x) .

(e) Suppose pi > 0 (1 � i � n) . Then

limk→∞

Tk = liml→∞

Al = n!∫Xn

⎛⎝∫

En

h(t1, . . . ,tn−1,x1, . . . ,xn)dλ n−1 (t)

⎞⎠dμn (x) , (10)

where

En :=

{(t1, . . . ,tn−1) ∈ R

n−1 |n−1

∑j=1

t j � 1, t j � 0, j = 1, . . . ,n−1

},

the function h defined on En×Xn by

h(t1, . . . ,tn−1,x1, . . . ,xn) :=

(n

∑j=1

t j p j

)f

⎛⎜⎜⎜⎝ 1

n

∑j=1

t j p j

(n

∑j=1

t j p jg(x j)

)⎞⎟⎟⎟⎠

with the notation tn := 1−n−1

∑j=1

t j .

Proof. (a) and (b) come from Theorem 4.(c) Theorem 1 (b) in [12] can be applied.(d) According to Proposition 2 in [14], the sequence (Al)l∈N+

is decreasing and

f

⎛⎝∫

X

gdμ

⎞⎠� Al �

∫X

f ◦ gdμ , l ∈ N+.


Define for all (x1, . . . ,xn) ∈ Xn the expressions

Gk,n (x1, . . . ,xn) :=1(n+k−1

k−1

) ∑(i1,...,in)∈Sk

(n

∑j=1

i j p j

)f

⎛⎜⎜⎜⎝ 1

n

∑j=1

i j p j

n

∑j=1

i j p jg(x j)

⎞⎟⎟⎟⎠ , k∈N+,

and

Bl,n (x1, . . . ,xn) :=1(n+l−1

l−1

) ∑i1+...+in=l

i j∈N; 1� j�n

(n

∑j=1

i j p j

)f

⎛⎜⎜⎜⎝ 1

n

∑j=1

i j p j

n

∑j=1

i j p jg(x j)

⎞⎟⎟⎟⎠ , l ∈N+.

Theorem 1 (a) in [12] shows that the sequence(Gk,n (x1, . . . ,xn)

)k∈N+

(11)

is increasing for all (x1, . . . ,xn) ∈ Xn . By Example 3 in [10], the sequence(Bl,n (x1, . . . ,xn)

)l∈N+

(12)

is decreasing for all (x1, . . . ,xn) ∈ Xn . It follows from Theorem 3 in [12] that

limk→∞

Gk,n (x1, . . . ,xn) = liml→∞

Bl,n (x1, . . . ,xn)

=∫En

h(t1, . . . ,tn−1,x1, . . . ,xn)dλ n−1 (t) , (x1, . . . ,xn) ∈ Xn. (13)

Putting all this together gives that for each k ∈ N+ and for each l ∈ N+

Gk,n (x1, . . . ,xn) � Bl,n (x1, . . . ,xn) , (x1, . . . ,xn) ∈ Xn.

By integrating both sides over Xn , we have Tk � Al (k ∈ N+, l ∈ N+) .(e) Assuming that the function h is λ n−1 × μn -integrable over En ×Xn for the

present, the monotonicity properties of the sequences (11) and (12), the limit formula(13), and the Fubini’s theorem imply (10).

The measurability of h is obvious. To justify the supposed integrability condition,choose a fixed interior point a of I . Since f is convex

f (t) � f (a)+ f ′+ (a)(t−a) , t ∈ I,

where f ′+ (a) means the right-hand derivative of f at a . By using this and the discreteJensen’s inequality, we have(

n

∑j=1

t j p j

)f (a)+ f ′+ (a)

(n

∑j=1

t j p jg(x j)−a

(n

∑j=1

t j p j

))

� h(t1, . . . ,tn−1,x1, . . . ,xn) �n

∑j=1

t j p j f (g(x j)) (14)

for all (t1, . . . ,tn−1,x1, . . . ,xn) ∈ En ×Xn . It is enough to prove that the functions onthe left hand side and the right hand side of the previous inequalities are λ n−1 × μn -integrable over En ×Xn . We consider only the function on the right hand side of (14),the other case can be handled similarly. In proving this, we may suppose that f isnonnegative on I , and therefore by the Fubini’s theorem, and then by Lemma 2.1 (a) in[8]

∫En×Xn

(n

∑j=1

t j p j f (g(x j))

)dλ n−1× μn (t1, . . . ,tn−1,x1, . . . ,xn)

=∫En

⎛⎝∫

Xn

n

∑j=1

t j p j f (g(x j))dμn (x)

⎞⎠dλ n−1 (t)

=

⎛⎝∫

X

f ◦ gdμ

⎞⎠⎛⎝∫

En

(n

∑j=1

t j p j

)dλ n−1 (t)

⎞⎠< ∞.

The proof is complete. �

REMARK 2. We stress that the sequences (Tk)k∈N+and (Al)l∈N+

, compared inpart (d) of the previous result, are generated from such refinements of the discreteJensen’s inequality which have been obtained by essentially different methods.

Now, the integral variant of Theorem 1 (a) is obtained.

COROLLARY 2. Assume (H1 ), (H3 ), and (H4 –H6 ). By choosing α = 1 , a = 0and b j = 1

λ j−1 (1 � j � n) , where λ j > 1 (1 � j � n) in (8), we have with the notation

d (λ ) :=n

∑j=1

1λ j−1

(a)

f

⎛⎝∫

X

gdμ

⎞⎠� T1 � . . . � Tk � Tk+1 � . . . �

∫X

f ◦ gdμ ,

where for every k ∈ N+

Tk = Tk,n( f ;g;μ ; pi)

=1

(d (λ )+1)k−1 ∑(i1,...,in)∈Sk

(k−1)!(i1 −1)! . . .(in −1)!

×n

∏j=1

1

(λ j −1)i j−1

(n

∑j=1

λ i j−1j p j

)∫Xn

f

⎛⎜⎜⎜⎝

n

∑j=1

λ i j−1j p jg(x j)

n

∑j=1

λ i j−1j p j

⎞⎟⎟⎟⎠dμn (x) .



f

⎛⎝∫

X

gdμ

⎞⎠= Hk (0) � Hk (t) � Tk,

where

Hk (t) = Hk,n (t; f ;g;μ ; pi)

=1

(d (λ )+1)k−1 ∑(i1,...,in)∈Sk

(k−1)!(i1 −1)! . . .(in −1)!

n

∏j=1

1

(λ j −1)i j−1

×(

n

∑j=1

λ i j−1j p j

)∫Xn

f

⎛⎜⎜⎜⎝

n

∑j=1


n

∑j=1

λ i j−1j p j

+(1− t)∫X

gdμ

⎞⎟⎟⎟⎠dμn (x) .

(c)

limk→∞

Tk =∫X

f ◦ gdμ .

Proof. We have only to apply Theorem 4 to get (a) and (b).(c) Let for all (x1, . . . ,xn) ∈ Xn

Dk,n (λ ;x1, . . . ,xn) =1

(d (λ )+1)k−1 ∑(i1,...,in)∈Sk

(k−1)!(i1−1)! . . . (in−1)!

×n

∏j=1

1

(λ j −1)i j−1

(n

∑j=1

λ i j−1j p j

)f

⎛⎜⎜⎜⎝

n

∑j=1


n

∑j=1

λ i j−1j p j

⎞⎟⎟⎟⎠ .

By Theorem 1 (a) in [11], the sequence(Dk,n (λ ;x1, . . . ,xn)

)k∈N+

is increasing, and by (b) of the same theorem

limk→∞

Dk,n (λ ;x1, . . . ,xn) =n

∑j=1

p j f (g(x j)) , (x1, . . . ,xn) ∈ Xn.

It follows from these facts that

limk→∞

Tk = limk→∞

∫Xn

Dk,n (λ ;x1, . . . ,xn)dμn (x)

=n

∑j=1

p j

∫Xn

f (g(x j))μn (x) =∫X

f ◦ gdμ .

The proof is now complete. �


4. Means generated by the expressions in the new refinements

In this section we introduce some new integral means (generalized quasi-arithmeticmeans) and study their properties.

DEFINITION 1. Assume (H1 –H3 ) and(H4 ) Let (Xi,Bi,μi) (i = 1, . . . ,n) be probability spaces.Assume further(H7 ) For each i = 1, . . . ,n , let gi be a measurable function on Xi taking values in

an interval I ⊂ R .(H8 ) Let ϕ , ψ : I → R be continuous and strictly monotone functions.(a) For each k ∈ N+ , we define integral means with respect to (5) by

Mψ,ϕ (k)= Mψ,ϕ(gi;μi; pi;ai;k)

:= ψ−1

(∑

(i1,...,in)∈Sk

uk (i1, . . . , in)

×(

n

∑j=1

a j (i j) p j

) ∫XTn

(ψ ◦ϕ−1)

⎛⎜⎜⎜⎝

n

∑j=1

a j (i j) p jϕ (g j (x j))

n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎠dμTn (x)

⎞⎟⎟⎟⎠ , (15)

if the integrals exist and finite.(b) For each k ∈ N+ and for all t ∈ [0,1] , integral means can be defined with

respect to (7) by

Mψ,ϕ (t;k)= Mψ,ϕ (t;gi;μi; pi;ai;k)

:= ψ−1

⎛⎝ ∑

(i1,...,in)∈Sk

uk (i1, . . . , in)

(n

∑j=1

a j (i j) p j

) ∫XTn

(ψ ◦ϕ−1)

×

⎛⎜⎜⎜⎜⎜⎝t

n

∑j=1

a j (i j) p jϕ (g j (x j))

n

∑j=1

a j (i j) p j

+(1− t)

n

∑j=1

a j (i j) p j

∫Xj

ϕ ◦ g jdμ j

n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎟⎟⎠dμTn (x)

⎞⎟⎟⎟⎟⎟⎠ , (16)

if the integrals exist and finite.

It has been shown in [13] that for any j = 1, . . . ,n

∑(i1,...,in)∈Sk

uk (i1, . . . , in)a j (i j) = 1, (17)


and therefore by (H3 )

∑(i1,...,in)∈Sk

uk (i1, . . . , in)

(n

∑j=1

a j (i j) p j

)= 1, k ∈ N+. (18)

This implies that

Mψ,ϕ (k) ∈ I and Mψ,ϕ (t;k) ∈ I, k ∈ N+, t ∈ [0,1] ,

that is they really define means.By Remark 1, if ϕ ◦ gi and ψ ◦ gi are μi -integrable on Xi (i = 1, . . . ,n) , and

ψ ◦ϕ−1 is either convex or concave, then the integrals in (15) and (16) exist and finite.The following integral mean is also needed: if (H3 –H4 ) and (H7 ) are satisfied,

and χ : I → R is a continuous and strictly monotone function, then define

Mχ = Mχ(gi;μi; pi) := χ−1

⎛⎝ n

∑i=1

pi

∫Xi

χ ◦ gidμi

⎞⎠ , (19)

if the integrals exist and finite.Let q , g : [a,b] → R be positive and Lebesgue-integrable functions, and let χ :

]0,∞[ → R be a continuous and strictly monotone function. The so called generalizedweighted quasi-arithmetic mean of g with respect to the weight function q

Mχ = Mχ(g;q) := χ−1

⎛⎜⎜⎜⎜⎜⎜⎝

b∫a

q(x)χ (g(x))dx

b∫a

q(x)dx

⎞⎟⎟⎟⎟⎟⎟⎠ (20)

is a special case of (19), and it contains different remarkable means (for example,weighted arithmetic, harmonic and geometric means). The properties of means (20) arestudied intensively, we just mention two papers dealing with integral means: Haluskaand Hutnik [6] and Sun, Long and Chu [19].

We continue this section with a discussion on the monotonicity of the introducedmeans.

THEOREM 5. Assume (H1 –H4 ), (H7 –H8 ), and assume that ϕ ◦gi and ψ ◦gi areμi -integrable on Xi (i = 1, . . . ,n) .Then

(a)Mϕ � Mψ,ϕ (1) � . . . � Mψ,ϕ (k) � . . . � Mψ , k ∈ N+, (21)

andMϕ � Mψ,ϕ (0;k) � Mψ,ϕ (t;k) � Mψ,ϕ (k), k ∈ N+, t ∈ [0,1] ,

if either ψ ◦ϕ−1 is convex and ψ is increasing or ψ ◦ϕ−1 is concave and ψ is de-creasing.


(b)

Mϕ � Mψ,ϕ (1) � . . . � Mψ,ϕ (k) � . . . � Mψ , k ∈ N+, (22)

and

Mϕ � Mψ,ϕ (0;k) � Mψ,ϕ (t;k) � Mψ,ϕ (k), k ∈ N+, t ∈ [0,1] ,

if either ψ ◦ ϕ−1 is convex and ψ is decreasing or ψ ◦ ϕ−1 is concave and ψ isincreasing.

Proof. (a) and (b) can be obtained by applications of Theorem 2 to the functionsψ ◦ϕ−1 and ϕ ◦ gi (i = 1, . . . ,n) (ϕ (I) is an interval), if ψ ◦ϕ−1 is convex, and tothe functions −ψ ◦ϕ−1 and ϕ ◦gi (i = 1, . . . ,n) , if ψ ◦ϕ−1 is concave, and then upontaking ψ−1 . �

Recently, in [17] by Khuram Ali Khan and Pecaric the inequalities (21) and (22)have been proved for the mean Mψ,ϕ(1) . It can be seen that our approach allows us toessentially generalize and extend some of the results from [17].

5. Connections to Hermite-Hadamard inequality

Different refinements of the left hand side of Hermite-Hadamard inequality can begot from Theorem 4.

THEOREM 6. Assume (H1 –H3 ), and let f be a convex function on [a,b] . Then(a)

f

(a+b

2

)� T1 � . . . � Tk � Tk+1 � . . . � 1

b−a

b∫a

f ,

where

Tk = Tk,n( f ; pi;ai)

=1

(b−a)n ∑(i1,...,in)∈Sk

uk (i1, . . . , in)

(n

∑j=1

a j (i j) p j

)

×∫

[a,b]n

f

⎛⎜⎜⎜⎝

n

∑j=1

a j (i j) p jx j

n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎠dλ n (x) , k ∈ N+.


f

(a+b

2

)= Hk (0) � Hk (t) � Hk (1) = Tk,


where

Hk (t) = Hk,n (t; f ; pi;ai)

=1

(b−a)n ∑(i1,...,in)∈Sk

uk (i1, . . . , in)

(n

∑j=1

a j (i j) p j

)

×∫

[a,b]n

f

⎛⎜⎜⎜⎝t

n

∑j=1

a j (i j) p jx j

n

∑j=1

a j (i j) p j

+(1− t)a+b

2

⎞⎟⎟⎟⎠dλ n (x) .

(c) Hk is convex and increasing for each k ∈ N+ . If f is continuous, then Hk isalso continuous.

Proof. We can apply Theorem 4 and Theorem 3, when the probability space is([a,b] ,B, 1

b−aλ)

(B now means the σ -algebra of Borel sets of [a,b]), I := [a,b] , gis the identity function on [a,b] , and f is a convex function on [a,b] . �

The investigation of functions like Hk , seems to be due to Dragomir, who hasintroduced and studied among others the function H1,1 in [4]. Many papers deal withsimilar functions, for example see Abdallah El Farissi [1], Dragomir and Agarwal [5],Yang and Wang [20] and Yang and Tseng [21]. Our result gives a new approach intreating the problem.

6. Proofs of the main inequalities

We need the following well known result:

LEMMA 1. (see [2], 16.1 Lemma) Let (Ω,A ,μ) be a measure space. Let E bea metric space, and f : E ×Ω → R a function with the properties

(i) ω → f (x,ω) is μ -integrable for each x ∈ E ,(ii) x → f (x,ω) is continuous at x0 ∈ E for every ω ∈ Ω ,(iii) there is a nonnegative μ -integrable function h on Ω such that

| f (x,ω)| � h(ω) , (x,ω) ∈ E ×Ω.

Then the function ϕ defined on E by

ϕ (x) =∫Ω

f (x,ω)dμ (ω)

is continuous at x0 .

Proof of Theorem 3. Fix k ∈ N+ .(a) Since convexity is invariant under affine maps, the integral is monotonic, and

the sum of convex functions is also convex, Hk is convex on [0,1] .

By applying the classical Jensen’s inequality, we get for all t ∈ [0,1] that

Hk (t) � ∑(i1,...,in)∈Sk

uk (i1, . . . , in)

(n

∑j=1

a j (i j) p j

)

× f

⎛⎜⎜⎜⎜⎜⎝∫

XTn

⎛⎜⎜⎜⎜⎜⎝t

n

∑j=1


n

∑j=1

a j (i j) p j

+(1− t)

n

∑j=1

a j (i j) p j

∫Xj

g jdμ j

n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎟⎟⎠dμTn (x)

⎞⎟⎟⎟⎟⎟⎠

= ∑(i1,...,in)∈Sk

uk (i1, . . . , in)

(n

∑j=1

a j (i j) p j

)f

⎛⎜⎜⎜⎜⎜⎝

n

∑j=1

a j (i j) p j

∫Xj

g jdμ j

n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎟⎟⎠

= Hk (0) . (23)

Suppose 0 � t1 < t2 � 1. The convexity of Hk , and Hk (t) � Hk (0) (t ∈ [0,1])mean that

Hk (t2)−Hk (t1)t2− t1

� Hk (t2)−Hk (0)t2

� 0,

and thus

Hk (t2) � Hk (t1) .

(b) When (18) is combined with (23) and with the discrete Jensen’s inequality, itfollows that

Hk (0) � f

⎛⎜⎝ ∑

(i1,...,in)∈Sk

uk (i1, . . . , in)

⎛⎜⎝ n

∑j=1

a j (i j) p j

∫Xj

g jdμ j

⎞⎟⎠⎞⎟⎠

= f

⎛⎜⎝ n

∑j=1

p j

∫Xj

g jdμ j

(∑

(i1,...,in)∈Sk

uk (i1, . . . , in)a j (i j)

)⎞⎟⎠= f

⎛⎜⎝ n

∑j=1

p j

∫Xj

g jdμ j

⎞⎟⎠ .

Hk (1) = Tk is obvious.

(c) It follows from (a).

(d) It remains only to show that Hk is continuous at 1 . We check the conditionsof Lemma 1.

(i) See Remark 1.


(ii) Since f is continuous, the function

t → f

⎛⎜⎜⎜⎜⎜⎝t

n

∑j=1


n

∑j=1

a j (i j) p j

+(1− t)

n

∑j=1

a j (i j) p j

∫Xj

g jdμ j

n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎟⎟⎠

is continuous at 1 for every x ∈ XTn .(iii) By applying the discrete Jensen’s inequality, we have

f

⎛⎜⎜⎜⎜⎜⎝t

n

∑j=1


n

∑j=1

a j (i j) p j

+(1− t)

n

∑j=1

a j (i j) p j

∫Xj

g jdμ j

n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎟⎟⎠

� t f

⎛⎜⎜⎜⎝

n

∑j=1


n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎠+(1− t) f

⎛⎜⎜⎜⎜⎜⎝

n

∑j=1

a j (i j) p j

∫Xj

g jdμ j

n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎟⎟⎠

� max

⎛⎜⎜⎜⎜⎜⎝ f

⎛⎜⎜⎜⎝

n

∑j=1


n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎠ , f

⎛⎜⎜⎜⎜⎜⎝

n

∑j=1

a j (i j) p j

∫Xj

g jdμ j

n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎠

for all t ∈ [0,1] and x ∈ XTn .Choose a fixed interior point a of I . Since f is convex

f (t) � f (a)+ f ′+ (a)(z−a) , z ∈ I,

where f ′+ (a) means the right-hand derivative of f at a . It follows from this that

f

⎛⎜⎜⎜⎜⎜⎝t

n

∑j=1


n

∑j=1

a j (i j) p j

+(1− t)

n

∑j=1

a j (i j) p j

∫Xj

g jdμ j

n

∑j=1

a j (i j) p j

⎞⎟⎟⎟⎟⎟⎠

� f (a)+ f ′+ (a)

⎛⎜⎜⎜⎜⎜⎝t

n

∑j=1


n

∑j=1

a j (i j) p j

+(1− t)

n

∑j=1

a j (i j) p j

∫Xj

g jdμ j

n

∑j=1

a j (i j) p j

−a

⎞⎟⎟⎟⎟⎟⎠


� min

⎛⎜⎜⎜⎝ f (a)+ f ′+ (a)

⎛⎜⎜⎜⎝

n

∑j=1


n

∑j=1

a j (i j) p j

−a

⎞⎟⎟⎟⎠

× f (a)+ f ′+ (a)

⎛⎜⎜⎜⎜⎜⎝

n

∑j=1

a j (i j) p j

∫Xj

g jdμ j

n

∑j=1

a j (i j) p j

−a

⎞⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎠

for all t ∈ [0,1] and x ∈ XTn .The result now follows from Lemma 1.The proof is complete. �

Proof of Theorem 2. (a) Since S1 = {(1, . . . ,1)} , (3), (2) and (H3 ) give that

T1 =∫

XTn

f

(n

∑j=1

p jg j (x j)

)dμTn (x) .

From the classical Jensen’s inequality we therefore have

T1 � f

⎛⎝∫

XTn

n

∑j=1

p jg j (x j)dμTn (x)

⎞⎠= f

⎛⎝ n

∑i=1

pi

∫Xi

gidμi

⎞⎠ .

According to Theorem 1

Tk,n(g(x1) , . . . ,g(xn) ; p1, . . . , pn;a1, . . . ,an)� Tk+1,n(g(x1) , . . . ,g(xn) ; p1, . . . , pn;a1, . . . ,an), k ∈ N+

for all fixed (x1, . . . ,xn) ∈ XTl , and hence

Tk � Tk+1, k ∈ N+.

Finally, it follows from the discrete Jensen’s inequality that

Tk � ∑(i1,...,in)∈Sk

uk (i1, . . . , in)∫

XTn

n

∑j=1

a j (i j) p j f (g j (x j))dμTn (x)

= ∑(i1,...,in)∈Sk

uk (i1, . . . , in)n

∑j=1

a j (i j) p j

∫Xj

f ◦ g jdμ j

=n

∑j=1

(∑

(i1,...,in)∈Sk

uk (i1, . . . , in)a j (i j)

)p j

∫Xj

f ◦ g jdμ j. (24)


This and (17) imply that

Tk �n

∑j=1

p j

∫Xj

f ◦ g jdμ j, k ∈ N+.

(b) Apply Theorem 3 (b).The proof is complete. �

Acknowledgements. The research of the 1st author has been supported by Hun-garian National Foundations for Scientific Research Grant No. K120186. The researchof the 2nd author has been fully supported by Croatian Science Foundation under theproject 5435.

RE F ER EN C ES

[1] ABDALLAH EL FARISSI, Simple proof and refinement of Hermite–Hadamard inequality, J. Math.Inequal. 4 (2010), no. 3, 365–369.

[2] H. BAUER, Measure and Integration Theory, Walter de Gruyter, Berlin, New York, 2001.[3] I. BRNETIC, C. E. M. PEARCE, J. PECARIC, Refinements of Jensen’s inequality, Tamkang J. Math.

31 (2000), no. 1, 63–69.[4] S. S. DRAGOMIR, Two mappings in connection to Hadamard’s inequalities, J. Math. Anal. Appl. 167

(1992), no. 1, 49–56.[5] S. S. DRAGOMIR, P. AGARWAL, Two new mappings associated with Hadamard’s inequalities for

convex functions, Appl. Math. Lett. 11 (1998), no. 3, 33–38.[6] J. HALUSKA, O. HUTNIK, Some inequalities involving integral means, Tatra Mt. Math. Publ. 35

(2007), no. 1, 131–146.[7] G. H. HARDY, J. E. LITTLEWOOD, G. POLYA, Inequalities, Cambridge University Press, Cambridge,

1978.[8] L. HORVATH, Inequalities corresponding to the classical Jensen’s inequality, J. Math. Inequal. 3

(2009), no. 2, 189–200.[9] L. HORVATH, A refinement of the integral form of Jensen’s inequality, J. Inequal. Appl. (2012)

2012:178, 19 pp.[10] L. HORVATH, J. PECARIC, A refinement of the discrete Jensen’s inequality, Math. Inequal. Appl. 14

(2011), no. 4, 777–791.[11] L. HORVATH, A new refinement of the discrete Jensen’s inequality depending on parameters, J. In-

equal. Appl. (2013) 2013:551, 16 pp.[12] L. HORVATH, Weighted form of a recent refinement of the discrete Jensen’s inequality, Math. Inequal.

Appl. 17 (2014), no. 3, 947–961.[13] L. HORVATH, Infinite refinements of the discrete Jensen’s inequality defined by recursion, J. Math.

Inequal. 9 (2015) no. 4, 1115–1132.[14] L. HORVATH, J. PECARIC, Refinements of the classical Jensen’s inequality coming from refinements

of the discrete Jensen’s inequality, Adv. Inequal. Appl., Vol. 2015 (2015), Art ID 8, 17 pp.[15] L. HORVATH, KHURAM ALI KHAN, J. PECARIC, Combinatorial Improvements of Jensen’s Inequal-

ity, Monographs in Inequalities 8, Element, Zagreb, 2014.[16] J. L. W. V. JENSEN, Sur les fonctions convexes et les inegalites entre les valeurs moyennes, Acta

Math. 30 (1906), 175–193.[17] KHURAM ALI KHAN, J. PECARIC, Mixed symmetric means related to the classical Jensen’s inequal-

ity, J. Math. Inequal. 7 (2013) no. 1, 43–62.[18] J. ROOIN, A refinement of Jensen’s inequality, JIPAM. J. Inequal. Pure Appl. Math. 6 (2), Article 38

(2005).[19] H. SUN, B. LONG, Y. CHU, On the generalized weighted quasi-arithmetic integral mean, Int. Journal

of Math. Analysis 7 (2013) no. 41, 2039–2048.


[20] G. S. YANG, C. S. WANG, Some refinements of Hadamard’s inequality, Tamkang J. Math. 28 (1997),87–92.

[21] G. S. YANG, K. L. TSENG, On certain integral inequalities related to Hermite-Hadamard inequali-ties, J. Math. Anal. Appl. 239 (1999), no. 1, 180–187.

(Received February 6, 2016) Laszlo HorvathDepartment of Mathematics, University of Pannonia

Egyetem u. 10, 8200 Veszprem, Hungarye-mail: [email protected]


Pierottijeva 6, 10000 Zagreb, Croatiae-mail: [email protected]



& Applications


GENERALIZATION OF POPOVICIU TYPE INEQUALITIES

VIA GREEN’S FUNCTION AND FINK’S IDENTITY

RAVI P. AGARWAL, SAAD IHSAN BUTT, JOSIP PECARIC AND ANA VUKELIC


Abstract. We obtain some useful identities via Green’s function and Fink’s identity, and applythem to generalize the known Popoviciu’s inequality for convex functions to higher order convexfunctions. Then we investigate the bounds for the identities related to the generalization of thePopoviciu inequality by using inequalities for the Cebysev functional. Some results relating tothe Gruss and Ostrowski type inequalities are also obtained. Finally, we construct new familiesof exponentially convex functions and Cauchy-type means by exploring at linear functionalsassociated with the obtained inequalities.

1. Introduction

Many areas in modern analysis directly or indirectly involve the applications ofconvex functions; further, convex functions are closely related to the theory of inequal-ities and many important inequalities are consequences of the applications of convexfunctions (see [10]). Divided differences are found to be very helpful when we aredealing with functions having different degrees of smoothness. The following defini-tion of divided difference is given in [10, p. 14].

DEFINITION 1. The n th-order divided difference of a function f : [a,b] → R atmutually distinct points x0, ...,xn ∈ [a,b] is defined recursively by

[xi; f ] = f (xi) , i = 0, . . . ,n,

[x0, . . . ,xn; f ] =[x1, . . . ,xn; f ]− [x0, . . . ,xn−1; f ]

xn − x0. (1)

It is easy to see that (1) is equivalent to

[x0, . . . ,xn; f ] =n

∑i=0

f (xi)q′ (xi)

, where q(x) =n

∏j=0

(x− x j) .

The following definition of a real valued convex function is characterized by n th-orderdivided difference (see [10, p. 15]).

Mathematics subject classification (2010): Primary 26D07, 26D15, 26D20, 26D99.Keywords and phrases: Convex function, divided difference, Fink’s identity, Cebysev functional,

Gruss inequality, Ostrowski inequality, exponential convexity.


1247


1248 R. P. AGARWAL, S. IHSAN BUTT, J. PECARIC AND A. VUKELIC

DEFINITION 2. A function f : [a,b] → R is said to be n -convex (n � 0) if andonly if for all choices of (n+1) distinct points x0, . . . ,xn ∈ [a,b] , [x0, . . . ,xn; f ] � 0holds.

If this inequality is reversed, then f is said to be n -concave. If the inequality isstrict, then f is said to be a strictly n -convex (n -concave) function.

REMARK 1. Note that 0-convex functions are non-negative functions, 1-convexfunctions are increasing functions, and 2-convex functions are simply the convex func-tions.

The following theorem gives an important criteria to examine the n -convexity ofa function f (see [10, p. 16]).

THEOREM 1. If f (n) exists, then f is n-convex if and only if f (n) � 0 .

In 1965, T. Popoviciu introduced a characterization of convex functions [11]. Theinequality of Popoviciu as given by Vasic and Stankovic in [12] can be written in thefollowing form (see [10, p. 173]):

THEOREM 2. Let m,k ∈ N , m � 3 , 2 � k � m−1 , [α,β ]⊂ R , x = (x1, ...,xm) ∈[α,β ]m , p = (p1, ..., pm) be a positive m-tuple such that ∑m

i=1 pi = 1 . Also let f :[α,β ] → R be a convex function. Then

pk,m(x,p; f ) � m− km−1

p1,m(x,p; f )+k−1m−1

pm,m(x,p; f ), (2)

where

pk,m(x,p; f ) = pk,m(x,p; f (x)) :=1

Cm−1k−1

∑1�i1<...<ik�m

(k

∑j=1

pi j

)f

⎛⎜⎜⎜⎝

k∑j=1

pi j xi j

k∑j=1

pi j

⎞⎟⎟⎟⎠

is the linear functional with respect to f .

In what follows in inequality (2), we will write

ϒ(x,p; f ) :=m− km−1

p1,m(x,p; f )+k−1m−1

pm,m(x,p; f )− pk,m(x,p; f ). (3)

REMARK 2. It is important to note that under the assumptions of Theorem 2, ifthe function f is convex then ϒ(x,p; f ) � 0, and ϒ(x,p; f ) = 0 for f (x) = x or whenf is a constant function.

Consider the Green function G : [α,β ]× [α,β ]→ R defined as

G(t,s) =

⎧⎨⎩

(t−β )(s−α)β−α , α � s � t;

(s−β )(t−α)β−α , t � s � β .

(4)

GENERALIZATION OF POPOVICIU TYPE INEQUALITIES 1249

The function G is convex and continuous w.r.t s and due to symmetry also w.r.t t .For any function λ : [α,β ] → R , λ ∈C2([α,β ]) , we have

λ (x) =β − xβ −α

λ (α)+x−αβ −α

λ (β )+∫ β

αG(x,s)λ ′′(s)ds, (5)

where the function G is defined in (4) (see [13]).In the present paper, we use A. M. Fink’s identity and prove many interesting

results. The following theorem is proved by A. M. Fink in [6].

THEOREM 3. Let a,b ∈ R , λ : [a,b] → R , n � 1 and λ (n−1) is absolutely con-tinuous on [a,b] . Then

λ (x) =n

b−a

∫ b

aλ (t)dt

−n−1

∑w=1

(n−ww!

)(λ (w−1) (a)(x−a)w −λ (w−1) (b)(x−b)w

b−a

)

+1

(n−1)!(b−a)

∫ b

a(x− t)n−1 w[a,b] (t,x)λ (n) (t)dt, (6)

where

w[a,b] (t,x) ={

t−a, a � t � x � b,t−b, a � x < t � b.

(7)

The organization of the paper is as follows: In Section 2, we use Fink’s identity andthe n -convexity of the function λ to establish a generalization of Popoviciu’s inequal-ity. In Section 3, we present some interesting results by employing Cebysev functionaland Gruss-type inequalities, also results relating to the Ostrowski-type inequality. Atthe end we study the functional defined as the difference between the R.H.S. and theL.H.S. of the generalized inequality. Here our objective is to investigate the propertiesof the functional, such as n -exponential and logarithmic convexity.

2. Generalization of Popoviciu’s inequality for n -convex functionsvia Green function and A. M. Fink’s identity

Motivated by the identity (3), we use (5) and Fink’s identity to prove the followinggeneralized identity.

THEOREM 4. Let λ : [α,β ]→R be such that for n � 3 , λ (n−1) is absolutely con-tinuous and let m,k ∈ N , m � 3 , 2 � k � m−1 , [α,β ]⊂R , x = (x1, ...,xm) ∈ [α,β ]m ,p = (p1, ..., pm) be a real m-tuple such that ∑k

j=1 pi j �= 0 for any 1 � i1 < ... < ik � m

and ∑mi=1 pi = 1 . Also let

k∑j=1

pi j xi j

k∑j=1

pi j

∈ [α,β ] for any 1 � i1 < ... < ik � m with G and


w[α ,β ](t,x) be the same as defined in (4) and (7) respectively. Then we have the follow-ing identity:

ϒ(x,p;λ (x))

= (n−2)

(λ (1) (β )−λ (1) (α)

β −α

)∫ β

αϒ(x,p;G(x,s))ds+

1(β −α)

∫ β

αϒ(x,p;G(x,s))

×( n−3

∑w=1

(n−2−w

w!

)(λ (w+1) (β )(s−β )w −λ (w+1) (α) (s−α)w

))ds

+1

(n−3)!(β −α)

∫ β

αλ (n) (t)

(∫ β

αϒ(x,p;G(x,s))(s− t)n−3 w[α ,β ] (t,s)ds

)dt. (8)

Proof. Using (5) in (3) and following the linearity of ϒ(x,p;λ (x)) , we have

ϒ(x,p;λ (s)) =∫ β

αϒ(x,p;G(x,s))λ

′′(s)ds. (9)

Differentiating (6), twice with respect variable s , we get

λ ′′ (s) =n−3

∑w=0

(n−2−w

w!

)(λ (w+1) (β )(s−β )w −λ (w+1) (α) (s−α)w

β −α

)

+1

(n−3)!(β −α)

∫ β

α(s− t)n−3 w[α ,β ] (t,s)λ (n) (t)dt

=n−2

∑w=1

(n−1−w(w−1)!

)(λ (w) (β )(s−β )w−1−λ (w) (α)(s−α)w−1

β −α

)

+1

(n−3)!(β −α)

∫ β

α(s− t)n−3 w[α ,β ] (t,s)λ (n) (t)dt

= (n−2)

(λ (1) (β )−λ (1) (α)

β −α

)

+n−2

∑w=2

(n−1−w(w−1)!

)(λ (w) (β )(s−β )w−1−λ (w) (α) (s−α)w−1

β −α

)

+1

(n−3)!(β −α)

∫ β

α(s− t)n−3 w[α ,β ] (t,s)λ (n) (t)dt. (10)

Using (10) in (9) and applying Fubini’s Theorem in the last term we get (8).


Alternatively, we use formula (6) for the function λ ′′ and replace n by n− 2(n � 3) , to get

λ ′′ (s) = (n−2)

(λ (1) (β )−λ (1) (α)

β −α

)

+n−3

∑w=1

(n−2−w

w!

)(λ (w+1) (β ) (s−β )w−λ (w+1) (α) (s−α)w

β −α

)

+1

(n−3)!(β −α)

∫ β

α(s− t)n−3 w[α ,β ] (t,s)λ (n) (t)dt (11)

= (n−2)

(λ (1) (β )−λ (1) (α)

β −α

)

+n−2

∑w=2

(n−1−w(w−1)!

)(λ (w) (β )(s−β )w−1−λ (w) (α) (s−α)w−1

β −α

)

+1

(n−3)!(β −α)

∫ β

α(s− t)n−3 w[α ,β ] (t,s)λ (n) (t)dt.

Now using (11) in (9) and applying Fubini’s Theorem in the last term we get (8). �The following theorem gives a generalization of Popoviciu’s inequality for n -

convex functions.

THEOREM 5. Let all the assumptions of Theorem 4 be satisfied and let for n � 3∫ β

αϒ(x,p;G(x,s))(s− t)n−3 w[α ,β ] (t,s)ds � 0, t ∈ [α,β ]. (12)

If λ is n-convex function such that λ (n−1) is absolutely continuous, then we have

ϒ(x,p;λ (x))

� (n−2)

(λ (1) (β )−λ (1) (α)

β −α

)∫ β

αϒ(x,p;G(x,s))ds+

1(β −α)

∫ β

αϒ(x,p;G(x,s))

×( n−3

∑w=1

(n−2−w

w!

)(λ (w+1) (β )(s−β )w −λ (w+1) (α) (s−α)w

))ds. (13)

Proof. Since λ (n−1) is absolutely continuous on [α,β ] , λ (n) exists almost every-where. As λ is n -convex, applying Theorem 1, we have, λ (n) � 0 for all x ∈ [α,β ] .Hence we can apply Theorem 4 to obtain (13) . �

Now we obtain a generalization of Popoviciu’s inequality for m-tuples.

THEOREM 6. Let in addition to the assumptions of Theorem 4, p = (p1, ..., pm) bea positive m-tuple such that ∑m

i=1 pi = 1 , and λ : [α,β ] → R be an n-convex function.

(i) If n is even and n > 3 , then (13) holds.

(ii) Let the inequality (13) be satisfied and

n−3

∑w=0

(n−2−w

w!

)(λ (w+1) (β )(s−β )w−λ (w+1) (α)(s−α)w

)� 0. (14)

Then we have

ϒ(x,p;λ (x)) � 0. (15)

Proof.

(i) Since Green’s function G(x,s) is convex and the weights are positive.

So ϒ(x,p;G(x,s)) � 0 by virtue of Remark 2. Also, since

ϑ (s) := (s− t)n−3w[α ,β ] (t,s) ={

(s− t)n−3 (t−α) , α � t � s � β ,

(s− t)n−3 (t−β ) , α � s < t � β ,

ϑ is positive for even n , where n > 3. So, (12) holds for even n . Now followingTheorem 5, we can obtain (13).

(ii) Using (14) in (13), we get (15). �

3. Bounds for identities related to generalization of Popoviciu’s inequality

In this section we present some interesting results by using Cebysev functionaland Gruss type inequalities. For two Lebesgue integrable functions f ,h : [α,β ] → R ,we consider the Cebysev functional

Δ( f ,h) =1

β −α

∫ β

αf (t)h(t)dt − 1

β −α

∫ β

αf (t)dt.

1β −α

∫ β

αh(t)dt.

The following Gruss type inequalities are given in [5].

THEOREM 7. Let f : [α,β ]→ R be a Lebesgue integrable function and h : [α,β ]→ R be an absolutely continuous function with (.−α)(β − .)[h′]2 ∈ L[α,β ]. Then wehave the inequality

|Δ( f ,h)| � 1√2[Δ( f , f )]

12

1√β −α

(∫ β

α(x−α)(β − x)[h′(x)]2dx

) 12

. (16)

The constant 1√2

THEOREM 8. Assume that h : [α,β ] → R is monotonic nondecreasing on [α,β ]and f : [α,β ] → R be an absolutely continuous with f ′ ∈ L∞[α,β ]. Then we have theinequality

|Δ( f ,h)| � 12(β −α)

|| f ′||∞∫ β

α(x−α)(β − x)dh(x). (17)


In the sequel, we consider above theorems to derive generalizations of the resultsproved in the previous section. In what follows we let

O(t) =∫ β

αϒ(x,p;G(x,s))(s− t)n−3 w[α ,β ] (t,s)ds � 0, t ∈ [α,β ]. (18)

THEOREM 9. Let λ : [α,β ] → R be such that for n � 3 , λ (n) is absolutely con-tinuous with (.−α)(β − .)[λ (n+1)]2 ∈ L[α,β ] . Let m,k ∈ N , m � 3 , 2 � k � m− 1 ,[α,β ] ⊂ R , x = (x1, ...,xm) ∈ [α,β ]m , p = (p1, ..., pm) be a real m-tuple such that

∑kj=1 pi j �= 0 for any 1 � i1 < ... < ik � m and ∑m

i=1 pi = 1 . Also let

k∑j=1

pi j xi j

k∑j=1

pi j

∈ [α,β ]

for any 1 � i1 < ... < ik � m with O defined in (18) . Then the remainder Kn(α,β ;λ )given in the following identity

ϒ(x,p;λ (x)) =∫ β

αϒ(x,p;G(x,s))

×n−3

∑w=0

(n−2−w

w!

)(λ (w+1) (β )(s−β )w −λ (w+1) (α)(s−α)w

β −α

)ds

+λ (n−1)(β )−λ (n−1)(α)

(β −α)2 (n−3)!

∫ β

αO(t)dt +Kn(α,β ;λ ), (19)

satisfies the bound

|Kn(α,β ;λ )| � 1√2(n−3)!

[Δ(O,O)]12

1√β −α

∣∣∣∣∫ β

α(t−α)(β − t)[λ (n+1)(t)]2dt

∣∣∣∣12

.

Proof. The proof is the direct application of Theorem 7 by making substitutionsf → O and h → λ (n) . �

The following Gruss type inequalities can be obtained by using Theorem 8.

THEOREM 10. Let λ : [α,β ] → R be such that for n � 3 , λ (n) is absolutely con-tinuous and let λ (n+1) � 0 on [α,β ] with O defined in (18) . Then in the representation(19) the remainder Kn(α,β ;λ ) satisfies the estimate

|Kn(α,β ;λ )| � ||O′||∞(n−3)!

[λ (n−1)(β )+ λ (n−1)(α)

2− λ (n−2)(β )−λ (n−2)(α)

β −α

].

Proof. Applying Theorem 8 for f → O and h → λ (n) and following the steps ofTheorem 3.4 in [3] (see also [4]), we get above result. �

Next we present an Ostrowski type inequality related to generalizations of Popovi-ciu’s inequality.

THEOREM 11. Suppose all the assumptions of Theorem 4 be satisfied. Moreover,assume that (p,q) is a pair of conjugate exponents, that is p,q∈ [1,∞] such that 1/p+1/q = 1 . Let |λ (n)|p : [α,β ] → R be a R-integrable function for some n � 3 . Then, wehave∣∣∣∣ϒ(x,p;λ (x))−

∫ β

αϒ(x,p;G(x,s))

×n−3

∑w=0

(n−2−w

w!

)(λ (w+1) (β ) (s−β )w−λ (w+1) (α) (s−α)w

β −α

)ds

∣∣∣∣� 1

(n−3)!β−α||λ (n)||p

(∫ β

α

∣∣∣∣∫ β

αϒ(x,p;G(x,s))(s−t)n−3 w[α ,β ] (t,s)ds

∣∣∣∣qdt

)1/q

. (20)

The constant on the R.H.S. of (20) is sharp for 1 < p � ∞ and the best possible forp = 1 .

Proof. The proof is similar to the Theorem 3.5 in [3] (see also [4]). �

4. Mean value theorems and n -exponential convexity

In the present section, we construct a positive linear functional and then give meanvalue theorems of Lagrange and Cauchy type.

REMARK 3. In virtue of Theorem 5, we can define the positive linear functionalwith respect to n -convex function λ as follows

Γ(λ ) := ϒ(x,p;λ (x))−∫ β

αϒ(x,p;G(x,s))

×n−3

∑w=0

(n−2−w

w!

)(λ (w+1) (β )(s−β )w−λ (w+1) (α) (s−α)w

β−α

)ds � 0. (21)

Lagrange and Cauchy type mean value theorems related to above functional aregiven in the following theorems.

THEOREM 12. Let λ : [α,β ] → R be such that λ ∈Cn[α,β ] . If the inequality in(12) holds, then there exist ξ ∈ [α,β ] such that

Γ(λ ) = λ (n)(ξ )Γ(ϕ),

where ϕ(x) = xn

n! and Γ(·) is defined by (21).


Proof. Similar to the proof of Theorem 4.1 in [8] (see also [1]). �

THEOREM 13. Let λ ,ψ : [α,β ] → R be such that λ ,ψ ∈ Cn[α,β ] . If the in-equality in (12) holds, then there exist ξ ∈ [α,β ] such that

Γ(λ )Γ(ψ)

=λ (n)(ξ )ψ(n)(ξ )

,

provided that the denominators are non-zero, where Γ(·) is defined by (21).

Proof. Similar to the proof of Corollary 4.2 in [8] (see also [1]). �Theorem 13 enables us to define Cauchy means, in fact

ξ =

(λ (n)

ψ(n)

)−1(Γ(λ )Γ(ψ)

),

means that ξ is the mean of α , β for given functions λ and ψ .We conclude our paper with the following remark.

REMARK 4. One can construct the non trivial examples of n−exponentially andexponentially convex functions from positive linear functional Γ(·) by following then−exponentially method introduced by Pecaric et.al. in [7] and [9] (see also [2], [3]and [4]). As an application to Cauchy means, it enables us to construct a large familiesof functions which are exponentially convex.

Acknowledgements. The research of 2nd author has been fully supported by H. E.C. Pakistan. The research of 3rd and 4th author has been fully supported by CroatianScience Foundation under the project 5435.

RE F ER EN C ES

[1] S. I. BUTT AND J. PECARIC, Generalized Hermite-Hadamard’s Inequality, Proc. A. Razmadze Math.Inst. 163 (2013), 9–27.

[2] S. I. BUTT, J. PECARIC AND A. VUKELIC, Generalization of Popoviciu type inequalities Via Fink’sidentity, Mediterr. J. Math. (to appear).

[3] S. I. BUTT, K. A. KHAN AND J. PECARIC, Further generalization of Popoviciu inequality for higherorder convex functions via Taylor polynomial, Turkish J. Math. 40 (2016), 333–349.

[4] S. I. BUTT, K. A. KHAN AND J. PECARIC, Popoviciu type inequalities Via Green function andgeneralized Montgomery identity, Math. Inequal. Appl. 18 (4) (2015), 1519–1538.

[5] P. CERONE AND S. S. DRAGOMIR, Some new Ostrowski-type bounds for the Cebysev functional andapplications, J. Math. Inequal. 8 (1) (2014), 159–170.

[6] A. M. FINK, Bounds of the deviation of a function from its avereges, Czechoslovak Math. J. 42 (117)(1992), 289–310.

[7] J. JAKSETIC AND J. PECARIC, Exponential Convexity Method, J. Convex Anal. 20 (1) (2013), 181–197.

[8] J. JAKSETIC, J. PECARIC AND A. PERUSIC, Steffensen inequality, higher order convexity and expo-nential convexity, Rend. Circ. Mat. Palermo. 63 (1) (2014), 109–127.


[9] J. PECARIC AND J. PERIC, Improvement of the Giaccardi and the Petrovic Inequality and RelatedStolarsky Type Means, An. Univ. Craiova Ser. Mat. Inform. 39 (1) (2012), 65–75.

[10] J. PECARIC, F. PROSCHAN AND Y. L. TONG, Convex functions, Partial Orderings and StatisticalApplications, Academic Press, New York, (1992).

[11] T. POPOVICIU, Sur certaines inegalites qui caracterisent les fonctions convexes, Analele StiintificeUniv. Al. I. Cuza, Iasi, Sectia Mat. 11 (1965), 155–164.

[12] P. M. VASIC AND LJ. R. STANKOVIC, Some inequalities for convex functions, Math. Balkanica. 6(44) (1976), 281–288.

[13] D. V. WIDDER, Completely convex function and Lidstone series, Trans. Am. Math. Soc. 51 (1942),387–398.

(Received March 12, 2016) Ravi P. AgarwalDepartment of Mathematics, Texas A&M University – Kingsville

700 University Blvd, Kingsville, USAe-mail: [email protected]

Saad Ihsan ButtDepartment of Mathematics, COMSATS

Institute of Information TechnologyLahore, Pakistan



10000 Zagreb, Croatiae-mail: [email protected]

Ana VukelicFaculty of Food Technology and BiotechnologyMathematics Department, University of Zagreb




& Applications


GENERALIZATION OF MAJORIZATION

THEOREM VIA TAYLOR’S FORMULA

SAAD IHSAN BUTT, LJILJANKA KVESIC AND JOSIP PECARIC


Abstract. We give generalization of majorization theorem for the class of n-convex functionsby using Taylor’s formula. We use inequalities for the Cebysev functional to obtain boundsfor the identities related to generalizations of majorization inequalities. We present mean valuetheorems and n−exponential convexity for the functional obtained from the generalized ma-jorization inequalities. At the end we discuss the results for particular families of function andgive means.

1. Introduction

Majorization gives us the precise answer about the location of the components ofthe vector x respected to that of vector y . The well known Majorization theorem givenby Marshall and Olkin [11] (see also [15], p. 320):

THEOREM 1. Let I be an interval in R and let x,y be two n-tuples such thatxi,yi ∈ I (i = 1, ...,n) . Then

n

∑i=1

φ(yi) �n

∑i=1

φ(xi)

holds for every continuous convex function φ : I → R iff

m

∑i=1

y[i] �m

∑i=1

x[i]

holds for m = 1,2, ...,n−1 and

n

∑i=1

y[i] =n

∑i=1

x[i].

The generalization of Theorem 1 was given by Fuchs in [6] as weighted Majoriza-tion Theorem (see also [15], p. 323):

Mathematics subject classification (2010): Primary 26D07, 26D15, 26D20, 26D99.Keywords and phrases: Convex function, divided difference, Taylor’s formula, Cebysev functional,

Gruss inequality, Ostrowski inequality, exponential convexity.


1257


1258 S. IHSAN BUTT, LJ. KVESIC AND J. PECARIC

THEOREM 2. Let x,y be two decreasing n-tuples from an interval I , let w =(w1, ...,wn) be a real n-tuple such that

k

∑i=1

wiyi �k

∑i=1

wixi, for k = 1, ...,n−1; (1)

andn

∑i=1

wiyi =n

∑i=1

wixi. (2)

Then for every continuous convex function φ : I → R , we have

n

∑i=1

wiφ(yi) �n

∑i=1

wiφ(xi). (3)

The following integral version of Theorem 2 is a simple consequence of TheoremA in [13] (see also [15], p. 328):

THEOREM 3. Let x,y : [a,b] → [α,β ] such that [α,β ] ⊂ I be decreasing andw : [a,b] → R be continuous functions. If∫ v

aw(t)y(t)dt �

∫ v

aw(t)x(t)dt for every v ∈ [a,b] (4)

and ∫ b

aw(t)y(t)dt =

∫ b

aw(t)x(t)dt (5)

hold, then for every continuous convex function φ : I → R , we have∫ b

aw(t)φ((y(t))dt �

∫ b

aw(t)φ(x(t))dt. (6)

For other integral version and generalization of majorization theorem see ([11], p.583) (see also [12], [1], [10]). The classical Taylor’s formula with integral remaindercan be stated as:

THEOREM 4. Let n be a positive integer and φ : [a,b]→ R be such that φ (n−1) isabsolutely continuous, then for all x ∈ [a,b] the Taylor’s formula at the point c ∈ [a,b]is

φ(x) = Tn−1(φ ;c,x)+Rn−1(φ ;c,x),

where Tn−1(φ ;c,x) is a Taylor’s polynomial of degree n−1 , i.e.

Tn−1(φ ;c,x) =n−1

∑k=0

φ (k)(c)k!

(x− c)k

and the remainder is given by

Rn−1(φ ;c,x) =1

(n−1)!

∫ x

cφ (n)(t)(x− t)n−1dt.

GENERALIZATION OF MAJORIZATION THEOREM VIA TAYLOR’S FORMULA 1259

In rest of the paper, we need the following real valued function of our interestdefined as:

(x− t)+ ={

x− t, t � x,0, t > x.

For two Lebesgue integrable functions f ,h : [α,β ] → R , we consider the Cebysevfunctional

Δ( f ,h) =1

β −α

∫ β

αf (t)h(t)dt− 1

β −α

∫ β

αf (t)dt · 1

β −α

∫ β

αh(t)dt.

In [5] the authors proved the following theorems:

THEOREM 5. Let f : [α,β ]→R be a Lebesgue integrable function and h : [α,β ]→R be an absolutely continuous function with (.−α)(β − .)[h′]2 ∈L[α,β ]. Then we havethe inequality

|Δ( f ,h)| � 1√2[Δ( f , f )]

12

1√β −α

(∫ β

α(x−α)(β − x)[h′(x)]2dx

) 12

. (7)

The constant 1√2



|Δ( f ,h)| � 12(β −α)

|| f ′||∞∫ β

α(x−α)(β − x)dh(x). (8)


2. Main results

We start the section with the proof of identities obtained by using Taylor’s formula.

THEOREM 7. Let φ : [α,β ]→ R be such that φ (n−1) is absolutely continuous forsome n � 1 and let w = (w1, ...,wm) , x = (x1, ...,xm) and y = (y1, ...,ym) be m-tuplessuch that xi,yi ∈ [α,β ] , wi ∈ R (i = 1, ...,m) . Then

m

∑i=1

wiφ(xi)−m

∑i=1

wiφ(yi)

=n−1

∑k=1

φ (k)(α)k!

( m

∑i=1

wi(xi−α)k −m

∑i=1

wi(yi −α)k)

+1

(n−1)!

∫ β

α

[ m

∑i=1

wi((xi − t)+)n−1−m

∑i=1

wi((yi − t)+)n−1]

φ (n)(t)dt, (9)


and

m

∑i=1

wiφ(xi)−m

∑i=1

wiφ(yi)

=n−1

∑k=1

φ (k)(β )k!

( m

∑i=1

wi(β − xi)k −m

∑i=1

wi(β − yi)k)

(−1)k

− 1(n−1)!

∫ β

α(−1)n−1

[ m

∑i=1

wi((t − xi)+)n−1−m

∑i=1

wi((t − yi)+)n−1]

φ (n)(t)dt. (10)

Proof. Using Taylor’s formula at point α in ∑mi=1 wiφ(xi)−∑m

i=1 wiφ(yi) , we have

m

∑i=1

wiφ(xi)−m

∑i=1

wiφ(yi)

=m

∑i=1

wi

( n−1

∑k=0

φ (k)(α)k!

(xi−α)k +1

(n−1)!

∫ xi

αφ (n)(t)(xi − t)n−1dt

)

−m

∑i=1

wi

( n−1

∑k=0

φ (k)(α)k!

(yi −α)k +1

(n−1)!

∫ yi

αφ (n)(t)(yi − t)n−1dt

)

=n−1

∑k=1

φ (k)(α)k!

( m

∑i=1

wi(xi−α)k −m

∑i=1

wi(yi −α)k)

+1

(n−1)!

∫ xi

α

m

∑i=1

wi(xi − t)n−1φ (n)(t)dt− 1(n−1)!

∫ yi

α

m

∑i=1

wi(yi− t)n−1φ (n)(t)dt

=n−1

∑k=1

φ (k)(α)k!

( m

∑i=1

wi(xi−α)k −m

∑i=1

wi(yi −α)k)

+1

(n−1)!

∫ β

α

m

∑i=1

wi((xi−t)+)n−1φ (n)(t)dt− 1(n−1)!

∫ β

α

m

∑i=1

wi((yi−t)+)n−1φ (n)(t)dt

where ∫ β

α

m

∑i=1

wi((xi − t)+)n−1φ (n)(t)dt =∫ xi

α

m

∑i=1

wi(xi − t)n−1φ (n)(t)dt +∫ β

xi

0

and ∫ β

α

m

∑i=1

wi((yi − t)+)n−1φ (n)(t)dt =∫ yi

α

m

∑i=1

wi(yi − t)n−1φ (n)(t)dt +∫ β

yi

0

So by using above result we will get (9) .Similarly using Taylor’s formula at point β in ∑m

i=1 wiφ(xi)−∑mi=1 wiφ(yi) , we

get (10) . �

Integral version of the above theorem can be stated as:


THEOREM 8. Let φ : [α,β ]→ R be such that φ (n−1) is absolutely continuous forsome n � 1 and let x,y : [a,b]→ [α,β ] , w : [a,b] → R be continuous functions. Then∫ b

aw(τ)φ(x(τ))dτ −

∫ b

aw(τ)φ(y(τ))dτ

=n−1

∑k=1

φ (k)(α)k!

(∫ b

aw(τ)

[(x(τ)−α)k − (y(τ)−α)k

]dτ)

+1

(n−1)!

∫ β

α

(∫ b

aw(τ)

[((x(τ)− t)+)n−1− ((y(τ)− t)+)n−1

]dτ)

φ (n)(t)dt,

(11)

and∫ b

aw(τ)φ(x(τ))dτ −

∫ b

aw(τ)φ(y(τ))dτ

=n−1

∑k=1

φ (k)(β )k!

(∫ b

aw(τ)

[(β − x(τ))k − (β − y(τ))k

]dτ)

(−1)k

− 1(n−1)!

∫ β

α(−1)n−1

(∫ b

aw(τ)

[((t−x(τ))+)n−1−((t−y(τ))+)n−1

]dτ)

φ (n)(t)dt.

(12)

In the following theorem we obtain generalizations of majorization inequality forn -convex functions.

THEOREM 9. Let φ : [α,β ]→ R be such that φ (n−1) is absolutely continuous forsome n � 1 and let w = (w1, ...,wm) , x = (x1, ...,xm) and y = (y1, ...,ym) be m-tuplessuch that xi,yi ∈ [α,β ] , wi ∈ R (i = 1, ...,m) . Then

(i) if φ is n-convex function andm

∑i=1

wi((xi − t)+)n−1−m

∑i=1

wi((yi − t)+)n−1 � 0, t ∈ [α,β ], (13)

thenm

∑i=1

wiφ(xi)−m

∑i=1

wiφ(yi) �n−1

∑k=1

φ (k)(α)k!

( m

∑i=1

wi(xi −α)k −m

∑i=1

wi(yi −α)k)

.

(14)

(ii) If φ is n-convex function and

(−1)n−1( m

∑i=1

wi((t − xi)+)n−1−m

∑i=1

wi((t − yi)+)n−1)

� 0, t ∈ [α,β ], (15)

thenm

∑i=1

wiφ(xi)−m

∑i=1

wiφ(yi) �n−1

∑k=1

φ (k)(β )k!

( m

∑i=1

wi(β −xi)k−m

∑i=1

wi(β −yi)k)

(−1)k.

(16)


Proof. Since the function φ is n -convex, therefore without loss of generality wecan assume that φ is n-times differentiable and φ (n) � 0 (see [15], p. 16). Hence wecan apply Theorem 7 to obtain (14) and (16) respectively. �

In the following Corollary, we give generalization of Fuch’s majorization theorem.

COROLLARY 1. Let all the assumptions of Theorem 7 be satisfied, x = (x1, ...,xm) ,y = (y1, ...,ym) be decreasing m-tuples and w = (w1, ...,wm) be any m-tuple such thatxi,yi ∈ [α,β ] , wi ∈ R (i = 1, ...,m) which satisfies (1) , (2) and φ : [α,β ] → R isn-convex function. Then

(i) for n � 1 , (14) holds. Moreover, let the inequality (14) be satisfied. If the function

F1(x) :=n−1

∑k=1

φ (k)(α)k!

(x−α)k. (17)

is convex, the R.H.S. of (14) is non negative, that is (3) holds.

(ii) If n is even, then (16) holds. Moreover, let the inequality (16) be satisfied. If thefunction

F2(x) :=n−1

∑k=1

(−1)kφ (k)(β )k!

(β − x)k. (18)

is convex, the R.H.S. of (16) is non negative, that is (3) holds.

Proof. (i) On account of given m-tuples satisfying (1) , (2) and the function x �→((x− t)+)n−1 being convex for given n , (13) holds by virtue of Theorem 2. Thereforeby following Theorem 9 we can obtain (14). Moreover, we can rewrite the R.H.S. of(14) in the form of the L.H.S. with φ = F1 , where F1 is defined in (17) and will beobtained after reorganization of this side. Since F1 is assumed to be convex, thereforeusing the given conditions on m-tuples and by following Theorem 2 the non negativityof R.H.S. of (14) is immediate, that is (3) holds.

Similarly we can prove the part (ii) . �

3. New upper bounds via Cebysev functional

In the sequel, we consider Theorems 5 and 6 to derive generalizations of theresults proved in the previous section. Let w = (w1, ...,wm) , x = (x1, ...,xm) andy = (y1, ...,ym) be m-tuples such that xi,yi ∈ [α,β ] , wi ∈ R (i = 1, ...,m) , denote

R(t) =m

∑i=1

wi((xi − t)+)n−1−m

∑i=1

wi((yi − t)+)n−1, t ∈ [α,β ], (19)

B(t) = (−1)n−1( m

∑i=1

wi((t− xi)+)n−1−m

∑i=1

wi((t − yi)+)n−1)

, t ∈ [α,β ]. (20)


THEOREM 10. Let φ : [α,β ] → R be such that φ (n) is absolutely continuous forsome n � 1 with (.− α)(β − .)[φ (n+1)]2 ∈ L[α,β ] and let w = (w1, ...,wm) , x =(x1, ...,xm) and y = (y1, ...,ym) be m-tuples such that xi,yi ∈ [α,β ] , wi ∈ R (i =1, ...,m) and let the functions R , B be defined by (19) , (20) respectively . Then

(i) the remainder K1n(α,β ;φ) given in the following identity

m

∑i=1

wiφ(xi)−m

∑i=1

wiφ(yi)

=n−1

∑k=1

φ (k)(α)k!

( m

∑i=1

wi(xi −α)k −m

∑i=1

wi(yi −α)k)

+φ (n−1)(β )−φ (n−1)(α)

(β −α)(n−1)!

∫ β

αR(t)dt +K1

n(α,β ;φ), (21)


|K1n(α,β ;φ)| � 1

(n−1)![Δ(R,R)]

12

√β −α

2

∣∣∣∣∫ β

α(t −α)(β − t)[φ (n+1)(t)]2dt

∣∣∣∣12

.

(ii) The remainder K2n(α,β ;φ) given in the following identity

m

∑i=1

wiφ(xi)−m

∑i=1

wiφ(yi)

=n−1

∑k=1

φ (k)(β )k!

( m

∑i=1

wi(β − xi)k −m

∑i=1

wi(β − yi)k)

(−1)k

+φ (n−1)(β )−φ (n−1)(α)

(α −β )(n−1)!

∫ β

αB(t)dt−K2

n(α,β ;φ), (22)


|K2n(α,β ;φ)| � 1

(n−1)![Δ(B,B)]

12

√β −α

2

∣∣∣∣∫ β

α(t−α)(β − t)[φ (n+1)(t)]2dt

∣∣∣∣12

.

Proof. Applying Theorem 5 for f �→R and h �→ φ (n) and employ similar methodas in Theorem 16 [9]. �

The following Gruss type inequalities can be obtained by using Theorem 6

THEOREM 11. Let φ : [α,β ] → R be such that φ (n) (n � 1) is absolutely con-tinuous function and φ (n+1) � 0 on [α,β ] and let the functions R , B be defined by(19) , (20) respectively. Then, we have

(i) the representation (21) and the remainder K1n(α,β ;φ) satisfies the bound

|K1n(α,β ;φ)| � 1

(n−1)!||R′||∞

[φ (n−1)(β )+ φ (n−1)(α)

2− φ (n−2)(β )−φ (n−2)(α)

β −α

].

(ii) The representation (22) and the remainder K2n(α,β ;φ) satisfies the bound

|K2n(α,β ;φ)| � 1

(n−1)!||B′||∞

[φ (n−1)(β )+ φ (n−1)(α)

2− φ (n−2)(β )−φ (n−2)(α)

β −α

].

Proof. Applying Theorem 6 for f �→R and h �→ φ (n) and employ similar methodas in Theorem 17 [9]. �

Now we intend to give the Ostrowski type inequalities related to generalizationsof majorization’s inequality.

THEOREM 12. Assume that all the assumptions of Theorem 7 hold. Moreover,assume (p,q) is a pair of conjugate exponents, that is 1 � p,q � ∞ , 1/p+ 1/q = 1 .Let |φ (n)|p : [α,β ] → R be a R-integrable function for some n � 1 . Then, we have:

(i)∣∣∣∣ m

∑i=1

wiφ(xi)−m

∑i=1

wiφ(yi)

−n−1

∑k=1

φ (k)(α)k!

( m

∑i=1

wi(xi −α)k −m

∑i=1

wi(yi −α)k)∣∣∣∣

� 1(n−1)!

||φ (n)||p(∫ β

α

∣∣∣∣ m

∑i=1

wi((xi − t)+)n−1−m

∑i=1

wi((yi − t)+)n−1

∣∣∣∣qdt

)1/q

.

(23)

(ii)∣∣∣∣ m

∑i=1

wiφ(xi)−m

∑i=1

wiφ(yi)

−n−1

∑k=1

φ (k)(β )k!

( m

∑i=1

wi(β − xi)k −m

∑i=1

wi(β − yi)k)

(−1)k

∣∣∣∣� 1

(n−1)!||φ (n)||p

(∫ β

α

∣∣∣∣(−1)n−1[ m

∑i=1

wi((t−xi)+)n−1−m

∑i=1

wi((t−yi)+)n−1]∣∣∣∣qdt

)1/q

.

(24)

The constant on the R.H.S. of (23) and (24) are sharp for 1 < p � ∞ and the bestpossible for p = 1 .

Proof. To prove above results, we employ similar method adopted in Theorem 19[9]. �


4. Associated linear functionals and exponential convexity

In the present section we will construct some linear functionals as differences ofthe L. H. S and R. H. S. of some of the inequalities derived earlier. The obtained linearfunctionals will be used in the construction of new families of exponentially convexfunctions and some related results will be derived.

Some definitions and basic results regarding exponentially convex functions canbe seen from [2], [7] and [14] which are used in sequel.

REMARK 1. By the virtue of Theorem 9, we define the positive linear functionalswith respect to n -convex function φ as follows

Ω1(φ) :=m

∑i=1

wiφ(xi)−m

∑i=1

wiφ(yi)

−n−1

∑k=1

φ (k)(α)k!

( m

∑i=1

wi(xi−α)k −m

∑i=1

wi(yi −α)k)

� 0, (25)

Ω2(φ) :=m

∑i=1

wiφ(xi)−m

∑i=1

wiφ(yi)

−n−1

∑k=1

φ (k)(β )k!

( m

∑i=1

wi(β − xi)k −m

∑i=1

wi(β − yi)k)

(−1)k � 0. (26)

The Lagrange and Cauchy type mean value theorems related to defined functionalsare in the following theorems.

THEOREM 13. Let φ : [α,β ] → R be such that φ ∈ Cn[α,β ] . If the inequalitiesin (14) and (16) are valid, then there exist ξi ∈ [α,β ] such that

Ωi(φ) = φ (n)(ξi)Ωi(ϕ); i = 1,2,

where ϕ(x) = xn

n! and Ωi(·) are defined in Remark 1.

Proof. Similar to the proof of Theorem 4.1 in [8] (see also [3]). �

THEOREM 14. Let φ ,λ : [α,β ]→R be such that φ ,λ ∈Cn[α,β ] . If the inequal-ities in (14) and (16) are valid, then there exist ξi ∈ [α,β ] such that

Ωi(φ)Ωi(λ )

=φ (n)(ξi)λ (n)(ξi)

; i = 1,2,

provided that the denominators are non-zero and Ωi(·) are defined in Remark 1.


Proof. Similar to the proof of Corollary 4.2 in [8] (see also [3]). �

Theorem 14 enables us to define Cauchy means, because if

ξi =

(φ (n)

λ (n)

)−1(Ωi(φ)Ωi(λ )

),

which show that ξi ( i = 1,2) are means of α , β for given functions φ and λ .Next we construct the non trivial examples of n−exponentially and exponentially

convex functions from positive linear functionals Ωi(·) ( i = 1,2). We use the ideagiven in [14].

THEOREM 15. Let Θ = {φt : t ∈ J} , where J is an interval in R , be a family offunctions defined on an interval I in R such that the function t �→ [x0, . . . ,xn;φt ] isn−exponentially convex in the Jensen sense on J for every (n+ 1) mutually differentpoints x0, . . . ,xn ∈ I . Then for the linear functionals Ωi(·) ( i = 1,2 ) as defined inRemark 1, the following statements are valid for each i = 1,2 :

(i) The function t → Ωi(φt) is n−exponentially convex in the Jensen sense on Jand the matrix [Ωi(φ t j+tl

2)]mj,l=1 is a positive semi-definite for all m ∈ N,m � n,

t1, ..,tm ∈ J . Particularly,

det[Ωi(φ t j+tl2

)]mj,l=1 � 0

for all m ∈ N , m = 1,2, ...,n.

(ii) If the function t → Ωi(φt ) is continuous on J , then it is n−exponentially convexon J .

Proof. Similar to the proof of Theorem 23 [9]. �

The following corollary is an immediate consequence of the above theorem.

COROLLARY 2. Let Θ = {φt : t ∈ J} , where J is an interval in R , be a familyof functions defined on an interval I in R , such that the function t �→ [x0, . . . ,xn;φt ]is exponentially convex in the Jensen sense on J for every (n + 1) mutually differentpoints x0, . . . ,xn ∈ I . Then for the linear functional Ωi(·) ( i = 1,2 ), the followingstatements hold:

(i) The function t → Ωi(φt ) is exponentially convex in the Jensen sense on J and thematrix [Ωi(φ t j+tl

2)]mj,l=1 is a positive semi-definite for all m∈N,m � n, t1, ..,tm ∈

J . Particularly,det[Ωi(φ t j+tl

2)]mj,l=1 � 0

for all m ∈ N , m = 1,2, ...,n.

(ii) If the function t → Ωi(φt) is continuous on J , then it is exponentially convex onJ .

COROLLARY 3. Let Θ = {φt : t ∈ J} , where J is an interval in R , be a familyof functions defined on an interval I in R , such that the function t �→ [x0, . . . ,xn;φt ] is2−exponentially convex in the Jensen sense on J for every (n+ 1) mutually differentpoints x0, . . . ,xn ∈ I . Let Ωi(·) ( i = 1,2 ) be linear functionals. Then the followingstatements hold:

(i) If the function t �→ Ωi(φt ) is continuous on J , then it is 2−exponentially convexfunction on J . If t �→ Ωi(φt ) is additionally strictly positive, then it is also log-convex on J . Furthermore, the following inequality holds true:

[Ωi(φs)]t−r � [Ωi(φr)]t−s [Ωi(φt)]s−r ,

for every choice r,s,t ∈ J , such that r < s < t .

(ii) If the function t �→ Ωi(φt) is strictly positive and differentiable on J, then forevery p,q,u,v ∈ J , such that p � u and q � v, we have

μp,q(Ωi,Θ) � μu,v(Ωi,Θ), (27)

where

μp,q(Ωi,Θ) =

⎧⎪⎪⎨⎪⎪⎩(

Ωi(φp)Ωi(φq)

) 1p−q

, p �= q,

exp

(ddp Ωi(φp)Ωi(φp)

), p = q,

(28)

for φp,φq ∈ Θ .

Proof. Similar to the proof of Corollary 2 [9]. �

5. Applications to Cauchy means

In the running section, we use a family of functions which fulfil the conditions ofTheorem 15, Corollary 2 and Corollary 3.

EXAMPLE 1. Let us consider a family of functions

Θ = {φt : [0,∞) → R : t > 0}defined by

φt(x) =

⎧⎨⎩

xt

t(t−1)···(t−n+1) , t /∈ {1, . . . ,n−1},x j logx

(−1)n−1− j j!(n−1− j)! , t = j ∈ {1, . . . ,n−1},

with 0 log0 = 0. Since dnφtdxn (x) = xt−n > 0, the function φt is n -convex for x > 0 and

t �→ dnφtdxn (x) is exponentially convex by definition. Using analogous arguing as in the


proof of Theorem 15 we also have that t �→ [x0, . . . ,xn;φt ] is exponentially convex (andso exponentially convex in the Jensen sense). Now, using Corollary 2 we conclude thatt �→ Ωi(φt) (i = 1,2) are exponentially convex in the Jensen sense. Hence, for thisfamily of functions, it is easy to give explicitly μt,q(Ωi,Θ) (i = 1) from (28),

μt,q(Ω1,Θ) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎛⎝ m

∑i=1

wixti−

m∑i=1

wiyti

m∑

i=1wix

qi −

m∑i=1

wiyqi

× t(t−1)···(t−n+1)q(q−1)···(q−n+1)

⎞⎠

1t−q

, t �= q,

exp

⎛⎝ m

∑i=1

wixti logxi−

m∑

i=1wiyt

i logyi

m∑

i=1wixt

i−m∑

i=1wiyt

i

+n−1∑

k=0

1k−t

⎞⎠ , t = q /∈ {1, . . . ,n−1},

exp

⎛⎝ m

∑i=1

wixti log2 xi−

m∑i=1

wiyti log2 yi

2

(m∑

i=1wixt

i logxi−m∑

i=1wiyt

i logyi

) +n−1∑

k=0k �=t

1k−t

⎞⎠ , t = q ∈ {1, . . . ,n−1}.

Similarly, one can give μt,q(Ωi,Θ) (i = 2) from (28). Now using Theorem 14 weconclude that

α �(

Ωi(φt)Ωi(φq)

) 1t−q

� β , i = 1,2.

Hence μt,q(Ωi,Θ) (i = 1,2) are means and their monotonicity is followed by (27).

We conclude the paper with the following remarks:

REMARK 2. One can also consider families of functions given in the last sectionof [4] to construct a large families of functions which are exponentially convex and newmonotonic means.

REMARK 3. All the results given above can also be given in integral versionsusing Theorem 8.

Acknowledgements. The research of 1st author has been fully supported by H. E.C. Pakistan. The research of 3rd author has been fully supported by Croatian ScienceFoundation under the project 5435.

RE F ER EN C ES

[1] N. S. BARNETT, P. CERONE AND S. S. DRAGOMIR Majorisation inequalities for Stieltjes integrals,Appl. Math. Lett. 22 (2009), 416–421.

[2] S. N. BERNSTEIN, Sur les fonctions absolument monotones, Acta Math. 52 (1929), 1–66.[3] S. I. BUTT AND J. PECARIC, Generalized Hermite-Hadamard’s Inequality, Proc. A. Razmadze Math.

Inst. 163 (2013), 9–27.[4] S. I. BUTT, K. A. KHAN AND J. PECARIC, Popoviciu type inequalities Via Green function and

generalized Montgomery identity, Math. Inequal. Appl. 18 (4) (2015), 1519–1538.[5] P. CERONE AND S. S. DRAGOMIR, Some new Ostrowski-type bounds for the Cebysev functional and

applications, J. Math. Inequal. 8 (1) (2014), 159–170.


[6] L. FUCHS, A new proof of an inequality of Hardy-Littlewood-Polya, Mat. Tidsskr. B (1947), 53–54.[7] J. JAKSETIC AND J. PECARIC, Exponential Convexity Method, J. Convex Anal. 20 (1) (2013), 181–

197.[8] J. JAKSETIC, J. PECARIC AND A. PERUSIC, Steffensen inequality, higher order convexity and expo-

nential convexity, Rend. Circ. Mat. Palermo. 63 (1) (2014), 109–127.[9] M. ADIL KHAN, NAVEED LATIF AND J. PECARIC, Generalization of majorization theorem, J. Math.

Inequal. 9 (3) (2015), 847–872.[10] N. LATIF, J. PECARIC AND I. PERIC, On Majorization, Favard and Berwald’s Inequalities, Ann.

Funct. Anal. 2 (2011), 31–50.[11] A. W. MARSHALL, I. OLKIN AND BARRY C. ARNOLD, Inequalities: Theory of Majorization and

Its Applications (Second Edition), Springer Series in Statistics, New York (2011).[12] L. MALIGRANDA, J. PECARIC AND L. E. PERSSON, Weighted Favard’s and Berwald’s Inequalities,

J. Math. Anal. Appl. 190 (1995), 248–262.[13] J. PECARIC, On some inequalities for functions with nondecrasing increments, J. Math. Anal. Appl.

98 (1984), 188–197.[14] J. PECARIC AND J. PERIC, Improvement of the Giaccardi and the Petrovic Inequality and Related

Stolarsky Type Means, An. Univ. Craiova Ser. Mat. Inform. 39 (1) (2012), 65–75.[15] J. PECARIC, F. PROSCHAN AND Y. L. TONG, Convex functions, Partial Orderings and Statistical

Applications, Academic Press, New York, (1992).

(Received March 14, 2016) Saad Ihsan ButtDepartment of Mathematics, COMSATS



Ljiljanka KvesicFaculty of Science and Education, University of Mostar

Matice hrvatske bb, 88000 Mostar, Bosnia and Herzegovinae-mail: [email protected]



& Applications


CONVERSES OF JESSEN’S INEQUALITY ON TIME SCALES II

J. BARIC, R. JAKSIC AND J. PECARIC


Abstract. We obtain new refinements of converse Jessen’s inequality with respect to the multipleLebesgue delta integral. The applicability of our results is illustrated in refinements of converseinequalities regarding monotonicity properties of generalized means, power means and somerefinements of converse Holder’s inequality, which are all proved in the time scale setting.

1. Introduction

1.1. On time scale calculus

The theory of time scales was introduced by Stefan Hilger in his PhD thesis [13]in 1988 as a unification of the theory of difference equations with that of differentialequations, unifying integral and differential calculus with the calculus of finite differ-ences, extending to cases “in between” and offering a formalism for studying hybriddiscrete-continuous dynamic systems. It has applications in any field that requires si-multaneous modelling of discrete and continuous time. Now, we briefly introduce thetime scales calculus and refer to [1, 14, 15] and the books [6, 19] for further details.

By a time scale T we mean any closed subset of R . The two most popular exam-ples of time scales are the real numbers R and the integers Z . Since the time scale T

may or may not be connected, we need the concept of jump operators.For t ∈ T , we define the forward jump operator σ : T → T by

σ(t) = inf{s ∈ T : s > t}and the backward jump operator by

ρ(t) = sup{s ∈ T : s < t}.In this definition, the convention is inf /0 = sup T (i.e., σ(t) = t if T has a maximumt ) and sup /0 = inf T (i.e., ρ(t) = t if T has a minimum t ). If σ(t) > t , then we saythat t is right-scattered, and if ρ(t) < t , then we say that t is left-scattered. Pointsthat are right-scattered and left-scattered at the same time are called isolated. Also, ifσ(t) = t , then t is said to be right-dense, and if ρ(t) = t , then t is said to be left-dense.

Mathematics subject classification (2010): 26D15, 34A40, 34N05.Keywords and phrases: Time scale, linear functional, Jessen’s inequality, converses, means, Holder’s

inequality.


1271

1272 J. BARIC, R. JAKSIC AND J. PECARIC

Points that are simultaneously right-dense and left-dense are called dense. The mappingμ : T → [0,∞) defined by

μ(t) = σ(t)− t

is called the graininess function. If T has a left-scattered maximum M , then we denoteT

κ = T \ {M} ; otherwise Tκ = T . If f : T → R is a function, then we define the

function f σ : T → R by

f σ (t) = f (σ(t)) for all t ∈ T.

In the following considerations, T will denote a time scale, IT = I ∩T will denote atime scale interval (for any open or closed interval I in R), and [0,∞)T will be usedfor the time scale interval [0,∞)∩T .

DEFINITION 1. Assume f : T → R is a function and let t ∈ Tκ . Then we define

f Δ(t) to be the number (provided it exists) with the property that given any ε > 0, thereis a neighborhood U of t such that∣∣∣ f (σ(t))− f (s)− f Δ(t)(σ(t)− s)

∣∣∣� ε |σ(t)− s| for all s ∈UT.

We call f Δ(t) the delta derivative of f at t . We say that f is delta differentiable onT

κ provided f Δ(t) exists for all t ∈ Tκ .

For all t ∈ Tκ , we have the following properties:

(i) If f is delta differentiable at t , then f is continuous at t .

(ii) If f is continuous at t and t is right-scattered, then f is delta differentiable at twith f Δ(t) = f (σ(t))− f (t)

μ(t) .

(iii) If t is right-dense, then f is delta differentiable at t iff the limit lims→t

f (t)− f (s)t−s

exists as a finite number. In this case, f Δ(t) = lims→t

f (t)− f (s)t−s .

(iv) If f is delta differentiable at t , then f (σ(t)) = f (t)+ μ(t) f Δ(t) .

DEFINITION 2. A function f : T → R is called rd-continuous if it is continuousat all right-dense points in T and its left-sided limits are finite at all left-dense pointsin T . We denote by Crd the set of all rd-continuous functions. We say that f is rd-continuously delta differentiable (and write f ∈ C1

rd ) if f Δ(t) exists for all t ∈ Tκ and

f Δ ∈ Crd .

DEFINITION 3. A function F : T→R is called a delta antiderivative of f : T→R

if FΔ(t) = f (t) for all t ∈ Tκ . Then we define the delta integral by∫ t

af (s)Δs = F(t)−F(a).

CONVERSES OF JESSEN’S INEQUALITY ON TIME SCALES II 1273

The importance of rd-continuous function is revealed by the following result.

THEOREM 1. Every rd-continuous function has a delta antiderivative.

Now we give some properties of the delta integral.

THEOREM 2. If a,b,c ∈ T , α ∈ R and f ,g ∈ Crd , then

(i)b∫a

( f (t)+g(t))Δt =b∫a

f (t)Δt +b∫a

g(t)Δt ;

(ii)b∫a

α f (t)Δt = αb∫a

f (t)Δt ;

(iii)b∫a

f (t)Δt = −a∫b

f (t)Δt ;

(iv)b∫a

f (t)Δt =c∫a

f (t)Δt +b∫c

f (t)Δt ;

(v)a∫a

f (t)Δt = 0 ;

(vi) if f (t) � 0 for all t , thenb∫a

f (t)Δt � 0 .

1.2. On positive linear functionals and time scale integrals

First we recall the following definition from [22].

DEFINITION 4. Let E be a nonempty set and L be a linear class of real-valuedfunctions f : E → R having the following properties.

(L1 ) If f ,g ∈ L and α,β ∈ R , then (α f + βg) ∈ L .

(L2 ) If f (t) = 1 for all t ∈ E , then f ∈ L .

A positive linear functional is a functional A : L → R having the following properties.

(A1 ) If f ,g ∈ L and α,β ∈ R , then A(α f + βg) = αA( f )+ βA(g) .

(A2 ) If f ∈ L and f (t) � 0 for all t ∈ E , then A( f ) � 0.

In [2, 3, 4, 10], the authors presented a series of inequalities for the time scale inte-gral and showed that it is not necessary to prove that kind of inequalities “from scratch”in the time scale setting as they can be obtained easily from well-known inequalitiesfor positive linear functionals since the time scale integral is in fact a positive linearfunctional. Consequently, the results on classical inequalities, whose generalizationsfor the positive linear functionals are given in the monograph [22], could be used forobtaining new inequalities for the time scale integral.

Now we quote three theorems from [2] that we need in our research.

THEOREM 3. Let T be a time scale. For a,b ∈ T with a < b, let

E = [a,b)∩T and L = Crd (E,R) .

Then (L1) and (L2) are satisfied. Moreover, delta integral

b∫a

f (t)Δt,

is a positive linear functional which satisfies conditions (A1) and (A2) .

Corresponding versions of Theorem 3 for nabla and α -diamond integrals are alsogiven in [2].

Multiple Riemann integration and multiple Lebesgue integration on time scalewas introduced in [8] and [9], respectively, and both integrals are also a positive linearfunctionals.

THEOREM 4. Let T1, . . . ,Tn be time scales. For ai,bi ∈ Ti with ai < bi , 1 � i �n, let

E ⊂ ([a1,b1)∩T1)×·· ·× ([an,bn)∩Tn)

be Lebesgue Δ-measurable and let L be the set of all Δ-measurable functions from Eto R . Then (L1) and (L2) are satisfied. Moreover, multiple Lebesgue delta integral ontime scales ∫

E

f (t)Δt,

is positive linear functional and satisfies conditions (A1) and (A2) .

THEOREM 5. Under the assumptions of Theorem 4, delta integral∫E|h(t)| f (t)Δt∫E|h(t)|Δt

,

where h : E →R is Δ-integrable and∫E|h(t)|Δt > 0 , is also a positive linear functional

satisfying (A1) , (A2) and A(1) = 1 .

1.3. On Jessen and Lah–Ribaric inequalities

Using the known Jessen inequality for positive linear functionals ([22, Theorem2.4]) and Theorem 5, M. Anwar, R. Bibi, M. Bohner and J. Pecaric proved in [2] thefollowing generalization of Jessen’s inequality on time scales.

THEOREM 6. Assume φ ∈ C(I,R) is convex, where I ⊂ R is an interval. LetE ⊂ R

n be as in Theorem 4 and suppose f is Δ-integrable on E such that f (E ) = I .Moreover, let h : E → R be Δ-integrable such that

∫E|h(t)|Δt > 0 . Then

φ

⎛⎝∫E|h(t)| f (t)Δt∫E|h(t)|Δt

⎞⎠�

∫E|h(t)|φ( f (t))Δt∫

E|h(t)|Δt

.

Lah and Ribaric proved in [20] the converse of Jensen’s inequality for convexfunctions (see also [21]). Beesack and Pecaric gave in [7] the generalization of Lah–Ribaric’s inequality for positive linear functionals. Applying the fact that the multi-ple Lebesgue delta time scale integral is a positive linear functional (Theorem 5) toBeesack–Pecaric’s result from [7], the following theorem is proved in [2].

THEOREM 7. Assume φ ∈C(I,R) is convex, where I = [m,M]⊂R , with m < M.Let E ⊂R

n be as in Theorem 4 and suppose f is Δ-integrable on E such that f (E ) =I . Moreover, let h : E → R be Δ-integrable such that

∫E|h(t)|Δt > 0 . Then

∫E|h(t)|φ( f (t))Δt∫

E|h(t)|Δt

�M−

∫E|h(t)| f (t)Δt∫E|h(t)|Δt

M−mφ(m)+

∫E|h(t)| f (t)Δt∫E|h(t)|Δt −m

M−mφ(M). (1)

Recently, in their paper [16], R. Jaksic and J. Pecaric presented new converses ofthe Jessen and Lah-Ribaric inequalities. Now, we quote their main result.

THEOREM 8. Let φ be a continuous convex function on an interval of real num-bers I and m,M ∈ R , m < M, with [m,M] ⊂ Int(I) , where Int(I) is the interior of I .Let L satisfy conditions (L1) , (L2) on E and let A be any positive linear functionalon L with A(1) = 1 . If f ∈ L satisfies the bounds

−∞ < m � f (t) � M < ∞ for every t ∈ E

and φ ◦ f ∈ L, then

0 � A(φ( f ))−φ(A( f ))� (M−A( f ))(A( f )−m) sup

t∈〈m,M〉Ψφ (t;m,M)

� (M−A( f ))(A( f )−m)φ ′−(M)−φ ′

+(m)M−m

(2)

� 14(M−m)(φ ′

−(M)−φ ′+(m)),

where φ ′−(M) = limx→M−

φ(x)−φ(M)x−M is a left hand derivate of φ at M , and φ ′

+(M) =

limx→M+

φ(x)−φ(M)x−M is a right hand derivate of φ at M , x∈ I . We also have the inequalities

0 � A(φ( f ))−φ(A( f )) � 14(M−m)2Ψφ (A( f );m,M)

� 14(M−m)(φ ′

−(M)−φ ′+(m)), (3)

where ΨΦ(·;m,M) : 〈m,M〉 → R is defined by

ΨΦ(t;m,M) =1

M−m

(φ(M)−φ(t)M− t

− φ(t)−φ(m)t−m

).

If φ is concave on I , then all inequalities in (2) and (3) are reversed.

2. New results

In this section, we prove converses of Jessen’s inequality on time scales whichrefine the results given in [5] for multiple Lebesgue delta integral. For simplicity, weintroduce the following notations

LΔ( f ) =∫

Ef (t)Δt and LΔ( f ,h) =

∫E f (t)|h(t)|Δt∫

E |h(t)|Δt,

where f ,h : E → R are Δ-integrable and∫E |h(t)|Δt > 0.

THEOREM 9. Let φ ∈ C(I,R) be convex, where I = [m,M] ⊂ R , with m < M.Assume E ⊂ R

n is as in Theorem 4 and suppose f is Δ-integrable on E such thatf (E ) = I . Moreover, let h : E → R be Δ-integrable such that

∫E |h(t)|Δt > 0 . Then

0 � LΔ(φ( f ),h)−φ(LΔ( f ,h)

)�(M−LΔ( f ,h)

)(LΔ( f ,h)−m

)sup

t∈〈m,M〉Ψφ (t;m,M)

�(M−LΔ( f ,h)

)(LΔ( f ,h)−m

) · φ ′−(M)−φ ′+(m)

M−m(4)

� 14(M−m)(φ ′

−(M)−φ ′+(m)),

where Ψφ (·;m,M) : 〈m,M〉 → R is defined by

ΨΦ(t;m,M) =1

M−m

(φ(M)−φ(t)M− t


).

If φ is concave on I , then all inequalities in (4) are reversed.

Proof. Since φ is a convex function, first inequality in (4) follows from Theorem6. From Theorem 7, we have

LΔ(φ( f ),h)−φ(LΔ( f ,h)

)(5)

� M−LΔ( f ,h)M−m

φ(m)+LΔ( f ,h)−m

M−mφ(M)−φ

(LΔ( f ,h)

)


=1

M−m

(M−LΔ( f ,h)

)(LΔ( f ,h)−m

)×(

φ(M)−φ(LΔ( f ,h)

)M−LΔ( f ,h)

− φ(LΔ( f ,h)

)−φ(m)LΔ( f ,h)−m

)

=(M−LΔ( f ,h)

)(LΔ( f ,h)−m

)Ψφ(LΔ( f ,h);m,M

)�(M−LΔ( f ,h)

)(LΔ( f ,h)−m

)sup

t∈〈m,M〉Ψφ (t;m,M),

which is the second inequality in (4), provided that LΔ( f ,h) = m,M . When LΔ( f ,h) isequal to m or M then inequality (4) is obvious.

Since,

supt∈〈m,M〉

Ψφ (t;m,M) =1

M−msup

t∈〈m,M〉

{φ(M)−φ(t)M− t


}

� 1M−m

(sup

t∈〈m,M〉φ(M)−φ(t)

M− t+ sup

t∈〈m,M〉−(φ(t)−φ(m))

t−m

)

=1

M−m

(sup

t∈〈m,M〉φ(M)−φ(t)

M− t− inf

t∈〈m,M〉φ(t)−φ(m)

t−m

)=

φ ′−(M)−φ ′+(m)

M−m,

the third inequality in (4) is true. The last inequality in (4) follows from the elementaryestimate (M−x)(x−m)

M−m � 14 (M−m) , for every x ∈ R . If the function φ is concave, then

−φ is convex, so applying (4) to −φ gives the reversed inequalities in (4) . Thiscomplete the proof. �

REMARK 1. According to (5), with the same assumptions as in Theorem 9, fol-lowing inequalities are also true

0 � LΔ(φ( f ),h)−φ(LΔ( f ,h)

)� 1

4(M−m)2Ψφ

(LΔ( f ,h);m,M

)� 1

4(M−m)(φ ′

−(M)−φ ′+(m)).

3. Applications

In this section we use the results from Theorem 9 to get new converse inequalitiesfor generalized means, power means and the Holder inequality in the time scale setting.

3.1. Generalized means

Applying classical results to the monotonicity properties of generalized meanswith respect to the functional A , found in [12, p. 75, Theorem 92] and [22, p. 108,Theorem 4.3], R. Jaksic and J. Pecaric proved in [16] the following converse.

THEOREM 10. Let L satisfy (L1) , (L2) and A satisfy (A1) , (A2) and A(1) = 1 .Suppose ψ ,χ : I →R are continuous and strictly monotone and φ = χ ◦ψ−1 is convex,where I ⊃ [m,M] , −∞ < m < M < ∞ . Then, for every f ∈ L such that m � f (t) � M,t ∈ [m,M] and ψ( f ),χ( f ) ∈ L, we have

0 � χ(Mχ( f ,A))− χ(Mψ( f ,A))� (Mψ −A(ψ( f )))(A(ψ( f ))−mψ ) sup

t∈〈m,M〉Ψχ◦ψ−1(ψ(t);mψ ,Mψ)

� (Mψ −A(ψ( f )))(A(ψ( f ))−mψ )[χ ◦ψ−1]′−(Mψ )− [χ ◦ψ−1]′+(mψ )

Mψ −mψ(6)

� 14(Mψ −mψ)([χ ◦ψ−1]′−(Mψ)− [χ ◦ψ−1]′+(mψ)),

where [mψ ,Mψ ] = ψ([m,M]) and Mψ ( f ,A) = ψ−1 (A(ψ( f ))) is a generalized meanwith respect to the operator A and function ψ . If φ is concave, then all inequalities in(6) are reversed.

Using the generalized mean on time scales, with respect to the multiple Lebesguedelta integral ([5]), from Theorem 9 and Theorem 10 we deduce the following result.

THEOREM 11. Suppose I = [m,M] , −∞ < m < M < ∞ , ψ ,χ : I →R are continu-ous and strictly monotone and φ = χ ◦ψ−1 is convex. Assume E ⊂R

n is as in Theorem4 and f ,h : E → R are Δ-integrable on E such that f (E ) = I and

∫E |h(t)|Δt > 0 .

Then,

0 � χ(Mχ(f ,LΔ

))− χ(Mψ(f ,LΔ

))�(Mψ −LΔ(ψ( f ),h)

)(LΔ(ψ( f ),h)−mψ

)sup

t∈〈m,M〉Ψχ◦ψ−1(ψ(t);mψ ,Mψ )

�(Mψ −LΔ(ψ( f ),h)

)(LΔ(ψ( f ),h)−mψ

)× (χ ◦ψ−1)′−(Mψ )− (χ ◦ψ−1)′+(mψ )

Mψ −mψ(7)

� 14

(Mψ −mψ

)((χ ◦ψ−1)′−(Mψ )− (χ ◦ψ−1)′+(mψ )

),

where [mψ ,Mψ ] = ψ([m,M]) . If φ is concave, then all inequalities in (7) are reversed.

Proof. The claim follows from Theorem 4, Theorem 9 and Theorem 10. �

3.2. Power means

The following result on power means with respect to a positive linear functional isproved in [16].

THEOREM 12. Let L satisfy (L1) , (L2) and A satisfy (A1) , (A2) with A(1) = 1 .Let 0 < m � f (t) � M < ∞ for t ∈ E , f r, f s, log f ∈ L for r,s ∈ R , r < s and let

φ(t) =

⎧⎨⎩

ts/r : r = 0,s = 0,1r log t : r = 0,s = 0,est : r = 0,s = 0.

(8)

If 0 < r < s or r < 0 < s, then

0 � (M[s]( f ,A))s − (M[r]( f ,A))s

� (Mr −A( f r))(A( f r)−mr) supt∈〈m,M〉

ΨΦ(tr;mr,Mr)

� sr(Mr −A( f r))(A( f r)−mr)

Ms−r −ms−r

Mr −mr (9)

� s4r

(Mr −mr)(Ms−r −ms−r).

If r < s < 0 , then

0 � (M[s]( f ,A))s − (M[r]( f ,A))s

� (Mr −A( f r))(A( f r)−mr) supt∈〈m,M〉

ΨΦ(tr;mr,Mr)

� sr(Mr −A( f r))(A( f r)−mr)

Ms−r −ms−r

Mr −mr (10)

� s4r

(Mr −mr)(Ms−r −ms−r).

If s = 0 and r < 0 , then

0 � log(M[0]( f ,A))− log(M[r]( f ,A))� (Mr −A( f r))(A( f r)−mr) sup

t∈〈m,M〉ΨΦ(tr;Mr,mr)

� −1r· (Mr −A( f r))(A( f r)−mr)

Mrmr (11)

� 14r

(mr −Mr)(

1mr −

1Mr

).

If r = 0 and s > 0 , then

0 � (M[s]( f ,A))s − (M[0]( f ,A))s

� (logM−A(log f ))(A(log f )− logm) supt∈〈m,M〉

ΨΦ(log t; logm, logM)

� s(logM−A(log f ))(A(log f )− logm)Ms −ms

logM− logm(12)

� s(Ms −ms) logMm

.

According to definition of power mean on time scales with respect of the multipleLebesgue delta integral ([5]), from the Theorem 12 we derive the following result.

THEOREM 13. Suppose E ⊂ Rn is as in Theorem 4, f is Δ-integrable on E

such that f (E ) = I and 0 < m � f (t) � M < ∞ , for t ∈ E , m,M ∈ R . Let h : E → R

be Δ-integrable such that∫E |h(t)|Δt > 0 . For r,s ∈ R suppose f r , f s , (log f ) are

Δ-integrable on E .

(i) If 0 < r < s or r < 0 < s, then

0 �(M[s] ( f ,LΔ

))s−(M[r] ( f ,LΔ

))s

�(Mr −LΔ( f r ,h)

)(LΔ( f r,h)−mr) sup

t∈〈m,M〉Ψφ (tr;mr,Mr)

� sr

(Mr −LΔ( f r,h)

)(LΔ( f r ,h)−mr)Ms−r −ms−r

Mr −mr (13)

� s4r

(Mr −mr)(Ms−r −ms−r) .

(ii) If r < s < 0 , then

0 �(M[s] ( f ,LΔ

))s−(M[r] ( f ,LΔ

))s

�(Mr −LΔ( f r ,h)


t∈〈m,M〉Ψφ (tr;mr,Mr)

� sr

(Mr −LΔ( f r,h)

)(LΔ( f r ,h)−mr)Ms−r −ms−r

Mr −mr (14)

� s4r

(Mr −mr)(Ms−r −ms−r) .

(iii) If s = 0 and r < 0 , then

0 � log(M[0] ( f ,LΔ

))− log(M[r] ( f ,LΔ

))�(Mr −LΔ( f r ,h)


t∈〈m,M〉Ψφ (tr;Mr ,mr)

� −1r·(Mr −LΔ( f r ,h)

)(LΔ( f r,h)−mr

)Mrmr (15)

� − 14r

· (Mr −mr)2

Mrmr .

(iv) If r = 0 and s > 0 , then

0 �(M[s] ( f ,LΔ

))s −(M[0] ( f ,LΔ

))s

�(logM−LΔ(log f ,h)

)(LΔ(log f ,h)− logm

)× sup

t∈〈m,M〉Ψφ (log t; logm, logM)

�(logM−LΔ(log f ,h)

)(LΔ(log f ,h)− logm

) · s(Ms −ms)logM− logm

(16)

� s(Ms −ms) logMm

.

Proof. The claim follows from Theorem 4, Theorem 9 and Theorem 12. �

3.3. Holder’s inequality

The following theorem gives Holder’s inequality for delta time scale integralsproved in [2].

THEOREM 14. Assume E ⊂ Rn is as in Theorem 4 and |w|| f |p , |w||g|q , |w fg|

are Δ-integrable on E , where p > 1 and q = p/(p−1) . Then,

∫E

|w(t) f (t)g(t)|Δt �

⎛⎝∫

E

|w(t)|| f (t)|pΔt

⎞⎠

1p⎛⎝∫

E

|w(t)||g(t)|qΔt

⎞⎠

1q

.

This inequality is reversed if 0 0 , and it is also re-

versed if p < 0 and∫E |w(t)|| f (t)|pΔt > 0 .

Next, we refine the converse Holder’s inequalities on time scales, proved in [5].

THEOREM 15. Assume E ⊂ Rn is as in Theorem 4 and w, f ,g are real functions

on E such that w, f ,g � 0 . For m,M ∈ R such that −∞ < m < M < ∞ let m �f (t)gq(t) � M, t ∈ E . If w f p , wgq , w f g are Δ-integrable on E and

∫E

w(t)gq(t)Δt >

0 , where p > 1 and q = p/(p−1) , then

0 �∫E

w(t) f p(t)Δt ·⎛⎝∫

E

w(t)gq(t)Δt

⎞⎠

pq

−⎛⎝∫

E

w(t) f (t)g(t)Δt

⎞⎠p

�

⎛⎝M

∫E

w(t)gq(t)Δt−∫E

w(t) f (t)g(t)Δt

⎞⎠⎛⎝∫

E

w(t) f (t)g(t)Δt −m∫E

w(t)gq(t)Δt

⎞⎠

× supt∈〈m,M〉

Ψφ (t;m,M) ·⎛⎝∫

E

w(t)gq(t)Δt

⎞⎠p−2

(17)

�

⎛⎝M

∫E

w(t)gq(t)Δt−∫E

w(t) f (t)g(t)Δt

⎞⎠⎛⎝∫

E

w(t) f (t)g(t)Δt −m∫E

w(t)gq(t)Δt

⎞⎠

× p(Mp−1−mp−1)M−m

·⎛⎝∫

E

w(t)gq(t)Δt

⎞⎠p−2

� p4(M−m)(Mp−1−mp−1)

⎛⎝∫

E

w(t)gq(t)Δt

⎞⎠p

.

For p < 0 , inequalities (17) hold if∫E w(t) f (t)g(t)Δt > 0 , t ∈ E . In case 0 < p < 1 ,

all inequalities in (17) are reversed.

Proof. Inequalities (17) follow directly from Theroem 9 by taking the function φto be of the form φ(t) = t p and replacing h by wgq and f by f g−

qp . For p < 0 and

p > 1, the function t p is convex and inequalities (17) follow from inequalities (4). For0 < p < 1, the function t p is concave and, according to Theorem 9, all inequalities in(17) will be reversed. �

THEOREM 16. Assume E ⊂ Rn is as in Theorem 4 and f ,g � 0 such that f p ,

gq , f g are Δ-integrable on E and∫E

gq(t)Δt > 0 , where 0 < p < 1 and q = p/(p−1) .

For m,M ∈ R such that −∞ < m < M < ∞ , let m � f (t)g−q(t) � M, t ∈ E . Then,

0 �∫E

f (t)g(t)Δt −⎛⎝∫

E

f p(t)Δt

⎞⎠

1p⎛⎝∫

E

gq(t)Δt

⎞⎠

1q

� 1∫E gq(t)Δt

⎛⎝M

∫E

gq(t)Δt−∫E

f p(t)Δt

⎞⎠ ·⎛⎝∫

E

f p(t)Δt−m∫E

gq(t)Δt

⎞⎠

× supt∈〈m,M〉

Ψφ (t;m,M) (18)

� 1p· M

− 1q −m− 1

q

M−m· 1∫

E gq(t)Δt

⎛⎝M

∫E

gq(t)Δt−∫E

f p(t)Δt

⎞⎠

×⎛⎝∫

E

f p(t)Δt−m∫E

gq(t)Δt

⎞⎠

� 14p

(M−m)(M− 1

q −m− 1q

)∫E

gq(t)Δt.

For p < 0 , inequalities (18) hold if∫E f p(t)Δt > 0 , t ∈ E . In case p > 1 , all inequal-

ities in (18) are reversed.

Proof. Inequalities (18) follow directly from Theroem 9 by taking the function φto be of the form φ(t) = t

1p and replacing h by gq and f by f p

gq . Namely, when p < 1,

the function t1p is convex and inequalities (18) follow from inequalities (4). For p > 1,

the function t p is concave and, according to Theorem 9, all inequalities in (18) will bereversed. �

THEOREM 17. Assume E ⊂Rn is as in Theorem 4 and f ,g � 0 such that gq , f g

are Δ-integrable on E and∫E

gq(t)Δt > 0 , where p < 0 or p > 1 and q = p/(p−1) .

Let m,M ∈ R such that −∞ < m < M < ∞ and m � f (t)g1−q(t) � M, t ∈ E . Then,

0 �∫E

f p(t)Δt ·⎛⎝∫

E

gq(t)Δt

⎞⎠

pq

−⎛⎝∫

E

f (t)g(t)Δt

⎞⎠p

�

⎛⎝M

∫E

gq(t)Δt−∫E

f (t)g(t)Δt

⎞⎠⎛⎝∫

E

f (t)g(t)Δt−m∫E

gq(t)Δt

⎞⎠

×⎛⎝∫

E

gq(t)Δt

⎞⎠p−2

supt∈〈m,M〉

Ψφ (t;m,M) (19)

�

⎛⎝M

∫E

gq(t)Δt−∫E

f (t)g(t)Δt

⎞⎠⎛⎝∫

E

f (t)g(t)Δt−m∫E

gq(t)Δt

⎞⎠

× p(Mp−1−mp−1)M−m

·⎛⎝∫

E

gq(t)Δt

⎞⎠p−2

� p4(M−m)(Mp−1−mp−1)

⎛⎝∫

E

gq(t)Δt

⎞⎠p

.

In case 0 < p < 1 , all inequalities in (19) are reversed.

Proof. Inequalities (19) follow directly from Theroem 9 by taking the function φto be of the form φ(t) = t p and replacing h by gq and f by f g1−q . Namely, for p < 0or p > 1, the function t p is convex and inequalities (19) follow from inequalities (4).For 0 < p < 1, the function t p is concave and, according to Theorem 9, all inequalitiesin (19) will be reversed. �

3.4. Additional improvements

Recently, R. Jaksic and J. E. Pecaric proved in [17] new refinement of the converseJensen inequality for normalized positive linear functionals, given in Theorem 8. Usingthat result, we derive the following theorem which refines inequality (4) from Theorem9.

THEOREM 18. Let φ ∈ C(I,R) be convex, where I = [m,M] ⊂ R , with m < M.Assume E ⊂ R

n and L are as in Theorem 4 with additional property that for everyf ,g ∈ L we have that min{ f ,g} ∈ L and max{ f ,g} ∈ L. Let f be Δ-integrable on Esuch that f (E ) = I . Moreover, let h : E →R be Δ-integrable such that

∫E |h(t)|Δt > 0 .


Then,

0 � LΔ(φ( f ),h)−φ(LΔ( f ,h)

)�(M−LΔ( f ,h)

)(LΔ( f ,h)−m

)sup

t∈〈m,M〉Ψφ (t;m,M)−LΔ( f ,h)δφ

�(M−LΔ( f ,h)

)(LΔ( f ,h)−m

) · φ ′−(M)−φ ′+(m)

M−m−LΔ( f ,h)δφ (20)

� 14(M−m)(φ ′

−(M)−φ ′+(m))−LΔ( f ,h)δφ ,

where

f = 12 −

| f−m+M2 |

M−m , δφ = φ(m)+ φ(M)−2φ(

m+M2

)and Ψφ (·;m,M) : 〈m,M〉 → R is defined by

ΨΦ(t;m,M) = 1M−m

(φ(M)−φ(t)

M−t − φ(t)−φ(m)t−m

).

If φ is concave on I , then the above inequalities are reversed.

Proof. Inequality (20) follows directly from the main result of [17] and the factthat multiple Lebesgue delta integral is a positive linear functional. �

REMARK 2. Using the resoning shown in Theorem 18, the refinements of all in-equalities proved in this paper in Theorem 10, Theorem 11, Theorem 12 and Theorem13 can be obtained.

Acknowledgement. This work has been fully supported by Croatian Science Foun-dation under the project 5435.

RE F ER EN C ES

[1] R. P. AGARWAL, M. BOHNER AND A. PETERSON, Inequalities on time scale: a survey, Math.Inequal. Appl. 4, 4 (2001), 535–557.

[2] M. ANWAR, R. BIBI, M. BOHNER AND J. PECARIC, Integral inequalities on time scales via thetheory of positive linear functionals, Abstract and Applied Analysis 2011, (2011).

[3] R. BIBI, M. BOHNER AND J. PECARIC, Cauchy-type means and exponential and logaritheoremicconvexity for superquadratic functions on time scales, Ann. Funct. Anal. 6, 1 (2015), 59–83.

[4] J. BARIC, R. BIBI, M. BOHNER AND J. PECARIC, Time scales integral inequalities for su-perquadratic functions, J. Korean Math. Soc. 50, 3 (2013), 465–477.

[5] J. BARIC, M. BOHNER, R. JAKSIC AND J. E. PECARIC, Converses of Jessen’s inequality on timescales, Mathematical Notes 98, 1 (2015), 11–24.

[6] M. BOHNER AND A. PETERSON,Dynamic Equations on Time Scales, Biremarkhauser, Boston, Basel,Berlin, 2001.

[7] P. R. BEESACK AND J. E. PECARIC, On Jessen’s inequality for convex functions, J. Math. Anal. Appl.110, 2 (1985), 536–552.

[8] M. BOHNER AND G. SH. GUSEINOV, Multiple integration on time scales, Dynamic Systems andApplications 14, 3–4 (2005), 579–606.


[9] M. BOHNER AND G. SH. GUSEINOV, Multiple Lebesgue integration on time scales, Advances inDifference Equations 2006, Article ID 26391, (2006), 13 pages.

[10] M. BOHNER, A. NOSHEEN, J. PECARIC AND A. YOUNUS, Some dynamic Hardy-type inequalitieswith general kernel, J. Math. Inequal. 8, 1 (2014), 185–199.

[11] S. S. DRAGOMIR, Bounds for the deviation of a function from the chord generated by its extremities,Bull. Aust. Math. Soc. 78, 2 (2008), 225–248.

[12] G. H. HARDY, J. E. LITTLEWOOD, G. POLYA, Inequalities 1st ed. and 2nd ed. Cambridge UniversityPress, Cambridge, England, (1934, 1952).

[13] S. HILGER, Ein Maßkettenkalkul mit Anwendung auf Zentrumsmannigfaltigkeiten, Phd. D. thesis,Universitat Wurzburg, 1988.

[14] S. HILGER, Analysis on measure chains -a unified approach to continuous and discrete calculus,Results Math. 18, 1–2 (1990), 18–56.

[15] S. HILGER, Differential and difference calculus unified, Nonlinear Anal. 30, 1 (1997), 143–166.[16] R. JAKSIC AND J. E. PECARIC, New converses of the Jessen and Lah-Ribaric inequalities II, J. Math.

Inequal. 7, 4 (2013), 617–645.[17] R. JAKSIC AND J. E. PECARIC, New converses of the Jessen and Lah-Ribaric inequalities III, sub-

mitted.[18] M. KLARICIC, J. E. PECARIC, J. PERIC, On the converse Jensen inequality, Applied Mathematics

and Computation 218, (2012), 6566–6575.[19] B. KAYMAKCALAN, V. LAKSHMIKANTHAM AND S. SIVASUNDARAM, Dynamic Systems on Mea-

sure Chains, Kluwer Academic Publishers, Dordrecht, (1996).[20] P. LAH, M. RIBARIC, Converse of Jensen’s inequality for convex functions, Univ. Beograd. Publ.

Elektrotehn. Fak. Ser. Mat. Fiz. 412–460, (1973), 201–205.[21] J. E. PECARIC, Konveksne funkcije: Nejednakosti, Naucna knjiga, Beograd, (1987).[22] J. PECARIC, F. PROSCHAN AND Y. C. TONG, Convex functions, Partial Orderings and Statistical

Applications, Academic Press, Inc. (1992).[23] F. H. WONG, C. C. YEH AND W. C. LIAN, An extension of Jensen’s inequality on time scales, Adv.

Dyn. Syst. Appl. 1, 1 (2006), 113–120.

(Received March 16, 2016) J. BaricFaculty of Electrical Engineering

Mechanical Engineering and Naval ArchitectureUniversity of Split

Rudjera Boskovica 32, 21000 Split, Croatiae-mail: [email protected]

R. JaksicFaculty of Textile Technology, University of Zagreb

Prilaz baruna Filipovica 28a, 10000 Zagreb, Croatiae-mail: [email protected]

J. PecaricFaculty of Textile Technology, University of Zagreb



& Applications


ON A PROBLEM CONNECTED WITH STRONGLY CONVEX FUNCTIONS

MIROSŁAW ADAMEK


Abstract. In this paper we show that the result obtained by Nikodem and Pales in [3] can byextended to a more general case. In particular, for a non-negative function F defined on a realvector space we define F -strongly convex functions and show that such functions are in the formg+F∗ , where g is a convex function and F∗ is a function associated with function F , iff F∗ isa quadratic function. Using this result, we get a characterization of quadratic functions.

1. Introduction

Let (X ,‖·‖) be a real normed space, D stand for a convex subset of X and c be apositive constant. A function f : D → R is called strongly convex with modulus c if

f (tx+(1− t)y) � t f (x)+ (1− t) f (y)− ct(1− t)‖ x− y‖2 , (1)

for all x,y ∈ D and t ∈ (0,1) .Such functions have been introduced by Polyak in [4] and as in turns out they play

an important role in optimization theory. Strongly convex functions have also beenstudied by many authors, among others, see [1], [5], [6]. A function f : D → R iscalled strongly Jensen convex with modulus c if

f(x+ y

2

)� f (x)+ f (y)

2− c

4‖ x− y‖2 , (2)

for all x,y ∈ D .In [3] the authors present relations between strongly convex (strongly Jensen con-

vex) and convex (Jensen convex) functions. In particular, they show that each stronglyconvex function (strongly Jensen convex function) is in the form g+‖·‖2 , where g isa convex function (Jensen convex function) iff the space (X ,‖·‖) is an inner productspace.

Now, if in (1) and (2) we replace c‖·‖2 , with a non-negative function F definedon X we get the following inequalities, respectively.

f (tx+(1− t)y) � t f (x)+ (1− t) f (y)− t(1− t)F(x− y),

Mathematics subject classification (2010): Primary 46C15; Secondary 26B25, 39B62.Keywords and phrases: Strongly convex function, strongly F -convex function, quadratic function.


1287


1288 M. ADAMEK

for all x,y ∈ D and t ∈ (0,1) .

f( x+ y

2

)� f (x)+ f (y)

2− 1

4F(x− y),

for all x,y ∈ D .The main goal of this paper is to resolve a problem of whether for such functions

a similar result as Nikodem and Pales got in [3] is possible to obtain.

2. Main result

At the beginning we formally introduce two definitions of functions aforemen-tioned in the introduction.

DEFINITION 1. Let X be a real vector space, D be a nonempty convex subset ofX and F : X → [0,∞) be a given function. A function f : D → R will be called F -strongly convex if

f(tx+(1− t)y

)� t f (x)+ (1− t) f (y)− t(1− t)F(x− y)

for all x,y ∈ D and t ∈ (0,1).

DEFINITION 2. Let X be a real vector space, D be a nonempty convex subsetof X and F : X → [0,∞) be a given function. A function f : D → R we will call F -strongly J-convex if

f(x+ y

2

)� f (x)+ f (y)

2− 1

4F(x− y)

for all x,y ∈ D .

Notice that in Definition 1 parameter t is arbitrary from the segment (0,1) . Thus,function f is F-strongly convex if and only if

f(tx+(1− t)y

)� t f (x)+ (1− t) f (y)− t(1− t)F(y− x)

for all x,y ∈ D and t ∈ (0,1). So, defining the function F∗ by setting

F∗(x) := max{F(−x),F(x)} , x ∈ X ,

we have the following observation.

OBSERVATION 1. Let X be a real vector space, D be a nonempty convex subsetof X and F : X → [0,∞) be a given function. A function f : D → R is F -stronglyconvex (F -strongly J-convex) if and only if a function f : D→R is F∗ -strongly convex(F∗ -strongly J-convex).

ON A PROBLEM CONNECTED WITH STRONGLY CONVEX FUNCTIONS 1289

2.1. F -strongly J-convexity

In this section we will consider F -strongly J-convex functions and we shall startwith three useful lemmas.

LEMMA 1. Let X be a real vector space, D be a nonempty convex subset of Xand F : X → [0,∞) be a given quadratic function (i.e. F(x+ y)+F(x− y) = 2F(x)+2F(y)). A function f : D → R is F -strongly J-convex if and only if the function g =f −F is J-convex.

Proof. F is a quadratic function thus

14F(x− y) = −F

(x+ y2

)+

12F(x)+

12F(y).

Now, the inequality

f(x+ y

2

)� f (x)+ f (y)

2− 1

4F(x− y)

can by written in an equivalent form

f(x+ y

2

)−F

(x+ y2

)� f (x)−F(x)+ f (y)−F(y)

2.

Taking g := f −F we get

g(x+ y

2

)� g(x)+g(y)

2. �

LEMMA 2. Let X be a real vector space and F : X → [0,∞) be a function that iseven. If the function F is F -strongly J-convex, then F( 1

2x) = 14F(x) far all x ∈ X .

Proof. We are assuming the inequality

F(x+ y

2

)� F(x)+F(y)

2− 1

4F(x− y), x,y ∈ X . (3)

From the above inequality with x = y = 0 and non-negativity of F we get F(0) = 0.Now, taking y = 0 we have

F( x

2

)� 1

4F(x), x ∈ X .

Moreover, putting y = −x , substituting x with x2 in (3) and using evenness of F we

obtain14F(x) � F

( x2

), x ∈ X .

Thus

F( x

2

)=

14F(x), x ∈ X . (4)

�

1290 M. ADAMEK

LEMMA 3. Let X be a real vector space and F : X → [0,∞) be a function that iseven. The function F is F -strongly J-convex if and only if F is a quadratic function.

Proof. Suppose that F is F -strongly J-convex. From Definition 2 and Lemma 2we get

F(x+ y)+F(x− y) � 2F(x)+2F(y), x,y ∈ X .

Now, putting x+y = u and x−y = v in the above inequality a using once more Lemma2 we obtain

F(u+ v)+F(u− v) � 2F(u)+2F(v), u,v ∈ X .

ThusF(x+ y)+F(x− y) = 2F(x)+2F(y), x,y ∈ X .

The reverse implication is obviously true. �The next result gives the solution of the problem aforementioned in case of F -

strongly J-convexity. Moreover, we obtain a characterization of quadratic functions.

THEOREM 1. Let X be a real vector space, D be a nonempty convex subset of Xand F : X → [0,∞) be a given function. The following conditions are equivalent:

1. For all function f : D → R , f is F -strongly J-convex if and only if the functiong = f −F∗ is J-convex;

2. The function F∗ is F∗ -strongly J-convex;

3. The function F∗ is a quadratic function.

Proof. Assuming (1) and taking g = 0 we obtain that F∗ = f . Thus F∗ is F -strongly J-convex and from Observation 1, F∗ is F∗ -strongly J-convex. So, we have(2). Lemma 3 follows the implication (2)⇒(3) and from Lemma 1 and Observation 1we deduce the implication (3)⇒(1). �

It is well known, that each quadratic and continuous function F : Rn → R can be

written in the following form F(x) = xAxT , where A is a symmetric matrix of degreen . Therefore, from Theorem 1 we have the following corollary.

COROLLARY 1. For �np spaces where n � 2 we have

22−s2 ‖x+ y‖s

p +2s−22 ‖x− y‖s

p � 2s2 ‖x‖s

p +2s2 ‖y‖s

p , x,y ∈ Rn,

if and only if p = s = 2.

In order to substantiate this corollary let us observe that, multiplying the aboveinequality by 2−

s+22 , the function F(x) := ‖x‖s

p must be F -strongly convex and, ofcourse, F = F∗ . Therefore, from Theorem 1 the function F must be quadratic andconsequently we have that

F(x) = ax21 +bx1x2 + cx2

2, (5)


for x = (x1,x2,0, . . . ,0) ∈ Rn . From the definition of F and (5), it follows that a = c ,

because F(x1,x2,0, . . . ,0) = F(x2,x1,0, . . . ,0) . Now taking x1 = 1, x2 = 0 we con-clude that a = 1. Hence, for x2 = 0 with arbitrary x1 we get s = 2. Thus

p√|x1|p + |x2|p =

√x21 +bx1x2 + x2

2.

Dividing this equality by |x1| and tending with |x1| to infinity we obtain b = 0. Thus

p√|x1|p + |x2|p =

√x21 + x2

2.

Finally, taking in the above equality x1 = x2 we obtain p = 2.Observe that if we take n = 1 in the previous corollary we also get s = 2 and, of

course, the value of p is unimportant.At the end of this section, notice that if we additionally assume the continuity of

F in Theorem 1, then the function F∗ must also be homogeneous of degree 2 (i.e.F∗(tx) = t2F∗(x) for all x ∈ X and t ∈ R) and consequently, using the well knownJordan-von Neumann theorem presented in [2], defines a symmetric bilinear form, thusX is an inner product space.

2.2. Strongly F -convexity

In this section F -strongly convex functions will be considered. We start withthree lemmas which are analogous to the lemmas presented in the previous section,respectively.

LEMMA 4. Let X be a real vector space, D be a nonempty convex subset of Xand F : X → [0,∞) be a given F -strongly affine function (i.e. we have ”=” insteadof ”�” in Definition 1). A function f : D → R is F -strongly convex if and only if thefunction g = f −F is convex.

Proof. F is a F -strongly affine function thus

t(1− t)F(x− y) = −F(tx+(1− t)y)+ tF(x)+ (1− t)F(y).

Now, the inequality

f (tx+(1− t)y) � t f (x)+ (1− t) f (y)− t(1− t)F(x− y)

can be written in an equivalent form

f (tx+(1− t)y)−F(tx+(1− t)y) � t( f (x)−F(x))+ (1− t)( f (y)−F(y)).

Taking g := f −F we get

g(tx+(1− t)y) � tg(x)+ (1− t)g(y). �

LEMMA 5. Let X be a real vector space and F : X → [0,∞) be a function that iseven. If the function F is F -strongly convex, then F is homogeneous of degree 2 .

1292 M. ADAMEK

Proof. From the assumption the following inequality holds true

F(tx+(1− t)y) � tF(x)+ (1− t)F(y)− t(1− t)F(x− y), (6)

for all t ∈ (0,1) and x,y ∈ X . Putting in this inequality y = 0 and using the fact thatF(0) = 0 we get

F(tx) � tF(x)+ t(1− t)F(x) = t2F(x),

thusF(tx) � t2F(x), (7)

for all t ∈ (0,1) and x∈ X . In order to show the reverse inequality, we put x = (1− t)u ,y = −tu in (6) and using the fact that F(0) = 0 we have

0 = F(0) � tF((1− t)u)+ (1− t)F(−tu)− t(1− t)F(u).

Now, using the above inequality, (7) and evenness of F we obtain

t(1− t)F(u) � tF((1− t)u)+ (1− t)F(tu) � t(1− t)2F(u)+ (1− t)F(tu).

Dividing this inequality by 1− t we get

tF(u) � t(1− t)F(u)+F(tu),

thusF(tu) � t2F(u), (8)

for all t ∈ (0,1) and u ∈ X . From (7) and (8) we conclude that

F(tx) = t2F(x), (9)

for all t ∈ (0,1) and x ∈ X . Moreover, if we substitute in the above equality x with xt

we conclude that (9) holds also for t > 1. Which together with the evenness of F givesthe equality

F(tx) = t2F(x), (10)

for all t ∈ R and x ∈ X . �

LEMMA 6. Let X be a real vector space and F : X → [0,∞) be a function thatis even. The function F is F -strongly convex if and only if F is a F -strongly affinefunction.

Proof. Assume that F is F -strongly convex, i.e.

F(tx+(1− t)y) � tF(x)+ (1− t)F(y)− t(1− t)F(x− y), (11)

for all t ∈ (0,1) and x,y ∈ X . Now, putting tx+(1− t)y = u√

t and x−y = v√t

in (11)and using Lemma 5 we get

tF(u) � F(tu+(1− t)v)+ t(1− t)F(u− v)− (1− t)F(v)


or equivalently

F(tu+(1− t)v) � tF(u)+ (1− t)F(v)− t(1− t)F(u− v),

for all t ∈ (0,1) and u,v ∈ X . This together with (11) implies that

F(tu+(1− t)v) = tF(u)+ (1− t)F(v)− t(1− t)F(u− v),

for all t ∈ (0,1) and x,y ∈ X , i.e. the function F is F -strongly affine.The reverse implication is obvious. �The next result gives the solution of the problem aforementioned in case of F -

strongly convexity.

THEOREM 2. Let X be a real vector space, D be a nonempty convex subset of Xand F : X → [0,∞) be a given function. The following conditions are equivalent:

1. For all function f : D → R , f is F -strongly convex if and only if the functiong = f −F∗ is convex;

2. The function F∗ is F∗ -strongly convex;

3. The function F∗ is F∗ -strongly affine (and of course quadratic and homogeneousof degree 2 , and X is an inner product space).

Proof. The implication (1)⇒(2) we argue as in the proof of Theorem 1. By virtueof Lemma 6 we have the implication (2)⇒(3). Finally, using Lemma 4 and Observation1 we obtain the implication (3)⇒(1). �

RE F ER EN C ES

[1] J.-B. HIRIART-URRUTY AND C. LEMARECHAL,Fundamentals of Convex Analysis, Springer-Verlag,Berlin-Heidelberg, 2001.

[2] P. JORDAN AND J. V. NEUMANN, On inner products in linear metric spaces, Ann. Math. 36 (1935),719–723.

[3] K. NIKODEM, ZS. PALES, Characterizations of inner product spaces by strongly convex functions,Banach J. Math. Anal. 5 (2011), no. 1, 83–87.

[4] B. T. POLYAK, Existence theorems and convergence of minimizing sequences in extremum problemswith restrictions, Soviet Math. Dokl. 7 (1966), 72–75.

[5] A. W. ROBERTS AND D. E. VARBERG, Convex Functions, Academic Press, New York-London, 1973.[6] J.-P. VIAL, Strong convexity of sets and functions, J. Math. Economics 9 (1982), 187–205.

(Received March 16, 2016) Mirosław AdamekDepartment of Mathematics, University of Bielsko-Biała

ul. Willowa 2, PL-43-309 Bielsko-Biała, Polande-mail: [email protected]


& Applications


ON WEIGHTED INTEGRAL AND

DISCRETE OPIAL–TYPE INEQUALITIES

MAJA ANDRIC, JOSIP PECARIC AND IVAN PERIC


Abstract. In this paper some multidimensional integral and discrete Opial-type inequalities dueto Agarwal, Pang and Sheng are considered. Theirs generalizations and extensions using sub-multiplicative convex functions, appropriate integral representations of functions, appropriatesummation representations of discrete functions and inequalities involving means are presented.

1. Introduction

In 1960, Z. Opial [10] proved next integral inequality:

Let x(t)∈C1[0,h] be such that x(0) = x(h) = 0 and x(t) > 0 for t ∈ (0,h) .Then ∫ h

0

∣∣x(t)x′(t)∣∣dt � h

4

∫ h

0

(x′(t)

)2dt, (1)

where constant h4 is the best possible.

Over the last five decades, an enormous amount of work has been done on this integralinequality, dealing with new proofs, various generalizations, extensions and discreteanalogues. Opial’s inequality is recognized as fundamental result in the analysis ofqualitative properties of solution of differential equations (see [3, 9] and the referencescited therein).

The aim of this paper is to generalize and extend some integral and discrete Opial-type inequalities due to Agarwal, Pang and Sheng ([1, 2, 6]). To establish these in-equalities, we will use some elementary techniques such as appropriate integral rep-resentations of functions, appropriate summation representations of the discrete func-tions and inequalities involving means. We start each section with inequality involvinga submultiplicative convex function. Recall that function f : [0,∞) → [0,∞) is calledsubmultiplicative function if it satisfies the inequality

f (xy) � f (x) f (y) , for all x,y ∈ [0,∞)

Mathematics subject classification (2010): 26B25, 26D15.Keywords and phrases: Opial’s inequality, Jensen’s inequality, integral inequality, discrete inequality,

multidimensional inequality.


1295


1296 MAJA ANDRIC, JOSIP PECARIC AND IVAN PERIC

(see for example [8]).One of such submultiplicative functions, which is also convex and increasing, is

f (x) = xp log(e+ x) , where p � 1+√

52 . The obtained results will give in a special case

improvements of corresponding inequalities in [1, 2, 6], and, at the same time, they willsimplify proofs of the corresponding theorems in [4, 5, 7].

For the following inequalities we present obtained generalizations, extensions andimprovements: first is a result by Agarwal and Pang from [1], observed in Section 2.Recall, AC[0,h] is the space of all absolutely continuous functions on [0,h] . Also, letB denotes the beta function.

THEOREM 1. [1] Let λ � 1 be a given real number and let p be a nonnegativeand continuous function on [0,h] . Further, let x ∈ AC[0,h] be such that x(0) = x(h) =0 . Then the following inequality holds∫ h

0p(t)|x(t)|λ dt � 1

2

(∫ h

0

(t(h− t)

)λ−12 p(t)dt

)∫ h

0|x′(t)|λ dt . (2)

For a constant function p, the inequality (2) reduces to

∫ h

0|x(t)|λ dt � hλ

2B

(λ +1

2,

λ +12

)∫ h

0|x′(t)|λ dt . (3)

Next is a multidimensional Poincare-type inequality by Agarwal and Sheng from[6], observed in Section 3. This inequality involves a special class of continuous func-tions, a class G(Ω) , whose definition and properties are given at the beginning of Sec-tion 3.

THEOREM 2. [6] Let λ ,μ � 1 and let u ∈ G(Ω) . Then the following inequalityholds ∫

Ω|u(x)|λ dx � K(λ ,μ)

∫Ω‖gradu(x)‖λ

μ dx ,

where

K(λ ,μ) =1

2mB

(1+ λ

2,1+ λ

2

)C

(λμ

)Gm

((b−a)λ

), (4)

C(α) =

{1 , α � 1 ,

m1−α , 0 � α � 1 .(5)

Finally, a discrete inequality, observed in Section 4, is a result by Agarwal andPang from [2]. A definition of a class G(Ω) for the discrete case and a definition offorward difference operator Δi are given at the beginning of Section 4.

THEOREM 3. [2] Let λ � 1 and let u ∈ G(Ω) . Then the following inequalityholds

X−1

∑x=1

|u(x)|λ � K(λ )X−1

∑x=0

(m

∑i=1

|Δi u(x)|2) λ

2

,

ON WEIGHTED OPIAL-TYPE INEQUALITIES 1297

where

K(λ ) =1m

C

(λ2

) m

∏i=1

(Xi−1

∑xi=1

12

(xi(Xi − xi))λ−1

2

) 1m

(6)

and C is defined by (5).

2. Integral inequalities in one variable

First we give a generalization of Theorem 1 involving submultiplicative convexfunctions. In a special case (Corollary 4) this theorem will improve result from Theo-rem 1.

THEOREM 4. Let n ∈ N and let fi be increasing, submultiplicative convex func-tions on [0,∞) , i = 1, . . . ,n. Let p be a nonnegative and integrable function on [0,h] .Further, let xi ∈ AC[0,h] be such that xi(0) = xi(h) = 0 for i = 1, . . . ,n. Then thefollowing inequality holds

∫ h

0p(t)

n

∏i=1

fi (|xi(t)|)dt

�(∫ h

0p(t)

n

∏i=1

(t

fi(t)+

h− tfi(h− t)

)−1

dt

)n

∏i=1

∫ h

0fi(|x′i(t)|)dt . (7)

Proof. As in [1], for each fixed i , i = 1, . . . ,n , from the hypotheses of the theoremwe have

xi(t) =∫ t

0x′i(s)ds ,

xi(t) = −∫ h

tx′i(s)ds .

Since fi is an increasing and convex function, we use Jensen’s inequality to obtain

fi (|xi(t)|) � fi

(1t

∫ t

0t |x′i(s)|ds

)� 1

t

∫ t

0fi(t |x′i(s)|

)ds

and by submultiplicativity of fi follows

fi (|xi(t)|) � 1t

∫ t

0fi(t) fi(|x′i(s)|)ds =

fi(t)t

∫ t

0fi(|x′i(s)|)ds . (8)


Analogously we obtain

fi (|xi(t)|) � fi

(1

h− t

∫ h

t(h− t) |x′i(s)|ds

)

� 1h− t

∫ h

tfi((h− t) |x′i(s)|

)ds

� 1h− t

∫ h

tfi(h− t) fi(|x′i(s)|)ds

=fi(h− t)h− t

∫ h

tfi(|x′i(s)|)ds . (9)

Multiplying (8) by tfi(t)

and (9) by h−tfi(h−t) and adding these inequalities, we find(

tfi(t)

+h− t

fi(h− t)

)fi (|xi(t)|) �

∫ h

0fi(|x′i(s)|)ds ,

i.e.

fi (|xi(t)|) �(

tfi(t)

+h− t

fi(h− t)

)−1 ∫ h

0fi(|x′i(s)|)ds . (10)

This gives us

n

∏i=1

fi (|xi(t)|) �n

∏i=1

[(t

fi(t)+

h− tfi(h− t)

)−1 ∫ h

0fi(|x′i(s)|)ds

]. (11)

Now multiplying (11) by p and integrating on [0,h] we obtain∫ h

0p(t)

n

∏i=1

fi (|xi(t)|)dt

�∫ h

0p(t)

n

∏i=1

[(t

fi(t)+

h− tfi(h− t)

)−1 ∫ h

0fi(|x′i(s)|)ds

]dt ,

which is the inequality (7). �

REMARK 1. For a special class of a submultiplicative convex functions fi on[0,∞) with fi(0) = 0 (i = 1, . . . ,n) , Theorem 4 also holds. Namely, submultiplica-tivity of a function implies its positivity, and if fi is a convex, nonnegative function on[0,∞) with fi(0) = 0, then fi is obviously an increasing function.

COROLLARY 1. Let n ∈ N and let fi be increasing, submultiplicative convexfunctions on [0,∞) , i = 1, . . . ,n. Let p be a nonnegative and integrable function on[0,h] . Further, let xi ∈ AC[0,h] be such that xi(0) = xi(h) = 0 for i = 1, . . . ,n. Thenthe following inequality holds∫ h

0p(t)

n

∏i=1

fi (|xi(t)|)dt

� 12n

(∫ h

0p(t)

n

∏i=1

(fi(t) fi(h− t)

t (h− t)

) 12

dt

)n

∏i=1

∫ h

0fi(|x′i(t)|)dt . (12)


Proof. The inequality (12) follows by the harmonic-geometric inequality

2

(t

fi(t)+

h− tfi(h− t)

)−1

�(

fi(t) fi(h− t)t (h− t)

) 12

. �

For n = 1 we have two following results.

COROLLARY 2. Let f be an increasing, submultiplicative convex function on[0,∞) and let p be a nonnegative and integrable function on [0,h] . Further, let x ∈AC[0,h] be such that x(0) = x(h) = 0 . Then the following inequality holds

∫ h

0p(t) f (|x(t)|)dt �

(∫ h

0p(t)

(t

f (t)+

h− tf (h− t)

)−1

dt

)∫ h

0f(|x′(t)|)dt . (13)

COROLLARY 3. Let f be an increasing, submultiplicative convex function on[0,∞) and let p be a nonnegative and integrable function on [0,h] . Further, let x ∈AC[0,h] be such that x(0) = x(h) = 0 . Then the following inequality holds

∫ h

0p(t) f (|x(t)|)dt � 1

2

(∫ h

0p(t)

(f (t) f (h− t)

t (h− t)

) 12

dt

)∫ h

0f(|x′(t)|)dt . (14)

Next result was proven by Brnetic and Pecaric in [7]. Here it is merely a conse-quence, a special case of Corollary 2 (as we can see from its proof). By the harmonic-geometric inequality, it is clear that (15) improves (2).

COROLLARY 4. Let λ � 1 be a given real number and let p be a nonnegative andcontinuous function on [0,h] . Further, let x ∈ AC[0,h] be such that x(0) = x(h) = 0 .Then the following inequality holds∫ h

0p(t)|x(t)|λ dt �

(∫ h

0

(t1−λ +(h− t)1−λ)−1

p(t)dt

)∫ h

0|x′(t)|λ dt . (15)

Proof. The inequality (15) will follow if we use the function f (t) = tλ and applyCorollary 2. �

3. Multidimensional integral inequalities

Let Ω be a bounded domain in Rm defined by Ω = ∏m

j=1[a j,b j] .Let x = (x1, . . . ,xm) be a general point in Ω and dx = dx1 . . .dxm . For any con-tinuous real-valued function u defined on Ω we denote

∫Ω u(x)dx the m-fold inte-

gral∫ b1a1

· · ·∫ bmam

u(x1, . . . ,xm)dx1 . . .dxm . Let Dku(x1, . . . ,xm) = ∂∂xk

u(x1, . . . ,xm) and

Dku(x1, . . . ,xm) = D1 · · ·Dku(x1, . . . ,xm) , 1 � k � m .We denote by G(Ω) the class of continuous functions u : Ω → R for which

Dmu(x) exists with u(x)|x j=a j = u(x)|x j=b j = 0, 1 � j � m .


Further, let u(x;s j) = u(x1, . . . ,x j−1,s j,x j+1, . . . ,xm) , and

‖gradu(x)‖μ =

(m

∑j=1

∣∣∣∣ ∂∂x j

u(x)∣∣∣∣μ) 1

μ

.

Also let α = (α1, . . . ,αm) and αλ = (αλ1 , . . . ,αλ

m) , λ ∈ R . In particular, (b− a) =(b1−a1, . . . ,bm−am) and (b−a)λ = ((b1−a1)λ , . . . ,(bm−am)λ ) . For the geometricand the harmonic means of α1, . . . ,αm we will use Gm(α) and Hm(α) , respectively.Let M[k](α) denote the mean of order k of α1, . . . ,αm .

We start with a weighted extension of Theorem 2 involving submultiplicative con-vex function. Again, in a special case (Corollary 6) this theorem will improve resultfrom Theorem 2.

THEOREM 5. Let f be an increasing, submultiplicative convex function on [0,∞) .Let p be a nonnegative and integrable function on Ω and u ∈ G(Ω) . Then the follow-ing inequality holds

∫Ω

p(x) f (|u(x)|)dx � 1m

Hm(α)∫

Ω

(m

∑i=1

f

(∣∣∣∣ ∂∂xi

u(x)∣∣∣∣))

dx, (16)

where α = (α1, . . . ,αm) and

αi =∫ bi

ai

(xi −ai

f (xi −ai)+

bi− xi

f (bi − xi)

)−1

p(x)dxi , i = 1, . . . ,m .

Proof. For each fixed i , i = 1, . . . ,m , we have

u(x) =∫ xi

ai

∂∂ si

u(x;si)dsi

and

u(x) = −∫ bi

xi

∂∂ si

u(x;si)dsi .

First we use Jensen’s inequality (since f is an increasing convex function) and thensubmultiplicativity of f , to obtain

f (|u(x)|) � f

(1

xi−ai

∫ xi

ai

(xi −ai)∣∣∣∣ ∂∂ si

u(x;si)∣∣∣∣dsi

)

� 1xi−ai

∫ xi

ai

f

((xi −ai)

∣∣∣∣ ∂∂ si

u(x;si)∣∣∣∣)

dsi

� 1xi−ai

∫ xi

ai

f (xi −ai) f

(∣∣∣∣ ∂∂ si

u(x;si)∣∣∣∣)

dsi

=f (xi −ai)xi−ai

∫ xi

ai

f

(∣∣∣∣ ∂∂ si

u(x;si)∣∣∣∣)

dsi (17)


and analogously

f (|u(x)|) � f (bi − xi)bi− xi

∫ bi

xi

f

(∣∣∣∣ ∂∂ si

u(x;si)∣∣∣∣)

dsi , (18)

for i = 1, . . . ,m . Multiplying (17) by xi−aif (xi−ai)

and (18) by bi−xif (bi−xi)

and adding theseinequalities, we find(

xi −ai

f (xi −ai)+

bi − xi

f (bi − xi)

)f (|u(x)|) �

∫ bi

ai

f

(∣∣∣∣ ∂∂ si

u(x;si)∣∣∣∣)

dsi ,

i.e.

f (|u(x)|) �(

xi −ai

f (xi −ai)+

bi− xi

f (bi − xi)

)−1∫ bi

ai

f

(∣∣∣∣ ∂∂ si

u(x;si)∣∣∣∣)

dsi , (19)

for i = 1, . . . ,m . Now multiplying (19) by p and integrating on Ω we obtain

∫Ω

p(x) f (|u(x)|)dx �∫ bi

ai

(xi −ai

f (xi −ai)+

bi− xi

f (bi − xi)

)−1

p(x)dxi

×∫

Ωf

(∣∣∣∣ ∂∂xi

u(x)∣∣∣∣)

dx , (20)

i.e. (∫ bi

ai

(xi −ai

f (xi −ai)+

bi− xi

f (bi − xi)

)−1

p(x)dxi

)−1∫Ω

p(x) f (|u(x)|)dx

�∫

Ωf

(∣∣∣∣ ∂∂xi

u(x)∣∣∣∣)

dx , (21)

for i = 1, . . . ,m . Notice that

α−1i =

(∫ bi

ai

(xi−ai

f (xi −ai)+

bi− xi

f (bi − xi)

)−1

p(x)dxi

)−1

, i = 1, . . . ,m .

Now, by summing these m inequalities (21), we find

m

∑i=1

α−1i

∫Ω

p(x) f (|u(x)|)dx �m

∑i=1

∫Ω

f

(∣∣∣∣ ∂∂xi

u(x)∣∣∣∣)

dx ,

which is the same as the inequality (16). �

COROLLARY 5. Let f be an increasing, submultiplicative convex function on[0,∞) . Let p be a nonnegative and integrable function on Ω and let u ∈ G(Ω) . Thenthe following inequality holds

∫Ω

p(x) f (|u(x)|)dx � 12m

Hm(β )∫

Ω

(m

∑i=1

f

(∣∣∣∣ ∂∂xi

u(x)∣∣∣∣))

dx , (22)


where β = (β1, . . . ,βm) and

βi =∫ bi

ai

(f (xi −ai) f (bi − xi)(xi −ai)(bi − xi)

) 12

p(x)dxi , i = 1, . . . ,m .

Proof. By harmonic-geometric inequality we have

2

(xi−ai

f (xi −ai)+

bi − xi

f (bi − xi)

)−1

�(

f (xi −ai) f (bi − xi)(xi −ai)(bi − xi)

) 12

.

Applying this and using Hm( 12 γ) = 1

2Hm(γ) , the inequality (22) follows. �Next result was proven by Agarwal, Brnetic and Pecaric in [4]. Here we use Theo-

rem 5 applied on a constant function p and the function f (t) = tλ to prove the inequal-ity (23). By the harmonic-geometric inequality, it follows that Corollary 6 improvesTheorem 2.

COROLLARY 6. Let λ ,μ � 1 and let u ∈ G(Ω) . Then the following inequalityholds ∫

Ω|u(x)|λ dx � K1(λ ,μ)

∫Ω‖gradu(x)‖λ

μ dx , (23)

where

K1(λ ,μ) =1m

I(λ )C(

λμ

)Hm

((b−a)λ

), (24)

I(λ ) =∫ 1

0

(t1−λ +(1− t)1−λ

)−1dt (25)


Proof. We follow steps from the proof of Theorem 5, using the function f (t) = tλ ,up to the inequality (20), which is now equal to

∫Ω|u(x)|λ dx �

∫ bi

ai

((xi −ai)1−λ +(bi− xi)1−λ

)−1dxi

∫Ω

∣∣∣∣ ∂∂xi

u(x)∣∣∣∣λ dx (26)

for i = 1, . . . ,m . However, since

∫ bi

ai

((xi −ai)1−λ +(bi− xi)1−λ

)−1dxi = (bi−ai)λ

∫ 1

0

(t1−λ +(1− t)1−λ

)−1dt

= (bi−ai)λ I(λ ) ,

the inequality (26) can be written as

∫Ω|u(x)|λ dx � (bi −ai)λ I(λ )

∫Ω

∣∣∣∣ ∂∂xi

u(x)∣∣∣∣λ dx . (27)


Multiplying both sides of the inequality (27) by (bi − ai)−λ , i = 1, . . . ,m , and thensumming these inequalities, we obtain

m

∑i=1

(bi −ai)−λ∫

Ω|u(x)|λ dx � I(λ )

∫Ω

(m

∑i=1

∣∣∣∣ ∂∂xi

u(x)∣∣∣∣λ)

dx ,

i.e. ∫Ω|u(x)|λ dx � 1

mI(λ )Hm

((b−a)λ

)∫Ω

(m

∑i=1

∣∣∣∣ ∂∂xi

u(x)∣∣∣∣λ)

dx . (28)

Our result now follows from (28) and the elementary inequality

m

∑i=1

aαi � C(α)

(m

∑i=1

ai

)α

, ai � 0 . (29)

�

4. Multidimensional discrete inequalities

Let x,X ∈ Nm0 be such that x � X , i.e., xi � Xi , i = 1, . . . ,m . Let Ω = [0,X ] ,

where [0,X ] ⊂ Nm0 . We denote by G(Ω) the class of functions u : Ω → R , which

satisfies conditions u(x)|xi=0 = u(x)|xi=Xi = 0, i = 1, . . . ,m . For u we define forwarddifference operators Δi , i = 1, . . . ,m , as

Δi u(x) = u(x1, . . . ,xi−1,xi +1,xi+1, . . . ,xm)−u(x) .

As in a previous section, let u(x;si) stand for u(x1, . . . ,xi−1,si,xi+1, . . . ,xm) , α =(α1, . . . ,αm) and let Hm(α) denote the harmonic mean of α1, . . . ,αm . Also, letX−1∑

x=1denote

m∑j=1

Xj−1

∑x j=1

.

First we present a weighted extension of Theorem 3 involving submultiplicativeconvex functions.

THEOREM 6. Let n∈N and let f j be submultiplicative convex functions on [0,∞)with f j(0) = 0 , j = 1, . . . ,n. Let p be a nonnegative function on Ω and let u j ∈ G(Ω)for j = 1, . . . ,n. Then the following inequality holds

X−1

∑x=1

p(x)n

∏j=1

f j(∣∣u j(x)

∣∣)� 1m

Hm (α)m

∑i=1

n

∏j=1

X−1

∑x=0

f j (|Δi u j(x)|) , (30)

where α = (α1, . . . ,αm) and

αi =Xi−1

∑xi=1

p(x)n

∏j=1

(xi

f j(xi)+

Xi− xi

f j(Xi − xi)

)−1

, i = 1, . . . ,m . (31)


Proof. For each fixed i (i = 1, . . . ,m) and each fixed j ( j = 1, . . . ,n) we have

u j(x) =xi−1

∑si=0

Δi u j(x;si) , u j(x) = −Xi−1

∑si=xi

Δi u j(x;si) .

From the discrete case of Jensen’s inequality (since f j is an increasing convex function)and the submultiplicativity of f j , we have

f j(|u j(x)|) � f j

(1xi

xi−1

∑si=0

xi∣∣Δi u j(x;si)

∣∣)

� 1xi

xi−1

∑si=0

f j(xi∣∣Δi u j(x;si)

∣∣)

� 1xi

xi−1

∑si=0

f j(xi) f j(∣∣Δi u j(x;si)

∣∣)

=f j(xi)

xi

xi−1

∑si=0

f j(∣∣Δi u j(x;si)

∣∣) (32)

and analogously

f j(|u j(x)|) � f j(Xi − xi)Xi − xi

Xi−1

∑si=xi


∣∣) (33)

for i = 1, . . . ,m . We multiply (32) by xif j(xi)

and (33) by Xi−xif j(Xi−xi)

. Then we add these

resulting inequalities, to obtain(xi

f j(xi)+

Xi− xi

f j(Xi− xi)

)f j(|u j(x)|) �

Xi−1

∑si=0


∣∣) ,i.e.

f j(|u j(x)|) �(

xi

f j(xi)+

Xi− xi

f j(Xi − xi)

)−1 Xi−1

∑si=0


∣∣) (34)

for i = 1, . . . ,m . This gives us

n

∏j=1

f j(|u j(x)|) �n

∏j=1

[(xi

f j(xi)+

Xi− xi

f j(Xi − xi)

)−1 Xi−1

∑si=0


∣∣)] (35)

for i = 1, . . . ,m . Now multiplying (35) by p we get

p(x)n

∏j=1

f j(|u j(x)|)

� p(x)

[n

∏j=1

(xi

f j(xi)+

Xi− xi

f j(Xi − xi)

)−1][

n

∏j=1

Xi−1

∑si=0


∣∣)]


for i = 1, . . . ,m . Summing from x = 1 to x = X −1, we get

X−1

∑x=1

p(x)n

∏j=1

f j(|u j(x)|)

�(

Xi−1

∑xi=1

p(x)n

∏j=1

(xi

f j(xi)+

Xi − xi

f j(Xi− xi)

)−1)

n

∏j=1

X−1

∑x=0

f j(∣∣Δi u j(x)

∣∣) (36)

for i = 1, . . . ,m . Multiplying both sides of the inequality (36) by α−1i and then adding

these m inequalities , we obtain

m

∑i=1

α−1i

X−1

∑x=1

p(x)n

∏j=1

f j(∣∣u j(x)

∣∣) �m

∑i=1

n

∏j=1

X−1

∑x=0

f j(∣∣Δi u j(x)

∣∣) ,which is the same as the inequality (30). �

COROLLARY 7. Let n ∈ N and let f j be submultiplicative convex functions on[0,∞) with f j(0) = 0 , j = 1, . . . ,n. Let p be a nonnegative function on Ω and letu j ∈ G(Ω) for j = 1, . . . ,n. Then the following inequality holds

X−1

∑x=1

p(x)n

∏j=1

f j(∣∣u j(x)

∣∣)� 12nm

Hm (β )m

∑i=1

n

∏j=1

X−1

∑x=0

f j (|Δi u j(x)|) , (37)

where β = (β1, . . . ,βm) and

βi =Xi−1

∑xi=1

p(x)n

∏j=1

(f j(xi) f j(Xi − xi)

xi (Xi − xi)

) 12

, i = 1, . . . ,m . (38)

Proof. By harmonic-geometric inequality we have

2

(xi

f j(xi)+

Xi − xi

f j(Xi − xi)

)−1

�(

f j(xi) f j(Xi− xi)xi (Xi− xi)

) 12

.

Applying this and using Hm( 12n β ) = 1

2n Hm(β ) , the inequality (37) follows. �

Next are special results for n = 1.

COROLLARY 8. Let f be a submultiplicative convex function on [0,∞) with f (0)=0 . Let p be a nonnegative function on Ω and u ∈ G(Ω) . Then the following inequalityholds

X−1

∑x=1

p(x) f (|u(x)|) � 1m

Hm (α)X−1

∑x=0

m

∑i=1

f (|Δi u(x)|) , (39)

where α = (α1, . . . ,αm) is defined by (31).


COROLLARY 9. Let f be a submultiplicative convex function on [0,∞) with f (0)=0 . Let p be a nonnegative function on Ω and u ∈ G(Ω) . Then the following inequalityholds

X−1

∑x=1

p(x) f (|u(x)|) � 12m

Hm (β )X−1

∑x=0

m

∑i=1

f (|Δi u(x)|) , (40)

where β = (β1, . . . ,βm) is defined by (38).

An improvement of Theorem 3 is the following result, given also Agarwal, Brneticand Pecaric in [5]. Here we use Corollary 8 applied on a constant function p and thefunction f (t) = tλ to prove the inequality (41). Thus again, by the harmonic-geometricinequality, it follows that Corollary 10 for μ = 2 improves Theorem 3.

COROLLARY 10. Let λ ,μ � 1 and let u ∈ G(Ω) . Then the following inequalityholds

X−1

∑x=1

|u(x)|λ � K1(λ ,μ)X−1

∑x=0

(m

∑i=1

|Δi u(x)|μ) λ

μ

, (41)

where

K1(λ ,μ) =1m

C

(λμ

)Hm (h(x,X ,λ )) , (42)

h(x,X ,λ ) = (h1(x,X ,λ ), . . . ,hm(x,X ,λ )) , (43)

hi(x,X ,λ ) =Xi−1

∑xi=1

(x1−λi +(Xi− xi)1−λ

)−1, i = 1, . . . ,m


Proof. From Corollary 8 using the function f (t) = tλ (and a constant function p )we have

X−1

∑x=1

|u(x)|λ � 1m

Hm (h(x,X ,λ ))X−1

∑x=0

m

∑i=1

|Δi u(x)|λ . (44)

The inequality (41) now follows from (44) and the elementary inequality (29). �

Acknowledgement. This work has been fully supported by Croatian Science Foun-dation under the project 5435.

RE F ER EN C ES

[1] R. P. AGARWAL AND P. Y. H. PANG, Remarks on the generalization of Opial’s inequality, J. Math.Anal. Appl., 190 (1995), 559–577.

[2] R. P. AGARWAL AND P. Y. H. PANG, Sharp discrete inequalities in n independent variables, Appl.Math. Comp., 72 (1995), 97–112.

[3] R. P. AGARWAL AND P. Y. H. PANG, Opial Inequalities with Applications in Differential and Differ-ence Equations, Kluwer Academic Publishers, Dordrecht, Boston, London, 1995.

[4] R. P. AGARWAL, J. PECARIC AND I. BRNETIC, Improved integral inequalities in n independentvariables, Computers Math. Applic., 33, 8 (1997), 27–38.


[5] R. P. AGARWAL, J. PECARIC AND I. BRNETIC, Improved discrete inequalities in n independentvariables, Appl. Math. Lett., 11, 2 (1998), 91–97.

[6] R. P. AGARWAL AND Q. SHENG, Sharp integral inequalities in n independent variables, NonlinearAnal., 26 (1996), 179–210.

[7] I. BRNETIC AND J. PECARIC, Some new Opial-type inequalities, Math. Inequal. Appl., 1, 3 (1998),385–390.

[8] J. GUSTAVSSON, L. MALIGRANDA AND J. PEETRE, A submultiplicative function, Nederl. Akad.Wetensch. Indag. Math., 51, 4 (1989), 435–442.

[9] D. S. MITRINOVIC, J. PECARIC AND A. M. FINK, Inequalities Involving Functions and Their Inte-grals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.

[10] Z. OPIAL, Sur une inegalite, Ann. Polon. Math., 8 (1960), 29–32.

(Received March 17, 2016) Maja AndricFaculty of Civil Engineering, Architecture and Geodesy

University of SplitMatice hrvatske 15, 21000 Split, Croatia




Ivan PericFaculty of Food Technology and Biotechnology

University of ZagrebPierottijeva 6, 10000 Zagreb, Croatia


& Applications


POPOVICIU TYPE INEQUALITIES VIA HERMITE’S POLYNOMIAL

SAAD IHSAN BUTT, KHURAM ALI KHAN AND JOSIP PECARIC


Abstract. We obtain useful identities via Hermite interpolation polynomial, by which the in-equality of Popoviciu for convex functions is generalized for higher order convex functions.We investigate the bounds for the identities extracted by the generalization of the Popoviciu in-equality using inequalities for the Cebysev functional. Some results relating to the Gruss andOstrowski type inequalities are constructed.

1. Introduction

A characterization of convex function established by T. Popoviciu [11] is studiedby many people (see [12, 10] and references with in). For recent work, we refer [5, 7, 8,9]. The following form of Popoviciu’s inequality is established by Vasic and Stankovicin [12] (see also page 173 [10]):

THEOREM 1. Let z,w ∈ N , z � 3 , 2 � w � z− 1 , [α,β ] ⊂ R , x = (x1, ...,xz) ∈[α,β ]z , p = (p1, ..., pz) be a positive z-tuple such that ∑z

u=1 pu = 1 . Also let f :[α,β ] → R be a convex function. Then

pw,z(x,p; f ) � z−wz−1

p1,z(x,p; f )+w−1z−1

pz,z(x,p; f ), (1)

where

pw,z(x,p; f ) = pw,z(x,p; f (x)) :=1( z−1

w−1

) ∑1�u1<...<ul�z

(w

∑v=1

puv

)f

⎛⎜⎜⎝

w∑

v=1puvxuv

w∑

v=1puv

⎞⎟⎟⎠

is the linear functional with respect to f .

By inequality (1), we write

P(x,p; f ) :=z−wz−1

p1,z(x,p; f )+w−1z−1

pz,z(x,p; f )− pw,z(x,p; f ); 2 � w � z−1.

(2)

Mathematics subject classification (2010): Primary 26D07, 26D15, 26D20, 26D99.Keywords and phrases: Convex function, divided difference, Hermite interpolation, Cebysev func-

tional, Gruss inequality, Ostrowski inequality, exponential convexity.


1309

1310 S. IHSAN BUTT, K. ALI KHAN AND J. PECARIC

REMARK 1. It is important to note that under the assumptions of Theorem 1, ifthe function f is convex then P(x,p; f ) � 0 and P(x,p; f ) = 0 if f is an identity orconstant function.

The mean value theorems and exponential convexity of the linear functionalP(x,p; f ) are given in [7] for positive tuples p . Some special classes of convex func-tions are considered to construct the exponential convexity of P(x,p; f ) in [7].

Let −∞ < α < β < ∞ and α = a1 < a2 . . . < ar = β , (r � 2) be the given points.For ψ ∈ Cn[α,β ] a unique polynomial ρH(s) of degree (n− 1) exists satisfying anyof the following conditions:Hermite conditions:

ρ (i)H (a j) = ψ(i)(a j); 0 � i � k j, 1 � j � r,

r

∑j=1

k j + r = n. (H)

It is of great interest to note that Hermite conditions include the following particularcases:Lagrange conditions: (r = n , k j = 0 for all j )

ρL(a j) = ψ(a j), 1 � j � n,

Type (m,n−m) conditions: (r = 2, 1 � m � n−1, k1 = m−1, k2 = n−m−1)

ρ (i)(m,n)(α) = ψ(i)(α), 0 � i � m−1,

ρ (i)(m,n)(β ) = ψ(i)(β ), 0 � i � n−m−1,

Two-point Taylor conditions: (n = 2m , r = 2, k1 = k2 = m−1)

ρ (i)2T (α) = ψ(i)(α), ρ (i)

2T (β ) = ψ(i)(β ), 0 � i � m−1.

We have the following result from [1].

THEOREM 2. Let −∞ < α < β < ∞ and α � a1 < a2 . . . < ar � β , (r � 2) bethe given points, and ψ ∈Cn([α,β ]) . Then we have

ψ(t) = ρH(t)+RH(ψ ,t) (3)

where ρH(t) is the Hermite interpolating polynomial, i.e.

ρH(t) =r

∑j=1

k j

∑i=0

Hi j(t)ψ(i)(a j);

the Hi j are fundamental polynomials of the Hermite basis defined by

Hi j(t) =1i!

ω(t)

(t−a j)k j+1−i

k j−i

∑k=0

1k!

dk

dtk

((t −a j)

k j+1

ω(t)

)∣∣∣∣∣t=a j

(t −a j)k, (4)

POPOVICIU TYPE INEQUALITIES VIA HERMITE’S POLYNOMIAL 1311

with

ω(t) =r

∏j=1

(t−a j)k j+1,

and the remainder is given by

RH(ψ ,t) =∫ β

αGH,n(t,s)ψ(n)(s)ds

where GH,n(t,s) is defined by

GH,n(t,s) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

l∑j=1

k j

∑i=0

(a j−s)n−i−1

(n−i−1)! Hi j(t); s � t,

−r∑

j=l+1

k j

∑i=0

(a j−s)n−i−1

(n−i−1)! Hi j(t); s � t,

(5)

for all al � s � al+1 ; l = 0, . . . ,r with a0 = α and ar+1 = β .

REMARK 2. In particular cases, for Lagrange conditions, from Theorem 2 wehave

ψ(t) = ρL(t)+RL(ψ ,t)

where ρL(t) is the Lagrange interpolating polynomial i.e,

ρL(t) =n

∑j=1

n

∏k=1k �= j

( t−ak

a j −ak

)ψ(a j)

and the remainder RL(ψ ,t) is given by

RL(ψ ,t) =∫ β

αGL(t,s)ψ(n)(s)ds

with

GL(t,s) =1

(n−1)!

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

l∑j=1

(a j − s)n−1n∏k=1k �= j

(t−aka j−ak

), s � t

−n∑

j=l+1(a j − s)n−1

n∏k=1k �= j

(t−aka j−ak

), s � t

(6)

al � s � al+1, l = 1,2, ...,n−1 with a1 = α and an = β .For type (m,n−m) conditions, from Theorem 2 we have

ψ(t) = ρ(m,n)(t)+R(m,n)(ψ ,t)

where ρ(m,n)(t) is (m,n−m) interpolating polynomial, i.e

ρ(m,n)(t) =m−1

∑i=0

τi(t)ψ(i)(α)+n−m−1

∑i=0

ηi(t)ψ(i)(β ),


with

τi(t) =1i!

(t −α)i( t−β

α −β

)n−m m−1−i

∑k=0

(n−m+ k−1

k

)( t −αβ −α

)k(7)

and

ηi(t) =1i!

(t−β )i( t−α

β −α

)m n−m−1−i

∑k=0

(m+ k−1

k

)( t−βα −β

)k. (8)

and the remainder R(m,n)(ψ ,t) is given by

R(m,n)(ψ ,t) =∫ β

αG(m,n)(t,s)ψ(n)(s)ds

with

G(m,n)(t,s) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

m−1∑j=0

[m−1− j∑

p=0

(n−m+p−1p

)(t−αβ−α

)p]× (t−α) j(α−s)n− j−1

j!(n− j−1)!

(β−tβ−α

)n−m, α � s � t � β

−n−m−1

∑i=0

[n−m−i−1∑

q=0

(m+q−1q

)( β−tβ−α

)q]× (t−β )i(β−s)n−i−1

i!(n−i−1)!

(t−αβ−α

)m, α � t � s � β .

(9)

For Type Two-point Taylor conditions, from Theorem 2 we have

ψ(t) = ρ2T (t)+R2T (ψ ,t)

where ρ2T (t) is the two-point Taylor interpolating polynomial i.e,

ρ2T (t) =m−1

∑i=0

m−1−i

∑k=0

(m+ k−1

k

)[ (t −α)i

i!

( t −βα −β

)m( t−αβ −α

)kψ(i)(α)

+(t−β )i

i!

( t −αβ −α

)m( t −βα −β

)kψ(i)(β )

]and the remainder R2T (ψ ,t) is given by

R2T (ψ ,t) =∫ β

αG2T (t,s)ψ(n)(s)ds

with

G2T (t,s) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(−1)m

(2m−1)! pm(t,s)

m−1∑j=0

(m−1+ jj

)(t − s)m−1− jq j(t,s), s � t;

(−1)m

(2m−1)!qm(t,s)

m−1∑j=0

(m−1+ jj

)(s− t)m−1− j p j(t,s), s � t;

(10)

where p(t,s) = (s−α)(β−t)β−α , q(t,s) = p(t,s),∀t,s ∈ [α,β ].

The following Lemma describes the positivity of Green’s function (5) see (Beesack[2] and Levin [6]).

LEMMA 1. The Green’s function GH,n(t,s) has the following properties:

(i)GH,n(t,s)

w(t) > 0 , a1 � t � ar , a1 � s � ar ;

(ii) GH,n(t,s) � 1(n−1)!(β−α) |w(t)|;

(iii)∫ β

α GH,n(t,s)ds = w(t)n! .

The organization of the paper is as follows: In Section 2, we use Hermite inter-polating polynomial and the n -convexity of the function ψ (defined in Theorem 2) toestablish a generalization of Theorem 1 for real weights. We discuss the results forparticular cases namely, Lagrange interpolating polynomial, (m,n−m) interpolatingpolynomial, two-point Taylor interpolating polynomial. In Section 3, we present someinteresting results about upper bounds by employing Cebysev functional and Gruss-type inequalities, also results relating to the Ostrowski-type inequality.

2. Popoviciu’s inequality by Hermite interpolating polynomial

We begin this section with the proof of our main identity related to generalizationsof Popoviciu’s inequality.

THEOREM 3. (Main) Let z,w∈N , z � 3 , 2 � w� z−1 , [α,β ]⊂R , x = (x1, ...,xz)∈ [α,β ]z , p = (p1, ..., pz) be a real z-tuple such that ∑w

v=1 puv �= 0 for any 1 � u1 <

... < uw � z ∑zu=1 pu = 1 and

w∑

v=1puvxuv

w∑

v=1puv

∈ [α,β ] for any 1 � u1 < ... < uw � z. Also

let α = a1 < a2 . . . < ar = β , (r � 2) be the given points, and ψ ∈Cn([α,β ]) . More-over, Hi j be the fundamental polynomials of the Hermite basis and GH,n be the greenfunctions as defined by (4) and (5) respectively. Then we have the following identity:

P(x,p;ψ(x)) =r

∑j=1

k j

∑i=0

ψ(i)(a j)P(x,p;Hi j(x))+∫ β

αP(x,p;GH,n(x,s))ψ(n)(s)ds. (11)

Proof. Using (3) in (2) and following the linearity of P(x,p; ·) , we get (11). �

For n -convex functions, we can give the following form of new identity (11).

THEOREM 4. Let all the assumptions of Theorem 3 be satisfied and ψ be an n-convex function. Then we have the following result:


IfP(x,p;GH,n(x,s)) � 0, s ∈ [α,β ] (12)

then

P(x,p;ψ(x)) �r

∑j=1

k j

∑i=0

ψ(i)(a j)P(x,p;Hi j(x)). (13)

Proof. Since the function ψ is n -convex, therefore without loss of generality wecan assume that ψ is n -times differentiable and ψ(n)(x) � 0 for all x ∈ [α,β ] (see[10], p. 16). Hence we can apply Theorem 3 to obtain (13) . �

REMARK 3. The inequality (13) hold in reverse directions if the inequality in (12)is reversed.

By using Lagrange conditions we can give the following result.

COROLLARY 1. Let all the assumptions of Theorem 3 be satisfied and GL be thegreen function as defined in (6). Also let ψ be n-convex function, then we have thefollowing result:

IfP(x,p;GL(x,s)) � 0, s ∈ [α,β ]

then

P(x,p;ψ(x)) �n

∑j=1

ψ(a j)P(x,p;

n

∏k=1k �= j

( x−ak

a j −ak

)). (14)

By using type (m,n−m) conditions we can give the following result.

COROLLARY 2. Let all the assumptions of Theorem 3 be satisfied and G(m,n) bethe Green function as defined by (9) and τi,ηi be as defined in (7) and (8) respectively.Also let ψ be n-convex function, then we have the following result:

IfP(x,p;G(m,n)(x,s)) � 0, s ∈ [α,β ]

then

P(x,p;ψ(x)) �m−1

∑i=0

ψ(i)(α)P(x,p;τi(x))+n−m−1

∑i=0

ψ(i)(β )P(x,p;ηi(x)).

By using Two-point Taylor conditions we can give the following result.

COROLLARY 3. Let all the assumptions of Theorem 3 be satisfied and G2T be thegreen function as defined by (10). Also let ψ be n-convex function, then we have thefollowing result:

IfP(x,p;G2T (x,s)) � 0, s ∈ [α,β ]


then

P(x,p;ψ(x)) �m−1

∑i=0

m−1−i

∑k=0

(m+k−1

k

)[ψ(i)(α)P

(x,p;

(x−α)i

i!

( x−βα−β

)m( x−αβ−α

)k)

+ψ(i)(β )P(x,p;

(x−β )i

i!

( x−αβ −α

)m( x−βα −β

)k)].

Now we obtain a generalization of Popoviciu’s type inequality for z-tuples.

THEOREM 5. Let in addition to the assumptions of Theorem 3, p = (p1, ..., pz) bea positive z-tuple such that ∑z

u=1 pu = 1 , and ψ : [α,β ] → R be an n-convex function.Assume further that the inequality (13) be satisfied and the function

F(x) =r

∑j=1

k j

∑i=0

ψ(i)(a j)Hi j(x) (15)

is convex. Then we haveP(x,p;ψ(x)) � 0. (16)

Proof. P is a linear functional, so we can rewrite the R.H.S. of (13) in the formP(x, p;F(x)) where F is defined in (15) and will be obtained after reorganization ofthis side. Since F is assumed to be convex, therefore using the given conditions and byfollowing Remark 1, the non negativity of the R.H.S. of (13) is immediate and we have(16) for positive z-tuples. �

3. Bounds for identities related to generalization of Popoviciu’s inequality

In this section we present some interesting results by using Cebysev functionaland Gruss type inequalities. For two Lebesgue integrable functions f ,h : [α,β ] → R ,we consider the Cebysev functional

Δ( f ,h) =1

β −α

∫ β

αf (t)h(t)dt − 1

β −α

∫ β

αf (t)dt.

1β −α

∫ β

αh(t)dt.

The following Gr uss type inequalities are given in [4].

THEOREM 6. Let f : [α,β ]→R be a Lebesgue integrable function and h : [α,β ]→R be an absolutely continuous function with (.−α)(β − .)[h′]2 ∈L[α,β ]. Then we havethe inequality

|Δ( f ,h)| � 1√2[Δ( f , f )]

12

1√β −α

(∫ β

α(x−α)(β − x)[h′(x)]2dx

) 12

. (17)

The constant 1√2


|Δ( f ,h)| � 12(β −α)

|| f ′||∞∫ β

α(x−α)(β − x)dh(x). (18)


In the sequel, we consider above theorems to derive generalizations of the resultsproved in the previous section. In order to avoid many notions let us denote

O(s) = P(x,p;GH(x,s)), s ∈ [α,β ]. (19)

THEOREM 8. Let all the assumptions of Theorem 3 be valid with −∞ < α < β <∞ and α = a1 < a2 . . . < ar = β , (r � 2) be the given points. Moreover, ψ ∈Cn([α,β ])such that ψ(n) is absolutely continuous with (.−α)(β − .)[ψ(n+1)]2 ∈ L[α,β ] . Also letHi j be the fundamental polynomials of the Hermite basis and the functions GH and Obe defined by (5) and (19) respectively. Then

P(x,p;ψ(x)) =r

∑j=1

k j

∑i=0

ψ(i)(a j)P(x,p;Hi j(x))

+ψ(n−1)(β )−ψ(n−1)(α)

(β −α)

∫ β

αO(s)ds+ Kn(α,β ;ψ). (20)

where the remainder Kn(α,β ;ψ) satisfy the bound

|Kn(α,β ;ψ)| � [Δ(O,O)]12

√β −α

2

∣∣∣∣∫ β

α(s−α)(β − s)[ψ(n+1)(s)]2ds

∣∣∣∣12

.

Proof. The proof is similar to the Theorem 9 in [3]. �The following Gruss type inequalities can be obtained by using Theorem 7.

THEOREM 9. Let all the assumptions of Theorem 3 be valid with −∞ < α < β <∞ and α = a1 < a2 . . . < ar = β , (r � 2) be the given points. Moreover, ψ ∈Cn([α,β ])such that ψ(n) is absolutely continuous and let ψ(n+1) � 0 on [α,β ] with O definedin (19) . Then the representation (20) and the remainder Kn(α,β ;ψ) satisfies theestimation

|Kn(α,β ;ψ)| � (β −α)||O′||∞[

ψ(n−1)(β )+ ψ(n−1)(α)2

− ψ(n−2)(β )−ψ(n−2)(α)(β −α)

].

Proof. The proof is similar to the Theorem [10] in [3]. �Now we intend to give the Ostrowski type inequalities related to generalizations

of Popoviciu’s inequality.

THEOREM 10. Suppose all the assumptions of Theorem 3 be satisfied. Moreover,assume (p,q) is a pair of conjugate exponents, that is p,q ∈ [1,∞] such that 1/p+1/q = 1 . Let |ψ(n)|p : [α,β ] → R be a R-integrable function for some n � 2 . Then, wehave ∣∣∣∣∣P(x,p;ψ(x))−

r∑j=1

k j

∑i=0

ψ(i)(a j)P(x,p;Hi j(x))

∣∣∣∣∣� ||ψ(n)||p

(∫ β

α

∣∣∣∣P(x,p;GH(x,s))∣∣∣∣qds

)1/q

. (21)

The constant on the R.H.S. of (21) is sharp for 1 < p � ∞ and the best possible forp = 1 , respectively.

Proof. The proof is similar to the Theorem 11 in [3]. �

REMARK 4. We can give all the above results of this sections for the Lagrangeconditions, Type (m,n−m) conditions, Two-point Taylor conditions.

REMARK 5. Analogous to Section 4 and Section 5 of [3], the n -exponential con-vexity, mean value theorems and related monotonic Cauchy means (along with exam-ples) can be constructed for the functional defined as the difference between the R.H.Sand the L.H.S of the generalized inequality (13).

Acknowledgements. The research of the first and second authors have been sup-ported by Higher Education Commission Pakistan. The research of the third author hasbeen fully supported by Croatian Science Foundation under the project 5435.

RE F ER EN C ES

[1] R. P. AGARWAL, P. J. Y. WONG, Error Inequalities in Polynomial Interpolation and Their Applica-tions, Kluwer Academic Publishers, Dordrecht/Boston/London, 1993.

[2] P. R. BEESACK, On the Green’s function of an N-point boundary value problem, Pacific J. Math. 12(1962), 801–812.

[3] S. I. BUTT, K. A. KHAN AND J. PECARIC, Popoviciu Type Inequalities via Green Function andGeneralized Montgomery Identity, Math. Inequal. Appl. 18 (4) (2015), 1519–1538.

[4] P. CERONE, S. S. DRAGOMIR, Some new Ostrowski-type bounds for the Cebysev functional andapplications, J. Math. Inequal. 8 (1) (2014), 159–170.

[5] L. HORVATH, K. A. KHAN AND J. PECARIC, Combinatorial Improvements of Jensens Inequality /Classical and New Refinements of Jensens Inequality with Applications, Monographs in inequalities8, Element, Zagreb, 2014, pp. 229.

[6] A. YU. LEVIN, Some problems bearing on the oscillation of solutions of linear differential equations,Soviet Math. Dokl., 4 (1963), 121–124.

[7] K. A. KHAN, J. PECARIC AND I. PERIC, Differences of weighted mixed symmetric means and relatedresults, Journal of Inequalities and Applications, 2010, Article ID 289730, 16 pages, (2010).

[8] K. A. KHAN, J. PECARIC AND I. PERIC,Generalization of Popoviciu Type Inequalities for SymmetricMeans Generated by Convex Functions, J. Math. Comput. Sci., 4 (6) (2014), 1091–1113.

[9] C. P. NICULESCU, The Integral Version of Popoviciu’s Inequality, J. Math. Inequal. 3 (3) (2009),323–328.

[10] J. PECARIC, F. PROSCHAN AND Y. L. TONG, Convex functions, Partial Orderings and StatisticalApplications, Academic Press, New York, 1992.


[11] T. POPOVICIU, Sur certaines inegalites qui caracterisent les fonctions convexes, Analele StiintificeUniv. “Al. I. Cuza”, Iasi, Sectia Mat. 11 (1965), 155–164.

[12] P. M. VASIC AND LJ. R. STANKOVIC, Some inequalities for convex functions, Math. Balkanica. 6(44) (1976), 281–288.

(Received March 21, 2016) Saad Ihsan ButtDepartment of Mathematics, COMSATS



Khuram Ali KhanDepartment of Mathematics, University of Sargodha

40100 Sargodha, Pakistane-mail: [email protected]





& Applications


APPLICATIONS OF THE HERMITE–HADAMARD INEQUALITY

MONIKA NOWICKA AND ALFRED WITKOWSKI


Abstract. We show how the recent improvement of the Hermite-Hadamard inequality can beapplied to some (not necessarily convex) planar figures and three-dimensional bodies satisfyingsome kind of regularity.

1. Introduction

The classical Hermite-Hadamard inequality [4] states that for a convex functionf : [a,b] → R

f

(a+b

2

)� 1

b−a

∫ b

af (t)dt � f (a)+ f (b)

2. (1)

Due to its simple and elegant form it became a natural object of investigations. Neumanand Bessenyei [6, 1] proved the version for simplices saying that if Δ ⊂Rn is a simplexwith barycenter b and vertices x0, . . . ,xn and f : Δ → R is convex, then

f (b) � 1Vol(Δ)

∫Δ

f (x)dx � f (x0)+ . . .+ f (xn)n+1

. (2)

The following generalizations for convex function on disk and ball can be found in [3]:If D(O,R) ⊂ R2 is a disk and f : D → R is convex and continuous, then

f (O) � 1πR2

∫∫D(O,R)

f (x,y)dxdy � 12πR

∫∂D(O,R)

f (x,y)ds

and If B(O,R) ⊂ R3 is a ball and f : B → R is convex and continuous, then

f (O) � 34πR3

∫∫∫B(O,R)

f (x,y,z)dxdydz � 14πR2

∫∫∂B(O,R)

f (x,y,z)dS.

The stronger version of the right-hand side of (1) ([9, page 140])

1b−a

∫ b

af (t)dt � 1

2

(f

(a+b

2

)+

f (a)+ f (b)2

)Mathematics subject classification (2010): 26D15, 26B15.Keywords and phrases: Hermite-Hadamard inequality, convex function, polygon, polyhedron, annu-

lus.


1319


1320 M. NOWICKA AND A. WITKOWSKI

also received generalizations for simplices [10], disks, 3-balls and regular n -gons P[2]:

1Vol(Δ)

∫Δ

f (x)dx � 1n+1

f (b)+n

n+1f (x0)+ . . .+ f (xn)

n+1, (3)

1πR2

∫∫D(O,R)

f (x,y)dxdy � 13

f (O)+23· 12πR

∫∂D(O,R)

f (x,y)ds,

34πR3

∫∫∫B(O,R)

f (x,y,z)dxdydz � 14

f (O)+34· 14πR2

∫∫∂B(O,R)

f (x,y,z)dS,

1Area(P)

∫∫P

f (x,y)dxdy � 13

f (O)+2

3Perim(P)

∫∂P

f (x,y)ds.

In this paper we use the lower and upper estimates for the average of a convexfunction over a simplex obtained by the authors in [7, 8]. We provide the alternateproof of the above results and then generalize them to figures and bodies satisfyingsome regularity conditions and to a broader class of functions.

2. Definitions and lemmas

Suppose x0, . . . ,xn ∈ Rn are the vertices of a simplex Δ ⊂ Rn .For a nonempty set K ⊂{0, . . . ,n} we denote by ΔK the simplex conv{xi : i∈K} .For every set K � {0, . . . ,n} we denote by Δ[K] the simplex with vertices

x[K]j =

1n+1 ∑

i∈Kxi +

n+1− kn+1

x j, j ∈ {0, . . . ,n} \K.

We shall denote by hλa the homothety with center a and scale λ , i.e. the mapping

defined by the formulahλ

a (x) = a+ λ (x−a).

By ∂B we shall denote the boundary of the set B .

REMARK 2.1. In the plane the simplices Δ[K] are: the triangle (if K = /0), in-tersection of the triangle and a line parallel to one of its sides and passing through itsbarycenter (if K has one element) and the barycenter itself if K has two elements.

In case of three dimensions we have respectively: the tetrahedron, triangles paral-lel to its faces, segments parallel to its edges; all of them having the same barycenter.

Note that the simplices Δ[K] can be obtained by applying homotheties to the facesof Δ . The details are explained in [7].

If U ⊂ Rk and f : U → R is a Riemann integrable function, then by

Avg( f ,U) =1

Vol(U)

∫U

f (x)dx

APPLICATIONS OF THE HERMITE-HADAMARD INEQUALITY 1321

we shall denote its average value over U . For simplicity of notation if A,B, . . . ,Z arepoints and U = conv{A,B, . . . ,Z} we shall write Avg( f ,AB . . .Z) .

The following results provide the main tools for our investigations:

THEOREM 2.1. ([8]) Suppose f : Δ→R is a convex function and K,L⊂{0, . . . ,n}are disjoint, nonempty sets. Then

Avg( f ,ΔK∪L) � cardKcardK ∪L

·Avg( f ,ΔK)+cardL

cardK ∪L·Avg( f ,ΔL).

THEOREM 2.2. ([7]) If K ⊂ L are proper subsets of {0, . . . ,n} and f : Δ → R isa convex function, then

f (b) � Avg( f ,Δ[L]) � Avg( f ,Δ[K]) � Avg( f ,Δ).

The above theorems were proven by Chen [2] in case Δ is a triangle.In the sequel we shall apply both theorems to some planar and 3-dimensional

bodies.

3. Quadrilateral

Bessenyei in [1] proved that if ABCD is a parallelogram and f is a convex func-tion, then Avg( f ,ABCD) � 1

4( f (A)+ f (B)+ f (C)+ f (D)) .We will try to generalize and improve this result.Consider a quadrilateral ABCD such that the segment AC divides its area evenly

(see Figure 1a).

A

B

C

D

a)

A

B

C

D

O

b)

A

B

C

D

P

Q

Rc)

Figure 1: Quadrilateral with equal halves

We can apply Theorem 2.1 to both triangles ABC and ADC to obtain

Avg( f ,ABC) � 13

f (B)+23

Avg( f ,AC), (4)

Avg( f ,ACD) � 13

f (D)+23

Avg( f ,AC), (5)


which yields

Avg( f ,ABCD) � 13

(f (B)+ f (D)

2+2Avg( f ,AC)

)(6)

� 13

(f (B)+ f (D)

2+ f (A)+ f (C)

).

By adding a midpoint O of the segment AC we can get another upper bound (seeFigure 1b):

Avg( f ,AOB) � 13

f (O)+23

Avg( f ,AB)

Avg( f ,BOC) � 13

f (O)+23

Avg( f ,BC)

Avg( f ,COD) � 13

f (O)+23

Avg( f ,CD)

Avg( f ,DOA) � 13

f (O)+23

Avg( f ,DA)

and since the four triangles have the same area this produces

Avg( f ,ABCD)� f (O)3

+23

Avg( f ,AB)+Avg( f ,BC)+Avg( f ,CD)+Avg( f ,DA)4

(7)

� f (O)3

+23

f (A)+ f (B)+ f (C)+ f (D)4

. (8)

Thus we have proven the following

THEOREM 3.1. Let ABCD be a quadrilateral such that the segment AC dividesit into two triangles of equal area and O be the midpoint of AC. If f : ABCD → R issuch that its restrictions to triangles ABC and ACD are convex, then the inequalities(6), (7) and (8) hold.

To obtain the lower bound we apply Theorem 2.2 to both triangles ABC andADC . By Remark 2.1 we have four reasonable choices for each triangle, so we canproduce 16 inequalities. An example is shown on the Figure 1c: the segments PQ =h2/3C (DA) and QR = h2/3

C (AB) pass through the barycenters of both triangles and there-fore Avg( f ,PQ) � Avg( f ,ACD) and Avg( f ,QR) � Avg( f ,ABC) , which leads to

Avg( f ,PQ)+Avg( f ,QR)2

� Avg( f ,ABCD).

The reader will easily find two other pairs of segments for which Theorem 2.2 can beapplied.

A parallelogram offers more opportunities: firstly, we obtain inequalities (4) and(5) with BD and AC swapped thus obtaining an improvement of Bessenyei’s result


THEOREM 3.2. Let ABCD be a parallelogram with center O and f : ABCD→ R

be such that its restrictions to triangles AOB, BOC, COD and DOA are convex, thenthe inequalities (7) and (8) hold and additionally


min{

f (B)+ f (D)2 +2Avg( f ,AC), f (A)+ f (C)

2 +2Avg( f ,BD)}

� 13

min{

f (B)+ f (D)2 + f (A)+ f (C), f (A)+ f (C)

2 + f (B)+ f (C)}

� f (A)+ f (B)+ f (C)+ f (D)4 .

The fact that the parallelogram can be divided into four triangles of equal areaopens new opportunities. For example we can apply Theorem 2.1 to AOB (and thencyclically to others) as follows:

Avg( f ,AOB) � 13

f (A)+23

Avg( f ,OB).

Summing and taking into account that the point O halves both diagonals we get

THEOREM 3.3. Under the assumption of Theorem 3.2


f (A)+ f (B)+ f (C)+ f (D)4

+23

Avg( f ,AC)+Avg( f ,BD)2

.

The reader will find more estimates applicable to parallelograms and rhombus inSection 4 devoted to polygons.

As above using Theorem 2.2 we can produce 64 different lower bounds. Figure 2illustrates two, probably the most spectacular, inequalities:

THEOREM 3.4. Under the assumption of Theorem 3.2 let A′B′C′D′ = h2/3O (ABCD) ,

KL = h2/3D (AO) , LM = h2/3

D (OC) , PQ = h2/3B (AO) , QR = h2/3

B (OC) . Then the follow-ing inequalities hold (see Figure 2)

Avg( f ,KL)+Avg( f ,LM)+Avg( f ,PQ)+Avg( f ,QR)4

� Avg( f ,ABCD),

A B

CD

K

L

M

R

Q

Pa)

O

A B

CD

A′B′

C′D′

b)

O

Figure 2: Lower bounds for parallelogram


Avg( f ,A′B′)+Avg( f ,B′C′)+Avg( f ,C′D′)+Avg( f ,D′A′)4

� Avg( f ,ABCD).

REMARK 3.1. If f is convex on the parallelogram, then obviously f is convexon all four triangles. The converse does not hold. In both cases one can easily concludethe inequality

14

4

∑k=1

f (Ok) � Avg( f ,ABCD),

where Ok ’s are the barycenters of the triangles. This inequality in case of convex fyields f (O) � Avg( f ,ABCD) . In the case of convexity on triangles only this may notbe true (consider the quadrilateral {(x,y) : |x|+ |y|= 1} and f (x,y) = −|y|).

4. Fans and n -gons

We shall call nice a star-shaped polygon P with vertices A0, . . . ,An−1 satisfyingthe following condition: there exists a point O called center in the kernel of P suchthat all triangles OAkAk+1 , k = 0, . . . ,n− 1 are of the same area. If additionally allsegments AkAk+1 are of the same length, then we shall call it very nice.

Regular n -gons are very nice, but the class of nice and very nice polygons is muchbroader.

THEOREM 4.1. If P is a nice polygon and f : P → R is convex on every triangleformed by its center O and two consecutive vertices, then

Avg( f ,P) � 13

f (O)+23· 1n

n−1

∑k=0

Avg( f ,AkAk+1), (9)

Avg( f ,P) � 13· 1n

n−1

∑k=0

f (Ak)+23· 1n

n−1

∑k=0

Avg( f ,OAk), (10)

Avg( f ,P) � 13

f (O)+23· 1n

n−1

∑k=0

f (Ak). (11)

If additionally A′k = h2/3

O (Ak) for k = 0, . . . ,n−1 , then

1n

n−1

∑k=0

Avg( f ,A′kA

′k+1) � Avg( f ,P) (12)

(see Figure 3).

Proof. To obtain (9) apply Theorem 2.1 to vertex O and side AkAk+1 , then add upthe inequalities. Similarly, for (10) use vertex Ak and side OAk+1 . The inequality (11)can be obtained from (9) or (10) by applying standard Hermite-Hadamard inequalities.

Finally (12) is consequence of Theorem 2.2. �

REMARK 4.1. Every triangle is a nice polygon with its barycenter as O .

REMARK 4.2. In case of a very nice polygon, the inequalities (9) and (12) can bewritten as

Avg( f ,∂h2/3O (P)) � Avg( f ,P) � 1

3f (O)+

23

Avg( f ,∂P).

Suppose n is even. Then we can group the triangles in pairs to get

Avg( f ,OAkAk+1) � 13

f (Ak+1)+23

Avg( f ,OAk)

Avg( f ,OAk−1Ak) � 13

f (Ak−1)+23

Avg( f ,OAk).

This shows that the following result holds true.

THEOREM 4.2. Under assumptions of Theorem 4.1 if the number of vertices of Pis even, then

Avg( f ,P) � 13

1n/2 ∑

k odd

f (Ak)+23

1n/2 ∑

k even

Avg( f ,OAk),

Avg( f ,P) � 13

1n/2 ∑

k even

f (Ak)+23

1n/2 ∑

k odd

Avg( f ,OAk),

(see Figure 3).

Theorem 4.1 (9) Theorem 4.1 (10) Theorem 4.1 (12) Theorem 4.2

Figure 3: Hermite-Hadamard inequalities for n-gon

REMARK 4.3. Note that every nice n -gon can be considered a nice 2n -gon byadding the midpoints of its sides to the set of vertices (see Figure 3).

Remark 3.1 remains valid also in this case.

A fan is a polygon with vertices O,A1, . . . ,An such that all triangles OAkAk+1 ,k = 1, . . . ,n− 1 are of the same orientation and ∑n−1

k=1 ∠AkOAk+1 < 2π . For nice fanswe obtain similar results as for nice n -gons. We encourage the reader to formulate anequivalent of Theorems 4.1 and 4.2.

5. Annulus

In [2] the following version of Hermite-Hadamard inequality can be found

THEOREM 5.1. ([2], Th. 2.2) Let U be a convex subset of a plane, and D⊂U bean annulus with radii r < R and C(s) denotes a co-centric circle with radius s, thenfor a convex function f : U → R hold

Avg(

f ,C(

2(r2+rR+R2)3(r+R)

))� Avg( f ,D)

and

Avg( f ,D) � 2r+R3(r+R)

Avg( f ,C(r))+r+2R

3(r+R)Avg( f ,C(R)).

We shall improve this result. For n > 4 let Ank , k = 0, . . . ,n− 1 be the vertices

of a regular n -gon inscribed in C(R) and Bnk , k = 0, . . . ,n− 1 be the vertices of a

regular n -gon inscribed in C(r) and rotated anticlockwise by πn . The two polygons

bound the area Dn , and divide it into 2n isosceles triangles K nk = An

kAnk+1B

nk and

L nk = Bn

kBnk+1A

nk+1 . We have

AreaK nk = Rsin

πn

(Rcos

πn− r)

, AreaL nk = r sin

πn

(R− rcos

πn

), (13)

AreaDn =n−1

∑k=0

(AreaK nk +AreaL n

k ) = nsinπn

cosπn

(R2− r2). (14)

Denote by Knk ,Ln

k the barycenters of K nk and L n

k . Applying the Hermite-Hadamardinequality, equations (13) and (14) we obtain

Avg( f ,Dn) =1

AreaDn

n−1

∑k=0

(∫K n

k

f (x)dx+∫L n

k

f (x)dx)

=n−1

∑k=0

(AreaK n

k

AreaDnAvg( f ,K n

k )+AreaL n

k

AreaDnAvg( f ,L n

k ))

�R(Rcos π

n − r)

cos πn (R2− r2)

1n

n−1

∑k=0

f (Knk )+

r(R− rcos π

n

)cos π

n (R2− r2)1n

n−1

∑k=0

f (Lnk). (15)

As n tends to infinity the two regular polygons with vertices Knk and Ln

k respectivelyapproach the circles of radii 1

3 (2R+r) and 13 (R+2r) , and the arithmetic means in (15)

tend to averages of f over these circles, so we have proven the following fact.

THEOREM 5.2. Under the assumptions of Theorem 5.1 the inequality

Rr+R

Avg

(f ,C

(r+2R

3

))+

rr+R

Avg

(f ,C

(2r+R

3

))� Avg( f ,D)

holds.


Similar reasoning and the strengthened version of the Hermite-Hadamard inequal-ity (3) applied to K n

k and L nk produce a better right bound.

THEOREM 5.3. Under the assumptions of Theorem 5.1 the inequality

Avg( f ,D) � 13

[R

r+RAvg

(f ,C

(r+2R

3

))+

rr+R

Avg

(f ,C

(2r+R

3

))]

+23

[r+2R

3(r+R)Avg( f ,C(R))+

2r+R3(r+R)

Avg( f ,C(r))]

holds.

REMARK 5.1. Suppose the function f is such there exist a point O ∈U and half-lines starting from O such that f is convex in each sector bounded by them. Then,if O is the center of D , the inequalities in (15) are valid for all triangles except theseintersecting with the sectors’ boundaries. Thus they can be neglected as n tends toinfinity, and the Theorems 5.2 and 5.3 remain valid for f .

6. Platonic bodies and related polytopes

Let B ⊂ R3 be a platonic body inscribed in a sphere with center O . Define thefollowing sets:

• S – set of segments joining O with vertices of B

• O – set of segments joining O with centers of faces

• E – set of edges of B

• D – set of segments joining centers of faces with their vertices.

THEOREM 6.1. Let f : B→R be a function such that its restriction to every pyra-mid formed by a face as a base and O as its apex is convex. Then with the abovenotation the following inequalities hold:

Avg( f ,B) � 14

f (O)+34

Avg( f ,∂B), (16)

Avg( f ,B) � 12

Avg( f ,O)+12

Avg( f ,E ), (17)

Avg( f ,B) � 12

Avg( f ,S )+12

Avg( f ,D), (18)

and

Avg( f ,∂h3/4O (B)) � Avg( f ,B). (19)


Proof. Let F be a face of B with vertices A0, . . . ,An−1 and O′ be its circumcenter.Split the pyramid FO into simplices OO′AkAk+1 . Applying Theorem 2.1 we obtain

Avg( f ,OO′AkAk+1) � 14

f (O)+34

Avg( f ,O′AkAk+1).

Summing over k and F we obtain (16). The inequalities


Avg( f ,OO′)+12

Avg( f ,AkAk+1)

lead to (17), while


Avg( f ,OAk)+12

Avg( f ,O′Ak+1)

give (18). Finally Theorem 2.2 leads to the inequalities

Avg( f ,h3/4O (O′AkAk+1)) � Avg( f ,OO′AkAk+1)

that finally yield (19). �Based on a platonic body B we can build a new polytope B∗ in the following way:

on every face F of B we build or excavate a regular pyramid of the same height withapex OF . If we denote by

• S – set of segments joining O with vertices of B ,

• O∗ – set of segments joining O with OF ’s,

• E – set of edges of B ,

• D∗ – set of segments joining OF ’s of the pyramids with vertices of F ,

then the same reasoning as above shows that the next theorem is valid.

THEOREM 6.2. Let f : B∗ → R be a function such that its restriction to everytetrahedron formed by O, OF and two adjacent vertices of F . Then with the abovenotation the following inequalities hold:

Avg( f ,B∗) � 14

f (O)+34

Avg( f ,∂B∗),

Avg( f ,B∗) � 12

Avg( f ,O∗)+12

Avg( f ,E ),

Avg( f ,B∗) � 12

Avg( f ,S )+12

Avg( f ,D∗),

and

Avg( f ,∂h3/4O (B∗)) � Avg( f ,B∗).


7. Dipyramid and dicone

Let P be a regular, convex n -gon with vertices A0, . . . ,An−1 . Suppose X is apoint on the line l perpendicular to the plane containing P and passing through itscenter. The set UX =

⋃n−1k=0 XAkAk+1 will be called a Chinese umbrella with vertex X .

The umbrella’s scaffold will be denoted by SX =⋃n−1

k=0 XAk .By a dipyramid with vertices O0,O1 we mean the body D bounded by two Chi-

nese umbrellas UO0 and UO1 . The dipyramid may be convex or not, depending on theposition of its vertices with respect to the plane of the polygon.

The following result is a consequence of Theorems 2.1 and 2.2.

THEOREM 7.1. Let D be a dipyramid with vertices O0 and O1 , and let f : D →R be a function that is convex on every simplex O0O1AkAk+1, k = 0, . . . ,n−1 . Then

Avg( f ,D) � 14 f (O0)+ 3

4 Avg( f ,UO1 ), (20)

Avg( f ,D) � 14 f (O1)+ 3

4 Avg( f ,UO0 ), (21)

Avg( f ,h3/4O0

(UO1)) � Avg( f ,D), (22)

Avg( f ,h3/4O1

(UO0)) � Avg( f ,D), (23)

Avg( f ,D) � 12 (Avg( f ,O0O1)+Avg( f ,∂P)) , (24)

Avg( f ,D) � 12

(Avg( f ,SO0 )+Avg( f ,SO1 )

), (25)

(see Figure 4).

Proof. Grouping the vertices of the simplex O0O1AkAk+1 into {O0},{O1AkAk+1}one gets the inequalities (20) and (22). Similar split {O1},{O0AkAk+1} gives (21) and(23).

Inequalities (24) and (25) follow by grouping them into {O0O1} , {AkAk+1} and{O0Ak} , {O1Ak+1} respectively. �

O0

O1

Theorem 7.1 (21)

O0

O1

Theorem 7.1 (23)

O0

O1

Theorem 7.1 (24)

O0

O1

Theorem 7.1 (25)

Figure 4: Hermite-Hadamard inequalities for dipyramid

Denote by ηi , i = 0,1 the angle between the line l and the plane of a side of UOi

and for 0 < s < 1 let Os = (1− s)O0 + sO1 . The umbrella UOs splits the dipyramidD into two dipyramids D0

s and D1s with vertices O0,Os and O1,Os respectively. It is

clear, thatVol(D0

s ) = sVol(D) and Vol(D1s ) = (1− s)Vol(D). (26)

Note also thatArea(UO0)Area(UO1)

=sinη1

sinη0

which gives

Area(∂D) =sinη0 + sinη1

sinη1Area(UO0) =

sinη0 + sinη1

sinη0Area(UO1). (27)

Now we are ready to generalize the results of Theorem 7.1.

THEOREM 7.2. Under the assumptions of Theorem 7.1 for all 0 < s < 1 the in-equalities

Avg( f ,D) � 14 f (Os)+ 3

4

(sAvg( f ,UO0 )+ (1− s)Avg( f ,UO1 )

), (28)

sAvg( f ,h3/4Os

(UO0 ))+ (1− s)Avg( f ,h3/4Os

(UO1 )) � Avg( f ,D), (29)

Avg( f ,∂h3/4Os

(D)) � sinη1 Avg( f ,D0s )

sinη0 + sinη1+

sinη0 Avg( f ,D1s )

sinη0 + sinη1(30)

are valid.

Proof. To prove (28) we apply inequality (20) to D0s and to D1

s and obtain

Avg( f ,D0s ) � 1

4 f (Os)+ 34 Avg( f ,UO0 ),

Avg( f ,D1s ) � 1

4 f (Os)+ 34 Avg( f ,UO1 ).

Now we multiply the first inequality by s , the second by 1−s and add up both inequal-ities taking into account equalities (26).

The proof of (29) is similar but uses (22) and (23).And finally from (22), (23) and (27) (which remains valid for homothetic images

also) it follows that

sinη0 + sinη1

sinη1−i

1

Area(∂h3/4Os

(D))

∫h3/4Os

(UOi)f (x)dx � Avg( f ,D i

s), i = 0,1.

We complete the proof by dividing these inequalities by the first term and adding side

by side, because ∂h3/4Os

(D) = h3/4Os

(UO0)∪h3/4Os

(UO1) . �


COROLLARY 7.3. Let s∗ = sinη0sinη0+sinη1

. The equations (27), (26) and Theorem7.2 imply that

Avg( f ,D) � 14 f (Os∗)+ 3

4 Avg( f ,∂D),

Avg( f ,∂h3/4Os∗ (D)) � Avg( f ,D).

With n growing to infinity our dipyramids approximate a dicone, where the n -gonP gets replaced by a circle of radius R . All formulas (20)–(24), (28)–(30) remain valid,while the formula (25) needs a modification.

Let us introduce a coordinate system in the most natural way (center O at thecenter of P , z-axis along the line l and x -axis along OA0 ). Then we have

Avg( f ,SO0) =1

n|O0A0|n−1

∑k=0

∫O0Ak

f (x,y,z)dl

with x = rcosϕ , y = r sinϕ , z = (R− r)cotη0, 0 � r � R

=1nR

n−1

∑k=0

∫ R

0f(rcos 2πk

n ,r sin 2πkn ,(R− r)cotη0

)dr

→ 12πR

∫ R

0

∫ 2π

0f (rcosϕ ,r sinϕ ,(R− r)cotη0) dϕ dr

=1

2πR

∫∫x2+y2�R2

f(x,y,(R−

√x2 + y2)cotη0

) 1√x2 + y2

dxdy

=sinη0

2πR

∫∫UO0

f (x,y,z)√x2 + y2

dS.

Therefore the following result holds.

THEOREM 7.4. If f is a convex function defined on a dicone D , and g(x) =f (x)

dist(x,l) (dist(x, l) denotes the distance from x to the line l ), then

Avg( f ,D) � R4

(Avg(g,UO0)+Avg(g,UO1)

).

8. Cube

A cube being a Platonic body enjoys all properties discussed in Section 6. In thissection we present a handful of other applications of Theorems 2.1 and 2.2.

THEOREM 8.1. Let C be a cube, and f : C → R be a function convex on everypyramid formed by a face and the center O of the cube. Fix two opposite vertices of acube and let P be a set being the sum of six diagonals of faces meeting at these vertices.Let Q be the set of three main diagonals joining the remaining six vertices. Then

Avg( f ,C ) � 12

Avg( f ,P)+12

Avg( f ,Q).


Proof. Let ABCD be the face containing diagonal AC in P . The pyramid ABCDOis the sum of two simplices ABCO and ACDO . Splitting vertices of each of them intogroups {AC},{BO} and {AC},{DO} respectively one gets

Avg( f ,ABCO) � 12

(Avg( f ,AC)+Avg( f ,BO))

Avg( f ,ACDO) � 12

(Avg( f ,AC)+Avg( f ,DO))

which gives

2Avg( f ,ABCDO) � Avg( f ,AC)+12(Avg( f ,BO)+Avg( f ,DO)).

We complete the proof in usual way, applying the same process to all diagonals inP . �

THEOREM 8.2. Let C be a cube, and f : C → R be a function convex on everypyramid formed by a face and the center O of the cube. Fix two opposite vertices of Cand let S be a set being the sum of edges meeting at these vertices. Let Q be the set ofthree main diagonals joining the remaining six vertices. Then

Avg( f ,C ) � 12

Avg( f ,S)+12

Avg( f ,Q).

Proof. The proof goes exactly the same way as the previous one, but this time wesplit the vertices of simplices into groups {CO},{AB} and {CO},{AD} respectivelywhich leads to

2Avg( f ,ABCDO) � Avg( f ,OC)+12(Avg( f ,AB)+Avg( f ,AD)). �

The cube can be split into six simplices of equal volumes in different ways. Oneof them is particularly interesting – we shall call it a diagonal split. Select two oppositevertices, say O1 and O2 . The remaining vertices can be connected by edges of the cubeso that they form a closed polygonal line L =V0 . . .V5V0 . The diagonal split consists ofsix simplices O1O2VkVk+1 . Note that O1Vk and O2Vk+1 are of the same length – theyare both edges of the cube or diagonals of its faces. The next theorem shows how thediagonal split can be explored.

THEOREM 8.3. Consider a diagonal split of a cube C . Denote by S the set of sixedges adjacent to O1 and O2 and by P the set of six diagonals of faces adjacent to O1

and O2 . If f : C → R is convex on each simplex of the split, then

Avg( f ,C ) � 12

Avg( f ,O1O2)+12

Avg( f ,L), (31)

Avg( f ,C ) � Avg( f ,P), (32)

Avg( f ,C ) � Avg( f ,S), (33)

(see Figure 5).


Proof. To prove (31) use Theorem 2.1 dividing the vertices of O1O2VkVk+1 intogroups {O1O2} and {VkVk+1} . Two other splits lead to (32) and (33). �

O1

O2

Theorem 8.3 (31)

O1

O2

Theorem 8.3 (33)

O1

O2

Theorem 8.3 (32)

Figure 5: Hermite-Hadamard inequalities for cube

Next theorem presents an interesing asymmetric case:

THEOREM 8.4. Let A be a vertex of the cube C and let S be the set consisting ofits faces nonadjacent to A. If f : C → R is convex, then

Avg( f ,C ) � 14

f (A)+34

Avg( f ,S), (34)

Avg( f ,h3/4A (S)) � Avg( f ,C ), (35)

(see Figure 6).

We leave the obvious proof to the reader.

A

Theorem 8.4 (34) Theorem 8.4 (35)

Figure 6: Hermite-Hadamard inequalities for cube ctd.


RE F ER EN C ES

[1] M. BESSENYEI, The Hermite-Hadamard Inequality on Simplices, Amer. Math. Monthly 115, 4(2008), 339–345.

[2] Y. CHEN, Multi-dimensional Hadamard’s inequalities, Tamkang J. Math. 43, 1 (2012), 1–10.[3] S. S. DRAGOMIR AND C. E. M. PEARCE, Selected topics on Hermite-Hadamard inequalities and

applications, RGMIA Monographs, Victoria University, Melbourne (2000), http://ajmaa.org/RGMIA/monographs.php.

[4] J. HADAMARD, Etude sur les proprietes des fonctions entieres et en particulier d’une fonction con-sideree par Riemann, J. Math. Pures Appl. 58, (1893), 171–215.

[5] P. C. HAMMER, The midpoint method of numerical integration, Math. Mag. 31, (1958), 193–195.[6] E. NEUMAN, Inequalities involving multivariate convex functions II, Proc. Amer. Math. Soc. 109, 4

(1990), 965–974.[7] M. NOWICKA AND A. WITKOWSKI, A refinement of the left-hand side of Hermite-Hadamard in-

equality for simplices, J. Inequal. Appl. (2015) 2015:373, DOI 10.1186/s13660-015-0904-0[8] M. NOWICKA AND A. WITKOWSKI, A refinement of the right-hand side of the Hermite-Hadamard

inequality for simplices, Aequationes Math. (2016) DOI: 10.1007/s00010-016-0433-z[9] J. PECARIC, F. PROSCHAN AND Y. L. TONG, Convex Functions, Partial Orderings, and Statistical

Applications, Academic Press, Inc., 1992.[10] S. WASOWICZ AND A. WITKOWSKI, On some inequality of Hermite-Hadamard type, Opuscula

Math. 32, 3 (2012), 591–600.

(Received March 22, 2016) Monika NowickaInstitute of Mathematics and Physics

UTP University of Science and Technologyal. prof. Kaliskiego 7

85-796 Bydgoszcz, Polande-mail: [email protected]

Alfred WitkowskiInstitute of Mathematics and Physics

UTP University of Science and Technologyal. prof. Kaliskiego 7

85-796 Bydgoszcz, Polande-mail: [email protected]



& Applications


ON CONVERGENCE PROPERTIES OF GAMMA–STANCU

OPERATORS BASED ON q–INTEGERS

OZGE DALMANOGLU AND MEDIHA ORKCU


Abstract. In this paper we introduce Stancu type generalization of Gamma operators based onthe concept of q -integers. We first establish local approximation theorems for these operators.Next, we investigate the weighted approximation properties and give an estimate for the rateof convergence using classical modulus of continuity. Lastly, we obtain a Voronovskaya typetheorem.

1. Introduction

Let f be a function defined on [0,∞) and satisfies the growth condition:

f (t) � Meβ t (M � 0; β � 0; t → ∞).

In 2005, Zeng [20] defined the following Gamma operators

Gn ( f ;x) =1

xnΓ(n)

∞∫0

f( t

n

)tn−1e−

tx dt, x > 0, (1)

for functions satisfying exponential growth condition. He studied the approximationproperties of these operators to the locally bounded functions and the absolutely con-tinuous functions. One of the important generalizations of the Gamma operators is dueto Mazhar [13], namely

Fn( f ;x) :=∞∫

0

∞∫0

gn(x,u)gn−1(u,t) f (t)dudt =(2n)!xn+1

n!(n−1)!

∞∫0

tn−1

(x+ t)2n+1 f (t)dt,

where gn(x,u) = xn+1

n! e−xuun , n > 1, x > 0. In [8] Karslı considered the followingGamma type linear and positive operators

Ln( f ;x) :=∞∫

0

∞∫0

gn+2(x,u)gn(u,t) f (t)dudt =(2n+3)!xn+3

n!(n+2)!

∞∫0

tn

(x+ t)2n+4 f (t)dt, (2)

Mathematics subject classification (2010): 41A25, 41A36.Keywords and phrases: Gamma operators, q -calculus, rate of convergence, weighted approximation,

Voronovskaya type theorem.


1335


1336 O. DALMANOGLU AND M. ORKCU

for x > 0, and obtained some approximation results. Karslı, Gupta and Izgi [9] gavean estimate for the rate of convergence of these operators on a Lebesgue point of afunction f of bounded variation defined on the interval (0,∞) . In [10], Karslı andOzarslan also gave some local and global approximation results for the same operators.

In the last two decades q -calculus are intensively used in the area of approximationtheory. Pioneer work is due to Lupas [12] and Phillips [14], as they introduce the q -analogue of well-known Bernstein polynomials. After q -Bernstein operators, severaloperators’ q -generalizations are defined and approximation properties are investigated.We now recall some concepts from q -calculus. Details can be found in [7].

For any real number q > 0, the q -integer and the q -factorial of a nonnegativeinteger k are defined as

[k]q := [k] =

⎧⎨⎩

1−qk

1−q, q �= 1

k, q = 1.

[k]q! := [k]! ={

[k] [k−1] . . . [1] , k = 1,2, . . .1, k = 0

respectively. For the integers n and k, the q -binomial coefficients are also defined as[nk

]q:=[

nk

]=

[n]![k]! [n− k]!

(n � k � 0).

The q -integral and the q -improper integral are defined as

a∫0

f (x)dqx = (1−q)∞

∑j=0

aq j f (aq j).

∫ ∞/A

0f (x)dqx = (1−q)

∞

∑n=−∞

f

(qn

A

)qn

A, A > 0, (3)

respectively, provided the sums converge absolutely. The classical exponential functionex has the following two q -analogues:

eq(x) =∞

∑k=0

xk

[k]!=

1(1− (1−q)x)∞

q

Eq(x) =∞

∑k=0

qk(k−1)/2 xk

[k]!= (1+(1−q)x)∞

q , (4)

where (1+ x)∞q =

∞∏j=0

(1 + q jx). The q -Gamma function was introduced by Thomae

[18] and later by Jackson [6] as the infinite product

Γq(t) = (1−q)1−t∞

∏n=0

1−qn+1

1−qn+t .

ON CONVERGENCE PROPERTIES OF GAMMA-STANCU OPERATORS BASED ON q -INTEGERS 1337

The integral representation of q -Gamma function [15] is given by

Γq(t) =

11−q∫0

xt−1Eq(−qx)dqx (5)

which satisfies the functional equations

Γq(t +1) = [t]Γq(t), t > 0, Γq(1) = 1,

Γq(t) = [t−1]!, t > 0. (6)

Note that (5) can also be rewritten via an improper integral as

Γq(t) =

∞/1−q∫0

xt−1Eq(−qx)dqx (7)

since Eq

(− qn

1−q

)= 0 for n � 0. For more detailed information, see [1] and [16].

As the q -generalizations of linear positive operators have been introduced by sev-eral authors, similar studies are also valid for the Gamma type operators. In [4], Caiintroduced a q -analogue of the Gamma operators defined by (1) as

Gn,q ( f ;x) =1

xnΓq (n)

∫ ∞/A

0f

(t

[n]

)tn−1Eq

(−qt

x

)dqt (8)

and investigated the approximation properties of these operators. In [3], Cai and Zengintroduced a q -generalization of gamma type operators given by (2) using the con-cept of q -integral. They estimated the rate of convergence and examined the weightedapproximation properties. Later then Zhao et. al. [21] proposed the Stancu type gen-eralization of the same q -Gamma operators and studied similar concepts. Stancu [17]was the first to modify Bernstein operators as

Pα ,βn ( f ;x) =

n

∑k=0

(nk

)xk(1− x)n−k f

(k+ αn+ β

)

for x ∈ [0,1] and α,β are any numbers satisfying 0 � α � β . For more studies onq -Gamma type operators see also [11] and references therein. Another operator that isrelated to our study is the Post-Widder operators. Unal et. al. [19] studied the statisticalapproximation properties of real and complex Post-Widder operators based on the q -integers. Recently Aydın et. al. [2] also introduced a generalization of q -Post-Widderoperators and studied approximation properties.

In the present paper, we introduce the Stancu type modification of the q -Gammaoperators. We study the approximation theorems for these operators. Local approxima-tion theorems, weighted approximation and rate of convergence results are investigated.Voronovskaya type theorem is also obtained in the last section.

2. Construction of the operators

By CB(0,∞), we denote the space of real valued functions defined on (0,∞) whichare bounded and continuous with the norm ‖ f‖ = sup

x>0| f (x)| .

We define the Stancu type generalization of the q -Gamma operators for n ∈ N,0 < q < 1 as

Gα ,βn,q ( f ;x) =

1xnΓq (n)

∫ ∞/ 1−qx

0f

(t + α[n]+ β

)tn−1Eq

(−qt

x

)dqt, x > 0. (9)

where α and β are two numbers satisfying 0 � α � β . It is easy to check that (9)is linear and positive. By putting α = β = 0, the operators reduces to the Gammaoperators defined by (8) with A = 1−q

x .We first give the following Lemma in order to investigate the approximation theo-

rems in the proceeding sections.

REMARK 1. Note that we take 1−qx instead of A in the definition of the improper

integral, so that we are able to get the function Γq (n) after making the change of vari-able when finding the test functions. We have to be careful when writing the upperlimit of the integral after making the change of variable as it differs from the ordinarycalculus.

LEMMA 1. For q ∈ (0,1) , x ∈ (0,∞) , we have the following test functions forthe operator defined in (9).

Gα ,βn,q (1;x) = 1

Gα ,βn,q (t;x) =

[n][n]+ β

x+α

[n]+ β

Gα ,βn,q

(t2;x)

=[n]2

([n]+ β )2

(1+

qn

[n]

)x2 +

2 [n]α([n]+ β )2

x+α2

([n]+ β )2

Gα ,βn,q

(t3;x)

=[n]3

([n]+ β )3

(1+

qn (2+q)[n]

+[2]q2n

[n]2

)x3

+3 [n]2 α

([n]+ β )3

(1+

qn

[n]

)x2 +

3 [n]α2

([n]+ β )3x+

α3

([n]+ β )3

Gα ,βn,q

(t4;x)

=[n]4

([n]+β )4

(1+

(1+[2]+[3])qn

[n]+

([2]+[3]+[2] [3])q2n

[n]2+

[2] [3]q3n

[n]3

)x4

+4 [n]3 α

([n]+ β )4

(1+

qn (2+q)[n]

+[2]q2n

[n]2

)x3

+6 [n]2 α2

([n]+ β )4

(1+

qn

[n]

)x2 +

4 [n]α3

([n]+ β )4x+

α4

([n]+ β )4.

ON CONVERGENCE PROPERTIES OF GAMMA-STANCU OPERATORS BASED ON q-INTEGERS 1339

Proof. We prove the first equality by making the change of variable t = ux .

Gα ,βn,q (1;x) =

1xnΓq (n)

∫ ∞/ 1−qx

0tn−1Eq

(−qt

x

)dqt

=1

xnΓq (n)

∫ ∞/1−q

0un−1xn−1Eq (−qu)xdqu

= 1.

For α = 0, β �= 0, and k = 0,1,2, . . . , using the identity (6), we can write

G0,βn,q

(tk;x)

=1

xnΓq (n)

∫ ∞/ 1−qx

0

(t

[n]+ β

)k

tn−1Eq

(−qt

x

)dqt

=1

xnΓq (n)

∫ ∞/1−q

0

ukxk

([n]+ β )k un−1xn−1Eq (−qu)xdqu

=xkΓq (n+ k)

([n]+ β )kΓq (n)=

[n+ k−1]![n−1]!

xk

([n]+ β )k .

Now in the light of the above equality, we can write Gα ,βn,q(tk;x)

in terms of G0,βn,q(tk;x)

as,

Gα ,βn,q

(tk;x)

=1

xnΓq (n)

∫ ∞/ 1−qx

0

(t + α[n]+ β

)k

tn−1Eq

(−qt

x

)dqt

=k

∑j=0

(kj

)(α

[n]+ β

) j 1xnΓq (n)

∫ ∞/ 1−qx

0

tk− j

([n]+ β )k− j tn−1Eq

(−qtx

)dqt

=k

∑j=0

(kj

)(α

[n]+ β

) j

G0,βn,q

(tk− j;x

).

Using the identity [n+ k] = [n]+qn[k] , for k � 0, one can obtain the desired equalitiesafter simple calculations. �

COROLLARY 1. For every q ∈ (0,1) , x ∈ (0,∞) , we have

Gα ,βn,q (t− x;x) =

([n]

[n]+ β−1

)x+

α[n]+ β

Gα ,βn,q

((t− x)2 ;x

)=

([n]2

([n]+ β )2

(1+

qn

[n]

)− 2 [n]

[n]+ β+1

)x2

+2α

([n]

([n]+ β )2− 1

[n]+ β

)x+

α2

([n]+ β )2(10)


Gα ,βn,q

((t− x)4 ;x

)=

{[n]4

([n]+ β )4+

[n]3

([n]+ β )4(1+[2]+ [3])qn

+[n]2 ([2]+ [3]+ [2] [3])q2n

([n]+ β )4

+[n] [2] [3]q3n

([n]+ β )4− 4 [n]3

([n]+ β )3

(1+

qn (2+q)[n]

+[2]q2n

[n]2

)

+6[n]2

([n]+ β )2

(1+

qn

[n]

)−4

[n][n]+ β

+1

}x4

+

{4 [n]3 α

([n]+ β )4

(1+

qn (2+q)[n]

+[2]q2n

[n]2

)

− 12 [n]2 α([n]+ β )3

(1+

qn

[n]

)+

12 [n]α([n]+ β )2

− 4α[n]+ β

}x3

+

{6 [n]2 α2

([n]+ β )4

(1+

qn

[n]

)− 12 [n]α2

([n]+ β )3+

6α2

([n]+ β )2

}x2

+

{4 [n]α3

([n]+ β )4− 4α3

([n]+ β )3

}x+

α4

([n]+ β )4(11)

THEOREM 1. Let q = (qn) be a sequence satisfing

(qn) ∈ (0,1), limn→∞

qn = 1 and limn→∞

qnn = a, a �= 0. (12)

For each f ∈CB(0,∞) , the sequence(Gα ,β

n,qn ( f ; .))

converges uniformly to the function

f on every compact subset of (0,∞).

Proof. Let limn→∞

qn = 1. Since limn→∞

qnn = a , a �= 0, we see that

Gα ,βn,qn

(1; .) ⇒ 1,

Gα ,βn,qn

(t; .) ⇒ x,

Gα ,βn,qn

(t2; .)

⇒ x2

from Lemma 1. Therefore from the well known Korovkin’s Theorem we get the desiredresult. �

3. Local approximation

Recall that the first-order and second-order modulus of continuities of the functionf ∈ CB(0,∞) is defined by for δ > 0

w( f ;δ ) = sup{| f (x+h)− f (x)| : x > 0, 0 � h � δ} ,

w2( f ;δ ) = sup{| f (x+2h)−2 f (x+h)+ f (x)| : x > 0, 0 � h � δ} .

The Peetre’s K-functional of the function f ∈ CB(0,∞) is defined by

K2( f ;δ ) = infg∈C2

B(0,∞)

{‖ f −g‖+ δ∥∥g′′∥∥} .

Here C2B(0,∞) is the space of functions f such that f , f ′, f ′′ ∈ CB(0,∞). The norm

on C2B is defined as

‖g‖C2B

= ‖g‖CB+∥∥g′∥∥CB

+∥∥g′′∥∥CB

.

It is known that there exists a positive constant C > 0 such that

K2( f ;δ ) � Cw2( f ;√

δ ). (13)

LEMMA 2. For f ∈CB(0,∞) one has∣∣∣Gα ,βn,q ( f ;x)

∣∣∣� ‖ f‖ .

Proof. The proof follows from the linearity of the operator Gα ,βn,q and from the

first identity of Lemma 1. �

Here is our direct local approximation theorem for the operators Gα ,βn,q .

THEOREM 2. Let f ∈CB(0,∞) and 0 < q < 1. For each x ∈ (0,∞)∣∣∣Gα ,βn,q ( f ;x)− f (x)

∣∣∣� Cw2( f ;√

δn,q(x))+w( f ;μn,q(x))

for some positive constant C, where

δn,q(x) = Gα ,βn,q

((t − x)2;x

)+(

[n][n]+ β

x+α

[n]+ β− x

)2

and

μn,q(x) =∣∣∣∣ [n][n]+ β

x+α

[n]+ β− x

∣∣∣∣Proof. For x∈ (0,∞) consider the following auxiliary operator Gα ,β

n,q ( f ;x) definedby

Gα ,βn,q ( f ;x) = Gα ,β

n,q ( f ;x)+ f (x)− f

([n]

[n]+ βx+

α[n]+ β

). (14)


From Corollary 1, Gα ,βn,q reproduce linear functions, i.e.

Gα ,βn,q (t − x;x) = 0.

Let x ∈ (0,∞) and g ∈C2B(0,∞). By Taylor’s Theorem we have

g(t) = g(x)+g′(x)(t − x)+t∫

x

(t −u)g′′(u)du.

Applying Gα ,βn,q to the both side of the above equality, we get

Gα ,βn,q (g(t);x)−g(x) = Gα ,β

n,q

⎛⎝ t∫

x

(t−u)g′′(u)du;x

⎞⎠

= Gα ,βn,q

⎛⎝ t∫

x

(t−u)g′′(u)du;x

⎞⎠

−

⎛⎜⎜⎝

[n][n]+β x+ α

[n]+β∫x

([n]

[n]+ βx+

α[n]+ β

−u

)g′′(u)du

⎞⎟⎟⎠

∣∣∣Gα ,βn,q (g;x)−g(x)

∣∣∣ � ∥∥g′′∥∥⎧⎨⎩Gα ,β

n,q

⎛⎝ t∫

x

(t −u)du;x

⎞⎠

+

⎛⎜⎜⎝

[n][n]+β x+ α

[n]+β∫x

∣∣∣∣ [n][n]+ β

x+α

[n]+ β−u

∣∣∣∣du

⎞⎟⎟⎠⎫⎪⎪⎬⎪⎪⎭

�∥∥g′′∥∥

{Gα ,β

n,q

((t − x)2;x

)+(

[n][n]+ β

x+α

[n]+ β− x

)2}

= δn,q(x)∥∥g′′∥∥

On the other hand from (14) and Lemma 2, we have

∣∣∣Gα ,βn,q ( f ;x)

∣∣∣ �∣∣∣Gα ,β

n,q ( f ;x)∣∣∣+2‖ f‖

� 3‖ f‖ .

ON CONVERGENCE PROPERTIES OF GAMMA-STANCU OPERATORS BASED ON q-INTEGERS 1343

Thus, from (14) we can write∣∣∣Gα ,βn,q ( f ;x)− f (x)

∣∣∣� ∣∣∣Gα ,βn,q ( f ;x)− f (x)

∣∣∣+ ∣∣∣∣ f (x)− f

([n]

[n]+ βx+

α[n]+ β

)∣∣∣∣�∣∣∣Gα ,β

n,q ( f −g;x)∣∣∣+ |( f −g)(x)|+

∣∣∣Gα ,βn,q (g;x)−g(x)

∣∣∣+∣∣∣∣ f(

[n][n]+ β

x+α

[n]+ β

)− f (x)

∣∣∣∣� 4‖ f −g‖+ δn,q(x)

∥∥g′′∥∥+∣∣∣∣ f (x)− f

([n]

[n]+ βx+

α[n]+ β

)∣∣∣∣Taking infimum on both side of the above inequality over all g ∈C2

B(0,∞), we get∣∣∣Gα ,βn,q ( f ;x)− f (x)

∣∣∣� 4K2( f ;δn)+w

(f ;

∣∣∣∣ [n][n]+ β

x+α

[n]+ β− x

∣∣∣∣)

from which we have the desired result by (13) . �Note that if q = (qn) is a sequence satisfying the conditions given in (12), then we

have limn→∞

δn,qn(x) = 0 and limn→∞

μn,qn(x) = 0 which gives us the pointwise rate of con-

vergence of the sequence(Gα ,β

n,qn ( f ;x))

to f (x) for every x∈ (0,∞) and f ∈CB(0,∞).

4. Weighted approximation

Let Bx2(0,∞) be the set of all functions defined on (0,∞) satisfying | f (x)| �Mf (1+ x2). Here Mf is a constant depending only on f . We also have the followingsubspaces of Bx2(0,∞) :

Cx2(0,∞) = { f ∈ Bx2(0,∞) : f is continuous on (0,∞)}C∗

x2(0,∞) ={

f ∈Cx2(0,∞) : limx→∞

f (x)1+ x2 = Kf < ∞

}

The norm on C∗x2(0,∞) is ‖ f‖x2 = sup

x>0

| f (x)|1+ x2 .

In this section we give the weighted approximation theorem for functions f inC∗

x2(0,∞) using the Korovkin type approximation theorems proved by Gadjiev [5].

THEOREM 3. Let q = (qn) be a sequence satisfing the conditions given in (12) .Then for each f ∈C∗

x2(0,∞) , we have

limn→∞

∥∥∥Gα ,βn,qn

( f ; .)− f (.)∥∥∥

x2= 0.

Proof. In order to prove the theorem we need to show that

limn→∞


(tk; .)− f (.)

∥∥∥x2

= 0, for k = 0,1,2. (15)

Then from the Korovkin’s type Theorem the proof is obvious.Since Gα ,β

n,qn (1;x) = 1, condition (15) holds for k = 0.For k = 1, we have∥∥∥Gα ,β

n,qn(t; .)− x

∥∥∥x2

=∥∥∥∥(

[n][n]+ β

−1

)x+

α[n]+ β

∥∥∥∥x2

�∣∣∣∣ [n][n]+ β

−1

∣∣∣∣supx>0

x1+ x2 +

α[n]+ β

supx>0

11+ x2

�∣∣∣∣ [n][n]+ β

−1

∣∣∣∣+ α[n]+ β

Thuslimn→∞


(t; .)− x∥∥∥

x2= 0.

Similarly for k = 2,


(t2; .)− x2

∥∥∥x2

= supx>0

∣∣∣Gα ,βn,qn

(t2;x)− x2

∣∣∣1+ x2

�[

[n]2

([n]+ β )2

(1+

qn

[n]

)−1

]supx>0

x2

1+ x2

+2 [n]α

([n]+ β )2supx>0

x1+ x2 +

α2

([n]+ β )2

�∣∣∣∣∣ [n] [n+1]([n]+ β )2

−1

∣∣∣∣∣+ 2 [n]α([n]+ β )2 +

α2

([n]+ β )2

Since limn→∞

qn = 1 we get

limn→∞


(t2; .)− x2

∥∥∥x2

= 0.

Hence the proof is completed from the Korovkin’s Theorem given by Gadjiev. �

5. Rate of convergence

Let wx0,c( f ;δ ) denote the modulus of continuity of f on the closed interval [x0,c] ,0 < x0 < c with

wx0,c( f ;δ ) = sup{| f (t)− f (x)| : x,t ∈ [x0,c],0 � |t− x| � δ} , δ > 0.

THEOREM 4. Let n ∈ N, q ∈ (0,1) and 0 < x0 < c. For every f ∈Cx2(0,∞),∥∥∥Gα ,βn,q ( f ; .)− f

∥∥∥[x0,c]

� Kγn,q +2wx0,c+1( f ;√γn,q), (16)

where γn,q is given by (10) and K is a positive constant depending on f and c. ‖.‖[x0,c]denotes the classical sup-norm on the space C [x0,c] .

Proof. For x ∈ [x0,c] and t > c+1, since t− x > 1

| f (t)− f (x)| � Mf(2+ t2 + x2)

= Mf

(2+ x2 +(t− x+ x)2

)� Mf

(2+2x2 +(t− x)2 +2x(t− x)2

)� Mf

((2+2x2)(t− x)2 +(2x+1)(t − x)2

)� Mf

(3+2c+2c2)(t− x)2 . (17)

For x ∈ [x0,c] , x0 � t � c+1, and δ > 0

| f (t)− f (x)| � wx0,c+1 ( f ; |t− x|) �(

1+|t− x|

δ

)wx0,c+1 ( f ;δ ) , δ > 0. (18)

From (17) and (18) , we get for all x ∈ [x0,c] and t � x0

| f (t)− f (x)| � K (t− x)2 +(

1+|t− x|

δ

)wx0,c+1 ( f ;δ )

where K = Mf(3+2c+2c2

). Thus we have∣∣∣Gα ,β

n,q ( f ;x)− f (x)∣∣∣� KGα ,β

n,q

((t−x)2 ;x

)+wx0,c+1 ( f ;δ )

[1+

1δ

(Gα ,β

n,q

((t−x)2 ;x

))1/2]

Using the identity for the second central moment of the operator Gα ,βn,q in (10) and

taking supremum over the interval x ∈ [x0,c] , the proof is completed. �

COROLLARY 2. Let q = (qn) be a sequence satisfing (12) and 0 < x0 < c. Forevery f ∈ Cx2(0,∞), we have

limn→∞


( f ; .)− f∥∥∥

[x0,c]= 0.

Proof. Now taking a sequence q = (qn) satisfing (12) instead of a fixed num-ber q ∈ (0,1) in Theorem 4, we obtain γn,qn , given by (10) tends to zero as n → ∞which gives us lim

n→∞wx0,c+1( f ;

√γn,qn) = 0 since f is continuous on [x0,c+ 1], x0 >

0. Consequently, it follows from Theorem 4 that for every f ∈ Cx2(0,∞), we get

limn→∞

∥∥∥Gα ,βn,qn ( f ; .)− f

∥∥∥[x0,c]

= 0. �

6. A Voronovskaya type theorem

In this section we give a Voronovskaya Type asymptotic formulas for the operatorsGα ,β

n,q with the help of the second and fourth central moments. Before the main theoremit is meaningful to give the following Lemma.

LEMMA 3. Let Gα ,βn,q be the operator defined by (9) . Let (qn) be a sequence such

that the conditions in (12) are satisfied. For all x > 0 we have,

limn→∞

[n]Gα ,βn,qn

(t− x;x) = α −βx (19)

limn→∞

[n]Gα ,βn,qn

((t− x)2 ;x

)= ax2 (20)

andlimn→∞

[n]2 Gα ,βn,qn

((t− x)4 ;x

)= 3a2x4. (21)

Proof. The first equality is obvious. For the second one we have,

limn→∞

[n]Gα ,βn,qn

((t− x)2 ;x

)= lim

n→∞

(qn

n [n]2 + β 2 [n]

([n]+ β )2x2− 2αβ [n]

([n]+ β )2x+

α2 [n]

([n]+ β )2

)

= ax2

For the last one, making some computations, one can easily see that the limit of thecoefficients of the terms xi, i = 0,1,2,3 tends to zero as n→∞. We have the coefficientof x4 as

qn(1−q)2 [n]5

([n]+ β )4 +[(−3[2]+ [3]+ [2][3])q2n +4βqn(1−q)

] [n]4

([n]+ β )4. (22)

Using the identity [n] = 1−qn

1−q , and applying it to the term [n]2 , we can rewrite (22) as

qn(1−qn)2 [n]3

([n]+ β )4+[(−3[2]+ [3]+ [2][3])q2n +4βqn(1−q)

] [n]4

([n]+ β )4,

from which, we get the limit 3a2 for n → ∞ . Consequently we have

limn→∞

[n]2 Gα ,βn,qn

((t− x)4 ;x

)= 3a2x4

as desired. �

Now we present the Voronovskaya type result for the operator (9) .

THEOREM 5. Let f , f ′, f ′′ ∈Cx2 (0,∞) and (qn) be a sequence such that (12) issatisfied, then we have

limn→∞

[n]{

Gα ,βn,qn

( f ;x)− f (x)}

= (α −βx) f ′ (x)+ax2

2f ′′ (x) ,

uniformly with respect to x ∈ [x0,c] , 0 < x0 < c.


Proof. For f , f ′, f ′′ ∈Cx2 (0,∞) and x > 0, we define a function as

r (t) = r(t,x) =

{f (t)− f (x)−(t−x) f ′(x)− 1

2 f ′′(x)(t−x)2

(t−x)2 , if t �= x

0, if t = x.

From the definition, r(x,x) = 0 and the function r(.,x)∈Cx2 (0,∞) . So, by the Taylor’sformula we write

f (t) = f (x)+ (t− x) f ′ (x)+12

f ′′ (x) (t− x)2 + r(t,x)(t− x)2 (23)

where r(t,x) is the Peano form of the remainder and limt→x

r(t,x)= 0. Applying Gα ,βn,qn ( f ;x)

to the both side of (23), we have

[n][Gα ,β

n,qn( f ;x)− f (x)

]= [n] f ′ (x)Gα ,β

n,qn(t− x;x) (24)

+12

[n] f ′′ (x)Gα ,βn,qn

((t− x)2 ;x

)+[n]Gα ,β

n,qn

((t − x)2r(t,x);x

)If we apply the Cauchy-Schwarz inequality for the last term on the right hand side ofthe equality (24) , we get

[n]Gα ,βn,qn

((t − x)2r(t,x);x

)�√

Gα ,βn,qn (r2(t,x);x)

√[n]2Gα ,β

n,qn ((t − x)4;x). (25)

Observe that r2(.,x) ∈Cx2 (0,∞) and r2(x,x) = 0. Also, from Corollary 2, we have

limn→∞

Gα ,βn,qn

(r2(t,x);x

)= r2(x,x) = 0 (26)

uniformly with respect to x ∈ [x0,c]. In the view of (25), (26) and the Lemma 3, weobtain

limn→∞

[n]Gα ,βn,qn

((t − x)2r(t,x);x

)= 0. (27)

Combining (19) , (20) and (27) we get the desired result. �

RE F ER EN C ES

[1] A. ARAL, V. GUPTA, R. P. AGARWAL, Applications of q -Calculus in Operator Theory, Springer,2013.

[2] D. AYDIN, A. ARAL AND G. BASCANBAZ-TUNCA, A Generalization of Post-Widder OperatorsBased on q-Integers, Annals of the Alexandru Ioan Cuza University-Mathematics.

[3] Q. B. CAI AND X. M. ZENG, On the convergence of a kind of q -gamma operators, J. Inequal. Appl.2013.1 (2013), 1–9.

[4] Q. B. CAI, Properties of Convergence for a Class of Genralized q-Gamma Operators, J. Math. Study47, 4 (2014), 388–395.

[5] A. D. GADJIEV, A problem on the convergence of a sequence of positive linear operators on un-bounded sets and theorems that are analogous to PP Korovkin’s theorem, Dokl. Akad. Nauk SSSR.218 (1974), 1001–1004.


[6] F. H. JACKSON, A generalization of the functions Γ(n) and xn , Proc. Roy. Soc. London 74, (1904),64–72.

[7] V. G. KAC AND P. CHEUNG, Quantum Calculus, Universitext. Springer, New York, 2002.[8] H. KARSLI, Rate of convergence of a new gamma type operators for functions with derivatives of

bounded variation, Math. Comput. Model. 45, 5–6 (2007), 617–624.[9] H. KARSLI, V. GUPTA AND A. IZGI, Rate of pointwise convergence of a new kind of gamma operators

for functions of bounded variation, Appl. Math. Letters 22, 4 (2009), 505–510.[10] H. KARSLI AND M. A. OZARSLAN, Direct local and global approximation results for operators of

gamma type, Hacet. J. Math. Stat. 39 (2010), 241–253.[11] H. KARSLI, P. N. AGRAWAL AND M. GOYAL, General Gamma type operators based on q-integers,

Applied Mathematics and Computation 251 (2015), 564–575.[12] A. LUPAS, A q-analogue of the Bernstein operator, Seminar on Numerical and Statistical Calculus,

University of Cluj-Napoca, 9 (1987), 85–92.[13] S. M. MAZHAR, Approximation by positive operators on infinite intervals, Mathematica Balkanica 5,

2 (1991), 99–104,[14] G. M. PHILLIPS, Bernstein polynomials based on the q-integers, The heritage of P. L. Chebyshev,

Ann. Numer. Math. 4 (1997), 511–518.[15] A. D. SOLE AND V. KAC, On the integral representation of the q-gamma and the q-beta functions,

Rendiconti Lincei: Matematica e Applicazioni 9, 16, 1 (2005), 11–19.[16] A. D. SOLE AND V. KAC, On integral representations of q -gamma and q-beta functions, arXiv

preprint math/0302032 (2003).[17] D. D. STANCU, Approximation of functions by a new class of linear polynomial operators, Rev.

Roumaine Math. Pures Appl. 13 (1968), 1173–1194.[18] J. THOMAE, Beitrage zur Theorie der durch die Heinesche Reihe, J. Reine Angew. Math, 70 (1869),

258–281.[19] Z. UNAL, M. A. OZARSLAN, O. DUMAN, Approximation properties of real and complex Post-Widder

operators based on q-integers, Miskolc Math. Notes 13 (2012), 581–603.[20] X. M. ZENG, Approximation properties of Gamma operators, J. Math. Anal. Appl., 311 (2005), 389–

401.[21] C. ZHAO, W. T. CHENG, X. M. ZENG, Some approximation properties of a kind of q -Gamma Stancu

operators, J. Inequal. Appl. (94) (2014).

(Received March 29, 2016) Ozge DalmanogluBaskent University, Faculty of Education

Department of Mathematics EducationBaglica, Ankara, Turkey


Mediha OrkcuGazi University, Faculty of Sciences

Department of MathematicsTeknikokullar, 06500 Ankara, Turkey



& Applications


CAUCHY’S ERROR REPRESENTATION OF HERMITE

INTERPOLATING POLYNOMIAL AND RELATED RESULTS

GORANA ARAS-GAZIC, JOSIP PECARIC AND ANA VUKELIC


Abstract. In this paper we consider convex functions of higher order. Using the Cauchy’s er-ror representation of Hermite interpolating polynomial the results concerning to the Hermite-Hadamard inequalities are presented. As a special case, generalizations for the zeros of orthog-onal polynomials are considered.

1. Introduction

We follow here notations and terminology about Hermite interpolating polynomialfrom [1, p. 62]:

Let −∞ < a < b < ∞ , and a � a1 < a2 . . . < ar � b , (r � 2) be given. Forf ∈ Cn[a,b] a unique polynomial PH(t) of degree (n− 1) , exists, fulfilling one of thefollowing conditions:

Hermite conditions

P(i)H (a j) = f (i)(a j); 0 � i � k j, 1 � j � r,

r

∑j=1

k j + r = n,

in particular:Simple Hermite or Osculatory conditions (n = 2m , r = m , k j = 1 for all j )

PO(a j) = f (a j), P′O(a j) = f ′(a j), 1 � j � m,

Lagrange conditions (r = n , k j = 0 for all j )

PL(a j) = f (a j), 1 � j � n,

Type (m,n−m) conditions (r = 2, 1 � m � n−1, k1 = m−1, k2 = n−m−1)

P(i)mn(a) = f (i)(a), 0 � i � m−1,

P(i)mn(b) = f (i)(b), 0 � i � n−m−1,

Two-point Taylor conditions (n = 2m , r = 2, k1 = k2 = m−1)

P(i)2T (a) = f (i)(a), P(i)

2T (b) = f (i)(b), 0 � i � m−1.

Mathematics subject classification (2010): Primary 26D15; Secondary 26D07, 26A51.Keywords and phrases: n -convex function, Hermite interpolating polynomial, Hermite-Hadamard in-

equality, orthogonal polynomials.


1349

1350 G. ARAS-GAZIC, J. PECARIC AND A. VUKELIC

DEFINITION 1. Let f be a real-valued function defined on the segment [a,b] .The divided difference of order n of the function f at distinct points x0, . . . ,xn ∈ [a,b] ,is defined recursively (see [7]) by

f [xi] = f (xi), (i = 0, . . . ,n)

and

f [x0, . . . ,xn] =f [x1, . . . ,xn]− f [x0, . . . ,xn−1]

xn− x0.

The value f [x0, . . . ,xn] is independent of the order of the points x0, . . . ,xn .The definition may be extended to include the case that some (or all) of the points

coincide. Assuming that f ( j−1)(x) exists, we define

f [x, . . . ,x︸︷︷︸j−times

] =f ( j−1)(x)( j−1)!

.

The notion of n-convexity goes back to Popoviciu ([8]). We follow the definitiongiven by Karlin ([6]):

DEFINITION 2. A function f : [a,b]→ R is said to be n -convex on [a,b] , n � 0,if for all choices of (n+1) distinct points in [a,b], n -th order divided difference of fsatisfies

f [x0, . . . ,xn] � 0.

In fact, Popoviciu proved that each continuous n -convex function on [0,1] is theuniform limit of the sequence of corresponding Bernstein’s polynomials (see for ex-ample [7, p. 293]). Also, Bernstein’s polynomials of continuous n -convex function arealso n -convex functions. Therefore, when stating our results for a continuous n -convexfunction f , without any loss in generality we assume that f (n) exists and is continuous.

In [5] M. Bessenyei and Zs. Pales were investigating the case of higher orderconvexity. The base points of the Hermite-Hadamard type inequalities turn out to bethe zeros of certain orthogonal polynomials. The main tools of the paper are basedon some methods of numerical analysis, like Gauss quadrature formula and Hermiteinterpolation. They considered the following Gauss type quadrature formulae wherethe coefficients and the base points are to be determined so that be exact when f is apolynomial of degree at most 2n−1, 2n , 2n and 2n+1, respectively:∫ b

aρ(t) f (t)dt =

n

∑k=1

ck f (ξk), (1)

∫ b

aρ(t) f (t)dt = c0 f (a)+

n

∑k=1

ck f (ξk), (2)

∫ b

aρ(t) f (t)dt =

n

∑k=1

ck f (ξk)+ cn+1 f (b), (3)

∫ b

aρ(t) f (t)dt = c0 f (a)+

n

∑k=1

ck f (ξk)+ cn+1 f (b). (4)

CAUCHY’S ERROR REPRESENTATION OF HERMITE INTERPOLATING POLYNOMIAL 1351

Using this formulae and the remainder term of the Hermite interpolation they provedHermite-Hadamard type inequalities in cases of odd and even higher order convexityseparately in the subsequent theorems:

THEOREM 1. Let ρ : [a,b] → R be a positive integrable function. Denote thezeros of Pm by ξ1, . . . ,ξm where Pm is the m-th degree member of the orthogonal poly-nomial system on [a,b] with respect to the weight function (x− a)ρ(x) , furthermoredenote the zeros of Qm by η1, . . . ,ηm where Qm is the m-th degree member of the or-thogonal polynomial system on [a,b] with respect to the weight function (b− x)ρ(x) .Define the coefficients α0, . . . ,αm and β1, . . . ,βm+1 by the formulae

α0 :=1

P2m(a)

∫ b

aP2

m(x)ρ(x)dx, αk :=1

ξk −a

∫ b

a

(x−a)Pm(x)(x− ξk)P′

m(ξk)ρ(x)dx

and

βk :=1

b−ηk

∫ b

a

(b− x)Qm(x)(x−ηk)Q′

m(ηk)ρ(x)dx, βm+1 :=

1Q2

m(b)

∫ b

aQ2

m(x)ρ(x)dx.

If a function f : [a,b] → R is (2m+1)-convex, then it satisfies the following Hermite-Hadamard type inequality

α0 f (a)+m

∑k=1

αk f (ξk) �∫ b

aρ(x) f (x)dx �

m

∑k=1

βk f (ηk)+ βm+1 f (b).

THEOREM 2. Let ρ : [a,b] → R be a positive integrable function. Denote thezeros of Pm by ξ1, . . . ,ξm where Pm is the m-th degree member of the orthogonalpolynomial system on [a,b] with respect to the weight function ρ(x) , furthermore de-note the zeros of Qm−1 by η1, . . . ,ηm−1 where Qm−1 is the (m− 1)-st degree mem-ber of the orthogonal polynomial system on [a,b] with respect to the weight function(b− x)(x−a)ρ(x) . Define the coefficients α1, . . . ,αm and β0, . . . ,βm by the formulae

αk :=∫ b

a

Pm(x)(x− ξk)P′

m(ξk)ρ(x)dx

and

β0 :=1

(b−a)Q2m−1(a)

∫ b

a(b− x)Q2

m−1(x)ρ(x)dx,

βk :=1

(b−ηk)(ξk −a)

∫ b

a

(b− x)(x−a)Qm−1(x)(x−ηk)Q′

m−1(ηk)ρ(x)dx,

βm :=1

(b−a)Q2m−1(b)

∫ b

a(x−a)Q2

m−1(x)ρ(x)dx.

If a function f : [a,b] → R is (2m)-convex, then it satisfies the following Hermite-Hadamard type inequality

m

∑k=1

αk f (ξk) �∫ b

aρ(x) f (x)dx � β0 f (a)+

m−1

∑k=1

βk f (ηk)+ βm f (b).


In this paper we obtain generalizations of above inequalities for convex functionsof higher order by using the Cauchy’s error representation of Hermite interpolatingpolynomial. As a special case, generalizations of Hermite-Hadamard type inequalities,where the base points turn out to be the zeros of orthogonal polynomials will be con-sidered. Similar results for Lidstone’s polynomial can be found in [4]. See also [2] and[3].

2. Cauchy’s error representation

In [1, p. 71] the following theorem is proved:

THEOREM 3. Let F(t) ∈ Cn−1([a,b]) and suppose that F (n)(t) exists at eachpoint of (a,b) . Then

F(t)−r

∑j=1

k j

∑i=0

Hi j(t)F(i)(a j) =1n!

ω(t)F(n)(ξ ), (5)

where ξ ∈ (a,b) and Hi j are fundamental polynomials of the Hermite basis defined by

Hi j(t) =1i!

ω(t)(t−a j)k j+1−i

k j−i

∑k=0

1k j!

[(t−a j)k j+1

ω(t)

](k)

t=a j

(t−a j)k, (6)

where

ω(t) =r

∏j=1

(t −a j)k j+1. (7)

Motivated by (5) and formulae (1), (2), (3) and (4) we define functionals Φ1( f ) ,Φ2( f ) , Φ3( f ) and Φ4( f ) respectively, by

Φ1(F) = F(t)−r

∑j=1

k j

∑i=0

Hi j(t)F (i)(a j), (8)

Φ2(F) = F(t)−k1

∑i=0

Hi1(t)F (i)(a)−r

∑j=2

k j

∑i=0

Hi j(t)F (i)(a j), (9)

Φ3(F) = F(t)−r−1

∑j=1

k j

∑i=0

Hi j(t)F (i)(a j)−kr

∑i=0

Hir(t)F (i)(b), (10)

Φ4(F) = F(t)−k1

∑i=0

Hi1(t)F(i)(a)−r−1

∑j=2

k j

∑i=0

Hi j(t)F (i)(a j)−kr

∑i=0

Hir(t)F (i)(b). (11)

Now, using Theorem 3 we get the following corollaries:


COROLLARY 1. Let F : [a,b] → R is n-convex function, and Hi j are defined on[a,b] by (6), such that k j is odd for all j = 1, . . . ,r . Then we have

Φ1(F) � 0. (12)

Proof. Since k j is odd for all j = 1, . . . ,r , then using (7), we get that ω(t) � 0.By using (5) for n -convex function F , (12) obviously holds. �

REMARK 1. If we put that n = 2m , r = m and k j = 1 for all j we get Hermiteinterpolating polynomial with simple Hermite or Osculatory conditions and then

F(t)−m

∑j=1

H0 j(t)F(a j)−m

∑j=1

H1 j(t)F ′(a j) � 0.

COROLLARY 2. Let F : [a,b] → R is n-convex function, and Hi j are defined on[a,b] by (6), such that a1 = a and k j is odd for all j = 2, . . . ,r . Then we have

Φ2(F) � 0. (13)

Proof. Now ω(t) = (t − a)k1+1 ∏rj=2(t − a j)k j+1 . Since k j is odd for all j =

2, . . . ,r , we get that ω(t)� 0. So, by using (5) for n -convex function F , (13) obviouslyholds. �

COROLLARY 3. Let F : [a,b] → R is n-convex function and Hi j are defined on[a,b] by (6), such that ar = b. Then

(a) If k j is odd for all j = 1, . . . ,r , we have

Φ3(F) � 0. (14)

(b) If k j is odd for all j = 1, . . . ,r−1 and kr is even, we have

Φ3(F) � 0. (15)

Proof. Now ω(t) = (t−b)kr+1 ∏r−1j=1(t−a j)k j+1 .

(a) Since k j is odd for all j = 1, . . . ,r , we get that ω(t) � 0.(b) Since k j is odd for all j = 1, . . . ,r−1 and kr is even, we get that ω(t) � 0.So, by using (5) for n -convex function F , (14) and (15) obviously hold. �

COROLLARY 4. Let F : [a,b] → R is n-convex function and Hi j are defined on[a,b] by (6), such that a1 = a and ar = b. Then

(a) If k j is odd for all j = 2, . . . ,r , we have

Φ4(F) � 0. (16)

(b) If k j is odd for all j = 2, . . . ,r−1 and kr is even, we have

Φ4(F) � 0. (17)


Proof. Now ω(t) = (t−a)k1+1(t−b)kr+1 ∏r−1j=2(t−a j)k j+1 .

(a) Since k j is odd for all j = 2, . . . ,r , we get that ω(t) � 0.(b) Since k j is odd for all j = 2, . . . ,r−1 and kr is even, we get that ω(t) � 0.So, by using (5) for n -convex function F , (16) and (17) obviously hold. �

REMARK 2. If we put r = 2, 1 � m � n−1, k1 = m−1, k2 = n−m−1 and k2

is odd then we get Hermite interpolating polynomial with (m,n−m) type conditionsand then

F(t)−m−1

∑i=0

Hi1(t)F (i)(a)−n−m−1

∑i=0

Hi2(t)F (i)(b) � 0.

For k2 even, the above inequality is reversed.If we put n = 2m , r = 2, k1 = k2 = m− 1 and m is even then we get Hermite

interpolating polynomial with two-point Taylor conditions and then

F(t)−m−1

∑i=0

Hi1(t)F(i)(a)−m−1

∑i=0

Hi2(t)F(i)(b) � 0.

For m odd, the above inequality is reversed.

REMARK 3. Similarly as in [3] we can construct new families of exponentiallyconvex function and Cauchy type means by looking at linear functionals (8), (9), (10)and (11). The monotonicity property of the generalized Cauchy means obtained viathese functionals can be prove by using the properties of the linear functionals associ-ated with this error representation, such as n -exponential and logarithmic convexity.

3. Generalization of the Hermite-Hadamard type inequalities

The classical Hermite-Hadamard inequality states that for a convex function F :[a,b]→ R the following estimation holds:

F

(a+b

2

)� 1

b−a

∫ b

aF(t)dt � F(a)+F(b)

2. (18)

As a consequences of our results given in Section 2, here we give the generalizedHermite-Hadamard type inequalities.

THEOREM 4. Let F : [a,b]→ R is n-convex function, ρ : [a,b]→ R is a positiveintegrable function and Hi j and Hi j are defined on [a,b] by

Hi j(t) =1i!

ω(t)(t−a j)k j+1−i

k j−i

∑k=0

1k j!

[(t−a j)k j+1

ω(t)

](k)

t=a j

(t−a j)k, (19)

and

Hi j(t) =1i!

ω(t)(t−b j)l j+1−i

l j−i

∑k=0

1l j!

[(t−b j)l j+1

ω(t)

](k)

t=b j

(t −b j)k, (20)

where

ω(t) =r

∏j=1

(t−a j)k j+1, ω(t) =r

∏j=1

(t −b j)l j+1

for a � a1 < a2 . . . < ar � b, a � b1 < b2 . . . < br � b,(r, r � 2) and ∑rj=1 k j + r =

∑rj=1 l j + r = n.

Then, if a1 = a, br = b, k j odd for all j = 2, . . . ,r , l j odd for all j = 1, . . . , r −1and lr even, we have

k1

∑i=0

F (i)(a)∫ b

aρ(t)Hi1(t)dt +

r

∑j=2

k j

∑i=0

F(i)(a j)∫ b

aρ(t)Hi j(t)dt

�∫ b

aρ(t)F(t)dt

�r−1

∑j=1

l j

∑i=0

F (i)(b j)∫ b

aρ(t)Hi j(t)dt +

l r

∑i=0

F (i)(b)∫ b

aρ(t)Hir(t)dt.

If F is n-concave, the inequalities are reversed.

Proof. We use Corollary 2 and Corollary 3(b). �

COROLLARY 5. Let F : [a,b]→R is (2r−1)-convex function and ρ : [a,b]→ R

is a positive integrable function. Then, we have

F(a)∫ b

aρ(t)H01(t)dt +

r

∑j=2

F(a j)∫ b

aρ(t)H0 j(t)dt +

r

∑j=2

F ′(a j)∫ b

aρ(t)H1 j(t)dt

�∫ b

aρ(t)F(t)dt (21)

�r−1

∑j=1

F(b j)∫ b

aρ(t)H0 j(t)dt +

r−1

∑j=1

F ′(b j)∫ b

aρ(t)H1 j(t)dt +F(b)

∫ b

aρ(t)H0r(t)dt,

where

H01(t) =P2

r−1(t)P2

r−1(a),

H0 j(t) =(t−a)P2

r−1(t)

(t −a j)2[P′

r−1(a j)]2 (a j −a)

(1− P′

r−1(a j)+ (a j −a)P′′r−1(a j)

(a j −a)P′r−1(a j)

(t−a j))

,

H1 j(t) =(t −a)P2

r−1(t)

(t−a j)(a j −a)[P′

r−1(a j)]2 ,

H0 j(t) =(b− t)P

2r−1(t)

(t −b j)2[P′r−1(b j)

]2(b−b j)

(1+

P′r−1(b j)− (b−b j)P

′′r−1(b j)

(b−b j)P′r−1(b j)

(t −b j)

),

H1 j(t) =(b− t)P

2r−1(t)

(t−b j)(b−b j)[P′r−1(b j)

]2 , H0r(t) =P

2r−1(t)

P2r−1(b)

,

and

Pr−1(t) =r

∏j=2

(t−a j), Pr−1(t) =r−1

∏j=1

(t−b j)

for a < a2 . . . < ar � b, a � b1 < b2 . . . < br−1 < b,(r � 2) .If F is (2r−1)-concave, the inequalities are reversed.

Proof. We put k1 = 0, k j = 1 for j = 2, . . . ,r and l j = 1 for j = 1, . . . ,r−1,lr = 0 in Theorem 4 and then calculate

H01(t) =(t−a)P2

r−1(t)(t −a)

·[

(t−a)(t −a)P2

r−1(t)

]t=a

=P2

r−1(t)P2

r−1(a),

H0 j(t) =(t−a)P2

r−1(t)(t−a j)2

⎧⎨⎩[

(t−a j)2

(t−a)P2r−1(t)

]t=a j

+

[(t−a j)2

(t −a)P2r−1(t)

]′t=a j

(t−a j)

⎫⎬⎭

=(t −a)P2

r−1(t)

(t−a j)2[P′

r−1(a j)]2 (a j −a)

(1− P′

r−1(a j)+ (a j −a)P′′r−1(a j)

(a j −a)P′r−1(a j)

(t−a j))

and

H1 j(t) =(t−a)P2

r−1(t)(t−a j)

[(t −a j)2

(t−a)P2r−1(t)

]t=a j

=(t−a)P2

r−1(t)

(t −a j)(a j −a)[P′

r−1(a j)]2 .

Coefficients H0 j , H1 j and H0r we get similarly. �

REMARK 4. If we choose Pr−1 and Pr−1 such that they are orthogonal withweight (t − a)ρ(t) and (b− t)ρ(t) respectively, to all polynomials of lower degree,i.e. ∫ b

a(t −a)ρ(t)Pr−1(t)tkdt = 0, k = 0,1, . . . ,r−2 (22)

and ∫ b

a(b− t)ρ(t)Pr−1(t)tldt = 0, l = 0,1, . . . ,r−2,

we get that ∫ b

aρ(t)H1 j(t)dt = 0 and

∫ b

aρ(t)H1 j(t)dt = 0.


Now, using the relation for coefficient H1 j(t) , we get

∫ b

aρ(t)H0 j(t)dt

=∫ b

a

ρ(t)(t−a)P2r−1(t)

(a j −a)(t−a j)2[P′

r−1(a j)]2 dt− P′

r−1(a j)+ (a j −a)P′′r−1(a j)

(a j −a)P′r−1(a j)

∫ b

aρ(t)H1 j(t)dt.

Now, using (22), we have

∫ b

a

ρ(t)(t−a)Pr−1(t)(t −a j)P′

r−1(a j)

(Pr−1(t)

(t −a j)P′r−1(a j)

−1

)dt = 0

becausePr−1(t)

(t −a j)P′r−1(a j)

−1 = (t −a j)Q(t),

where Q(t) is polynomial of degree r−3. So,

∫ b

a

ρ(t)(t−a)P2r−1(t)

(t−a j)2[P′

r−1(a j)]2 dt =

∫ b

a

ρ(t)(t−a)Pr−1(t)(t−a j)P′

r−1(a j)dt.

Similarly, we calculate∫ ba ρ(t)H0 j(t)dt and get the following relations for coeffi-

cients in (21): ∫ b

aρ(t)H01(t)dt =

1

P2r−1(a)

∫ b

aρ(t)P2

r−1(t)dt,

∫ b

aρ(t)H0 j(t)dt =

1(a j −a)P′

r−1(a j)

∫ b

a

ρ(t)(t−a)Pr−1(t)(t −a j)

dt,

∫ b

aρ(t)H1 j(t)dt = 0,

∫ b

aρ(t)H0 j(t)dt =

1

(b−b j)P′r−1(b j)

∫ b

a

ρ(t)(b− t)Pr−1(t)(t−b j)

dt,

∫ b

aρ(t)H1 j(t)dt = 0,

∫ b

aρ(t)H0r(t)dt =

1

P2r−1(b)

∫ b

aρ(t)P

2r−1(t)dt,

which is result proved by M. Bessenyei and Zs. Pales in [5] (see Theorem 1).

THEOREM 5. Let F : [a,b]→ R is n-convex function, ρ : [a,b]→ R is a positiveintegrable function and Hi j and Hi j are defined on [a,b] by (19) and (20) respectively.


Then, if b1 = a, br = b, k j odd for all j = 1, . . . ,r , l j odd for all j = 2, . . . , r −1 andlr even, we have

r

∑j=1

k j

∑i=0

F(i)(a j)∫ b

aρ(t)Hi j(t)dt

�∫ b

aρ(t)F(t)dt

�l1

∑i=0

F (i)(a)∫ b

aρ(t)Hi1(t)dt +

r−1

∑j=2

l j

∑i=0

F(i)(b j)∫ b

aρ(t)Hi j(t)dt

+l r

∑i=0

F (i)(b)∫ b

aρ(t)Hir(t)dt.

If F is n-concave, the inequalities are reversed.

Proof. We use Corollary 1 and Corollary 4(b). �

COROLLARY 6. Let F : [a,b] → R is (2r)-convex function and ρ : [a,b] → R isa positive integrable function. Then, we have

r

∑j=1

F(a j)∫ b

aρ(t)H0 j(t)dt +

r

∑j=1

F ′(a j)∫ b

aρ(t)H1 j(t)dt

�∫ b

aρ(t)F(t)dt

� F(a)∫ b

aρ(t)H01(t)dt +

r

∑j=2

F(b j)∫ b

aρ(t)H0 j(t)dt (23)

+r

∑j=2

F ′(b j)∫ b

aρ(t)H1 j(t)dt +F(b)

∫ b

aρ(t)H0(r+1)(t)dt,

where

H0 j(t) =P2

r (t)

(t −a j)2 [P′r(a j)]

2

(1− P′′

r (a j)P′

r(a j)(t−a j)

),

H1 j(t) =P2

r (t)

(t−a j) [P′r(a j)]

2 , H01(t) =(b− t)P

2r−1(t)

(b−a)P2r−1(a)

,

H0 j(t) =(t −a)(b− t)P

2r−1(t)

(b j −a)(b−b j)(t −b j)2[P′r−1(b j)

]2×(

1+(2b j −a−b)P

′r−1(b j)− (b−b j)(b j −a)P

′′r−1(b j)

(b−b j)(b j −a)P′r−1(b j)

(t −b j)

),

H1 j(t) =(t−a)(b− t)P

2r−1(t)

(t −b j)(b j −a)(b−b j)[P′r−1(b j)

]2 , H0(r+1)(t) =(t−a)P

2r−1(t)

(b−a)P2r−1(b)

and

Pr(t) =r

∏j=1

(t−a j), Pr−1(t) =r

∏j=2

(t−b j)

for a � a1 < a2 . . . < ar � b, a < b2 . . . < br < b,(r � 2) . If F is (2r)-concave, theinequalities are reversed.

Proof. We put k j = 1 for j = 1, . . . ,r and l j = 1 for j = 2, . . . ,r , l1 = lr+1 = 0 inTheorem 5 and then calculate

H0 j =P2

r (t)(t−a j)2

{[(t−a j)2

P2r (t)

]t=a j

+[(t−a j)2

P2r (t)

]′t=a j

(t−a j)

}

=P2

r (t)

(t−a j)2 [P′r(a j)]

2

(1− P′′

r (a j)P′

r(a j)(t −a j)

),

H1 j(t) =P2

r (t)t−a j

[(t−a j)2

P2r (t)

]t=a j

=P2

r (t)

(t−a j) [P′r(a j)]2

,

H01(t) =(t−a)(t−b)P

2r−1(t)

t−a

[t −a

(t −a)(t−b)P2r−1(t)

]t=a

=(b− t)P

2r−1(t)

(b−a)P2r−1(a)

,

H0 j(t) =(t −a)(t−b)P

2r−1(t)

(t−b j)2

×⎧⎨⎩[

(t−b j)2

(t −a)(t−b)P2r−1(t)

]t=b j

+

[(t−b j)2

(t−a)(t−b)P2r−1(t)

]′t=b j

(t −b j)

⎫⎬⎭

=(t −a)(b− t)P

2r−1(t)

(b j −a)(b−b j)(t −b j)2[P′r−1(b j)

]2×(

1+(2b j −a−b)P

′r−1(b j)− (b−b j)(b j −a)P

′′r−1(b j)

(b−b j)(b j −a)P′r−1(b j)

(t−b j)

),

H1 j(t) =(t−a)(t−b)P

2r−1(t)

(t−b j)

[(t−b j)2

(t −a)(t−b)P2r−1(t)

]t=b j

=(t −a)(b− t)P

2r−1(t)

(t−b j)(b j −a)(b−b j)[P′r−1(b j)

]2 .

Coefficient H0(r+1) we get similarly as coefficient H01(t) . �


REMARK 5. If we choose Pr such that it is orthogonal with weight ρ(t) to allpolynomials of lower degree, i.e.

∫ b

aρ(t)Pr(t)tkdt = 0, k = 0,1, . . . ,r−1

we get that ∫ b

aρ(t)H1 j(t)dt = 0.

Now, similar as in Remark 4 we get

∫ b

aρ(t)H0 j(t)dt =

1P′

r(a j)

∫ b

a

ρ(t)Pr(t)(t−a j)

dt.

Also, if we choose Pr−1 such that it is orthogonal with weight (t−a)(b− t)ρ(t) , to allpolynomials of lower degree, i.e.

∫ b

aρ(t)(t−a)(b− t)Pr−1(t)tldt = 0, l = 0,1, . . . ,r−2,

we get that ∫ b

aρ(t)H1 j(t)dt = 0

and then

∫ b

aρ(t)H0 j(t)dt =

1

(b j −a)(b−b j)P′r−1(b j)

∫ b

a

ρ(t)(t−a)(b− t)Pr−1(t)(t−b j)

dt,

which is result proved by M. Bessenyei and Zs. Pales in [5] (see Theorem 2).

REMARK 6. If we put r = 1 and ρ(t) = 1 in Corollary 6, we get n = 2 and fora1 = a+b

2 calculate

H01(t) = 1 ⇒∫ b

aH01(t)dt = b−a,

H11(t) = t− a+b2

⇒∫ b

aH11(t)dt = 0,

H01(t) =b− tb−a

⇒∫ b

aH01(t)dt =

b−a2

,

H02(t) =t−ab−a

⇒∫ b

aH02(t)dt =

b−a2

.

So, using (23) for n = 2 we get classical Hermite-Hadamard inequality (18).


COROLLARY 7. Let F : [a,b]→ R is (2m)-convex function and ρ : [a,b]→ R isa positive integrable function. Then, if m is odd, we have

m

∑j=1

F(a j)∫ b

aρ(t)H0 j(t)dt +

m

∑j=1

F ′(a j)∫ b

aρ(t)H1 j(t)dt

�∫ b

aρ(t)F(t)dt

�m−1

∑i=0

F(i)(a)∫ b

aρ(t)Hi1(t)dt +

m−1

∑i=0

F (i)(b)∫ b

aρ(t)Hi2(t)dt,

where H0 j and H1 j as in Corollary 6 with r = m,

Hi1(t) =(t −a)i(t−b)m

i!

m−1−i

∑k=0

(−1)k(m+ k−1)![(m−1)!]2 (a−b)m+k

(t−a)k, and

Hi2(t) =(t−a)m(t −b)i

i!

m−1−i

∑k=0

(−1)k(m+ k−1)!

[(m−1)!]2 (b−a)m+k(t−b)k.

If F is (2m)-concave, the inequalities are reversed.

Proof. We use Remark 1 and 2 and then calculate

Hi1(t) =1i!

(t −a)m(t −b)m

(t −a)m−i

m−1−i

∑k=0

1(m−1)!

[(t −a)m

(t−a)m(t −b)m

](k)

t=a(t −a)k

=(t−a)i(t −b)m

i!

m−1−i

∑k=0

(−1)k(m+ k−1)![(m−1)!]2 (a−b)m+k

(t−a)k.

Coefficient Hi2(t) we get similarly. �

Acknowledgements. The research of the authors has been fully supported by Croa-tian Science Foundation under the project 5435.

RE F ER EN C ES

[1] R. P. AGARWAL AND P. J. Y. WONG, Error Inequalities in Polynomial Interpolation and Their Ap-plications, Kluwer Academic Publishers, Dordrecht/Boston/London, 1993.

[2] G. ARAS-GAZIC, V. CULJAK, J. PECARIC AND A. VUKELIC, Generalization of Jensen’s inequalityby Lidstone’s polynomial and related results, Math. Ineq. Appl. 16 (4) (2013), 1243–1267.

[3] G. ARAS-GAZIC, V. CULJAK, J. PECARIC AND A. VUKELIC, Generalization of Jensen’s inequalityby Hermite polynomials and related results, Math. Rep. 17 (67) (2) (2015), Article 4.

[4] G. ARAS-GAZIC, V. CULJAK, J. PECARIC AND A. VUKELIC, Cauchy’s error representation ofLidstone interpolating polynomial and related results, J. Math. Inequal. 9 (4) (2015), 1207–1225.

[5] M. BESSENYEI AND ZS. PALES, Higher-order generalizations of Hadamard’s inequality, Publ. Math.Debrecen 61 (3–4) (2002), 623–643.


[6] S. KARLIN, Total Positivity, Stanford Univ. Press, Stanford, 1968.[7] J. E. PECARIC, F. PROSCHAN AND Y. L. TONG, Convex functions, partial orderings, and statistical

applications, Mathematics in science and engineering 187, Academic Press, 1992.[8] T. POPOVICIU, Sur l’approximation des fonctions convexes d’ordre superieur, Mathematica 10,

(1934), 49–54.

(Received March 30, 2016) Gorana Aras-GazicFaculty of Architecture, University of Zagreb

Fra Andrije Kacica Miosica 26, 10000 Zagreb, Croatiae-mail: [email protected]



Ana VukelicFaculty of Food Technology and Biotechnology

University of ZagrebPierottijeva 6, 10000 Zagreb, Croatia



& Applications


BOAS–TYPE INEQUALITY FOR 3–CONVEX FUNCTIONS AT A POINT

JOSIP PECARIC, DORA POKAZ AND MARJAN PRALJAK


Abstract. Starting from a very general form of Boas-type inequality from [5] we get Boas-typeinequality for 3 -convex functions at a point. For special λ -balanced sets, weight functions andmeasures we derive various examples.

1. Introduction

In [2], R. P. Boas proved that the inequality

∫ ∞

0Φ(

1M

∫ ∞

0f (tx)dm(t)

)dxx

�∫ ∞

0Φ( f (x))

dxx

(1)

holds for all continuous convex functions Φ : [0,∞) → R , measurable non–negativefunctions f : R+ → R , and non–decreasing bounded functions m : [0,∞) → R , whereM = m(∞)−m(0)> 0 and the inner integral on the left-hand side of (1) is the Lebesgue–Stieltjes integral with respect to m . This inequality represent one direction of gener-alization of the famous Hardy inequality. After its author, relation (1) was named theBoas inequality. In the case of a concave function Φ , (1) holds with the sign of inequal-ity reversed.

S. Kaijser et al. [6] (see also the paper [7] of N. Levinson) established the so-calledgeneral Hardy-Knopp-type inequality

∫ ∞

0Φ(

1x

∫ x

0f (t)dt

)dxx

�∫ ∞

0Φ( f (x))

dxx

, (2)

for positive measurable functions f : R+ → R , and a real convex function Φ on R+ .Later on, A. Cizmesija et al. [4] generalized relation (2) to the so-called strengthenedHardy-Knopp-type inequality by inserting a weight function and integrating over in-tervals of non-negative real numbers. Further, in [3] A. Cizmesija et al. considered a

general Borel measure λ on R+ , such that L = λ (R+) =∫ ∞

0dλ (t) < ∞ , and proved

Mathematics subject classification (2010): Primary 26D10; Secondary 26D15.Keywords and phrases: Boas inequality, 3 -convex function at a point.This work has been fully supported by Croatian Science Foundation under the project 5435.


1363

1364 J. PECARIC, D. POKAZ AND M. PRALJAK

that for a convex function Φ on an interval I ⊆ R and a weight function u on R+ theinequality ∫ ∞

0u(x)Φ(A f (x))

dxx

� 1L

∫ ∞

0w(x)Φ( f (x))

dxx

holds for all measurable functions f : R+ →R such that f (x)∈ I for all x∈R+ , where

A f (x) =1L

∫ ∞

0f (tx)dλ (t) and w(x) =

∫ ∞

0u(x

t

)dλ (t) < ∞ , x ∈ R+ .

Observe that a non-decreasing and bounded function m : [0,∞) → R such thatM = m(∞)−m(0) > 0 induces a finite Borel measure λ on R+ and vice versa. Forsuch a function and measure, related Lebesgue and Lebesgue-Stieltjes integrals areequivalent. Thus, all the above results can be stated for A f (x) defined by

A f (x) =1M

∫ ∞

0f (tx)dm(t), x ∈ R+,

so they refine and generalize inequality (1).Another generalization of (1) was given by D. Luor [8] in a setting with σ -finite

Borel measures μ and ν on a topological space X and a Borel probability measure λon R+ . The weighted version of that Luor’s result is obtained in [5] in a setting with atopological space and σ -finite Borel measures as following.

Let λ be a finite Borel measure on R+ . By supp λ we denote its support, that is,the set of all t ∈ R+ such that λ (Nt) > 0 holds for all open neighbourhoods Nt of t .Hence,

L =∫

supp λdλ (t) =

∫ ∞

0dλ (t) = λ (R+) < ∞. (3)

On the other hand, let X be a topological space equipped with a continuous scalarmultiplication (a,x) �→ ax ∈ X , for a ∈ R+ , x ∈ X , such that

1x = x, a(bx) = (ab)x, x ∈ X , a,b ∈ R+.

Further, let the Borel set Ω ⊆ X be λ -balanced, that is, tΩ = {tx : x ∈ Ω} ⊆ Ω , forall t ∈ supp λ . For a Borel measurable function f : Ω → R , we define its Hardy–Littlewood average A f as

A f (x) =1L

∫ ∞

0f (tx)dλ (t), x ∈ Ω. (4)

Finally, suppose that μ and ν are σ –finite Borel measures on X . For t > 0 anda Borel set S ⊆ X we define

μt(S) = μ(

1tS

). (5)

Obviously, μt is a σ -finite Borel measure on X for each t ∈ R+ . Throughout thispaper, we suppose that the measures μt are absolutely continuous with respect to the

measure ν , that is, μt � ν for each t ∈ supp λ . As usual, bydμt

dνwe denote the

related Radon–Nikodym derivative.Thus, the following weighted general Boas-type inequality is given in [5].

BOAS-TYPE INEQUALITY FOR 3 -CONVEX FUNCTIONS AT A POINT 1365

THEOREM 1. Let λ be a finite Borel measure on R+ and L be defined by (3).Let μ and ν be σ -finite Borel measures on a topological space X , μt be defined by(5) and such that μt � ν for all t ∈ supp λ . Further, let Ω ⊆ X be a λ -balanced setand u be a non-negative function on X , such that

v(x) =∫ ∞

0u

(1tx)

dμt

dν(x)dλ (t) < ∞, x ∈ Ω. (6)

Suppose Φ : I → R is a non-negative convex function on an interval I ⊆ R . If f : Ω →R is a Borel measurable function, such that f (x) ∈ I for all x ∈ Ω , and A f is definedby (4), then A f (x) ∈ I for all x ∈ Ω and the inequality∫

Ωu(x)Φ(A f (x))dμ(x) � 1

L

∫Ω

v(x)Φ( f (x))dν(x) (7)

holds. For a non-positive concave function Φ , the sign of inequality in (7) is reversed.

Notice that the condition on non-negativity of the convex function Φ in Theorem1 can be omitted only in a particular setting with cones in X . More precisely, thefollowing corollary holds.

COROLLARY 1. If in Theorem 1 we have tΩ = Ω for λ -a.e. t ∈ supp λ , then (7)holds for all convex functions Φ on an interval I ⊆ R . In that case, for all concavefunctions Φ relation (7) holds with the sign of inequality reversed.

We will make further generalization based on the inequality (7) . Instead of convexfunctions we will introduce a different class of functions, following the idea of I. A.Baloch et al. [1] and J. Pecaric et al. [10].

This paper is organized in following way: after the Introduction, in Section 2 wedefine class of functions K c

1 (I) and prove the Boas inequality of Levinson type forsuch functions. We point out the dual class of functions and corresponding dual in-equality. We discuss 3-convexity at the point and give several one-dimensional results.In Section 3 we obtain multidimensional results and examples of Levinson type con-cerning balls in R

n centred at the origin and their dual sets.

CONVENTIONS. An interval I in R is any convex subset of R , while by IntI wedenote its interior. By R+ we denote the set of all positive real numbers i.e. R+ =(0,∞) . A k th order divided difference of a function f : I → R , where I is an intervalin R , at distinct points x0, . . . ,xk ∈ I is defined recursively by

[xi] f = f (xi), for i = 0, . . . ,k

and

[x0, . . . ,xk] f =[x1, . . . ,xk] f − [x0, . . . ,xk−1] f

xk − x0.

A function f : I → R is called k -convex if [x0, . . . ,xk] f � 0 for all choices of k + 1distinct points x0, . . . ,xk ∈ I . If the k th derivative f (k) of a k -convex function exists,then f (k) � 0, but f (k) may not exist (for properties of divided differences and k -convex

functions see [11]). For R > 0 we denote by B(R) a ball in Rn centred at the origin and

of radius R , that is, B(R) = {x ∈ Rn : |x| � R} , where |x| denotes the Euclidean norm

of x ∈ Rn . By its complementary set we mean the set R

n \B(R) = {x ∈ Rn : |x| > R} .

2. Main results

In order to make generalization of the inequality (7) in Levinson’s sense we willreplace convex functions with the following class of functions.

DEFINITION 1. Let f : I → R , where I is an interval in R , be a function and c ∈IntI . We say that f ∈ K c

1 (I) (resp. f ∈ K c2 (I)) if there exists a constant α such that

the function F(x) = f (x)− α2 x2 is concave (resp. convex) on I ∩ (−∞,c] and convex

(resp. concave) on I∩ [c,∞) .

REMARK 1. If f ∈ K ci (a,b) , i = 1,2, and f ′′(c) exists, then f ′′(c) = α . Let

f ∈ K c1 (a,b) . Due to the concavity and convexity of F for every distinct points x j ∈

(a,c] and y j ∈ [c,b) , j = 1,2,3, we have

[x1,x2,x3]F = [x1,x2,x3] f −α � 0 � [y1,y2,y3] f −α = [y1,y2,y3]F.

Therefore, if f ′′−(c) and f ′′+(c) exist, letting x j ↗ c and y j ↘ c , we get

f ′′−(c) � α � f ′′+(c).

Similary, for f ∈ K c2 (a,b) , we have f ′′+(c) � α � f ′′−(c) . �

REMARK 2. Function f : I → R is 3-convex (resp. 3-concave) if and only if f ∈K c

1 (I) (resp. f ∈ K c2 (I)) for every c ∈ IntI . In other words, a function is 3-convex

on an interval if and only if it is 3-convex at every point of its interior, so the propertyfrom the definition of K c

1 (I) can be described as “3-convexity at point c”.

For the main result we need another set of measures, sets and functions that alsosatisfying Theorem1. So, let λ be a finite Borel measure on R+ such that

L =∫ ∞

0dλ (t) =

∫suppλ

dλ(t) < ∞. (8)

Let Ω ⊆ X be a λ -balanced Borel set, let measures μ , μt ,t ∈ R+ and ν be σ -finite Borel measures on X such that μt(S) = μ

( 1t S), for t ∈ R+ and S ⊆ X Borel set

and μt � ν , t ∈ supp λ . Finally, let u be a non-negative function on X , such that

v(x) =∫ ∞

0u

(1tx)

dμt

dν(x)dλ(t) < ∞, x ∈ Ω. (9)

For a Borel measurable function g : Ω → R the Hardy-Littlewood average Ag ofg defined by

Ag(x) =1

L

∫ ∞

0g(tx)dλ (t), x ∈ Ω.


THEOREM 2. Let X , Ω , λ , μ , ν , μt , L , u , v be as in Theorem 1. Furthermore,let Ω , λ , μ , ν , μt , L , u , v be another set of measures and functions that satisfyTheorem 1. If f : Ω → I ∩ (−∞,c] and g : Ω → I ∩ [c,∞) are measurable functionssatisfying

∫Ω

u(x)(A f (x))2dμ(x)− 1L

∫Ω

v(x) f 2(x)dν(x)

=∫

Ωu(x)(Ag(x))2dμ(x)− 1

L

∫Ω

v(x)g2(x)dν(x) (10)

then for every Φ ∈ K c1 (I) the following inequality holds

∫Ω

u(x)Φ(Ag(x))dμ(x)− 1

L

∫Ω

v(x)Φ(g(x))dν(x)

�∫

Ωu(x)Φ(A f (x))dμ(x)− 1

L

∫Ω

v(x)Φ( f (x))dν(x). (11)

If Φ ∈ K c2 (I) in the above setting, then (11) holds with the sign of inequality reversed.

Proof. From Definition 1 there exists a constant α such that F(x) = Φ(x)− α2 x2

is concave on I∩ (−∞,c] so we can apply Theorem 1 on the function F and get

∫Ω

u(x)F(A f (x))dμ(x)− 1L

∫Ω

v(x)F( f (x))dν(x) � 0

By the definition of the function F we have

∫Ω

u(x)[Φ(A f (x))− α

2(A f (x))2

]dμ(x)− 1

L

∫Ω

v(x)[Φ( f (x))− α

2f 2(x)

]dν(x) � 0.

Since integral is a linear functional we can write

∫Ω

u(x)Φ(A f (x))dμ(x)− 1L

∫Ω

v(x)Φ( f (x))dν(x)

� α2

[∫Ω

u(x)(A f (x))2dμ(x)− 1L

∫Ω

v(x) f 2(x)dν(x)]. (12)

For the same constant α , the second part of Definition 1 gives us a convex functionF(x) = Φ(x)− α

2 x2 on I∩ [c,∞) . Now, from Theorem 1 we have

∫Ω

u(x)F(Ag(x))dμ(x)− 1

L

∫Ω

v(x)F(g(x))dν(x) � 0.

Similarly as in the first part of the proof, we obtain

∫Ω

u(x)[Φ(Ag(x))− α

2(Ag(x))2

]dμ(x)− 1

L

∫Ω

v(x)[Φ(g(x))− α

2g2(x)

]dν(x) � 0


and also ∫Ω

u(x)Φ(Ag(x))dμ(x)− 1

L

∫Ω

v(x)Φ(g(x))dν(x)

� α2

[∫Ω

u(x)(Ag(x))2dμ(x)− 1

L

∫Ω

v(x)g2(x)dν(x)]. (13)

Due to assumption (10) the right hand sides of inequalities (12) and (13) are equal.Hence, we obtain (11). In the case Φ ∈ K c

2 (I) , the function F(x) = Φ(x)− α2 x2 is

convex on I∩ (−∞,c] and concave on I∩ [c,∞) . Following the idea of the first part ofthe proof we get our statement. �

Similarly as in [9] we analize α from the Definition 1.

REMARK 3. The assumption of equality (10) in Theorem 2 can be weakened.More concretely, if

(a) α � 0 and ∫Ω

u(x)(A f (x))2dμ(x)− 1L

∫Ω

v(x) f 2(x)dν(x)

�∫


L

∫Ω

v(x)g2(x)dν(x), (14)

or

(b) α � 0 and ∫Ω

u(x)(A f (x))2dμ(x)− 1L

∫Ω

v(x) f 2(x)dν(x)

�∫


L

∫Ω

v(x)g2(x)dν(x), (15)

then (11) holds. Indeed, if we multiply (14) with α2 � 0 we get

α2

[∫Ω

u(x)(A f (x))2dμ(x)− 1L

∫Ω

v(x) f 2(x)dν(x)]

� α2

[∫Ω

u(x)(Ag(x))2dμ(x)− 1

L

∫Ω

v(x)g2(x)dν(x)]

(16)

so we can chain inequalities (12) and (13) to get (11). In the case when we multiply(15) with α

2 � 0 we again get (16) and the conclusion is the same.

COROLLARY 2. Let X , Ω , Ω , λ , λ , μ , μ , ν , ν , μt , μt , L , L , u , u , v , v beas in Theorem 2 and assume that (10) holds. If Φ is 3 -convex on the interval I , then(11) holds. If Φ is 3 -concave, then (11) holds with the sign reversed.

Proof. If Φ is 3-convex, then by Remark 2 it also belongs to K c1 (I) for every

c ∈ Int I , so we can again apply Theorem 2. �For Lebesgue measures and some intervention on the weight functions, from The-

orem 2 we obtain the following result.

COROLLARY 3. Let λ and λ be finite Borel measures on R+ and L and L bedefined by (3) and (8) respectively. Let Ω ⊆R+ be a λ -balanced set such that tΩ = Ωfor λ -a.e. t ∈ supλ and Ω ⊆ R+ be a λ -balanced such that tΩ = Ω for λ -a.e.t ∈ sup λ . Suppose that u and u are non-negative functions on R+ , such that

w(x) =∫ ∞

0u(x

t

)dλ (t) < ∞, x ∈ Ω

andw(x) =

∫ ∞

0u(x

t

)dλ (t) < ∞, x ∈ Ω.

If f : Ω → I∩ (−∞,c] and g : Ω → I∩ [c,∞) are measurable functions satisfying∫Ω

u(x)(A f (x))2 dxx− 1

L

∫Ω

w(x) f 2(x)dxx

=∫

Ωu(x)(Ag(x))2 dx

x− 1

L

∫Ω

w(x)g2(x)dxx

,

the following inequality∫Ω

u(x)Φ(Ag(x))dxx− 1

L

∫Ω

w(x)Φ(g(x))dxx

�∫

Ωu(x)Φ(A f (x))

dxx− 1

L

∫Ω

w(x)Φ( f (x))dxx

(17)

holds for Φ ∈ K c1 (I) . If Φ ∈ K c

2 (I) in the above setting, then (17) holds with the signof inequality reversed.

Proof. It follows directly from Theorem 2 if we set X = R+ , the measures μ , ν ,μ and ν to be the Lebesgue measures and replace the weight functions u and u with

x �→ u(x)x

, x �→ u(x)x

respectively. For such measures we getdμt

dν(x) =

dμt

dν(x) =

1t

,

t ∈ R+ . In this setting, we have

v(x) =∫ ∞

0u(x

t

)· tx· 1t

dλ (t) =1x

∫ ∞

0u(x

t

)dλ (t) =

w(x)x

, x ∈ Ω,

and v(x) =w(x)

x, x ∈ Ω where the function v and v are defined by (6) and (9). �

EXAMPLE 1. Consider the Theorem 2 with X = Ω = R+ dλ (t) = χ(0,1)(t)dt .For 0 < b � ∞ , let dμ(x) = χ(0,b)(x)dx , and ν(x) = dx . Instead of the weight u

we take the function x �→ u(x)x

χ(0,b)(x) . Then supp λ = (0,1] , L = 1, tΩ = Ω for

t ∈ supp λ ,dμt

dν(x) =

1t

χ(0,tb)(x) ,

A f (x) =∫ 1

0f (tx)dt = H f (x),

and

v(x) =∫ 1

0

u(

1t x)

1t x

· 1t

χ(0,tb)(x)dt =1x

∫ 1

xb

u(x

t

)dt =

∫ b

xu(y)

dyy2 =

w(x)x

,

for x ∈ (0,b) .For measurable functions f : R+ → I∩ (−∞,c] and g : Ω → I∩ [c,∞) the condi-

tion (10) becomes∫ b

0u(x)(H f (x))2 dx

x−∫ b

0w(x) f 2(x)

dxx

=∫

Ωu(x)(Ag(x))2 dμ(x)− 1

L

∫Ω

v(x)g2(x)dν(x),

and for a function Φ from K c1 (I) the following inequality∫

Ωu(x)Φ(Ag(x))dμ(x)− 1

L

∫Ω

v(x)Φ(g(x))dν(x)

�∫ b

0u(x)Φ(H f (x))

dxx−∫ b

0w(x)Φ( f (x))

dxx

(18)

holds. If Φ is from K c2 (I) , then the sign of inequality (18) is reversed.

On the other hand, we have dual example.

EXAMPLE 2. Let X = Ω = R+ and dλ (t) = χ[1,∞)(t)dtt2

in the Theorem 2. For

0 � b < ∞ , let u : (b,∞) → R be a non-negative locally integrable function on its do-main. Let dμ(x) = χ(b,+∞)(x)dx and dν(x) = dx . Then we get a dual result to (18)

(see also [3, 4, 6]). So supp λ = [1,∞) , L = 1, w(x) =1x

∫ x

bu(t)dt and

Ag(x) =∫ ∞

1g(tx)

dtt2

= x∫ ∞

xg(t)

dtt2

= Hg(x), x ∈ (b,∞).

For measurable functions f : Ω → I∩(−∞,c] and g : Ω → I∩ [c,∞) the condition(10) becomes ∫

Ωu(x)(A f (x))2 dμ(x)− 1

L

∫Ω

v(x) f 2(x)dν(x)

=∫ ∞

0u(x)(Hg(x))2 dx

x−∫ ∞

0w(x)g2(x)

dxx

,

BOAS-TYPE INEQUALITY FOR 3-CONVEX FUNCTIONS AT A POINT 1371

and for Φ ∈ K c1 (I) the following inequality∫ ∞

0u(x)Φ(Hg(x))

dxx−∫ ∞

0w(x)Φ(g(x))

dxx

�∫


L

∫Ω

v(x)Φ( f (x))dν(x) (19)

holds. If Φ ∈ K c2 (I) , then the sign of inequality (19) is reversed.

3. Multidimensional examples

In Corollary 1 the condition tΩ = Ω , λ -a.e. t ∈ supp λ is emphasized, so thelogical choice of the multidimensional examples is setting with balls in R

n centred atthe origin.

COROLLARY 4. Suppose that 0 < b � ∞ and that a positive function ψ on [0,1]and a non-negative function u on R

n are such that

v(x) =∫ 1

|x|b

u

(1tx)

t−nψ(t)dt < ∞, x ∈ B(b) (20)

and

P1 =∫ 1

0ψ(t) dt < ∞. (21)

If f : B(b) → I∩ (−∞,c] and g : Ω → I∩ [c,∞) are measurable functions satisfying

1

P21

∫B(b)

u(x)(∫ 1

0ψ(t) f (tx)dt

)2

dx− 1P1

∫B(b)

v(x) f 2(x)dx

=∫


L

∫Ω

v(x)g2(x)dν(x), (22)

then for Φ ∈ K c1 (I) the inequality∫

Ωu(x)Φ(Ag(x))dμ(x)− 1

L

∫Ω

v(x)Φ(g(x))dν(x)

�∫

B(b)u(x)Φ

(1P1

∫ 1

0ψ(t) f (tx)dt

)dx− 1

P1

∫B(b)

v(x)Φ( f (x))dx (23)

holds.

Proof. Follows from Theorem 2 rewritten with X = Rn , Ω = B(b) , dλ (t) =

ψ(t)χ(0,1)(t)dt , dμ(x) = χB(b)(x)dx , and dν(x) = dx . Here we have supp λ = (0,1] ,dμt

dν(x) = t−nχB(tb)(x) , and A f (x) =

1P1

∫ 1

0ψ(t) f (tx)dt . It is easy to see that in this

setting (20) reduces to (6), and (10) and (11) becomes (22) and (23). �Applying Corollary 4 to some particular u and Φ we get the following result.

EXAMPLE 3. Apply Corollary 4 for u(x) ≡ 1 and the 3-convex function Φ(x) =xp , p > 2 or p ∈ (0,1) . In this setting, if f : B(b) → I∩ (−∞,c] and g : Ω → I∩ [c,∞)are measurable functions satisfying

1

P21

∫B(b)

(∫ 1

0ψ(t) f (tx)dt

)2

dx− 1P1

∫B(b)

v(x) f 2(x)dx

=∫


L

∫Ω

v(x)g2(x)dν(x),

then the following inequality∫Ω

u(x)(Ag(x))pdμ(x)− 1

L

∫Ω

v(x)gp(x)dν(x)

� 1Pp

1

∫B(b)

(∫ 1

0ψ(t) f (tx)dt

)p

dx− 1P1

∫B(b)

v(x) f p(x)dx

holds, where P1 is defined by (21). Notice that Φ(x) = xp , p ∈ (1,2) or p < 0 is a3-concave function.

Similarly, we get the dual result by using the set Rn \B(b) .

COROLLARY 5. Suppose that 0 � b < ∞ and that the positive function ψ on[1,∞) and the non-negative function u on R

n are such that

v(x) =∫ |x|

b

1u

(1tx)

t−nψ(t)dt < ∞, x ∈ Rn \B(b) (24)

and

P∞ =∫ ∞

1ψ(t) dt < ∞. (25)

If f : Ω→ I∩(−∞,c] and g : Rn\B(b)→ I∩[c,∞) are measurable functions satisfying

∫Ω

u(x)(A f (x))2dμ(x)− 1L

∫Ω

v(x) f 2(x)dν(x)

=1P2

∞

∫Rn\B(b)

u(x)(∫ ∞

1ψ(t)g(tx)dt

)2

dx− 1P∞

∫Rn\B(b)

v(x)g2(x)dx (26)

then for Φ ∈ K c1 (I) the following inequality

∫Rn\B(b)

u(x)Φ(

1P∞

∫ ∞

1ψ(t)g(tx)dt

)dx− 1

P∞

∫Rn\B(b)

v(x)Φ(g(x))dx

�∫


L

∫Ω

v(x)Φ( f (x))dν(x) (27)

holds.


Proof. The proof follows from Theorem 2 if we set dλ (t) = ψ(t)χ(1,∞)(t)dt , X =R

n , Ω = Rn \B(b) , dμ(x) = χRn\B(b)(x)dx and dν(x) = dx . Then we get supp λ =

[1,∞) ,dμt

dν(x) = t−nχRn\B(tb)(x) and Ag(x) =

1P∞

∫ ∞

1ψ(t)g(tx)dt . So (6), (10) and

(11) become (24), (26) and (27), respectively. �

EXAMPLE 4. Apply Corollary 5 for u(x) ≡ 1 and the 3-convex function Φ(x) =xp , p > 2 or p ∈ (0,1) . If f : Ω → I ∩ (−∞,c] and g : R

n \ B(b) → I ∩ [c,∞) aremeasurable functions satisfying

∫Ω

u(x)(A f (x))2dμ(x)− 1L

∫Ω

v(x) f 2(x)dν(x)

=1P2

∞

∫Rn\B(b)

(∫ ∞

1ψ(t)g(tx)dt

)2

dx− 1P∞

∫Rn\B(b)

v(x)g2(x)dx,

then the following inequality

∫Rn\B(b)

(1P∞

∫ ∞

1ψ(t)g(tx)dt

)p

dx− 1P∞

∫Rn\B(b)

v(x)gp(x)dx

�∫

Ωu(x)(A f (x))pdμ(x)− 1

L

∫Ω

v(x) f p(x)dν(x)

holds, where P∞ is defined by (25).

RE F ER EN C ES

[1] I. A. BALOCH, J. PECARIC, M. PRALJAK, Generalization of Levinson’s inequality, J. Math. Inequal.9 (2) (2015), 571–586

[2] R. P. BOAS, Some integral inequalities related to Hardy’s inequality, J. Anal. Math. 23 (1970), 53–63.

[3] A. CIZMESIJA, S. HUSSAIN, AND J. PECARIC, Some new refinements of Hardy and Polya–Knopp’sinequalities, J. Funct. Spaces Appl. 7 (2) (2009), 167–186.

[4] A. CIZMESIJA, J. PECARIC, AND L.-E. PERSSON, On strengthened Hardy and Polya-Knopp’s in-equalities, J. Approx. Theory 125 (2003), 74–84.

[5] A. CIZMESIJA, J. PECARIC, D. POKAZ, A new general Boas-type inequality and related Cauchy-typemeans, Math. Inequal. Appl., 12 (3) (2012), 599–617.

[6] S. KAIJSER, L. NIKOLOVA, L. E. PERSSON, AND A. WEDESTIG, Hardy-Type Inequalities via Con-vexity, Math. Inequal. Appl., 8 (2005), 403–417.

[7] N. LEVINSON, Generalizations of an Inequality of Hardy, Duke Math. J. 31 (1964), 389–394.

[8] D. LUOR, Modular inequalities for the Hardy-Littlewood averages, Math. Inequal. Appl. 13 (3)(2010), 635–642.

[9] K. KRULIC HIMMELREICH, J. PECARIC, D. POKAZ, M. PRALJAK, New results about Hardy-typeinequality, J. Math. Inequal., 9 (4) (2015), 1259-1269.


[10] J. PECARIC, M. PRALJAK, A. WITKOWSKI,Generalized Levinson’s inequality and exponential con-vexity, Opuscula Math., 35 (1) (2015), 397–410.

[11] J. PECARIC, FRANK PROSCHAN, Y. L. TONG, Convex Functions, Partial Orderings and StatisticalApplications, Academic Press, San Diego, 1992.

(Received April 1, 2016) Josip PecaricFaculty of Textile Technology, University of Zagreb


Dora PokazFaculty of Civil Engineering, University of Zagreb

Kaciceva 26, 10000 Zagreb, Croatiae-mail: [email protected]

Marjan PraljakFaculty of Food Technology and Biotechnology, University of Zagreb



& Applications


INEQUALITIES AND BOUNDS FOR A

CERTAIN BIVARIATE ELLIPTIC MEAN

EDWARD NEUMAN


Abstract. This paper deals with a new mean introduced recently by this author. This mean is adegenerate case of the completely symmetric elliptic integral of the second kind. In particularinequalities involving mean under discussion are obtained. Also, bounds in the mean in questionare obtained. Bounding expressions are convex combinations of some quantities depending onvariables of the mean.

1. Introduction and notation

In recent years certain bivariate means have been investigated extensively by sev-eral researchers. A complete list of research papers which deal with this subject is toolong to be included here even if we would restrict our attention to papers published inthe last ten years. The goal of this paper is to obtain inequalities and optimal boundsfor the particular mean introduced recently by this author (see [15]). Its definition isincluded below (see (2)). In what follows the letters a and b will always stand forpositive and unequal numbers.

First we recall definition of the Schwab-Borchardt mean of a and b :

SB(a,b)≡ SB =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

√b2−a2

cos−1(a/b)if a < b,

√a2−b2

cosh−1(a/b)if a > b

(1)

(see, e.g., [2], [3]). This mean has been studied extensively in [19], [20] and in [8]. It iswell known that the mean SB is strict, nonsymmetric and homogeneous of degree onein its variables.

Mean SB can also be expressed in terms of the degenerated completely symmetricelliptic integral of the first kind (see, e.g., [15]). It has been pointed out in [19] thatsome well known bivariate means such as logarithmic mean and two Seiffert means(see [23, 24]) can be represented by the Schwab-Borchardt mean of two simpler meanssuch as geometric and arithmetic means or as the Schwab-Borchardt mean of arithmeticand the square - mean root mean. This idea was utilized lately by this author and other

Mathematics subject classification (2010): 26E60, 26D05.Keywords and phrases: Bivariate elliptic means, inequalities, Ky Fan inequalities.


1375

1376 E. NEUMAN

researchers as well. For more details the interested reader is referred to [4, 5, 6, 7, 8, 9,10, 13, 14, 18, 21, 22, 25, 26]

The mean studied in this paper is defined as follows:

N(a,b) ≡ N =12

(a+

b2

SB(a,b)

)(2)

(see [15]). It’s easy to see that mean N is also strict, nonsymmetric and homogeneousofdegree one in its variables. Some authors call this mean, Neuman mean of the secondkind (see, e.g., [5, 7, 21, 22, 25, 26]). Mean N can also be represented in terms ofthe degenerated completely symmetric elliptic integral of the second kind (see, e.g.,[15]). By taking the N -mean of two other means one can generate several new bivariatemeans. This idea was utilized in [15].

This paper can be regarded as continuation of investigations initiated in author’searlier papers [18, 8, 15, 11, 10, 9, 13, 16, 14, 12, 17] and is organized as follows.Some preliminary results and formulas needed in this paper are given in Section 2.Inequalities involving mean N are derived in Section 3. Bounds for the mean underdiscussion are obtained in Section 4. The Ky Fan type inequalities are established inSection 5.

2. Preliminary results and formulas needed in this paper

First of all let us record another formulas for means SB and N . Those will beutilized frequently in susequent sections of this paper.

One can easily verify that (1) implies

SB(a,b)≡ SB =

⎧⎪⎪⎨⎪⎪⎩

bsinrr

= atanr

rif a < b,

bsinhs

s= a

tanhss

if b < a,

(3)

where

cosr = a/b if a b. (4)

Clearly

0 < r <π2

(5)

ands > 0. (6)

Corresponding formulas for the mean N , obtained with the aid of (2) and (3), readas follows:

N(a,b) ≡ N =12b(cosr+

rsinr

)=

12a(1+

rsinrcosr

)(7)

BIVARIATE ELLIPTIC MEAN 1377

provided a b , then

N(a,b) ≡ N =12b(coshs+

ssinhs

)=

12a(1+

ssinhscoshs

). (8)

Here the domains for r and s are the same as these in (5) and (6).For later use let

v =a−ba+b

. (9)

Clearly 0 < |v| < 1.The unweighted arithmetic mean A of a and b is defined as

A =a+b

2.

For the reader’s convenience let us recall definitions of the first and the secondSeiffert means, denoted by P and T , respectively, the Neuman-Sandor mean M , andthe logarithmic mean L :

P = Av

sin−1 v, T = A

vtan−1 v

,

M = Av

sinh−1 v, L = A

v

tanh−1 v,

(10)

(see [23], [24], [19]).We will also utilize the l’Hopital Monotone Rule [1]:Let c,d ∈ R (c < d) and let f ,g : [c,d]→ R be continuous functions that are dif-

ferentiable on (c,d) . Assume that g′(x) �= 0 for each x ∈ (c,d). If f ′/g′ is increasing

(decreasing) on (c,d) , then so aref (x)− f (c)g(x)−g(c)

andf (x)− f (d)g(x)−g(d)

. If mono-

tonicity of f ′/g′ is strict, then so is monotonicity of two functions represented by theabove quotients.

3. Inequalities involving mean N

The goal of this section is to establish two inequalities which connect the Schwab-Borchardt mean SB with the mean N . We have the following:

THEOREM 1. Let a,b > 0 , a �= b. Then

SB(a,b) <a+2b

3<

b+N(b,a)2

< N(a,b). (11)

If a > b, then

N(a,b) < A < SB(b,a). (12)

1378 E. NEUMAN

Proof. The first inequality in (11) has been established in [19]. In the proof of thesecond inequality in (11) we apply the following one

b+2a3

< N(b,a)

(see [15]) to the third member of (11) to obtain the desired result. The third inequalityin (11) appears in [15, Theorem 4.1]. It is included here for the sake of completeness.For the proof of the first inequality in (12) we use the first part of (8) together with theformula coshs = a/b to obtain

N(a,b) =12

(a+b

ssinhs

)Taking into account that (s/sinhs) < 1 we obtain the desired inequality N(a,b) < A .For the proof of the second inequality in (12) we will utilize the invariance property ofthe Schwab-Borchardt mean

SB(A,√

Aa) = SB(b,a) (13)

(see [2, 3]) and the inequality [19]:

(xy2)1/3 < SB(x,y) (14)

(x,y > 0,x �= y). In (14) we let x = A and y =√

Aa and next apply (13) to obtain

A2/3a1/3 < SB(b,a). (15)

The assumption that a > b yields A < a . This in conjunction with (15) gives the desiredinequality A < SB(b,a) . This completes the proof of the second inequality in (12). �

4. Bounds for N

For the sake of presentation we introduce two auxiliary functions

Φ1(r) =2sinr− sinrcosr− r

2(sinr)(1− cosr)(16)

(0 < r < π/2) and

Ψ1(s) =s+ sinhscoshs−2sinhs

2(sinhs)(coshs−1)(17)

(s > 0) . It is known [15] that the function Φ1(r) is strictly decreasing while the func-tion Ψ1(s) is strictly increasing. Moreover, Φ1(0+) = 1/3 and Φ1(π/2) = 1−π/4.Also, Ψ1(0+) = 1/3 and Ψ1(∞−) = 1/2.

We are in a position to establish the following:

THEOREM 2. If a < b, then the simultaneous inequality

αa+(1−α)b < N(a,b) < βa+(1−β )b (18)

holds true if

α � 13

and β � 1− π4

= 0.214 . . . . (19)

If a > b, then the inequality (18) is valid if

α � 13

and β � 12. (20)

Proof. We shall prove first the theorem in the case when a < b . It follows from(18) that

β <N/b−1a/b−1

< α. (21)

Utilizing (7) and the first formula in (4) we can write (21) as follows

β < 1− π4

� Φ1(r) � 13

< α.

Hence the assertion follows. Assume now that a > b . We follow the idea used in thefirst part of this proof. Let us note that in this case inequalities in (21) are reversed, i.e.we have α < Ψ1(s) < β . Combining this with relevant parts of l’Hopital MonotoneRule yields

α <13

� Ψ1(s) � 12

< β .

The proof is complete. �In the proof of the next result we will utilize the following function

Φ2(r) =r2 + r sinrcosr−2sin2 r

2(sinr)(r− sin r)(22)

(0 < r < π/2) .Our next task is to determine all the parameters α and β for which the following

inequality

αb+(1−α)SB(a,b)< N(a,b) < βb+(1−β )SB(a,b), (23)

is satisfied for positive numbers a and b which satisfy the condition a < b .We shall prove now the following:

THEOREM 3. Inequality (23) holds true if

α � 0 and β � π2−84π −8

= 0.409 . . . . (24)

1380 E. NEUMAN

Proof. Let a < b . Making use of (3) and (7) we can easily show that the two-sidedinequality (23) can be written in the form

α < Φ2(r) < β . (25)

Taking into account that the function Φ2(r) is strictly increasing (see [22, Lemma 2.4])and also that Φ2(0+) = 0 and Φ2(π/2) = (π2−8)/(4π−8)) we conclude, using (25),that conditions (24) must be satisfied in order for the inequality (23) to be valid. �

We shall now illustrate the last result with the following:

EXAMPLE 1. Let A and G stand for the unweighted arithmetic and geometricmeans of two positive unequal numbers, and also let P be the first Seiffert mean ofthe same numbers. Writing NGA for N(G,A) we obtain using (23) and the fact thatP = SB(G,A)

αA+(1−α)P < NGA < βA+(1−β )P,

where α and β must to satisfy conditions (24). In particular, with α = 0 and β = 12 ,

we obtain the inequality

P < NGA <12(A+P)

which is a possibly a new one.

We shall discuss now a problem of finding bounds for N(a,b) in the form ofgeometric means of a and b :

aαb1−α < N(a,b) < aβ b1−β . (26)

We have the following:

THEOREM 4. If a < b, then the inequality (26) is satisfied for all numbers α andβ such that α � 1/3 and β � 0 . Otherwise, if a > b, then (26) is valid if α � 1/3and β � 1 .

Proof. For the proof of the first part of the assertion we rewrite (26) a as follows

β <log(N/b)log(a/b)

< α, (27)

where N ≡ N(a,b) . Using (4) and (7) we write the above two-sided inequality as

β < Φ3(r) < α

where

Φ3(r) =log(sin2r+2r)− log(2sinr)− log2

log(cosr)

(0 < r < π/2) . It is known (see [6, Lemma 2.3]) that the function Φ3(r) is strictlydecreasing on (0,π/2) . Moreover, 0 � Φ3(r) � 1/3 on the stated domain. Hence theassertion follows.

The second part of the thesis can be established in a similar manner. Using (26)we have

α <log(N/b)log(a/b)

< β

Utilizing (4) and (8) we can write the above inequality in the form

α < Ψ3(s) < β ,

where

Ψ3(s) =log(sinh2s+2s)− log(2sinhs)− log2

log(coshs)

(s > 0) . Making use of Lemma 2.4 in [6] we conclude that the function Ψ3(s) isstrictly increasing provided s > 0 and also that 1/3 � Ψ3(s) � 1. The assertion nowfollows. The proof is complete. �

We shall now deal with problems of finding bounds for the reciprocals of the meanN in terms of reciprocals of its variables a and b . Let now a < b . More exactly we arelooking for all numbers α and β for which the inequality

α1a

+(1−α)1b

<1N

< β1a

+(1−β )1b

(28)

holds true.

THEOREM 5. If a < b, then the inequality (28) is satisfied provided α � 0 andβ � 1/3 . Otherwise, if a > b, then the inequalities (28) hold true if α � 1 and β �1/3 .

Proof. It is easy to see that (28) is equivalent to

α <

ab

1− ab

1− Nb

Nb

< β

provided a < b . Let us denote the second member of the above inequality by Φ4 . Thenutilizing (4) and (7) we get

Φ4 ≡ Φ4(r) =cosr

1− cosr2sinr− sinrcosr− r

sinrcosr+ r

(0 < r < π/2) . Making use of Lemma 2.8 in [4] we conclude that the function Φ4(r)is strictly decreasing on its domain and also that Φ4(0+) = 1/3 and Φ4(π/2)= 0. Thisyields

α < 0 � Φ4(r) � 1/3 < β (29)

This completes the proof of the first part of the thesis of our theorem.

1382 E. NEUMAN

Assume now that a > b . It is easy to see that the two-sided inequality (28) can bewritten as

β <a/b

1−a/b1−N/b

N/b< α.

Denote the middle member of the above inequality by Ψ4 ≡Ψ4(s) (s > 0) . Using(4) and (8) we get

Ψ4(s) =cosh(s)

1− coshs2sinh(s)− sinhscoshs− s

sinhscoshs+ s.

Making use of Lemma 2.6 in [4] we conclude that the function Ψ4(s) is strictly increas-ing for all s > 0 and also that Ψ4(0+) = 1/3. and Ψ4(∞−) = 1. Hence the assertionfollows. �

5. The Ky Fan inequalities involving mean N

Ky Fan inequalities for various pairs of means have been a subject of many re-search papers published in mathematical literature. The Ky Fan inequalities for theSchwab-Borchardt mean are derived in [19] while the Ky Fan inequalities for particularmeans derived from the N mean are established in [15].

The goal of this section is to establish Ky Fan inequalities for the means N(a,b)and N(b,a) . Before we will state and prove the main result of this section let us intro-

duce more notation. To this end we will assume that 0 < a,b � 12

. Also, let a′ = 1−a

and b′ = 1−b . Research in this section is motivated by validity of the inequality [15,Theorem 4.1]:

b < N(a,b) < N(b,a) < a

provided b < a . It is natural to ask whether this inequality has its counterpart in theform of Ky Fan inequality? The answer is provided in the following:

THEOREM 6. Let 0 < b < a � 12

. Then the inequalities

bb′

<N(a,b)N(a′,b′)

<N(b,a)N(b′,a′)

<aa′

(30)

are valid. Inequalities (30) are reversed if 0 < a < b � 12

.

Proof. It is elementary to show that assumption 0 < b < a � 12

implies the in-

equalitybb′

<aa′

. (31)

We shall establish now inequalities (30). For, let us write the leftmost inequality in (30)as follows

N(a′,b′)b′

<N(a,b)

b

and also introduce f1 , where

f1 =N(a,b)

b.

Application of (8) gives

f1 ≡ f1(s) =sinhscoshs+ s

2sinhs=:

n1(s)d1(s)

,

where coshs = a/b. It is known that the function f1(s) is even and strictly increasingfor s > 0 (see [15, p. 287]). Let s′ be defined implicitly as coshs′ = a′/b′ . Then (31)implies coshs′ < coshs and this in turn yields s′ < s . Further, monotonicity of f1 givesf1(s′) < f1(s) or what is the same that

N(a′,b′)b′

<N(a,b)

b.

Hence the first inequality in (30) follows. For the proof of the second inequality in (30)we introduce

f2 =N(b,a)N(a,b)

.

Using (7) and (8) we obtain

f2 =sinrcosr+ r

(sinr)(1+

ssinhscoshs

) . (32)

Taking into account that coshs = sec r we obtain sinhs = tanr and s = cosh−1(sec r) .With the aid of these formulas we can write (32) as

f2(r) =sinrcosr+ r

sin r+ cos2 rcosh−1(sec r)=:

n2(r)d2(r)

(0 < r < π/2) . We shall show now that the function f2(r) is strictly increasing on itsdomain. Differentiating we obtain(

n′2(r)d′

2(r)

)′=

cosh−1(sec r)[(sinr)cosh−1(sec r)−1]2

> 0.

Thus the function n′2(r)d′2(r)

is strictly increasing. Using l’Hopital Monotone Rule we con-

clude that the function f2 = n2(r)d2(r)

is also strictly increasing. Thus f2(s′) < f2(s). Thelast inequality can be written in terms of the mean N as

N(b′,a′)N(a′,b′)

<N(b,a)N(a,b)

which gives the second inequality in (30). In the proof of the third inequality in (30) weshall use quantity f3 , where

f3 =a

N(b,a).

1384 E. NEUMAN

Making use of (7) we obtain

f3 ≡ f3(r) =2sinr

sinrcosr+ r=:

n3(r)d3(r)

(0 < r < π/2) . It follows from the proof of Theorem 6.1 in [15] that f3(r) is strictlyincreasing on 0 < r < π/2. This in turn implies that f3(s′) < f3(s). Thus the lastinequality gives

a′

N(b′,a′)<

aN(b,a)

.

The third inequality in (30) now follows. The second assertion of the Theorem 6 canbe derived from the first one. It is easy to see that replacing a by b and b′ by a′ gives

the desired result. Let us note that the assumption 0 < a aa′

.

The proof is complete. �

RE F ER EN C ES

[1] G. D. ANDERSON, M. K. VAMANAMURTHY, M. VUORINEN, Monotonicity rules in calculus, Amer.Math. Monthly 133 (2006), 805–816.

[2] J. M. BORWEIN, P. B. BORWEIN, Pi and the AGM – A Study in Analytic Number Theory and Com-putational Complexity, Wiley, New York, 1987.

[3] B. C. CARLSON, Algorithms involving arithmetic and geometric means, Amer. Math. Monthly 78(1971), 496–505.

[4] S.-B. CHEN, Z.-Y. HE, Y.-M. CHU, Y.-Q. SONG, X.-J. TAO, Note on certain inequalities for Neu-man means, J. Inequal. Appl. 2014 2014:370, 10 pages.

[5] Z.-J. GUO, Y.-M. CHU, Y.-Q. SONG, X.-J. TAO, Sharp bounds for Neuman means by harmonic,arithmetic, and contra-harmonic means, Abstr. Appl. Anal. Volume 2014, Article ID914242, 8 pages.

[6] Z.-J. GUO, Y. ZHANG, Y.-M. CHU, Y.-Q. SONG, Sharp bounds for Neuman means in terms ofgeometric, arithmetic and quadratic means, arXiv: 1405, 4384v1, May 2014.

[7] Y.-M. LI, B.-Y. LONG, Y.-M. CHU, Sharp bounds for the Neuman-Sandor mean in terms of gener-alized logarithmic mean, J. Math. Inequal. 4, 4 (2012), 567–577.

[8] E. NEUMAN, Inequalities for the Schwab-Borchardt mean and their applications, J. Math. Inequal. 5,4 (2011), 601–609.

[9] E. NEUMAN, A note on a certain bivariate mean, J. Math. Inequal. 6, 4 (2012), 637–643.[10] E. NEUMAN, Sharp inequalities involving Neuman-Sandor and logarithmic means, J. Math. Inequal.

7, 3 (2013), 413–419.[11] E. NEUMAN, Inequalities involving certain bivariate means, J. Inequal. Spec. Functions 4, 4 (2013),

12–20.[12] E. NEUMAN, A one-parameter family of bivariate mean, J. Math. Inequal. 7, 3 (2013), 399–412.[13] E. NEUMAN, On some means derived from the Schwab-Borchardt mean, J. Math. Inequal. 8, 1 (2014),

171–183.[14] E. NEUMAN, On some means derived from the Schwab-Borchardt mean II, J. Math. Inequal. 8, 2

(2014), 361–370.[15] E. NEUMAN, On a new bivariate mean, Aequat. Math. 88 (2014), 277–289.[16] E. NEUMAN, Inequalities involving generalized trigonometric and hyperbolic functions, J. Math. In-

equal. 8, 4 (2014), 725–736.[17] E. NEUMAN, Optimal bounds for certain bivariate means, Issues of Analysis. 7 (21), 1 (2014), 35–43.


[18] E. NEUMAN, On a new bivariate mean II, Aequat. Math. 89 (2015), 1031–1040.[19] E. NEUMAN, J. SANDOR, On the Schwab-Borchardt mean, Math. Pannon. 14, 2 (2003), 253–266.[20] E. NEUMAN, J. SANDOR, On the Schwab-Borchardt mean II, Math. Pannon. 17, 1 (2006), 49–59.[21] W.-M. QIAN, Y.-M. CHU, Refinements and bounds for Neuman means in terms of arithmetic and

contra-harmonic means, J. Math. Inequal. 9, 3 (2015), 873–881.[22] W.-M. QIAN, Z.-H. SHAO, Y.-M. CHU, Sharp inequalities involving Neuman means of the second

kind and other bivariate means, J. Math. Inequal. 9, 2 (2015), 531–540.[23] H.-J. SEIFFERT, Problem 887, Nieuw. Arch. Wisk. 11 (1993), 176.[24] H.-J. SEIFFERT, Aufgabe 16, Wurzel 29 (1995), 87.[25] Y. ZHANG, Y.-M. CHU, Y.-L. JIANG, Sharp geometric mean bounds for Neuman mean, Abstr. Appl.

Anal., Volume 2014, Article ID 949815, 6 pages.[26] T.-H. ZHAO, Y.-M. CHU, B.-Y. LIU, Optimal bounds for Neuman-Sandor mean in terms of arith-

metic and contra-harmonic means, Abstr. Appl. Anal., Volume 2012, Article ID 302635, 9 pages.

(Received April 6, 2016) Edward NeumanMathematical Research Institute

144 Hawthorn Hollow, Carbondale, IL 62903e-mail: [email protected]


& Applications


GENERALIZATION OF THE JENSEN–MERCER

INEQUALITY BY TAYLOR’S POLYNOMIAL

A. MATKOVIC


Abstract. We present generalizations of the Jensen-Mercer inequality for the class of n -convexfunctions, obtained by using Taylor’s polynomial and Green function. By applying those in-equalities we obtain some results related to Cebysev functionals.

1. Introduction

In paper [1] the following integral version of the Jensen-Mercer inequality forconvex functions was proved.

THEOREM A. Let g : [a,b] → R be a continuous and monotonic function and[α,β ] be an interval such that the image of g is a subset of [α,β ] . Let functionλ : [a,b]→ R be either continuous or of bounded variation satisfying

λ (a) � λ (t) � λ (b) for all t ∈ [α,β ] , λ (b)−λ (a) > 0. (1)

Then for every continuous convex function ϕ : [α,β ] → R the inequality

ϕ

(α + β −

∫ ba g(x)dλ (x)∫ b

a dλ (x)

)� ϕ (α)+ ϕ (β )−

∫ ba ϕ (g(x))dλ (x)∫ b

a dλ (x)(2)

holds.

REMARK 1. Inequality (2) is also valid when the condition (1) is replaced withthe more strict condition that λ is a nondecreasing function such that λ (a) �= λ (b) .

Let us recall the definition of n -convex functions (see [7, pp. 14–15]).

DEFINITION 1. A function f : [a,b]→ R is said to be n -convex on [a,b] , n � 0,if for all choices of (n+1) distinct points in [a,b] , the n -th divided difference of fsatisfies f [x0, . . . ,xn] � 0. If this inequality is reversed, then f is said to be n -concave.If the inequality is strict, then f is said to be a strictly n -convex (n -concave) function.

Mathematics subject classification (2010): 26D15, 26D20.Keywords and phrases: Jensen-Mercer inequality, n -convex functions, exponential convexity.This work has been fully supported by Croatian Science Foundation under the project 5435.


1387


1388 A. MATKOVIC

The divided difference of order n of the function f : [a,b] → R at distinct pointsx0, . . . ,xn ∈ [a,b] is defined recursively by

f [xi] = f (xi) , (i = 0, . . . ,n)

and

f [x0, . . . ,xn] =f [x1, . . . ,xn]− f [x0, . . . ,xn−1]

xn − x0.

The value f [x0, . . . ,xn] is independent of the order of the points x0, . . . ,xn .Note that 0-convex functions are non-negative functions, 1-convex functions are

increasing functions and 2-convex functions are simply convex functions.Consider the Green function G defined on [α,β ]× [α,β ] by

G(t,s) =

{ (t−β )(s−α)β−α for α � s � t,

(s−β )(t−α)β−α for t � s � β .

(3)

The Green function is continuous and convex in s and, since it is symmetric, also in t .It can be easily shown by integrating by parts that every function ϕ : [α,β ] → R ,

ϕ ∈C2 ([α,β ]) can be represented in the form

ϕ (x) =β − xβ −α

ϕ (α)+x−αβ −α

ϕ (β )+∫ β

αG(x,s)ϕ ′′ (s)ds. (4)

LEMMA 1. Let g : [a,b]→R be a continuous and monotonic function, and [α,β ]be an interval such that the image of g is a subset of [α,β ] . Let function λ : [a,b]→ R

be either continuous or of bounded variation satisfying (1) , and G the Green functiondefined by (3) . Then for every function ϕ ∈C2 ([α,β ]) the identity

ϕ

(α + β −


a dλ (x)

)−(

ϕ (α)+ ϕ (β )−∫ ba ϕ (g(x))dλ (x)∫ b

a dλ (x)

)

=∫ β

α

[G

(α + β −


a dλ (x),s

)+∫ ba G(g(x) ,s)dλ (x)∫ b

a dλ (x)

]ϕ ′′ (s)ds (5)

holds.

Proof. Using (4) we obtain

ϕ

(α + β −


a dλ (x)

)−(

ϕ (α)+ ϕ (β )−∫ ba ϕ (g(x))dλ (x)∫ b

a dλ (x)

)

=

(∫ ba g(x)dλ (x)∫ b

a dλ (x)−α

)ϕ (α)β −α

+

(β −


a dλ (x)

)ϕ (β )β −α

+∫ β

αG

(α + β −


a dλ (x),s

)ϕ ′′ (s)ds− (ϕ (α)+ ϕ (β ))

+

∫ ba

[β−g(x)

β−α ϕ (α)+ g(x)−αβ−α ϕ (β )+

∫ βα G(g(x) ,s)ϕ ′′ (s)ds

]dλ (x)∫ b

a dλ (x).

JENSEN-MERCER INEQ. AND TAYLOR’S POLYNOMIAL 1389

Since(∫ ba g(x)dλ (x)∫ b

a dλ (x)−α

)ϕ (α)β −α

+

(β −


a dλ (x)

)ϕ (β )β −α

− (ϕ (α)+ ϕ (β ))

+

∫ ba

[β−g(x)

β−α ϕ (α)+ g(x)−αβ−α ϕ (β )

]dλ (x)∫ b

a dλ (x)

=1

β −α

[∫ ba g(x)dλ (x)∫ b

a dλ (x)ϕ (α)−αϕ (α)+β ϕ (β )−


a dλ (x)ϕ (β )

−β ϕ (α)−β ϕ (β )+ αϕ (α)+αϕ (β )+β ϕ (α)−∫ ba g(x)dλ (x)∫ b

a dλ (x)ϕ (α)

+∫ ba g(x)dλ (x)∫ b

a dλ (x)ϕ (β )−αϕ (β )

]= 0,

(5) immediately follows. �

In the rest of the paper, for the sake of simplicity, let us denote

G (s) = G

(α + β −


a dλ (x),s

)+∫ ba G(g(x) ,s)dλ (x)∫ b

a dλ (x). (6)

2. Generalizations by Taylor’s polynomial

In this section we generalize inequality (2) for n -convex functions using the fol-lowing Taylor’s formula with the integral remainder.

Let n be a positive integer, function ϕ : [α,β ] → R such that ϕ(n−1) is absolutelycontinuous, and c ∈ [α,β ] . Then for all x ∈ [α,β ]

ϕ (x) = Tn−1 (ϕ ;c,x)+Rn−1 (ϕ ;c,x) (7)

holds, where

Tn−1 (ϕ ;c,x) =n−1

∑k=0

ϕ(k) (c)k!

(x− c)k (8)

is Taylor’s polynomial of degree n−1, and the remainder is given by

Rn−1 (ϕ ;c,x) =1

(n−1)!

∫ x

cϕ(n) (t) (x− t)n−1 dt. (9)

Applying Taylor’s formula at the points α and β respectively we get

ϕ (x) =n−1

∑k=0

ϕ(k) (α)k!

(x−α)k +1

(n−1)!

∫ β

αϕ(n) (t)

((x− t)+

)n−1dt, (10)

1390 A. MATKOVIC

and

ϕ (x) =n−1

∑k=0

ϕ(k) (β )k!

(−1)k (β − x)k − 1(n−1)!

∫ β

α(−1)n−1 ϕ(n) (t)

((t− x)+

)n−1dt,

(11)where

(x− t)+ ={

x− t, t � x,0, t > x

. (12)

Note that for n � 1 function((x− t)+

)n−1is convex in x and in t .

Applying Taylor’s formula (7) for ϕ ′′ , we can get the following identities.

LEMMA 2. Let functions g : [a,b] → R , λ : [a,b] → R be as in Lemma 1, andG : [α,β ] → R defined by (6) . Then for every function ϕ : [α,β ] → R such thatϕ(n−1) is absolutely continuous for some n � 3 , the identities

ϕ

(α + β −


a dλ (x)

)−(

ϕ (α)+ ϕ (β )−∫ ba ϕ (g(x))dλ (x)∫ b

a dλ (x)

)

=n−3

∑k=0

ϕ(k+2) (α)k!

∫ β

αG (s) (s−α)k ds

+1

(n−3)!

∫ β

α

(∫ β

tG (s)(s− t)n−3 ds

)ϕ(n) (t)dt, (13)

and

ϕ

(α + β −


a dλ (x)

)−(

ϕ (α)+ ϕ (β )−∫ ba ϕ (g(x))dλ (x)∫ b

a dλ (x)

)

=n−3

∑k=0

ϕ(k+2)(β )k!

(−1)k∫ β

αG (s)(β − s)kds

− 1(n−3)!

∫ β

α

(∫ t

αG (s) (s− t)n−3 ds

)ϕ(n)(t)dt (14)

hold.

Proof. Applying Taylor’s formula (7) for ϕ ′′ , at the points α and β , respectively,and replacing n by n−2 (n � 3) we have

ϕ ′′ (s) =n−3

∑k=0

ϕ(k+2) (α)k!

(s−α)k +1

(n−3)!

∫ s

αϕ(n) (t)(s− t)n−3 dt, (15)

and

ϕ ′′ (s) =n−3

∑k=0

ϕ(k+2) (β )k!

(−1)k (β − s)k − 1(n−3)!

∫ β

sϕ(n) (t)(s− t)n−3 dt. (16)


Using (15) in (5) we get

ϕ

(α + β −


a dλ (x)

)−(

ϕ (α)+ ϕ (β )−∫ ba ϕ (g(x))dλ (x)∫ b

a dλ (x)

)

=n−3

∑k=0

ϕ(k+2) (α)k!

∫ β

αG (s) (s−α)k ds+

1(n−3)!

∫ β

αG (s)

(∫ s

αϕ(n) (t)(s−t)n−3 dt

)ds.

Applying Fubini’s theorem we obtain (13) . Analogously using (16) in (5) and apply-ing Fubini’s theorem we obtain (14) . �

LEMMA 3. Let g : [a,b]→R and λ : [a,b]→R be as in Lemma 1. Then for everyfunction ϕ : [α,β ] → R such that ϕ(n−1) is absolutely continuous for some n � 1 theidentities

ϕ

(α + β −


a dλ (x)

)−(

ϕ (α)+ ϕ (β )−∫ ba ϕ (g(x))dλ (x)∫ b

a dλ (x)

)

=n−1

∑k=1

ϕ(k) (α)k!

⎡⎣(β −


a dλ (x)

)k

− (β −α)k +∫ ba (g(x)−α)k dλ (x)∫ b

a dλ (x)

⎤⎦

+1

(n−1)!

∫ β

α

⎡⎣((α + β −


a dλ (x)− t

)+

)n−1

−(β − t)n−1 +∫ ba

((g(x)− t)+

)n−1 dλ (x)∫ ba dλ (x)

]ϕ(n) (t)dt, (17)

and

ϕ

(α + β −


a dλ (x)

)−(

ϕ (α)+ ϕ (β )−∫ ba ϕ (g(x))dλ (x)∫ b

a dλ (x)

)

=n−1

∑k=1

ϕ(k) (β )k!

(−1)k

⎡⎣(∫ b

a g(x)dλ (x)∫ ba dλ (x)

−α

)k

− (β −α)k +∫ ba (β −g(x))k dλ (x)∫ b

a dλ (x)

⎤⎦

− 1(n−1)!

∫ β

α(−1)n−1

⎡⎣((t−α −β +


a dλ (x)

)+

)n−1

−(t−α)n−1 +∫ ba

((t −g(x))+

)n−1dλ (x)∫ b

a dλ (x)

]ϕ(n) (t)dt (18)

hold.

Proof. Using formula (10) and the facts that (α − t)+ = 0 and (β − t)+ = β − t

1392 A. MATKOVIC

for t ∈ [α,β ] , we have

ϕ

(α + β −


a dλ (x)

)−(

ϕ (α)+ ϕ (β )−∫ ba ϕ (g(x))dλ (x)∫ b

a dλ (x)

)

=n−1

∑k=0

ϕ(k) (α)k!

(β −


a dλ (x)

)k

+1

(n−1)!

∫ β

αϕ(n) (t)

((α + β −


a dλ (x)− t

)+

)n−1

dt

−ϕ (α)−n−1

∑k=0

ϕ(k) (α)k!

(β −α)k − 1(n−1)!

∫ β

αϕ(n) (t)(β − t)n−1 dt

+n−1

∑k=0

ϕ(k) (α)k!

∫ ba (g(x)−α)k dλ (x)∫ b

a dλ (x)

+1

(n−1)!

∫ β

αϕ(n) (t)

∫ ba

((g(x)− t)+

)n−1dλ (x)∫ b

a dλ (x)dt.

By regrouping and canceling all the first members in the above sums we obtain (17) .Analogously using formula (11) we obtain (18) . �

Using Lemmas 2 and 3 we can get the following generalizations of the Jensen-Mercer inequality for n -convex functions.

THEOREM 1. Let g : [a,b] → R be a continuous and monotonic function, and[α,β ] be an interval such that the image of g is a subset of [α,β ] . Let λ : [a,b]→R beeither continuous or of bounded variation satisfying (1) , and G : [α,β ] → R definedby (6) . Let ϕ : [α,β ] → R be a n-convex function such that ϕ(n−1) is absolutelycontinuous for some n � 3 .

(i) Then the inequality

ϕ

(α + β −


a dλ (x)

)−(

ϕ (α)+ ϕ (β )−∫ ba ϕ (g(x))dλ (x)∫ b

a dλ (x)

)

�n−3

∑k=0

ϕ(k+2) (α)k!

∫ β

αG (s) (s−α)k ds (19)

holds. Moreover, if ϕ(k) (α) � 0 for k = 2,3, . . . ,n−1 , then the right hand sideof (19) is negative or equals zero and (2) holds.


(ii) If n is even then the inequality

ϕ

(α + β −


a dλ (x)

)−(

ϕ (α)+ ϕ (β )−∫ ba ϕ (g(x))dλ (x)∫ b

a dλ (x)

)

�n−3

∑k=0

ϕ(k+2) (β )k!

(−1)k∫ β

αG (s) (β − s)k ds (20)

holds. Moreover, if ϕ(k) (β ) � 0 for k = 2,4, . . . ,n− 2 and ϕ(k) (β ) � 0 fork = 3,5, . . . ,n−1 , then the right hand side of (20) is negative or equals zero and(2) holds.

(iii) If n is odd then the reversed inequality (20) holds. Moreover, if ϕ(k) (β ) � 0 fork = 2,4, . . . ,n− 1 and ϕ(k) (β ) � 0 for k = 3,5, . . . ,n− 2 , then the right handside of the reversed inequality (20) is nonnegative and reverse inequality in (2)holds.

Proof. Since the Green function G is convex and G(α,s) = G(β ,s) = 0, fromTheorem A follows

G (s) = G

(α + β −


a dλ (x),s

)+∫ ba G(g(x) ,s)dλ (x)∫ b

a dλ (x)� 0.

Hence, if n is even then

∫ β

tG (s)(s− t)n−3 ds � 0, for t � s � β , (21)

and ∫ t

αG (s) (s− t)n−3 ds � 0, for α � s � t. (22)

Also, if n is odd then the inequality in (21) remains the same while the inequality in(22) becomes reversed.

Since the function ϕ is n -convex, without loss of generality we can assume thatϕ is n -times differentiable and ϕ(n) � 0 (see [7, p. 16 and p. 293]). Therefore we have

∫ β

α

(∫ β

tG (s) (s− t)n−3 ds

)ϕ(n) (t)dt � 0. (23)

Analogously, for even n we have

∫ β

α

(∫ t

αG (s) (s− t)n−3 ds

)ϕ(n) (t)dt � 0, (24)

while for odd n we have reversed inequality in (24) . Now, applying Lemma 2 weconclude (i) , (ii) and (iii) . �

1394 A. MATKOVIC

REMARK 2. The right hand side of (19) can be written in the form

∫ β

αG (s)

(n−3

∑k=0

ϕ(k+2) (α)k!

(s−α)k)

ds.

Hence, in case Tn−3 (ϕ ;α,s) = ∑n−3k=0

ϕ(k+2)(α)k! (s−α)k � 0 it is negative or equals zero

and inequality (2) holds. Similarly, the right hand side of (20) can be written in theform ∫ β

αG (s)

(n−3

∑k=0

ϕ(k+2) (β )k!

(s−β )k)

ds.

Therefore, in case n is even and Tn−3 (ϕ ;β ,s) � 0 inequality (2) holds, while in casen is odd and Tn−3 (ϕ ;β ,s) � 0 reverse inequality in (2) holds.

THEOREM 2. Let g : [a,b] → R be a continuous and monotonic function, and[α,β ] be an interval such that the image of g is a subset of [α,β ] . Let λ : [a,b] → R

be either continuous or of bounded variation satisfying (1) , and ϕ : [α,β ] → R an-convex function such that ϕ(n−1) is absolutely continuous for some n � 1 .

(i) Then the inequality

ϕ

(α + β −


a dλ (x)

)−(

ϕ (α)+ ϕ (β )−∫ ba ϕ (g(x))dλ (x)∫ b

a dλ (x)

)

�n−1

∑k=1

ϕ(k) (α)k!

⎡⎣(β−


a dλ (x)

)k

−(β−α)k +∫ ba (g(x)−α)k dλ (x)∫ b

a dλ (x)

⎤⎦(25)

holds. Moreover, if ϕ(k) (α) � 0 for k = 2,3, . . . ,n−1 , then the right hand sideof (25) is negative or equals zero and (2) holds.

(ii) If n is even then the inequality

ϕ

(α + β −


a dλ (x)

)−(

ϕ (α)+ ϕ (β )−∫ ba ϕ (g(x))dλ (x)∫ b

a dλ (x)

)

�n−1

∑k=1

ϕ(k)(β )k!

(−1)k

⎡⎣(∫ b

a g(x)dλ (x)∫ ba dλ (x)

−α

)k

−(β−α)k+∫ ba (β−g(x))kdλ (x)∫ b

a dλ (x)

⎤⎦

(26)

holds. Moreover, if ϕ(k) (β ) � 0 for k = 2,4, . . . ,n− 2 and ϕ(k) (β ) � 0 fork = 3,5, . . . ,n−1 , then the right hand side of (26) is negative or equals zero and(2) holds.


(iii) If n is odd then the reversed inequality (26) holds. Moreover, if ϕ(k) (β ) � 0 fork = 2,4, . . . ,n− 1 and ϕ(k) (β ) � 0 for k = 3,5, . . . ,n− 2 , then the right handside of the reversed inequality (26) is nonnegative and reverse inequality in (2)holds.

Proof. Since((x− t)+

)n−1is convex function and (α − t)+ = 0 and (β − t)+ =

β − t for t ∈ [α,β ] , from Theorem A follows((α + β −


a dλ (x)− t

)+

)n−1

−(β − t)n−1+∫ ba

((g(x)− t)+

)n−1dλ (x)∫ b

a dλ (x)� 0.

Since the function ϕ is n -convex, without loss of generality we can assume that ϕis n -times differentiable and ϕ(n) � 0. Hence, applying Lemma 3 we conclude (i) .Analogously, we conclude (ii) and (iii) . �

REMARK 3. In case Tn−1 (ϕ ;α,x) is convex function the right hand side of (25)is negative or equals zero and inequality (2) holds. In case Tn−1 (ϕ ;β ,x) is convexfunction and n is even the right hand side of (26) is negative or equals zero and (2)holds. In case Tn−1 (ϕ ;β ,x) is convex function and n is odd the right hand side of thereversed inequality (26) is nonnegative and reverse inequality in (2) holds.

3. Related results

For two Lebesgue integrable functions f ,h : [α,β ] → R the Cebysev functional isgiven as

Δ( f ,h) =1

β −α

∫ β

αf (t)h(t)dt−

(1

β −α

∫ β

αf (t)dt

)·(

1β −α

∫ β

αh(t)dt

). (27)

In paper [3] the following theorems were proved.

THEOREM B. Let f : [α,β ]→ R be a Lebesgue integrable and h : [α,β ]→ R bean absolutely continuous function with (·−a)(b−·)[h′]2 ∈ L [α,β ] . Then the inequal-ity

|Δ( f ,h)| � 1√2

[Δ( f , f )]12

1√β −α

(∫ β

α(t −α)(β − t)

[h′ (t)

]2 dt

) 12

(28)

holds, where the constant 1√2

is the best possible.

THEOREM C. Assume that h : [α,β ] → R is monotonic nondecreasing on [α,β ]and f : [α,β ] → R is absolutely continuous with f ′ ∈ L∞ [α,β ] . Then the inequality

|Δ( f ,h)| � 12(β −α)

∥∥ f ′∥∥

∞

(∫ β

α(t−α)(β − t)dh(t)

) 12

(29)

1396 A. MATKOVIC

holds, where the constant 12 is the best possible.

Proofs of the following theorems utilize the main ideas from the proofs of thesimilar theorems in [2] and [6], so we omit them here.

For a continuous and monotonic function g : [a,b] → R , an interval [α,β ] suchthat the image of g is subset of [α,β ] , a function λ : [a,b] → R either continuous orof bounded variation satisfying (1) , and function G : [α,β ]→ R defined by (6) , let usdenote

R (t) =∫ β

tG (s) (s− t)n−3 ds, (30)

R (t) =∫ t

αG (s)(s− t)n−3 ds, (31)

B (t)=

((α+β−


a dλ (x)−t

)+

)n−1

−(β−t)n−1 +∫ ba

((g(x)−t)+

)n−1dλ (x)∫ b

a dλ (x),

(32)and

B (t)=

((t−α−β+


a dλ (x)

)+

)n−1

−(t−α)n−1 +∫ ba

((t−g(x))+

)n−1dλ (x)∫ b

a dλ (x).

(33)Considering the function R we have the following identity in which the remainder Kn

is estimated by using Theorem B.

THEOREM 3. Let g : [a,b] → R be a continuous and monotonic function, and[α,β ] be an interval such that the image of g is a subset of [α,β ] . Let functionλ : [a,b] → R be either continuous or of bounded variation satisfying (1) . Let func-tion ϕ : [α,β ] → R be such that ϕ(n) is absolutely continuous for some n � 3 with

(·−a)(b−·)[ϕ(n+1)

]2 ∈ L [α,β ] , and let the functions G and R be defined by (6)and (30) , respectively. Then the remainder Kn given in the following formula

ϕ

(α + β −


a dλ (x)

)−(

ϕ (α)+ ϕ (β )−∫ ba ϕ (g(x))dλ (x)∫ b

a dλ (x)

)

=n−3

∑k=0

ϕ(k+2) (α)k!

∫ β

αG (s) (s−α)k ds+

ϕ(n−1) (β )−ϕ(n−1) (α)(β −α)(n−3)!

∫ β

αR (t)dt

+Kn (ϕ ;α,β ) (34)


|Kn (ϕ ;α,β )| �√

β −α√2(n−3)!

[Δ(R,R)]12

∣∣∣∣∫ β

α(t−α)(β − t)

[ϕ(n+1) (t)

]2dt

∣∣∣∣12

.

(35)

Application of Theorem C gives the following Ostrowsky type inequality.

THEOREM 4. Let g : [a,b] → R be a continuous and monotonic function, and[α,β ] be an interval such that the image of g is a subset of [α,β ] . Let functionλ : [a,b] → R be either continuous or of bounded variation satisfying (1) , and letthe functions G and R be defined by (6) and (30) , respectively. Let functionϕ : [α,β ] → R be such that ϕ(n) is absolutely continuous for some n � 3 withϕ(n+1) � 0 on [α,β ] . Then the remainder Kn in (34) satisfies the estimation

|Kn (ϕ ;α,β )|� 1(n−3)!

∥∥R ′∥∥∞

[ϕ(n−1) (β )+ ϕ(n−1) (α)

2− ϕ(n−2) (β )−ϕ(n−2) (α)

β −α

].

(36)

If the function ϕ(n) belongs to Lp , then we have the following theorem.

THEOREM 5. Assume (p,q) is a pair of conjugate exponents, i.e. 1 � p,q � ∞ ,1p + 1

q = 1 . Let g : [a,b] → R be a continuous and monotonic function, and [α,β ] bean interval such that the image of g is a subset of [α,β ] . Let function λ : [a,b]→R beeither continuous or of bounded variation satisfying (1) , and let the functions G andR be defined by (6) and (30) , respectively. Let function ϕ : [α,β ] → R be such that

ϕ(n−1) is absolutely continuous and∣∣∣ϕ(n)

∣∣∣p is an R-integrable for some n � 3 . Then

∣∣∣∣∣ϕ(

α + β −∫ ba g(x)dλ (x)∫ b

a dλ (x)

)−(

ϕ (α)+ ϕ (β )−∫ ba ϕ (g(x))dλ (x)∫ b

a dλ (x)

)

−n−3

∑k=0

ϕ(k+2) (α)k!

∫ β

αG (s) (s−α)k ds

∣∣∣∣∣� 1

(n−3)!

(∫ β

α|R (t)|q dt

) 1q ∥∥∥ϕ(n)

∥∥∥p, (37)

where the constant on the right-hand side of (37) is sharp for 1 < p � ∞ and the bestpossible for p = 1 .

Analogous identities and Ostrowski type inequalities hold for the other three func-tions R , B and B .

REMARK 4. We can also obtain other related results, analogous to those in [2] and[6], using the main ideas from [4] and [5]. In particular, we can produce new familiesof n -exponentially convex and exponentially convex functions, applying functionals,constructed as differences of the right hand side and left hand side of some of theinequalities derived earlier, on some given families with the same property.

Acknowledgement. The author would like to thank the referee for his invaluablecomments and insightful suggestions

1398 A. MATKOVIC

RE F ER EN C ES

[1] J. BARIC, A. MATKOVIC, Bounds for the normalized Jensen-Mercer functional, J. Math. Inequal. 3(4) (2009), 529–541.

[2] S. I. BUTT, K. A. KHAN, J. PECARIC, Popovicu type inequalities via Green function and generalizedMontgomery identity, Math. Inequal. Appl. 18 (4) (2015), 1519–1538.

[3] P. CERONE, S. S. DRAGOMIR, Some new Ostrowski-type bounds for the Cebysev functional andapplications, J. Math. Inequal. 8 (1) (2014), 159–170.

[4] J. JAKSETIC, J. PECARIC, Exponential convexity method, J. Convex Anal. 20 (1) (2013), 181–197.[5] J. JAKSETIC, J. PECARIC, A. PERUSIC, Steffensen inequality, higher order convexity and exponential

convexity, Rend. Circ. Mat. Palermo. 63 (1) (2014), 109–127.[6] M. ADIL KHAN, N. LATIF AND J. PECARIC, Generalization of majorization theorem, J. Math. In-

equal. 9 (3) (2015), 847–872.[7] J. E. PECARIC, F. PROSCHAN AND Y. L. TONG, Convex Functions, Partial Orderings, and Statistical

Applications, Academic Press, Inc. 1992.

(Received April 11, 2016) A. MatkovicFaculty of Electrical Engineering

Mechanical Engineering and Naval ArchitectureUniversity of Split

Ru -dera Boskovica 32, 21000 Split, Croatiae-mail: [email protected]



& Applications


ASYMPTOTIC BEHAVIOR OF POWER MEANS

NEVEN ELEZOVIC AND LENKA MIHOKOVIC


Abstract. We consider asymptotic behavior of classical n -variable means. General expansionsof these means are known in the term of Bell polynomials. Here, simple recursive algorithmsare derived. The obtained coefficients are used in analysis of some inequalities between meanswhich include the first asymptotic term.

1. Introduction

In this paper we discuss relations between some classical means, using techniqueof asymptotic expansions. Let us fix the notation first. A letter M will denote any ofmeans mentioned below. For n -tuples of positive real numbers, a = (a1,a2, . . . ,an) andw = (w1,w2, . . . ,wn) , ∑n

i=1 wi = 1, let us denote weighted means:

Q(a) =√

w1a21 +w2a2

2 + . . .+wna2n, (quadratic mean)

A(a) = w1a1 +w2a2 + . . .+wnan, (arithmetic mean)

G(a) = aw11 aw2

2 · · ·awnn , (geometric mean)

H(a) =1

w1

a1+

w2

a2+ . . .+

wn

an

, (harmonic mean)

Mr(a) =[w1a

r1 +w2a

r2 + . . .+wna

rn

]1/r

, r �= 0. (power mean)

It is well known that it holds Q � A � G � H , and that all these means are con-tained in the family of power means, geometric mean corresponds to the limit caser → 0.

The inequality between arithmetic and geometric mean holds for all positive valuesof its arguments. Note that the values of A and G can be quite different and the quotient(A−G)/G can be arbitrary large.

What can be said if all arguments are taken from a subinterval I , far away from theorigin? Then all variables are of the same order of magnitude and relative differencebetween means is in principle much smaller.

Mathematics subject classification (2010): 26E60, 41A60.Keywords and phrases: Asymptotic expansion, power mean, asymptotic inequality.This work has been fully supported by Croatian Science Foundation under the project 5435.


1399


1400 N. ELEZOVIC AND L. MIHOKOVIC

For a fixed n -tuple (a1,a2, . . . ,an) a shift by positive variable x will move this n -tuple into a bounded subinterval In . Let us denote by xe+a the n -tuple with elements(x+a j) , where e = (1,1, . . . ,1) . The behaviour of M(xe+a) for large value of x willbe essential in our analysis.

In the recent papers [2, 7, 8, 10], a similar problem was analyzed for two-variablemeans. Many of results obtained there can be translated in a setting of this paper, buthere we discuss questions which are marginal in the case of two-parameter means.

Since for all xA(xe+a) = x+A(a),

the asymptotic expansion for this mean has only these two terms.Since

H(xe+a) = x · 1n∑j=1

wj

1+a j/x

= x · 1n∑j=1

wj

(1− a j

x+O(1/x2)

)= x · 1

1− A(a)x

+O(1/x2)= x

(1+

A(a)x

+O(1/x2))

= x+A(a)+O(1/x)

we can conclude that it holds

H(xe+a)− x→ A(a) as x → ∞.

Since H � G � A , the same conclusion holds also for geometric mean. It will be shownthat a power mean has the same property. Therefore, the asymptotic expansion of thefunction x �→ M(xe+a) should be of the form

M(xe+a) = x+A(a)+∞

∑k=2

ck(a)x−k+1

for any of means from the above-mentioned list. Here cn , n ∈ N0 , are homogenouspolynomials of order n which can be expressed as a function of

mk = w1ak1 +w2a

k2 + . . .+wna

kn = Mk

k (a).

Let us denote m0 = 1.The complete asymptotic expansion of power mean in terms of Bell polynomials

was proved in [1]:

THEOREM. For each p ∈ R , the power means Mp(xe+a) possess the completeasymptotic expansion

Mr(xe+a) = x+∞

∑k=0

ck(a)x−k, x → ∞.

ASYMPTOTIC BEHAVIOR OF POWER MEANS 1401

In the case p �= 0 the coefficients are given by

ck(r) =1

(k+1)!

k+1

∑j=1

j!

(1/rj

)Bk+1, j

[i!

(ri

)Mi

i (a)].

In case p = 0 coefficients are given by

ck(0) =(−1)k+1

(k+1)!Yk+1

[−(i−1)!Mii(a)].

Bm, j[yi] := Bm, j(y1, . . . ,ym− j+1) denotes partial Bell polynomials defined by

1j!

(∞

∑k=1

yktk

k!

) j

=∞

∑m= j

Bm, j[yi]tm

m!, j = 1,2, . . .

and Ym[yi] = Ym(y1, . . . ,yn) are complete Bell polynomials defined by

exp

(∞

∑j=1

y jt j

j!

)= 1+

∞

∑m=1

Ym[yi]tm

m!.

In this paper we derive an efficient recursive formula for such expansion and sim-plified verisons in the particular cases of the above-mentioned means. Our results arebased on the using of the following lemmas about functional transformations of asymp-totic series, see [3, 4, 8, 9].

LEMMA 1.1. Let a0 = 1 and g(x) be a function with the asymptotic expansion

g(x) ∼∞

∑n=0

anx−n.

Then for all real p it holds

g(x)p ∼∞

∑n=0

Pn(p)x−n,

where P0 = 1 and

Pn(p) =1n

n

∑k=1

[k(1+ p)−n]akPn−k(p).

Especially, for p = −1 we obtain formula for the reciprocal value of an asymptoticseries:

1g(x)

∼∞

∑n=0

Rnx−n,

where R0 = 1 and

Rn = −n

∑k=1

akRn−k.


LEMMA 1.2. Let functions f (x) and g(x) have the following asymptotic expan-sions (a0 �= 0, b0 �= 0 ) as x → ∞:

f (x) ∼∞

∑n=0

anx−n, g(x) ∼

∞

∑n=0

bnx−n.

Then the asymptotic expansion of their quotient f (x)/g(x) reads as

f (x)g(x)

∼∞

∑n=0

cnx−n,

where coefficients cn are defined by

cn =1b0

(an−

n

∑k=1

bkcn−k

).

LEMMA 1.3. Let

g(x) =∞

∑n=1

anx−n

be a given asymptotical expansion. Then the composition exp(g(x)) has asymptoticexpansion of the following form

exp(g(x)) =∞

∑n=0

bnx−n

where b0 = 1 and

bn =1n

n

∑k=1

kakbn−k, n � 1.

This paper is organized as follows. In the next section an asymptotic expansion ofpower mean was found. Coefficients depending on a real parameter r are defined byrecursive relation and some interesting properties are detected. As a consequence weobtained asymptotic expansions of some well known classical means. Section 3 beginswith reminder of the results from our previous paper in which we studied inequalitiesrelated to bivariate means and continues with the discussion of the known inequalitiesrelated to n -variable classical means. In Section 4, we established some asymptoticinequalities covering arithmetic, geometric and harmonic means.

2. Power mean

The power mean can be written in a way

Mr(xe+a) =[ n

∑j=1

wj(x+a j)r]1/r

= x

[ n

∑j=1

wj

∞

∑k=0

(rk

)ak

jx−k]1/r

= x

[ ∞

∑k=0

(rk

)1n

( n

∑j=1

wjakj

)x−k]1/r

= x

[ ∞

∑k=0

(rk

)mkx

−k]1/r

.

From Lemma 1.1 we get the following result.


THEOREM 2.1. General power mean has the following asymptotic expansion

Mr(xe+a) = x ·∞

∑k=0

ck(r)x−k,

where c0 = 1 and

ck(r) =1k

k

∑j=1

[j

(1+

1r

)− k

](rj

)mjck− j(r), k ∈ N.

The first few coefficients are

c0(r) = 1,

c1(r) = m1,

c2(r) = − 12(r−1)(m2

1−m2),

c3(r) = 16(r−1)

((2r−1)m3

1−3(r−1)m1m2 +(r−2)m3),

c4(r) = − 124(r−1)

((3r−1)(2r−1)m4

1−6(r−1)(2r−1)m21m2

+3(r−1)2m22 +4(r−2)(r−1)m1m3 − (r−3)(r−2)m4

).

(2.1)

THEOREM 2.2. Coefficients ck , k ∈ N0 , are homogeneous polynomials in vari-ables (a1, . . . ,an) and have the following form:

ck(r,a) = ∑α1,α2,...,αk�0

α1+2α2+···+kαk=k

qα1,...,αk(r)m1(a)α1 · · ·mk(a)αk , (2.2)

where

∑α1,α2,...,αk�0

α1+2α2+···+kαk=k

qα1,...,αk(r) = 0, k � 2. (2.3)

Proof. Using homogeneity of mean Mr , we have

λx∞

∑k=0

ck(r,a)x−k = λMr(xe+a) = Mr(λxe+ λa)

= λx∞

∑k=0

ck(r,λa)(λx)−k = λx∞

∑k=0

λ−kck(r,λa)x−k,

hence,ck(r,λa) = λ kck(r,a).

The form (2.2) can be deduced easily using induction. The property (2.3) followsfrom the special case:

x+a = Mr(xe+ae) =∞

∑k=0

ck(r,ae)x−k

=∞

∑k=0

ak

⎛⎜⎝ ∑

α1,α2,...,αk�0α1+2α2+···+kαk=k

qα1,...,αk(r)

⎞⎟⎠x−k. �


COROLLARY 2.3. Harmonic mean has the following expansion

H(xe+a) = x+m1 +∞

∑k=2

ckx−k+1,

where coefficients are given by c0 = 1 and

ck =k

∑j=1

(−1) j−1mjck− j.


c0 = 1,

c1 = m1,

c2 = m21−m2,

c3 = m31−2m1m2 +m3,

c4 = m41−3m2

1m2 +m22 +2m1m3 −m4,

c5 = m51−4m3

1m2 +3m21m3−2m2m3 +3m1m

22 −2m1m4 +m5.

COROLLARY 2.4. Quadratic mean has the following asymptotic expansion

Q(xe+a) = x ·∞

∑k=0

ckx−k,

where c0 = 1 , c1 = m1 and

ck =(

3k−2

)m1ck−1 +

(3k−1

)m2ck−2, k � 2.


c0 = 1,

c1 = m1,

c2 = 12 (−m2

1 +m2),

c3 = 12m1(m2

1−m2),

c4 = 18 (−5m4

1 +6m21m2−m2

2),

c5 = 18m1(7m2

1−3m2)(m21 −m2).

THEOREM 2.5. Geometric mean has the following asymptotic expansion

G(xe+a) = x ·∞

∑k=0

ckx−k,

where c0 = 1 and

ck =1k

k

∑j=1

(−1) j−1mjck− j, k � 2.


Proof.

log(G(xe+a)) =n

∑j=1

wj log(x+a j) = logx+n

∑j=1

wj

∞

∑k=1

(−1)k−1ak

j

xk

= logx+∞

∑k=1

(−1)k−1 mk

xk .

Now the desired result follows from

G(xe+a) = xexp

(∞

∑k=1

(−1)k−1 mk

kx−k

)

using Lemma 1.3. �The first few coefficients are

c0 = 1,

c1 = m1,

c2 = 12 (m2

1−m2),

c3 = 16 (m3

1−3m1m2 +2m3),

c4 = 124(m4

1 −6m21m2 +3m2

2 +8m1m3−6m4).

Notice that in the limit r → 0 coefficients (2.1) become equal to coefficients in theasymptotic expansion of geometric mean.

2.1. Ratio of power means.

Abel and Ivan [1] proved that ratio

Mp(xe+a)−Mq(xe+a)Mr(xe+a)−Ms(xe+a)

possesses an asymptotic expansion and the first few coefficients were written. Here, wegive recursive formula for all coefficients. The proof of this theorem follows immedi-ately using Lemma 1.2.

THEOREM 2.6. For all real p,q,r,s with r �= s, the asymptotic expansion of theratio of power means is given by

Mp(xe+a)−Mq(xe+a)Mr(xe+a)−Ms(xe+a)

∼∞

∑k=0

ckx−k,

where

ck =1

c2(r)− c2(s)[(ck+2(p)− ck+2(q))−

k

∑j=1

(c j+2(r)− c j+2(s))ck− j].


2.2. Gini mean

One natural generalization of a power mean is the Gini mean, defined as:

Gp,q(a) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

[ap1+···+ap

n

aq1+···+aq

n

] 1p−q

, p �= q,

exp(

∑nk=1 ap

k lnak

∑nk=1 ap

k

), p = q �= 0,

√a1 · · ·an, p = q = 0.

Using the same technique, one can prove the following theorem.

THEOREM 2.7. The Gini mean has the following asymptotic expansion

Gp,q(xe+a) = x ·∞

∑k=0

ckx−k,

where c0 = 1 ,

ck =1k

k

∑j=1

[j

(1+

1p−q

)− k

]b jck− j, k ∈ N,

and

bk =(

pk

)mk −

k

∑j=1

(qj

)mjbk− j, k � 0.


c0 = 1,

c1 = m1,

c2 = − 12(p+q−1)(m2

1−m2),

c3 = 16

(((p+q−1)(2q−1)+2p(p−1))m3

1

−3((p+q−1)(q−1)+ p(p−1))m1m2

+((p+q−1)(q−2)+ p(p−1))m3),

c4 = 124

(− ((p+q−1)(6q2−5q+6p2−9p+1)+4p(p−1))m41

+6(p+q−1)(2q2−3q+2p2−3p+1)m21m2

−3(p+q−1)(p(p−2)+(q−1)2)m22

−4(p+q−2)(p(p−2)+(q−1)2)m1m3

+(p+q−3)(q2−3q+ p2−3p+2)m4).

3. Inequalities between means

3.1. Bivariate means

Let us discuss briefly inequalities between bivariate means which follow from theirasymptotic expansions. Let us denote

α =t + s2

, β =t − s

2.

It is proved in [7] that

Q(x+ s,x+ t) = x+ α +β 2

2x− αβ 2

2x2 +β 2(4α2 −β 2)

8x3 +o(x−3),

A(x+ s,x+ t) = x+ α,

I(x+ s,x+ t) = x+ α − β 2

6x+

αβ 2

6x2 − β 2(60α2 +13β 2)360x3 +o(x−3),

L(x+ s,x+ t) = x+ α − β 2

3x+

αβ 2

3x2 − β 2(15α2 +4β 2)45x3 +o(x−3),

G(x+ s,x+ t) = x+ α − β 2

2x+

αβ 2

2x2 − β 2(4α2 + β 2)8x3 +o(x−3),

H(x+ s,x+ t) = x+ α − β 2

x+

αβ 2

x2 − β 2α3

8x3 +o(x−3).

As a consequence, the folowing inequalities are proved.

THEOREM 3.1. ([7]) Let 0 < s < t . Then for all x > 0 we have

Q(x+ s,x+ t) < x+ α +β 2

2x, (3.1)

I(x+ s,x+ t) > x+ α − β 2

6x, (3.2)

L(x+ s,x+ t) > x+ α − β 2

3x, (3.3)

G(x+ s,x+ t) > x+ α − β 2

2x, (3.4)

H(x+ s,x+ t) > x+ α − β 2

x. (3.5)

In the case s < t , s+ t < 0 the inequalities (3.2)–(3.5) hold with opposite sign, for eachx > −s.

Here, I is identric and L logarithmic mean. We shall interpret these inequalitiesin different settings, which is closer to the problem of comparison of power means.

3.2. n -variable means

Let a1 � a2 � . . . � an be fixed positive n -tuple. In the sequel we shall considerthe equal-weight case, w1 = . . . = wn = 1

n . Then, all means are symmetric and as aconsequence, all polynomials cr are symmetric polynomials.

We are dealing with asymptotic behaviour of means, so it is natural to suppose thata1 is sufficient large. Let us denote bi = ai− x , for each 1 � i � n . Then

M(a) = M(xe+b)

and we can use asymptotic expansion for the function on the right. All means consid-ered in this paper have asymptotic expansion of the form

M(a) = x+A(b)+ γ2m2

1−m2

x+ . . .

where γ2 is a constant and both of the following terms behave well under translation:

x+A(b) = A(a),

m2−m21 =

1n2 ∑

1�i< j�n

(bi−b j)2 =1n2 ∑

1�i< j�n

(ai−a j)2.

It is natural to pose the following problem: find the best constants γ and δ suchthat for given two means F1 and F2 , F1 � F2 , we have

m2−m21

γ< F2(a)−F1(a) <

m2−m21

δ.

Some partial answers to this problem are already known.For 0 � a1 � . . . � an the following inequalities for n -variable means A , G and

H , with equal weights are given in [11, p. 39]:

12n2

1an

∑1�i< j�n

(ai−a j)2 � A(a)−G(a) � 12n2

1a1

∑1�i< j�n

(ai−a j)2 (3.6)

and1

2n2

a31

a4n

∑1�i< j�n

(ai −a j)2 � G(a)−H(a) � 12n2

a3n

a41

∑1�i< j�n

(ai −a j)2.

Later on, Zhan, Xi and Chu [13] found improvements of these inequalities. If n � 2and 0 < b � a1 � . . . � an � B , then

∑1�i< j�n(ai −a j)2

2n2B(n−1)/nA1/n(a)� A(a)−M0(a) � ∑1�i< j�n(ai −a j)2

2n2b(n−1)/nA1/n(a)

and

b(n−1)/n

2n2B(2n−1)/n ∑1�i< j�n

(ai−a j)2 � M0(a)−M−1(a) � B(n−3)/n

2n2b(2n−3)/n ∑1�i< j�n

(ai −a j)2.

(3.7)

We shall show that these inequalities are not optimal. Let us begin with the A-Gcase for n = 2. We have

a1 +a2

2−√

a1a2 =(a2−a1)2

8ξ,

where

ξ =(√

a1 +√

a2

2

)2

.

Therefore, one has

(a2−a1)2

8γ< A(a)−G(a) <

(a2−a1)2

8δ

whenever δ < ξ < γ . This is consistent with (3.6) since ξ given above is a mean anda1 � ξ � an .

Similar inequality is true also in the A-H inequality, where the critical value isξ = (a1 +a2)/2.

In the case of G-H inequality, the critical value is

ξ =(√

a1 +√

a2

2

)2(a1 +a2

2

)1√a1a2

.

Therefore, at least for n = 2 better simple bounds can be found, for example:

δ =a1 +a2

2< ξ <

a1 +a2

2

√a2

a1= γ

which is better than bounds

δ =√

a1a2, γ = a2

√a2

a1

which follow from (3.7).

4. Asymptotic inequalities

DEFINITION 4.1. Let F(s,t) be any homogenous bivariate function such that

F(x+ s,x+ t) = ck(t,s)x−k+1 +O(x−k).

If ck(s, t) > 0 for all s and t , we say that F is asymptotically greater than zero, andwrite

F � 0.

Asymptotic inequalities between means impose necessary conditions for the properinequalities, see [4–7]. The sign of the first neglected coefficient is essential in suchanalysis. Let us describe the idea in the case of means G and H . It holds

G(xe+b)−H(xe+b)− m2 −m21

2x= − 1

6x2 (4m3−9m1m2 +5m31)+o(x−2).

In the general case, the sign of

Δ(b) = 4m3−9m1m2 +5m31

depends on the number of variables n .

THEOREM 4.2. Let n∈ {1,2,3,4,5} . In the case 0 � b1 � b2 � . . . � bn we have

G(xe+b)−H(xe+b)− m2−m21

2x≺ 0. (4.1)

In the case b1 � b2 � . . . � bn � 0 the opposite inequality holds in (4.1) .

Proof. Let 0� b1 � . . . � bn . According to Corollary 2.1. (1) from [12], inequality

Δ(b) � 0

of degree d = 3 holds for all b ∈ Rn+ if and only if it holds for every b ∈ Rn

+ such thatthe number of non-zero distinct components of b is less or equal to max( 3

2�,1) = 1.Let us take

b = (1, . . . ,1,0, . . . ,0) (4.2)

which consists of k units and n− k zeros. Since 0 � kn � 4

5 or kn = 1, it follows

Δ(b) = 5k3

n3 −9k2

n2 +4kn

= 5kn

(kn− 4

5

)(kn−1

)� 0. �

The positivity od Δ is not true for n � 6. Take k < n such that 45 < k

n . Thisis possible for each n � 6. Then, for b as in (4.2) we have Δ(b) < 0. Hence, theinequality

G(xe+b)−H(xe+b)− m2 −m21

2x≺ 0

cannot be true under conditions 0 � b1 � . . . � bn , for n � 6.Similar conclusions also hold for other means.

THEOREM 4.3. In the case 0 � b1 � b2 � . . . � bn we have

A(xe+b)−G(xe+b)− m2−m21

2x� 0,

A(xe+b)−H(xe+b)− m2−m21

x� 0,


Q(xe+b)−H(xe+b)− 3(m2−m21)

2x≺ 0,

Q(xe+b)−G(xe+b)− m2−m21

x� 0,

Q(xe+b)−A(xe+b)− m2−m21

x≺ 0.

In the case b1 � b2 � . . . � bn � 0 the opposite inequalities hold.

Proof. The proof follows as in the previous theorem since it holds:

A(xe+b)−G(xe+b)− m2−m21

2x=

m31 −3m1m2 +2m3

6x2 +O(x−3),

Q(xe+b)−H(xe+b)− 3(m2−m21)

2x= −m3

1−3m1m2 +2m3

2x2 +O(x−3),

Q(xe+b)−G(xe+b)− m2−m21

x= −m3

1−m3

3x2 +O(x−3),

Q(xe+b)−A(xe+b)− m2−m21

x= −m1(m2−m2

1)2x2 +O(x−3). �

Acknowledgement. Authors would like to thank the anonymous referee for valu-able corrections and comments which improved this paper considerably.

RE F ER EN C ES

[1] U. ABEL, M. IVAN, A complete asymptotic expansion of power means, J. Math. Anal. Appl. 325, 1(2007), 554–559.

[2] T. BURIC, N. ELEZOVIC, Asymptotic expansion of Arithmetic-Geometric mean, J. Math. Inequal. 9,4 (2015), 1181–1190.

[3] C.-P. CHEN, N. ELEZOVIC, L. VUKSIC, Asymptotic formulae associated with the Wallis power func-tion and digamma function, J. Class. Anal. 2, 2 (2013), 151–166.

[4] C.-P. CHEN, N. ELEZOVIC, L. VUKSIC, Asymptotic expansion of integral mean of polygamma func-tions, Math. Inequal. Appl. 18, 1 (2015), 255–266.

[5] N. ELEZOVIC, Asymptotic inequalities and comparison of classical means, J. Math. Inequal. 9, 1(2015), 177–196.

[6] N. ELEZOVIC, Asymptotic expansions of gamma and related functions, binomial coefficients, inequal-ities and means, J. Math. Inequal. 9, 4 (2015), 1001–1054.

[7] N. ELEZOVIC, L. VUKSIC, Asymptotic expansions of bivariate classical means and related inequali-ties, J. Math. Inequal. 8, 4 (2014), 707–724.

[8] N. ELEZOVIC, L. VUKSIC, Asymptotic expansions and comparison of bivariate parameter means,Math. Inequal. Appl. 17, 4 (2014), 1225–1244.

[9] N. ELEZOVIC, L. VUKSIC, Asymptotic expansions of integral means and applications to the ratio ofgamma functions, Appl. Math. Comput. 235 (2014), 187–200.

[10] N. ELEZOVIC, L. VUKSIC, Neuman-Sandor means, asymptotic expansions and related inequalities,J. Math. Inequal. 9, 4 (2015), 1337–1348.


[11] D. S. MITRINOVIC, J. E. PECARIC, A. M. FINK, Classical and New Inequalities in Analysis, KluwerAcademic Publishers, Dordrecht, 1993.

[12] V. TIMOFTE, On the positivity of symmetric polynomial functions. Part I: General results, J. Math.Anal. Appl. 284 (2003), 174–190.

[13] F. ZHANG, B.-Y. XI, Y.-M. CHU, A new method to prove and find analytic inequalities, Abstr. Appl.Anal. 2010, Art. ID 128934, 19 pages.

(Received April 15, 2016) Neven ElezovicFaculty of Electrical Engineering and Computing

University of ZagrebUnska 3, 10000 Zagreb, Croatia


Lenka MihokovicFaculty of Electrical Engineering and Computing

University of ZagrebUnska 3, 10000 Zagreb, Croatia



& Applications


ON SOME PROPERTIES OF STRICTLY CONVEX FUNCTIONS

WITOLD JARCZYK AND KAZIMIERZ NIKODEM


Abstract. We prove that some careless modification of the definition of strong convexity leadsto a condition which is equivalent to that one of strict convexity.

Given a convex subset D of a real linear space a function f : D → R is calledconvex if ∧

x,y∈D

∧t∈(0,1)

f (tx+(1− t)y) � t f (x)+ (1− t) f (y)

and strictly convex if the above inequality is strict whenever x �= y :∧x,y∈Dx�=y

∧t∈(0,1)

f (tx+(1− t)y) < t f (x)+ (1− t) f (y).

Replacing the signs ” � ” and ” < ” by ” � ” and ” > ” above we come to the notionsof concavity and strict concavity, respectively. Clearly every strictly convex function isconvex but the converse fails to be true: the function R � x �−→ x serves as an example.

If the considered space is endowed with a norm ‖·‖ we can introduce one notionrelated to the convexity more. Namely, a function f : D → R is said to be stronglyconvex with modulus c ∈ (0,+∞) if∧

x,y∈D

∧t∈(0,1)

f (tx+(1− t)y) � t f (x)+ (1− t) f (y)− ct(1− t)‖x− y‖2 .

We say that f is strongly convex if it is strongly convex with some positive modulus.Similarly we introduce the notion of strong concavity.

It seems that the notion of strong convexity was introduced by Polyak [6] in 1966.Strongly convex functions play an important role in optimization theory and mathemat-ical economics. Many properties and applications of them can be found in the literature.Let us mention here the papers by Vial [8], Montrucchio [2], Jovanovic [1], Polovinkin[5]. Also the classical book [7] due to Roberts and Varberg contains some informationon that notion. Finally let us mention the paper [4] by the second present author; that isa survey article entirely devoted to strongly convex functions.

Mathematics subject classification (2010): 26A51, 39B62.Keywords and phrases: Strict convexity, strong convexity.


1413

1414 W. JARCZYK AND K. NIKODEM

Evidently every strongly convex function is strictly convex. However, the converseis generally not the case: the exponential function R � x �−→ expx is strictly convexbut it is not strongly convex with any modulus c ∈ (0,+∞) ; as any two norms in R

(endowed with the usual linear operations) are equivalent, the latter does not depend onthe norm considered there.

The main aim of this note is to answer the following question: what can be said ifwe make a typical ”student” mistake and formally change the order of quantifiers in thedefinition of strong convexity: namely, instead of the condition∨

c∈(0,+∞)

∧x,y∈D

∧t∈(0,1)

f (tx+(1− t)y) � t f (x)+ (1− t) f (y)− ct(1− t)‖x− y‖2

we consider a weaker one∧x,y∈D

∨c∈(0,+∞)

∧t∈(0,1)

f (tx+(1− t)y) � t f (x)+ (1− t) f (y)− ct(1− t)‖x− y‖2 . (1)

We prove that, rather unexpectedly, the following result holds.

THEOREM. Let D be a convex subset of a real normed space and let f : D → R .Then condition (1) holds if and only if f is strictly convex.

Proof. Assume that the function f is strictly convex. To prove (1) fix points x,y ∈D . Without loss of generality we may assume that x �= y . It is well known that thefunction Fx,y : [0,1] → R , given by

Fx,y(t) = f (tx+(1− t)y),

is strictly convex (see for instance [3, Prop. 3.4.2]; the reader can also verify this factvia routine calculation). Define the function Gx,y : [0,1] → R by

Gx,y(t) = t f (x)+ (1− t) f (y)− f (tx+(1− t)y).

Then

Gx,y(t) = t f (x)+ (1− t) f (y)−Fx,y(t), t ∈ [0,1],

so Gx,y is strictly concave as the difference of an affine function and a strictly convexone. In what follows to simplify the notation write G instead of Gx,y . Observe that Gis continuous, G(0) = G(1) = 0 and, by the strict convexity of f , we have G(t) > 0for every t ∈ (0,1) .

For any c ∈ (0,+∞) define a function Rc : (0,1) → R by

Rc(t) = ct(1− t)‖x− y‖2 .

To get (1) it is enough to prove the existence of a c ∈ (0,+∞) satisfying

Rc(t) � G(t), t ∈ (0,1).

ON SOME PROPERTIES OF STRICTLY CONVEX FUNCTIONS 1415

Suppose on the contrary that this is not the case. Then there exists a sequence (tn)n∈N

of numbers from (0,1) such that

R1/n (tn) > G(tn) , n ∈ N.

Choose a subsequence (tkn)n∈Nof (tn)n∈N

converging to a t0 ∈ [0,1] . Then, sincet(1− t) � 1/4 for all t ∈ R , we have

G(tkn) < R1/kn (tkn) =1kn

tkn (1− tkn)‖x− y‖2 � ‖x− y‖2

4kn

for each n∈N . Therefore, by the continuity of G , we get G(t0) � 0, and thus G(t0) =0, whence either t0 = 0, or t0 = 1. Assume, for instance, that t0 = 0. Since G(0) =0 < G(t) for t ∈ (0,1) and G is concave, it follows that G has a positive derivative at0 . On the other hand

G(tkn)tkn

<R1/kn (tkn)

tkn

=‖x− y‖2

kn(1− tkn) <

‖x− y‖2

kn, n ∈ N,

whence

G′(0) = limn→∞

G(tkn)tkn

� 0,

a contradiction. Consequently, condition (1) holds.The converse implication is obvious. �It turns out that if we additionally assume the continuity of a strictly convex func-

tion f : D→R , then also c : D×D→ (0,+∞) provided by condition (1) can be chosenregular in a sense.

COROLLARY. Let D be a convex subset of a real normed space and let f : D→R .If the function f is continuous and strictly convex, then there exists an upper semicon-tinuous function c0 : D×D \Δ → (0,+∞) , where Δ = {(x,y) ∈ D×D : x = y} , suchthat

f (tx+(1− t)y) � t f (x)+ (1− t) f (y)− c0(x,y)t(1− t)‖x− y‖2

for all x,y ∈ D with x �= y and t ∈ (0,1) .

Proof. Since f is continuous it follows that for every t ∈ (0,1) the function

D×D � (x,y) �−→ Gx,y(t) = t f (x)+ (1− t) f (y)− f (tx+(1− t)y)

is continuous. Now putting

c0(x,y) := inf

{Gx,y(t)

t(1− t)‖x− y‖2 : t ∈ (0,1)

}

for each x,y ∈D, x �= y , we see that the function c0 : D×D\Δ → R is upper semicon-tinuous.

1416 W. JARCZYK AND K. NIKODEM

On account of the Theorem, for all x,y ∈ D with x �= y , there exists a c(x,y) ∈(0,+∞) such that

f (tx+(1− t)y) � t f (x)+ (1− t) f (y)− c(x,y)t(1− t)‖x− y‖2 , t ∈ (0,1),

that is

0 < c(x,y) � Gx,y(t)

t(1− t)‖x− y‖2 , t ∈ (0,1).

Thus

0 < c(x,y) � c0(x,y) � Gx,y(t)

t(1− t)‖x− y‖2 , t ∈ (0,1),

whenever x,y ∈ D and x �= y , which proves that the function c0 takes only positivevalues and satisfies the inequality stated in the assertion. �

RE F ER EN C ES

[1] M. V. JOVANOVIC, A note on strongly convex and strongly quasi-convex functions, Math. Notes 60(1996), 778–779.

[2] L. MONTRUCCHIO,Lipschitz continuous policy functions for strongly concave optimization problems,J. Math. Econom. 16 (1987), 259–273.

[3] C. P. NICULESCU AND L.-E. PERSSON, Convex Functions and their Applications. A ContemporaryApproach, CMS Books in Mathematics vol. 23, Springer-Verlag, New York, 2006.

[4] K. NIKODEM, On strongly convex functions and related classes of functions, in: T. M. Rassias (ed.)Handbook of functional equations, 365–405, Springer Optim. Appl. 95, Springer, New York, 2014.

[5] E. S. POLOVINKIN, Strongly convex analysis, Sb. Math. 187 (1996), 259–286.[6] B. T. POLYAK, Existence theorems and convergence of minimizing sequences in extremum problems

with restrictions, Sov. Math. Dokl. 7 (1966), 72–75.[7] A. W. ROBERTS AND D. E. VARBERG, Convex functions, Academic Press, New York-London, 1973.[8] J. P. VIAL, Strong and weak convexity of sets and functions, Math. Oper. Res. 8 (1983), 231–259.

(Received April 22, 2016) Witold JarczykInstitute of Mathematics and Informatics

John Paul II Catholic University of LublinKonstantynow 1h, PL-20-708 Lublin, Poland


andFaculty of Mathematics, Computer Science and Econometrics

University of Zielona GoraSzafrana 4a, PL-65-516 Zielona Gora, Poland


Kazimierz NikodemDepartment of Mathematics, University of Bielsko-Biała

Willowa 2, PL-43-309 Bielsko-Biała, Polande-mail: [email protected]


& Applications


CHEBYSHEV–GRUSS TYPE INEQUALITIES ON TIME

SCALES VIA TWO LINEAR ISOTONIC FUNCTIONALS

LUDMILA NIKOLOVA AND SANJA VAROSANEC


Abstract. We give a generalization of the Chebyshev-Gruss inequality by using the concept ofderivative on time scales combined with application of the Chebyshev inequality involving twolinear isotonic functionals. This approach covers integral case, discrete case, results from frac-tional and quantum calculus.

1. Introduction and preliminaries

The well-known classical Chebyshev inequality for Riemann integrals states thatif p, f and g are integrable real functions on [a,b] ⊂ R , p � 0, and if f and g aresimilarly ordered, then∫ b

ap(x)dx

∫ b

ap(x) f (x)g(x)dx �

∫ b

ap(x) f (x)dx

∫ b

ap(x)g(x)dx. (1)

If f and g are oppositely ordered then the reverse of the inequality in (1) is valid, [12,p. 239].

There is another inequality which is also joined with the name of Chebyshev. Inliterature it is known as the Chebyshev-Gruss inequality. It is an inequality whichgives an upper bound for the absolute value of the Chebyshev difference involving thesupremum of the first derivative of functions f and g . Precisely, the Chebyshev-Grussinequality is the following result, [12, p. 297].

THEOREM 1. Let f ,g : [a,b] → R be absolutely continuous functions. If f ′,g′ ∈L∞[a,b] , then∣∣∣∣ 1

b−a

∫ b

af (x)g(x)dx−

(1

b−a

∫ b

af (x)dx

)(1

b−a

∫ b

ag(x)dx

)∣∣∣∣� 1

12(b−a)2‖ f ′‖∞ · ‖g′‖∞. (2)

Mathematics subject classification (2010): 26E70, 26D15, 26A33.Keywords and phrases: The Chebyshev-Gruss inequality, fractional integral operator, isotonic linear

functional, time scale.


1417


1418 L. NIKOLOVA AND S. VAROSANEC

The difference on the right hand side, under the sign of absolute value, is calledthe Chebyshev difference or the Chebyshev functional. Usually it is given in weightedversion as follows

T ( f ,g, p) =∫ b

ap(x)dx

∫ b

ap(x) f (x)g(x)dx−

∫ b

ap(x) f (x)dx

∫ b

ap(x)g(x)dx.

There exist a lot of estimations for T , but the most known is the Gruss inequalitywhich states that

|T ( f ,g, p)| �√

T ( f , f , p)T (g,g, p)

and if numbers m,M,n,N are such that m � f (x) � M , n � g(x) � N for all x ∈ [a,b] ,then

|T ( f ,g, p)| � 14(M−m)(N−n)

(∫ b

ap(x)dx

)2

.

Until now, Gruss type inequalities are investigated in different settings. There existresults involving sequences, functions, functionals, matrices, operators etc. In the paper[13] authors open new direction of investigation using two linear functionals instead ofonly one.

The lower bound for the Chebyshev difference is given in the following theorem,[14].

THEOREM 2. Let f and g be two differentiable functions on [a,b] , monotonic inthe same direction and p � 0 . If | f ′(x)| � m � 0 and |g′(x)| � r � 0 on [a,b] , then

T ( f ,g, p) � mrT (x−a,x−a, p). (3)

Since discrete versions of inequalities (1), (2) and (3) are also known, see forexample [12, p. 240], [14], it is a natural question to ask: Does a general approachwhich covers integral and discrete versions of the above-mentioned inequalities exist?The answer is affirmative and it is given by a method of calculus on time scales. Inthis approach the Chebyshev inequality involving two isotonic linear functionals playsa main role. Let us mention some definitions and theorems related to that topic.

Let E be a non-empty set and L be a class of real-valued functions on E satisfyingthat a linear combination of functions from L is also in L and the function 1 belongs toL , (1(t) = 1 for t ∈ E ). A functional A : L → R is called an isotonic linear functionalif it is linear and has a property: if f ∈ L is non-negative, then A( f ) � 0.

The main subject in the Chebyshev inequality is a pair of similarly or oppositelyordered functions. We say that functions f and g on E are similarly ordered (or syn-chronous) if for each x,y ∈ E

( f (x)− f (y))(g(x)−g(y)) � 0.

If the reversed inequality holds, then we say that f and g are oppositely ordered orasynchronous.

Let us mention very recently proved result, [13].

CHEBYSHEV-GRUSS TYPE INEQUALITIES 1419

THEOREM 3. (The Chebyshev inequality for two isotonic linear functionals) Let Aand B be two isotonic linear functionals on L and let f ,g be two functions on E suchthat f , g , f g ∈ L. If f and g are similarly ordered functions, then

A( f g)B(1)+A(1)B( f g) � A( f )B(g)+A(g)B( f ). (4)

If f and g are oppositely ordered functions, then the reverse inequality in (4) holds.

Putting A = B in (4) and divided by 2 we get the Chebyshev inequality for oneisotonic positive functional. As we see, in the Chebyshev-Gruss inequality the firstderivative of functions f and g has appeared. In this paper we use a Δ-derivativeof function defined on a time scale set T . Let us mention here some definitions andproperties from the time scale theory which we use in our research. For more detailssee [1, 5, 6, 11, 18].

A time scale T is an arbitrary non-empty closed subset of the set R . A segment[a,b] in T is defined as [a,b] = {t ∈T : a � t � b} . Other kinds of intervals are definedsimilarly. On T we define two jump operators ρ and σ :

ρ(t) = sup{s ∈ T : s < t}, σ(t) = inf{s ∈ T : s > t}.A point t ∈ T is called left-dense if t > infT and ρ(t) = t , left scattered if ρ(t) <t , right-scattered if σ(t) > t and right-dense if t < supT and σ(t) = t . If T has aleft- scattered maximum M , then T

k = T\{M} , otherwise Tk = T . If T has a right-

scattered minimum m , then Tk = T\{m} , otherwise Tk = T .We say that f : T → R has the delta derivative f Δ(t) ∈ R at t ∈ T

k (provided itexists) if for each ε > 0 there exists a neighborhood U of t in T such that

| f (σ(t))− f (s)− f Δ(t)(σ(t)− s)| � ε|σ(t)− s| for all s ∈U.

On a similar way we define the nabla derivative (∇-derivative) f ∇(t) , [18]. For f :T → R and t ∈ Tk the nabla derivative at t is the number (provided it exists) such thatfor each ε > 0 there exists a neighborhood U of t in T such that

| f (ρ(t))− f (s)− f ∇(t)(ρ(t)− s)|� ε|ρ(t)− s| for all s ∈U.

A function f : T → R is called Δ-predifferentiable with region of differentiationD provided that the following conditions hold: f is continuous on T ; D⊂ T

k , Tk −D

is countable and contains no right-scattered elements of T and f is Δ-differentiableat each t ∈ D , ([6, 11, 18]). Similarly, a ∇-predifferentiable function is defined in[18]. As a consequence of the mean-value theorem we have the following statement,([6, 11]):

Let f : T → R be a Δ-predifferentiable function with region of differentiation D.If f Δ(t) � 0 for all t ∈ D, then f is increasing on T .

Similar statement holds for ∇-predifferentiable f , [18].The paper is organized in the following way. After this chapter with described

motivation, definitions and useful properties, we follow with Chebyshev-Gruss typeinequality involving two linear isotonic functionals in general settings - in time scalestheory. The third section is devoted to results involving lower bounds for the Chebyshevdifference and in the last chapter we give several examples.

2. Upper bound for the Chebyshev difference

As we say at the beginning of the paper we estimate a difference between twosides in the Chebyshev inequality for two functionals (4). For that difference we usethe abbreviation T ( f ,g) , i.e.

T ( f ,g) = A(1)B( f g)+B(1)A( f g)−A( f )B(g)−A(g)B( f ).

By linearity of functionals A and B we have that T is linear in each argument.In this section, set E is a time scale T and L is a set of real functions defined on

T . This section is devoted to generalization of the classical Chebyshev-Gruss inequality(2). The main theorem is the following.

THEOREM 4. Let A and B be two isotonic linear functionals on L. Let f , g ,h1 , h2 : T → R be Δ-predifferentiable functions with region of differentiation D, suchthat T ( f ,g) , T (h1,g) , T ( f ,h2) and T (h1,h2) exist and hΔ

1 , hΔ2 don’t change the sign,

hΔ1 (t) , hΔ

2 (t) �= 0 for t ∈ D. Then

|T ( f ,g)| �∥∥∥∥ f Δ

hΔ1

∥∥∥∥∞

∥∥∥∥gΔ

hΔ2

∥∥∥∥∞|T (h1,h2)|, (5)

where

∥∥∥∥ f Δ

hΔ1

∥∥∥∥∞

= supt∈D

∣∣∣∣ f Δ(t)hΔ

1 (t)

∣∣∣∣ .Proof. Let us suppose that hΔ

1 > 0, hΔ2 > 0. Denote by

F =∥∥∥∥ f Δ

hΔ1

∥∥∥∥∞

, G =∥∥∥∥gΔ

hΔ2

∥∥∥∥∞

.

Without loss of generality we may assume that F,G < ∞ . Then functions Fh1 + f ,

Gh2 + g are increasing. Namely, from assumptions we get∣∣∣ f Δ

hΔ1

∣∣∣ � F , i.e. −FhΔ1 �

f Δ � FhΔ1 , so (Fh1 + f )Δ � 0 on D and Fh1 + f is increasing on T . Similarly, we

get (Gh2 +g)Δ � 0. Using the same arguments we obtain that functions Fh1− f andGh2−g are increasing.

So we can use the Chebyshev inequality for two functionals, i.e. we have

T (Fh1 + f ,Gh2 +g) � 0 and T (Fh1− f ,Gh2−g) � 0.

By properties of T we get

T ( f ,g)+FG ·T (h1,h2)+F ·T (h1,g)+G ·T( f ,h2) � 0

T ( f ,g)+FG ·T (h1,h2)−F ·T (h1,g)−G ·T( f ,h2) � 0.

Adding these two inequalities we obtain

T ( f ,g) � −FG ·T (h1,h2).

Since G =∥∥∥ (−g)Δ

hΔ2

∥∥∥∞

we can write T ( f ,−g) � −FG ·T(h1,h2) and we get

T ( f ,g) � FG ·T(h1,h2).

Since h1 and h2 are similarly ordered, we have T (h1,h2) � 0, that means

|T ( f ,g)| � FG ·T (h1,h2) = FG · |T (h1,h2)|,which is in fact, inequality (5).

Suppose that hΔ1 > 0, hΔ

2 < 0. We apply the above-proved result replacing thefunction h2 by −h2 . Since ∣∣∣∣ gΔ(t)

(−h2)Δ(t)

∣∣∣∣=∣∣∣∣gΔ(t)hΔ

2 (t)

∣∣∣∣we get (5) also in this case. The other two possibilities for h1 and h2 can be regardedin the same way. �

In the previous theorem we can substitute Δ with ∇ . Then the previous theorembecomes as the following:

THEOREM 5. Let A and B be two isotonic linear functionals. Let f , g , h1 ,h2 : T → R be ∇-predifferentiable functions with region of differentiation D, such thatT ( f ,g) , T (h1,g) , T ( f ,h2) and T (h1,h2) exist and h∇

1 , h∇2 don’t change the sign,

h∇1 (t) , h∇

2 (t) �= 0 for t ∈ D. Then

|T ( f ,g)| �∥∥∥∥∥ f ∇

h∇1

∥∥∥∥∥∞

∥∥∥∥∥g∇

h∇2

∥∥∥∥∥∞

|T (h1,h2)|. (6)

3. Additional results for bounds of T ( f ,g)

The following theorem is a generalization of (3). In fact this result gives us ad-ditional information about bounds for T ( f ,g) together with the Chebyshev-Gruss in-equality.

THEOREM 6. Let A and B be two isotonic linear functionals. Let f , g , h1 ,h2 : T → R be Δ-predifferentiable functions with region of differentiation D, such thatT ( f ,g),T ( f ,h2) , T (h1,g) and T (h1,h2) exist.

(i) If hΔ1 ,hΔ

2�

(�)0 and if(f Δ �

(�)mhΔ1 and gΔ �

(�) rhΔ2

)or

(f Δ �

(�) −mhΔ1 and gΔ �

(�) − rhΔ2

)for some non-negative m,r , then

T ( f ,g) � mr ·T (h1,h2) � 0.

(ii) If hΔ1

�(�)0 , hΔ

2�

(�) 0 and if(f Δ �


(�) rhΔ2

)or

(f Δ �


(�) − rhΔ2

)for some m,r � 0 , then

T ( f ,g) � mr ·T (h1,h2) � 0.


(iii) If hΔ1 ,hΔ

2�

(�) 0 and if(f Δ �


(�) − rhΔ2

)or

(f Δ �


(�) rhΔ2

),

m,r � 0 , thenT ( f ,g) � −mr ·T (h1,h2) � 0.

(iv) If hΔ1

�(�) 0 , hΔ

2�

(�) 0 and if(f Δ �


(�) − rhΔ2

)or

(f Δ �


(�) rhΔ2

),

m,r � 0 , thenT ( f ,g) � −mr ·T (h1,h2) � 0.

Proof. Let us prove one case (among possible 16 cases) in details. Let hΔ1 ,hΔ

2 � 0and f Δ � mhΔ

1 , and gΔ � rhΔ2 . Considering Δ-derivatives of functions f −mh1 and

g− rh2 we find that( f −mh1)Δ � 0 and (g− rh2)Δ � 0,

hence f −mh1 and g−rh2 are increasing. Furthermore, from assumption f Δ � mhΔ1 �

0 we conclude that f is increasing. Applying the Chebyshev inequality (4) on twoincreasing functions f −mh1 and h2 we get

T ( f −mh1,h2) � 0, i.e. T ( f ,h2) � mT (h1,h2). (7)Similarly, applying the Chebyshev inequality for two functionals (4) on two increasingfunctions g− rh2 and f we get

T (g− rh2, f ) � 0, i.e. T ( f ,g) � rT ( f ,h2). (8)From (7) and (8) we get

T ( f ,g) � rT ( f ,h2) � mr ·T (h1,h2) � 0,where the last inequality is true since h1 and h2 are increasing. Other cases are provedin a similar manner. �

COROLLARY 1. Let A and B be isotonic linear functionals on L, and let f ,g :T → R be Δ-predifferentiable functions with region of differentiation D such that 0 �m � gΔ(x) � M for x ∈ D.

(i) If f Δ � 0 , thenm ·T ( f ,e1) � T ( f ,g) � M ·T ( f ,e1), (9)

where e1(x) = x .

(ii) If f Δ � 0 , then the reverse signs in the above inequality hold.

Proof. Putting in the previous theorem 6(i):f = f , g = g, h1 = f , h2 = e1, m = 1, r = m

we get the first inequality in (9). The second inequality is a consequence of Theorem 4.Of course, this corollary can be proved directly applying the Chebyshev inequality onpairs of functions f and g−me1 , or f and Me1 −g . �

REMARK 1. If in Theorem 6 and Corollary 1 we substitute Δ with ∇ we getcorresponding results from ∇ calculus.

4. Applications

In this section we give applications of the previous theorems for some particularcases. Also, we list papers in which particular cases of results from Sections 2 and 3appear.

4.1. Δ-integral

Let a,b ∈ T with a < b and let A and B be Cauchy Δ-integrals of f , i.e. A( f ) =B( f ) =

∫ ba w(x) f (x)Δx , w � 0. Definition and properties of it are given in [5] and [11].

Using the fact that A is an isotonic linear functional (see [2]), and if assumptions ofTheorem 4 are satisfied we get the following Chebyshev-Gruss inequality:∣∣∣∣∫ b

aw(x)Δx

∫ b

aw(x) f (x)g(x)Δx−

∫ b

aw(x) f (x)Δx

∫ b

aw(x)g(x)Δx

∣∣∣∣�∥∥∥∥ f Δ

hΔ1

∥∥∥∥∞

∥∥∥∥gΔ

hΔ2

∥∥∥∥∞

∣∣∣∣∫ b

aw(x)Δx

∫ b

aw(x)h1(x)h2(x)Δx

−∫ b

aw(x)h1(x)Δx

∫ b

aw(x)h2(x)Δx

∣∣∣∣ . (10)

Let us mention that, in general, Theorem 4 gives us a result for two different linearfunctionals, so we can, for example, use integrals with different weights.

REMARK 2. In paper [16, Theorem 9] another version of the Chebyshev-Grussinequality (10) is given. They used h1(t) = h2(t) = t and it is proved via the generalizedMontgomery identity.

Results of Theorem 6 and Corollary 1 for Δ-integral seem to be quite new.

4.2. ∇-integral

Let us define A as A( f ) =∫ ba w(x) f (x)∇x = B( f ) . More about ∇-integral, es-

pecially about its properties and connections with the theory of linear functionals canbe found in [2] and [5]. Applying Theorem 5 we get the following Chebyshev-Grussinequality:∣∣∣∣∫ b

aw(x)∇x

∫ b

aw(x) f (x)g(x)∇x−

∫ b

aw(x) f (x)∇x

∫ b

aw(x)g(x)∇x

∣∣∣∣�∥∥∥∥∥ f ∇

h∇1

∥∥∥∥∥∞

∥∥∥∥∥g∇

h∇2

∥∥∥∥∥∞

∣∣∣∣∫ b

aw(x)∇x

∫ b

aw(x)h1(x)h2(x)∇x

−∫ b

aw(x)h1(x)∇x

∫ b

aw(x)h2(x)∇x

∣∣∣∣ , (11)

where f ,g,h1,h2 satisfy assumptions of Theorem 5. We don’t find any results of thattype in literature and it seems new to us. Also, a ∇-analogue of Theorem 6 containtsnew results.

4.3. q -integral

Let 0 < q < 1, b > 0 and T = {0}∪ {bqn : n = 0,1,2, . . .} . For a function f :T → R we have

f ∇(t) =f (qt)− f (t)

(q−1)t, t �= 0,

and f ∇(0) = lims→0f (s)− f (0)

s if this limit exists. In q -calculus the number f ∇(t) isusually noted as Dq( f )(t) and called a q -derivative of a function f at a point t . TheJackson integral of f (or q -integral) is defined as

Iq( f ) =∫ b

0f (x)dq(x) := b(1−q)

∞

∑n=0

qn f (bqn).

In particular case, when h1(x) = h2(x) = x , A = B = Iq inequality (6) becomes

|bIq( f g)− Iq( f )Iq(g)| � ||Dq( f )|| ||Dq(g)|| qb4

(1+q+q2)(1+q)2 ,

with ||Dq( f )|| = supt∈T |Dq( f )(t)| .In [20] one can find a similar result proved by the Montgomery identity, while in

[17] an inequality of Chebyshev-Gruss type involving q -derivative and q -integral on ageneral interval [a,b] is given. The proof of the last-mentioned result is based on theproperties of Lipschitz functions.

4.4. Continuous case – isotonic functionals, the Riemann integral

If Δ-derivative coincides with classical derivative a result from Theorem 4 for onelinear functional, i.e. a case A = B is given in [15]. In that paper the proof is based onthe Cauchy mean-value theorem. A case when h1 = h2 = h is rediscovered in [10] butusing different proof. In fact, that proof is based on the Chebyshev inequality and weuse their idea in our proof of Theorem 4.

Furthermore, if A( f ) = B( f ) =∫ ba w(x) f (x)dx , w � 0, then inequality (5) be-

comes: ∣∣∣∣∫ b

aw(x)dx

∫ b

aw(x) f (x)g(x)dx−

∫ b

aw(x) f (x)dx

∫ b

aw(x)g(x)dx

∣∣∣∣ (12)

�∥∥∥∥ f ′

h′1

∥∥∥∥∞

∥∥∥∥ g′

h′2

∥∥∥∥∞

∣∣∣∣∫ b

aw(x)dx

∫ b

aw(x)h1(x)h2(x)dx −

∫ b

aw(x)h1(x)dx

∫ b

aw(x)h2(x)dx

∣∣∣∣ ,where f ,g,h1,h2 satisfy assumptions of Theorem 4 for this particular case. The casewhen h1(x) = h2(x) = x is proved in [9]. For w = 1 we get inequality (2).

Putting in Theorem 6(i): A( f ) = B( f ) =∫ ba w(t) f (t)dt , h1(x) = h2(x) = x−a , f

and g are two differentiable, monotonic functions in the same direction with | f ′(x)| �m and |g′(x)| � r on [a,b] , then we get result from [14].

If h1(x) = x−a , h2(x) = b− x , f and g are two differentiable, monotonic func-tions in the opposite direction with | f ′(x)| � m and |g′(x)|� r on [a,b] , then Theorem6(ii) gives result from [14], also.

4.5. Continuous case – fractional integral operators

Let us consider a fractional hypergeometric operator Iα ,β ,η,μt which covers several

types of well-known fractional operators: the Riemann-Liouville fractional integral op-erator (β = −α , η = μ = 0), the Erdelyi-Kober operator (β = 0,μ = 0) and Saigooperator (μ = 0).

If t > 0, α > max{0,−β −μ} , μ > −1, β −1 < η < 0, then a fractional hyper-geometric operator is defined as

A( f )= Iα ,β ,η,μt { f} :=

t−α−β−2μ

Γ(α)

∫ t

0σ μ(t−σ)α−1

2 F1

(α+β+μ ,−η ,α;1−σ

t

)f (σ)dσ

where the function 2F1(a,b,c,t) =∞

∑n=0

(a)n(b)n

(c)n

tn

n!is the Gaussian hypergeometric

function and (a)n is the Pochhammer symbol: (a)n = a(a+1) . . .(a+n−1) , (a)0 =1, [3]. A functional A( f ) = Iα ,β ,η,μ

t { f (t)} is isotonic linear, so we can write theChebyshev-Gruss type inequality for fractional hypergeometric operators.

THEOREM 7. Let p, q , f , g , h1 , h2 be functions on [0,∞) , p , q � 0 , f , gdifferentiable, h1 , h2 differentiable, strictly monotonic in the same direction. Let t >0 , α > max{0,−β − μ} , μ > −1 , β − 1 < η < 0 , γ > max{0,−δ − ν} , ν > −1 ,δ −1 < ζ < 0 .

If∥∥∥ f ′

h′1

∥∥∥∞

,∥∥∥ g′

h′2

∥∥∥∞

< ∞ , then

|Iα ,β ,η,μt {p}Iγ,δ ,ζ ,ν

t {q f g}+ Iα ,β ,η,μt {p f g}Iγ,δ ,ζ ,ν

t {q}− Iα ,β ,η,μt {p f}Iγ,δ ,ζ ,ν

t {qg}−Iα ,β ,η,μ

t {pg}Iγ,δ ,ζ ,νt {q f}|

�∥∥∥∥ f ′

h′1

∥∥∥∥∞

∥∥∥∥ g′

h′2

∥∥∥∥∞

∣∣∣Iα ,β ,η,μt {p}Iγ,δ ,ζ ,ν

t {qh1h2}+ Iα ,β ,η,μt {ph1h2}Iγ,δ ,ζ ,ν

t {q}

−Iα ,β ,η,μt {ph1}Iγ,δ ,ζ ,ν

t {qh2}− Iα ,β ,η,μt {ph2}Iγ,δ ,ζ ,ν

t {qh1}∣∣∣ .

Proof. It is a consequence of Theorem 4 for A( f ) = Iα ,β ,η,μt {p f} and B( f ) =

Iγ,δ ,ζ ,νt {q f} . �

A particular case of Theorem 7 for two Riemann-Liouville fractional operatorsA( f ) = Jα p f (t) , B( f ) = Jβ q f (t) and h(t) = t is given in [8]. An analogue result fortwo Riemann-Liouville q -integrals can be find in [7]. Furthermore, the Chebyshev-Gruss type inequality for the Saigo q -integral operators is given in [19].

Some results of the third section also can be found in recent literature. For exam-ple, if A and B are the same non-weighted Riemann-Liouville fractional integrals, i.e.A( f ) = B( f ) = Jα f (t) , then results from Corollary 1 are given in [4]. Results of thesame Corollary but for one fractional hypergeometric operator are given in [3].

Here we give results for only one class of fractional integral operators, i.e. for frac-tional hypergeometric operators. Of course, analogue results, which are consequencesof Theorems 4 and 6 and Corollary 1 hold for other types of fractional integral oper-ators which have isotonic property, for example, for the Hadamard operator, for theKatugampola operator, the Agrawal integral operator etc.


4.6. Discrete case

Let T ⊆ N . Then functions f ,g,h1,h2 become sequences ( fk)k , (gk)k , (h1k)k ,(h2k)k , and Δ-derivative becomes a difference Δ between two consecutive elements ofa sequence, i.e. Δak = ak+1 − ak . For this particular case a part (i) of Theorem 6 hasthe following form.

THEOREM 8. Let (pk)k and (qk)k be non-negative n-tuples. If real n-tuples(h1k)k , (h2k)k are monotonic in the same direction with property: Δhik �= 0 for allk = 1, . . . ,n, i = 1,2 , and if for some non-negative m,r(

Δ fkΔh1k

� m andΔgk

Δh2k� r

)or

(Δ fkΔh1k

� −m andΔgk

Δh2k� −r

)for k = 1, . . . ,n, then

∑ pk fkgk ∑qk +∑ pk ∑qk fkqk −∑ pk fk ∑qkgk −∑ pkgk ∑qk fk

� mr(∑ pkh1kh2k ∑qk +∑ pk ∑qkh1kh2k −∑ pkh1k ∑qkh2k −∑ pkh2k ∑qkh1k

).

The above result for same n -tuples p and q , h1k = h2k = k , and correspondingvariant of Theorem 4 is given in [14]. Other parts of Theorem 6 can also be given indiscrete form.

Acknowledgements. The research of the first author was partially supported by theSofia University SRF under contract No 146/2015. The research of the second authorwas supported by Croatian Science Foundation under the project 5435.

RE F ER EN C ES

[1] R. AGARWAL, M. BOHNER, A. PETERSON, Inequalities on time scales: A survey, Math. Inequal.Appl., 4 (4) (2001), 535–557.

[2] M. ANWAR, R. BIBI, M. BOHNER, J. PECARIC, Integral inequalities on time scales via the theoryof isotonic linear functionals, Abstr. Appl. Anal. 2001 (2011) Art. ID: 483595, 1–16.

[3] D. BALEANU, S. D. PUROHIT, PRAVEEN AGARWAL, On fractional integral inequalities involvinghypergeometric operators, Chin. J. Math. (N. Y.), Volume 2014 (2014), Article ID 609476.

[4] S. BELARBI, Z. DAHMANI, On some new fractional integral inequalities, JIPAM. J. Inequal. PureAppl. Math., 10 (3) (2009), Art. 86.

[5] M. BOHNER, A. PETERSON, Dynamic Equations on Time Scales: An Introduction with Applications,Birkhauser, Boston, 2001.

[6] M. BOHNER, G. GUSEINOV, A. PETERSON, Introduction to the time scales calculus, Chapter inbook: Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003, pp. 1–15, Eds: M.Bohner and A. Peterson.

[7] K. BRAHIM, S. TAF, On some fractional q -integral inequalities, Malaya Journal of Matematik, 3 (1)(2013), 21–26.

[8] Z. DAHMANI, Some results associated with fractional integrals involving the extended Chebyshevfunctional, Acta Univ. Apulensis Math. Inform., 27 (2011), 217–224.

[9] S. S. DRAGOMIR, Some integral inequalities of Gruss type, Indian J. Pure Appl. Math., 31 (4) (2000),397–415.

[10] H. GONSKA, I. RASA AND M. RUSU, Cebysev-Gruss-type inequalities revisited, Math. Slovaca, 63(5) (2013), 1007–1024.

[11] G. SH. GUSEINOV, Integration on time scales, J. Math. Anal. Appl., 285 (1) (2003), 107–127.


[12] D. S. MITRINOVIC, J. E. PECARIC AND A. M. FINK, Classical and new Inequalities in Analysis,Kluwer Academic Publishers, Dordrecht-Boston-London, 1993.

[13] L. NIKOLOVA, S. VAROSANEC,Chebyshev and Gruss type inequalities involving two linear function-als and applications, Math. Inequal. Appl., 19 (1) (2016), 127–143.

[14] J. PECARIC, On the Ostrowski generalization of Cebysev’s inequality, J. Math. Anal. Appl., 102 (2)(1984), 479–487.

[15] J. PECARIC, S. S. DRAGOMIR AND J. SANDOR, On some Gruss type inequalities for isotonic func-tionals, Rad Hrvat. Akad. Znan. Umj. Mat. Znan., [467] 11 (1994), 41–47.

[16] M. Z. SARIKAYA, N. AKTAN, H. YILDIRIM, On weighted Cebysev-Gruss type inequalities on timescales, J. Math. Inequal., 2 (2008), 185–195.

[17] J. TARIBOON, S. K. NTOUYAS, Quantum integral inequalities on finite intervals, J. Inequal. Appl.,2014, 2014:121.

[18] S. G. TOPAL, Rolle’s and generalized mean value theorems on time scales, J. Difference Equ. Appl.,8 (4) (2002), 333–344.

[19] W. YANG, Some new Chebyshev and Gruss-type integral inequalities for Saigo fractional integraloperators and their q -analogues, Filomat, 29 (6) (2015), 1269–1289.

[20] W. YANG, On weighted q-Cebysev-Gruss type inequalities, Comput. Math. Appl., 61 (2011), 1342–1347.

(Received April 26, 2016) Ludmila NikolovaDepartment of Mathematics and Informatics

Sofia UniversitySofia, Bulgaria


Sanja VarosanecDepartment of Mathematics, University of Zagreb

Zagreb, Croatiae-mail: [email protected]

& Applications


GENERALIZED INTEGRAL INEQUALITIES FOR CONVEX FUNCTIONS

M. EMIN OZDEMIR AND ALPER EKINCI


Abstract. In this paper, we prove some general inequalities for convex functions and give Os-trowski, Hadamard and Simpson type results for a special case of these inequalities.

1. Introduction

The function f : [a,b] → R, is said to be convex, if we have

f (tx+(1− t)y) � t f (x)+ (1− t) f (y)

for all x,y ∈ [a,b] and t ∈ [0,1] . For more information see the papers [3], [1], [2].Let f : I ⊆R→R be a convex function and let a,b∈ I, with a < b. The following

double inequality:

f

(a+b

2

)� 1

b−a

b∫a

f (x)dx � f (a)+ f (b)2

is known in the literature as Hadamard’s inequality.In 1928 Ostrowski proved the following famous inequality:

THEOREM 1. Let f : [a,b] → R be continuous on [a,b] and differentiable on(a,b) and its derivative f ′ : (a,b) → R be bounded on (a,b) , that is, || f ′||∞ :=sup

t∈(a,b)| f ′ (x)| < ∞. Then for any x ∈ [a,b] , the following inequality holds:

∣∣∣∣∣∣ f (x)− 1b−a

b∫a

f (t)dt

∣∣∣∣∣∣�∣∣∣∣∣14 +

(x− a+b

2

)2(b−a)2

∣∣∣∣∣(b−a)∣∣∣∣ f ′∣∣∣∣∞ .

The inequality is sharp in the sense that the constant 1/4 cannot be replaced by asmaller one. In the rest of this section we list known results which we will generalizein the following section.

Sarikaya et al. obtained following Simpson type inequalities in [5].

Mathematics subject classification (2010): 26D15, 26D10.Keywords and phrases: Convex functions, Hermite-Hadamard inequality, Simpson’s inequality,

power-mean inequality, Ostrowski’s inequality; Holder’s inequality.


1429

1430 M. E. OZDEMIR AND A. EKINCI

THEOREM 2. Let f : I ⊆ R → R be a differential mapping on Io such that f ′ ∈L1 [a,b] , where a,b ∈ I with a < b. If | f ′| is convex on [a,b] , then the following in-equality holds:∣∣∣∣∣∣16

[f (a)+4 f

(a+b

2

)+ f (b)

]− 1

b−a

b∫a

f (x)dx

∣∣∣∣∣∣� 5(b−a)72

[| f ′(a)|+ | f ′(b)|] .(1)

THEOREM 3. Let f : Io ⊆ R → R be a differential mapping on Io such that f ′ ∈L1 [a,b] , where a,b∈ I with a < b. If | f ′|q is convex on [a,b] , q � 1 , then the followinginequality holds: ∣∣∣∣∣∣16

[f (a)+4 f

(a+b

2

)+ f (b)

]− 1

b−a

b∫a

f (u)du

∣∣∣∣∣∣ (2)

� (b−a)72

(5)1− 1q

{[61 | f ′ (b)|q +29 | f ′ (a)|q

18

] 1q

+[61 | f ′ (a)|q +29 | f ′ (b)|q

18

] 1q}

.

Estimation for the difference between the middle and the leftmost term in theHadamard inequality was proved by Kirmaci in [8].

THEOREM 4. Let f : Io →R be a differential mapping on Io , a,b∈ Io with a < b.If | f ′| is convex on [a,b] , then we have∣∣∣∣∣∣ 1

b−a

b∫a

f (x)dx− f

(a+b

2

)∣∣∣∣∣∣� b−a4

( | f ′(a)|+ | f ′(b)|2

). (3)

Similarly, bound for the difference between the middle and the right most term inthe Hadamard inequality was considered by Dragomir and Agarwal in [7] and has thefollowing forms.

THEOREM 5. Let f : Io ⊆ R → R be a differential mapping on Io , a,b ∈ Io witha < b. If | f ′| is convex on [a,b] , then the following inequality holds:∣∣∣∣∣∣ f (a)+ f (b)

2− 1

b−a

b∫a

f (x)dx

∣∣∣∣∣∣ (4)

� (b−a)8

[∣∣ f ′(a)∣∣+ ∣∣ f ′(b)

∣∣] .

GENERALIZED INTEGRAL INEQUALITIES FOR CONVEX FUNCTIONS 1431

THEOREM 6. Let f : Io ⊆ R → R be a differential mapping on Io , a,b ∈ Io witha 1. If the new mapping | f ′|p/(p−1) is convex on [a,b] , then the fol-lowing inequality holds:∣∣∣∣∣∣ f (a)+ f (b)

2− 1

b−a

b∫a

f (x)dx

∣∣∣∣∣∣ (5)

� (b−a)

2(p+1)1/p

[| f ′(a)|p/p−1 + | f ′(b)|p/p−1

2

](p−1)/p

.

In this paper we give inequalities involving m harmonic polynomials and a func-

tion f such that∣∣∣ f (n)

∣∣∣q is convex for some q � 1 and n ∈ N. After each general

inequality we obtain a result related to particular case n = 1, m = 2 and point out thatfor some especial values of our variables h and c, these results become the knownresults given in the above text.

2. Main results

In the further text m and n are fixed integers, σ := {a = x0 < x1 < x2 < ... < xm =b} is a division of an interval [a,b] with m+1 nodes, σ ′ := {0 = s0 < s1 < ... < sm = 1}is a corresponding division of the interval [0,1] connectedwith σ by relation s j =

x j−ab−a .

Let{Pjk}

k∈N, j = 1, ...,m be harmonic sequences of polynomials, i.e. P′

jk = Pj,k−1 ,k ∈ N, Pj0 = 1, j = 1, ...,m. Let us define a kernel Sn as

Sn (t,σ) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

P1n (t) , t ∈ [a,x1]P2n (t) , t ∈ (x1,x2]

...Pmn (t) , t ∈ (xm−1,b] .

In paper [4] Pecaric and Varosanec gave the following identity for n -times differen-tiable function f on [a,b] :

(−1)nb∫

a

Sn (x,σ) f (n) (x)dx =b∫

a

f (t)dt +n

∑k=1

(−1)k[Pmk (b) f (k−1) (b) (6)

+m−1

∑j=1

[Pjk (x j)−Pj+1,k (x j)

]f (k−1) (x j)−P1k (a) f (k−1) (a)

].

For further applications of this identity see [4], [6].Let us denote the right-hand side of the above identity by Im. Using substitution

x = tb+(1− t)a on the left handside of (6) we get

(−1)nb∫

a

Sn (x,σ) f (n) (x)dx

= (−1)n (b−a)1∫

0

Sn (tb+(1− t)a,σ) f (n) (tb+(1− t)a)dt.

Using notation Cn (t,σ ′) = (b−a)Sn (tb+(1− t)a) identity (6) becomes

(−1)n1∫

0

Cn(t,σ ′) f (n) (tb+(1− t)a)dt = Im. (7)

Also, in the further text we use abbreviations Kjn (t) = (b−a)Pjn (tb+(1− t)a) , j =1, ...,m, i.e.

Cn(t,σ ′)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

K1n (t) , t ∈ [0,s1]K2n (t) , t ∈ (s1,s2]

...Kmn (t) , t ∈ (sm−1,1] .

THEOREM 7. Let f : I ⊆ R → R be an n-times differentiable function on I◦,a,b∈ I◦, a< b and

∣∣∣ f (n)∣∣∣q be convex on [a,b] for some q � 1 such that Sn (t,σ) f (n) (t)

is integrable on [a,b] . Then

|Im| �m

∑j=1

⎛⎝ s j∫

s j−1

∣∣Kjn (t)∣∣dt

⎞⎠1− 1

q⎧⎨⎩∣∣∣ f (n) (b)

∣∣∣q⎛⎝ s j∫

s j−1

t∣∣Kjn (t)

∣∣dt

⎞⎠ (8)

+∣∣∣ f (n) (a)

∣∣∣q⎛⎝ s j∫

s j−1

(1− t)∣∣Kjn (t)

∣∣dt

⎞⎠⎫⎬⎭

1q

.

Proof. Using (7) and the power-mean inequality for q � 1 we have

|Im| �1∫

0

∣∣Cn(t,σ ′)∣∣ ∣∣∣ f (n) (tb+(1− t)a)

∣∣∣dt

=m

∑j=1

s j∫s j−1

∣∣Kjn (t)∣∣ ∣∣∣ f (n) (tb+(1− t)a)

∣∣∣dt

�m

∑j=1

⎛⎝ s j∫

s j−1


⎞⎠1− 1

q⎛⎝ s j∫

s j−1

∣∣Kjn (t)∣∣ ∣∣∣ f (n) (tb+(1− t)a)

∣∣∣q dt

⎞⎠

1q

.


Since∣∣∣ f (n)

∣∣∣q is convex we have

|Im| �m

∑j=1

⎛⎝ s j∫

s j−1


⎞⎠1− 1

q

×⎛⎝ s j∫

s j−1

∣∣Kjn (t)∣∣(t ∣∣∣ f (n) (b)

∣∣∣q +(1− t)∣∣∣ f (n) (a)

∣∣∣q)dt

⎞⎠

1q

=m

∑j=1

⎛⎝ s j∫

s j−1


⎞⎠1− 1

q

×⎛⎝∣∣∣ f (n) (b)

∣∣∣qs j∫

s j−1

t∣∣Kjn (t)

∣∣dt +∣∣∣ f (n) (a)

∣∣∣qs j∫

s j−1

(1− t)∣∣Kjn (t)

∣∣dt

⎞⎠

1q

.

So, we deduce the desired result. �

The case when q = 1 is significant, so we write that result as the following corol-lary.

COROLLARY 1. If f satisfies assumptions of Theorem 7 with q = 1 which also

means∣∣∣ f (n)

∣∣∣ is convex on [a,b] , then

|Im| �∣∣∣ f (n) (b)

∣∣∣ m

∑j=1

s j∫s j−1

t∣∣Kjn (t)

∣∣dt +∣∣∣ f (n) (a)

∣∣∣ m

∑j=1

s j∫s j−1

(1− t)∣∣Kjn (t)

∣∣dt. (9)

COROLLARY 2. If f satisfies assumptions of Theorem 7 with n = 1, m = 2, then

∣∣∣∣∣∣h [ f (a)+ f (b)]+ (1−2h) f (x)− 1b−a

b∫a

f (u)du

∣∣∣∣∣∣ (10)

� (b−a)

⎧⎨⎩(

c2

2−hc+h2

)1− 1q [

λ1∣∣ f ′ (b)

∣∣q + λ2∣∣ f ′ (a)

∣∣q] 1q

+(

c2 +12

− (c+h)(1−h))1− 1

q [λ3∣∣ f ′ (b)

∣∣q + λ4∣∣ f ′ (a)

∣∣q] 1q ,


where x ∈ [a,b] , h ∈ [0, 12

], c = x−a

b−a such that h � c � 1−h and

λ1 =16

(2c3−3c2h+2h3)

λ2 =16

(−6ch+6h2−2h3−2c3 +3c2 +3c2h)

λ3 =16

(2c3 +3c2h−3c2−2h3 +6h2−3h+1

)λ4 =

16

(2−2c3 +6c2−3c2h+6ch−6c+2h3−3h

).

Proof. Let us consider a division σ = {a � x � b} of an interval [a,b] . Let usdefine the kernel S1(t,σ) as

S1(t,σ) ={

P11(t) = t− (1−h)a−hb, t ∈ [a,x]P21(t) = t−ha− (1−h)b, t ∈ (x,b] .

Polynomials P11 and P21 are obviously harmonic. Denote by c := x−ab−a . Then we

consider a corresponding division σ ′ = {0 � c � 1} of the interval [0,1] and the kernelC1(t,σ ′) is equal to

C1(t,σ ′) ={

K11(t) = (b−a)2(t−h), t ∈ [0,c]K21(t) = (b−a)2(t−1+h), t ∈ (c,1] .

Putting in Theorem 7 the kernel C1(t,σ ′) after simple calculation with taking intoaccount the condition h � c � 1−h that is,

c∫0

|t−h|dt =h∫

0

(h− t)dt +c∫

h

(t−h)dt =c2

2− ch+h2

and1∫

c

|t −1+h|dt =1−h∫c

(1−h− t)dt +1∫

1−h

(t−1+h)dt =c2 +1

2− (c+h)(1−h),

we get the desired inequality. �

REMARK 1. If we set h = 16 and c = 1

2 in (10) we obtain inequality (2) .

REMARK 2. For different selections of parameters h and c in (10) we obtain thefollowing Ostrowski, Simpson and Hadamard type inequalities.

(i) For the case q = 1, h = 0 we have∣∣∣∣∣∣ f (x)− 1b−a

b∫a

f (u)du

∣∣∣∣∣∣� b−a

6

[(1−3c2 +4c3)∣∣ f ′ (b)

∣∣+ (2−6c+9c2−4c3)∣∣ f ′ (a)∣∣]

which is an Ostowski type inequality. Furthermore, if c = 12 we get inequality

(3) .

(ii) For the case q = 1, h = 16 we obtain the following Simpson type inequality∣∣∣∣∣∣16 [ f (a)+4 f (x)+ f (b)]− 1

b−a

b∫a

f (u)du

∣∣∣∣∣∣� (b−a)

6

[(23−3c2 +4c3

)∣∣ f ′ (b)∣∣+(5

3−6c+9c2−4c3

)∣∣ f ′ (a)∣∣]

which for c = 12 collapses to inequality (1) .

(iii) If q = 1, h = c = 12 we obtain the following Hadamard type inequality (4) .

THEOREM 8. Let f : I ⊆ R → R be an n-times differentiable mapping on I◦,a,b ∈ I◦, a < b, and

∣∣∣ f (n)∣∣∣q is convex on [a,b] for some q > 1, Sn (t,σ) f (n) (t) is

integrable on [a,b] . Then the following inequality holds:

|Im| �⎛⎝ m

∑j=1

s j∫s j−1

∣∣Kjn (t)∣∣p dt

⎞⎠

1p⎛⎝∣∣∣ f (n) (a)

∣∣∣q +∣∣∣ f (n) (b)

∣∣∣q2

⎞⎠

1q

, (11)

where 1p + 1

q = 1 .

Proof. From (7) and property of modulus we have

|Im| =

∣∣∣∣∣∣1∫

0

Cn(t,σ ′) f (n) (tb+(1− t)a)dt

∣∣∣∣∣∣�

1∫0

∣∣∣Cn(t,σ ′) f (n) (tb+(1− t)a)

∣∣∣dt.

By using Holder inequality we have∣∣∣∣∣∣1∫

0

Cn(t,σ ′) f (n) (tb+(1− t)a)dt

∣∣∣∣∣∣�

⎛⎝ 1∫

0

∣∣Cn(t,σ ′)∣∣p dt

⎞⎠

1p⎛⎝ 1∫

0

∣∣∣ f (n) (tb+(1− t)a)∣∣∣q dt

⎞⎠

1q

.



∣∣∣q is convex, by Hadamard inequality we get

1∫0

∣∣∣ f (n) (tb+(1− t)a)∣∣∣q dt �

∣∣∣ f (n) (a)∣∣∣q +

∣∣∣ f (n) (b)∣∣∣q

2.

Also we have1∫

0

∣∣Cn(t,σ ′)∣∣p dt =

m

∑j=1

s j∫s j−1

∣∣Kjn (t)∣∣p dt.

By combining these we deduce the desired result. �

COROLLARY 3. If f satisfies assumptions of Theorem 8 with n = 1, m = 2, then∣∣∣∣∣∣h [ f (a)+ f (b)]+ (1−2h) f (x)− 1b−a

b∫a

f (u)du

∣∣∣∣∣∣ (12)

� (b−a)

(2h1+p +(c−h)1+p +(1− c−h)1+p

1+ p

) 1p ( | f ′ (a)|q + | f ′ (b)|q

2

) 1q

,

where x ∈ [a,b] , h ∈ [0, 12

], c = x−a

b−a such that h � c � 1−h.

Proof. Putting in Theorem 8 the kernel C1 (t,σ ′) defined as in the proof of Corol-lary 2, after simple calculation we get desired inequality. �

REMARK 3. For different selections of the parameters h and c in (12) we obtainthe following Ostrowski, Hadamard and Simpson type inequalities

(i) For the case h = 0 we have∣∣∣∣∣∣ f (x)− 1b−a

b∫a

f (u)du

∣∣∣∣∣∣� (b−a)

(c1+p+(1−c)1+p

1+p

) 1p ( | f ′ (a)|q + | f ′ (b)|q

2

) 1q

.

(ii) For the case h = 12 and c = 1

2 we obtain (5) .

(iii) For h = 16 and c = 1

2 we have∣∣∣∣∣∣16[

f (a)+4 f

(a+b

2

)+ f (b)

]− 1

b−a

b∫a

f (u)du

∣∣∣∣∣∣� (b−a)

6

(21+p +13(p+1)

) 1p( | f ′ (a)|q + | f ′ (b)|q

2

) 1q

.

THEOREM 9. Let f : I ⊆ R → R be an n-times differentiable mapping on I◦,a,b ∈ I◦, a < b, and

∣∣∣ f (n)∣∣∣q is convex on [a,b] for some q > 1, Sn (t,σ) f (n) (t) is

integrable on [a,b] . Then the following inequality holds:

|Im| �m

∑j=1

⎛⎝ s j∫

s j−1


⎞⎠

1p⎛⎝∣∣∣ f (n) (x j)

∣∣∣q +∣∣∣ f (n)

(x j−1

)∣∣∣q2

⎞⎠

1q

, (13)

where 1p + 1

q = 1 .

Proof. By using (7) , property of modulus and Holder Inequality we have

|Im| �1∫

0

∣∣∣Cn(t,σ ′) f (n) (tb+(1− t)a)

∣∣∣dt.

�m

∑j=1

s j∫s j−1

∣∣∣Kjn (t) f (n) (tb+(1− t)a)∣∣∣dt

�m

∑j=1

⎛⎝ s j∫

s j−1


⎞⎠

1p⎛⎝ s j∫

s j−1

∣∣∣ f (n) (tb+(1− t)a)∣∣∣q dt

⎞⎠

1q

.


∣∣∣q is convex, by using Hadamard inequality we get

s j∫s j−1

∣∣∣ f (n) (tb+(1− t)a)∣∣∣q dt �

∣∣∣ f (n) (x j)∣∣∣q +

∣∣∣ f (n) (x j−1)∣∣∣q

2.

So, this implies (13) . �

COROLLARY 4. If f satisfies assumptions of Theorem 8 with n = 1, m = 2, then∣∣∣∣∣∣h [ f (a)+ f (b)]+ (1−2h) f (x)− 1b−a

b∫a

f (u)du

∣∣∣∣∣∣ (14)

� (b−a)

⎧⎨⎩(

h1+p +(c−h)1+p

1+ p

) 1p ( | f ′ (a)|q + | f ′ (x)|q

2

) 1q

+

(h1+p +(1− c−h)1+p

1+ p

) 1p ( | f ′ (b)|q + | f ′ (x)|q

2

) 1q

⎫⎬⎭ ,

where x ∈ [a,b] , h ∈ [0, 12

], c = x−a

b−a such that h � c � 1−h.


Proof. Putting in Theorem 9 the kernel Cn (t,σ ′) defined as in the proof of Corol-lary 2, after simple calculation we get desired inequality. �

REMARK 4. For different selections of the parameters h and c in (14) we obtainthe following Ostrowski, Hadamard and Simpson type inequalities

(i) For the case h = 0 we have∣∣∣∣∣∣ f (x)− 1b−a

b∫a

f (u)du

∣∣∣∣∣∣� (b−a)

⎧⎨⎩(

c1+p

1+ p

) 1p( | f ′ (a)|q + | f ′ (x)|q

2

) 1q

+

((1− c)1+p

1+ p

) 1p ( | f ′ (b)|q + | f ′ (x)|q

2

) 1q

⎫⎬⎭ .

(ii) For the case h = 12 and c = 1

2 we get the following inequality∣∣∣∣∣∣ f (a)+ f (b)2

− 1b−a

b∫a

f (u)du

∣∣∣∣∣∣� (b−a)

2

(1

2(1+ p)

) 1p

⎧⎨⎩(| f ′ (a)|q +

∣∣ f ′ ( a+b2

)∣∣q2

) 1q

+

(| f ′ (b)|q +

∣∣ f ′ ( a+b2

)∣∣q2

) 1q

⎫⎬⎭ .

(iii) For h = 16 and c = 1

2 we have the following Simpson type inequality∣∣∣∣∣∣16[

f (a)+4 f (a+b

2)+ f (b)

]− 1

b−a

b∫a

f (u)du

∣∣∣∣∣∣� (b−a)

6

(21+p +16(p+1)

) 1p

⎧⎨⎩(| f ′ (a)|q +

∣∣ f ′ ( a+b2

)∣∣q2

) 1q

+

(| f ′ (b)|q +

∣∣ f ′ ( a+b2

)∣∣q2

) 1q

⎫⎬⎭ .


RE F ER EN C ES

[1] B. G. PACHPATTE, On some inequalities for convex functions, RGMIA Research Report Collection 6,E (1) (2003).

[2] G. H. TOADER, On a generalization of the convexity, Mathematica 30, 53 (1988), 83–87.[3] H. KAVURMACI, M. AVCI AND M. E. OZDEMIR, New inequalities of Hermite-Hadamard type for

convex functions with applications, Journal of Inequalities and Applications 86, 1 (2011).[4] J. PECARIC, S. VAROSANEC, Harmonic polynomials and generalization of Ostrowski inequality with

Applications in Numerical Integration, Nonlinear Analysis Series A: Theory, Methods and Applica-tions 47, 1 (2011), 2365–2374.

[5] M. Z. SARIKAYA, E. SET AND M. E. OZDEMIR, On new inequalities of Simpson type for convexfunctions, RGMIA Research Report Collection 13, 2 (2) (2010).

[6] S. KOVAC, J. PECARIC, Weighted version of general integral formula of Euler type, Math. Inequal.Appl. 13, 3 (2010), 579–599.

[7] S. S. DRAGOMIR, R. P. AGARWAL, Two inequalities for differentiable mappings and applications,Applied Mathematics Letters 11, 5 (1998), 91–95.

[8] U. S. KIRMACI, Inequalities for dfferentiable mappings and applications to special means of realnumbers and to midpoint formula, Appl. Math. Comp. 147, 1 (2004), 137–146.

(Received January 16, 2016) M. Emin OzdemirUludag University, Education Faculty

Bursa, Turkeye-mail: [email protected]

Alper EkinciAgrı Ibrahim Cecen UniversityFaculty of Science and Letters

Department of Mathematics04100 Agrı, Turkey


Publishe

d b

y

UDC 51 ISSN 1331-4343

CONTENTS

athematical nequalities pplications

MIA&

Zagreb, Croatia Volume 19, Number 4, October 2016

Vol 19, N

o 4 (2016) 1141-1440

Ravi P. AgarwalMartin BohnerJean-Christophe BourinYuri BuragoMaria A. Hernandez CifreZeev DitzianTamás ErdélyiKálmán GyoryFrank HansenDon HintonAndrea LaforgiaLászló LosoncziLech MaligrandaHorst MartiniMarko MatićRam N. Mohapatra

M. Sal MoslehianConstantin P. NiculescuBohumir OpicDonal O’ReganZsolt PálesLars Erik PerssonLubos PickIosif PinelisSimo PuntanenSaburou SaitohJavier SoriaHari M. SrivastavaStevo StevićKenneth B. StolarskyGeorge P. H. Styan

Ma

them

atic

al Ine

qua

lities &

Ap

plic

atio

ns

1141 Zsolt Pales and Pawe PasteczkaCharacterization of the Hardy property of means and the best Hardy constants

1159 Merve Avc Ard c, Ahmet Ocak Akdemir and Erhan SetNew Ostrowski like inequalities for GG -convex and GA -convex functions

1169 Josip Pecaric and Marjan PraljakHermite interpolation and inequalities involvingweighted averages of n -convex functions




Recursively de ned re nements of the integral form of Jensen’s inequality1247 Ravi P. Agarwal, Saad Ihsan Butt, Josip Pecaric and Ana Vukelic



Converses of Jessen’s inequality on time scales II1287 Miros aw Adamek – On a problem connected with strongly convex functions1295 Maja Andric, Josip Pecaric and Ivan Peric










EDITORS-IN-CHIEF

MANAGING EDITOR ASSOCIATE EDITORS

EDITORIAL BOARD

Josip Pečarić Ivan Perić Sanja Varošanec

Neven Elezović Iva Franjić Marjan Praljak

mathematical nequalities i pplications a - narod.hr · horst martini marko matić ... volume 19,...

Documents