some integral inequalities for interval-valued...

Comp. Appl. Math. (2018) 37:1306–1318https://doi.org/10.1007/s40314-016-0396-7

Some integral inequalities for interval-valued functions

H. Román-Flores1 · Y. Chalco-Cano1 · W. A. Lodwick2

Received: 18 October 2013 / Revised: 22 March 2015 / Accepted: 26 October 2016 /Published online: 16 November 2016© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016

Abstract In this paper, we explore some integral inequalities for interval-valued functions.More precisely, using the Kulisch–Miranker order on the space of real and compact intervals,we establishMinkowski’s inequality and thenwe derive Beckenbach’s inequality via an inter-val Radon’s inequality. Also, some examples and applications are presented for illustratingour results.

Keywords Interval-valued functions · Minkowski’s inequality · Radon’s inequality ·Beckenbach’ inequality

Mathematics Subject Classification 26D15 · 26E25 · 28B20

1 Introduction

The importance of the study of set-valued analysis from a theoretical point of view as well asfrom their application is well known (see Aubin and Cellina 1984; Aubin and Franskowska1990). Also, many advances in set-valued analysis have beenmotivated by control theory and

Communicated by Marko Rojas-Medar.

This work was supported in part by Conicyt-Chile through Projects Fondecyt 1120674 and 1120665. Also,W. A. Lodwick was supported in part by FAPESP 2011/13985.

B H. Romá[email protected]

Y. [email protected]

W. A. [email protected]

1 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile

2 Department of Mathematical and Statistical Sciences, University of Colorado,Denver, CO 80217, USA

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Some integral inequalities for interval-valued functions 1307

dynamical games and, in addition, optimal control theory and mathematical programmingwere a motivating force behind set-valued analysis since the sixties (see Aubin and Fran-skowska 2000). Interval Analysis is a particular case and it was introduced as an attempt tohandle interval uncertainty that appears in many mathematical or computer models of somedeterministic real-world phenomena. The first monograph dealing with interval analysis wasgiven by Moore (1966). Moore is recognized as the first to use intervals in computationalmathematics, now called numerical analysis. He also extended and implemented the arith-metic of intervals to computers. One of his major achievements was to show that Taylorseries methods for solving differential equations not only are more tractable, but also moreaccurate (see Moore 1985).On the other hand, several generalizations of classical integral inequalities were obtained inthe recent years by Agahi et al. (2010, 2011a, b, 2012), Flores-Franulic and Román-Flores(2007), Flores-Franulic et al. (2009), Mesiar and Ouyang (2009), Román-Flores and Chalco-Cano (2007), Román-Flores et al. (2007a, b, 2008, 2013), in the context of non-additivemeasures and integrals (also see the following related references: Pap 1995; Ralescu andAdams 1980; Román-Flores and Chalco-Cano 2006; Sugeno 1974; Wang and Klir 2009). Ingeneral, any integral inequality can be a very powerful tool for applications and, in particular,when we think an integral operator as a predictive tool then an integral inequality can be veryimportant in measuring, computing errors and delineating such processes.

Interval-valued functions (or fuzzy-interval valued functions) may provide an alternativechoice for considering the uncertainty into the prediction processes and, in connection withthis, the Aumann integral for interval-valued function is the natural-associated expectation(see for example Puri and Ralescu 1986).Also, several integral inequalities involving functions and their integrals and derivatives,such as Wirtinger’s inequality, Ostrowski’s inequality, and Opial’s inequality, among others,have been extensively studied during the past century (see for example Anastassiou 2011;Mitrinovic et al. 1991). All these studies have been fundamental tools in the development ofmany areas in mathematical analysis.Recently, some differential-integral inequalities have been extended to the set-valued context.For example Anastassiou (2011), using the Hukuhara derivative, extended an Ostrowskitype inequality to the context of fuzzy-valued functions. Chalco-Cano et al. (2012) usingthe concept of generalized Hukuhara differenciability (see Chalco-Cano et al. 2011, 2013)establish some Ostrowski type inequalities for interval-valued functions.This presentation generalizes Minkowski’s inequality for interval-valued functions and, asan application, establishes the Beckenbach’s inequality for interval-valued functions via aninterval Radon’s inequality.

2 Preliminaries

2.1 Interval operations

Let R be the one-dimensional Euclidean space. Following Diamond and Kloeden (1994), letKC denote the family of all non-empty compact convex subsets of R, that is,

KC = {[a, b] | a, b ∈ R and a ≤ b}. (1)

The Pompeiou–Hausdorffmetric onKC (Pompeiuwas the firstmathematician introducingthe concept of set distance, see Birsan and Tiba 2006), frequently called Hausdorff metric,is defined by

123

1308 H. Román-Flores et al.

H(A, B) = max {d(A, B), d(B, A)} , (2)

where d(A, B) = maxa∈A d(a, B) and d(a, B) = minb∈B d(a, b) = minb∈B |a − b|.Remark 1 An equivalent form for the Hausdorff metric defined in (2) is

M([a, a

],[b, b

]) = max{∣∣a − b

∣∣ ,

∣∣a − b

∣∣}

which is also known as the Moore metric on the space of intervals (see Moore and Kearfott2009, Eq. (6.3), pp. 52).

It is well known that (KC , H) is a complete metric space (see Aubin and Cellina 1984;Diamond and Kloeden 1994).If A ∈ KC then we define the norm of A as ‖A‖ = H (A, 0) = H ([0, 0]).The Minkowski sum and scalar multiplication are defined on KC by means

A + B = {a + b | a ∈ A, b ∈ B} and λA = {λa | a ∈ A}. (3)

Also, if A = [a, a] and B = [b, b] are two compact intervals then we define the difference

A − B = [a − b, a − b

], (4)

the product

A · B = [min

{ab, ab, ab, ab

}, max

{ab, ab, ab, ab

}], (5)

and the division

A

B=

[min

{a

b,a

b,a

b,a

b

}, max

{a

b,a

b,a

b,a

b

}], (6)

whenever 0 /∈ B.An order relation “≤ ” is defined on KC as follows (see Kulisch and Miranker 1981):

[a, a] ≤ [b, b] ⇔ a ≤ b and a ≤ b. (7)

Remark 2 We note that if [a, b], [c, d], and [x, y] are intervals with positive enpoints then

[a, b] ≥ [x, y] ⇔ [a, b][c, d] ≥ [x, y]

[c, d] (8)

[c, d] ≤ [x, y] ⇔ [a, b][c, d] ≥ [a, b]

[x, y] . (9)

The spaceKC is not a linear space since it does not possess an additive inverse and thereforesubtraction is not well defined (see Aubin and Cellina 1984; Markov 1979). However, onevery important property of interval arithmetic is that:

A, B,C, D ∈ KC , A ⊆ B,C ⊆ D ⇒ A ∗ C ⊆ B ∗ D (10)

where ∗ can be sum, subtraction, product or division. Property (10) is called the “inclusionisotony ”of interval operations and it is recognized as the fundamental principle of intervalanalysis (see Moore and Kearfott 2009).One consequence of this is that any function f (x) described by an expression in the variablex which can be evaluated by a programmable real calculation can be embedded in intervalcalculations using the natural correspondence between operation so that if x ∈ X ∈ KC thenf (x) ∈ f (X), where f (X) is interpreted as the calculation of f (x) with x replaced by X

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and the operations replaced by interval operations. The evaluation f (X) is called the naturalinterval extension of the expression f (x).Also it is necessary to observe that two different expressions for a same real function canresult in very different interval-valued functions. For instance, if we consider f (x) = x(1−x) and f (x) = x − x2and taking X = [3, 5] then, on the one hand we obtain f (X) =[3, 5] ([1, 1] − [3, 5]) = [−20,−6], and on the other we have f (X) = [3, 5] − [3, 5]2 =[−22,−4].

Denote the range of a function f (x) over an interval X as

R( f ) |X= { f (x) | x ∈ X} . (11)

Then from (10) it follows for the natural interval extension that

R( f ) |X⊆ f (X). (12)

As an example consider the function f (x) = x2 + x and X = [−1, 1]. Then

R( f ) |X=[−1

4, 2

],

whereas using the natural interval extension of f (x) we obtain

f (X) = [−1, 1]2 + [−1, 1] = [−2, 2] ,

which is a more larger interval but nevertheless contains R( f ) |X .For the particular case when f (x) is monotone and continuous over an interval X = [a, b],we can define

f ([a, b]) = [min { f (a), f (b)} ,max { f (a), f (b)}] (13)

and, in this case, f (X) = R( f ) |X .For example:

(a) if f (x) = xr , r > 0, and 0 ≤ a ≤ b then

f ([a, b]) = [a, b]r = [ar , br

]. (14)

(b) If g(x) = ex then the “exponential” of an interval [a, b] is defined asg ([a, b]) = e[a,b] =

[ea, eb

]. (15)

For more details on interval operations and interval analysis see Markov (1979), Moore(1966) and Rokne (2001).

2.2 Integral of interval-valued functions

If T = [a, b] is a closed interval and F : T → KC is an interval-valued function, then wewill denote

F(t) = [ f (t), f (t)],where f (t) ≤ f (t), ∀t ∈ T . The functions f and f are called the lower and the upper(endpoint) functions of F , respectively. For interval-valued functions it is clear that F : T →KC is continuous at t0 ∈ T if

limt→t0F(t) = F(t0), (16)

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where the limit is taken in the metric space (KC , H). Consequently, F is continuous at t0 ∈ Tif and only if its endpoint functions f and f are continuous functions at t0 ∈ T . We denoteby C ([a, b],KC ) the family of all continuous interval-valued functions.

Definition 1 Let M the class of all Lebesgue measurable sets of T , then

(a) the function f : T → R is measurable if and only if

f −1(C) ∈ Mfor all closed subset C of R.

(b) the interval-valued function F : T → KC is measurable if and only if

Fω(C) = {t ∈ T/F(t) ∩ C = ∅} ∈ M, ∀ C ⊆ R, C closed.

(c) Also, if F : T → KC is an interval-valued function and f : T → R, then we say thatf is a selector (or selection) of F if and only if f (t) ∈ F(t) for all t ∈ T . In this caseif, additionally f is a measurable function, then we say that f is a measurable selectorof F . Finally, an integrable selector of F is a measurable selector of F for which thereis

∫T f (t).

Definition 2 (Aubin and Cellina 1984) Let F : T → KC be an interval-valued function.The integral (Aumann integral) of F over T = [a, b] is defined as

∫ b

aF(t)dt =

{∫ b

af (t)dt | f ∈ S(F)

}, (17)

where S(F) is the set of all integrable selectors of F , i.e.,

S(F) = { f : T → R | f integrable and f (t) ∈ F(t) for all t ∈ T } .

If S(F) = ∅, then the integral exists and F is said to be integrable (Aumann integrable).

Note that if F is integrable then it has a measurable selector which is integrable and, conse-quently, S(F) = ∅.Also, in above definition, the integral symbol

∫ ba F(t)dt and/or

∫ ba f (t)dt denotes the integral

with respect to the Lebesgue measure.

Definition 3 We say that a mapping F : T → KC is integrally bounded if there exists apositive integrable function g : T → R such that ‖F(t)‖ ≤ g(t), for all t ∈ T .

Theorem 1 (Aubin and Cellina 1984) Let F : T → KC be a measurable and integrallybounded interval-valued function. Then it is integrable and

∫ ba F(t)dt ∈ KC .

Corollary 1 (Aubin and Cellina 1984; Diamond and Kloeden 1994) A continuous interval-valued function F : T → KC is integrable.

The Aumann integral satisfies the following properties.

Proposition 1 (Aubin and Cellina 1984; Diamond and Kloeden 1994) Let F,G : T → KC

be two measurable and integrally bounded interval-valued functions. Then

(i)∫ t2t1

(F(t) + G(t)) dt = ∫ t2t1

F(t)dt + ∫ t2t1G(t)dt , a ≤ t1 ≤ t2 ≤ b

(ii)∫ t2t1

F(t)dt = ∫ τ

t1F(t)dt + ∫ t2

τF(t)dt , a ≤ t1 ≤ τ ≤ t2 ≤ b.

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Theorem 2 (Bede and Gal 2005) Let F : T → KC be a measurable and integrally boundedinterval-valued function such that F(t) = [ f (t), f (t)]. Then f and f are integrable functionsand

∫ t2

t1F(t)dt =

[∫ t2

t1f (t)dt,

∫ t2

t1f (t)dt

]. (18)

Remark 3 Above Theorem 2 is a direct consequence of two relevant results:

(a) (Aumann 1965, Theorem 1, pp. 2)∫T F(t)dt is convex.

(b) (Aumann 1965, Theorem 4, pp. 2) If F is closed-valued then∫T F(t)dt is compact.

In fact, because f , f ∈ S(F) then, by convexity of∫T F(t)dt , we obtain [∫T f (t)dt,

∫T f (t)dt] ⊆ ∫

T F(t)dt .On the other hand, if f ∈ S(F) then f (t) ≤ f (t) ≤ f (t), for all t ∈ T , which implies

that∫

Tf (t)dt ∈

[∫

Tf (t)dt,

∫

Tf (t)dt

]

and, consequently,∫T F(t)dt ⊆ [∫T f (t)dt,

∫T f (t)dt]. Therefore equality (18) in Theorem

2 holds.

To finalize this section, we give an example of prediction under uncertainty using intervaltools (see Puri and Ralescu 1986).

Example 1 Toss a fair coin. Denote the outcomes Tail by T and Head byH. Suppose a playerloses approximately 10 EUR if the outcome is T, and wins an amount much larger than 100EUR but not much larger than 1000 EUR if the outcome is H.

The question here is: what is the expected value for the next outcome?To represent the uncertainty contained in the above linguistic descriptions, we can define

the interval random variable X : {T, H} → KC , where

(a) X (T ) = approximately − 10, and(b) X (H) = much larger than 100 but not much larger than 1000.

Furthermore, if E = {T, H} then we can consider the measure space (E,P(E), μ) takingas is usual: μ(T ) = μ(H) = 1

2 , μ(X) = 1 and μ(∅) = 0.Suppose we interpret the linguistic variables as X (T ) = [−12,−8] and X (H) =

[250, 1010].Now, we can write X (z) = [ f (z), f (z)] where f , f : {T, H} → R are defined by

f (T ) = −12, f (H) = 250, f (T ) = −8, and f (H) = 1010.So, using properties of the Aumann integral we have

∫

Ef dμ =

∫

{T }f dμ +

∫

{H}f dμ = 1

2(−12) + 1

2(250) = 119

and∫

Ef dμ =

∫

{T }f dμ +

∫

{H}f dμ = 1

2(−8) + 1

2(1010) = 501.

Thus, the expected value for the next outcome is the interval E(X) = [119, 501].

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3 Minkowski’s inequality

The well-known inequality due toMinkowski can be stated as follows (see Hardy et al. 1934,pp. 31):

Theorem 3 Let f (x), g(x) ≥ 0 and p ≥ 1, then

(∫( f (x) + g(x))p dx

) 1p ≤

(∫f (x)pdx

) 1p +

(∫g(x)pdx

) 1p

(19)

with equality if and only if f and g are proportional, and if 0 < p < 1, then

(∫( f (x) + g(x))p dx

) 1p ≥

(∫f (x)pdx

) 1p +

(∫g(x)pdx

) 1p

(20)

with equality if and only if f and g are proportional.

We recall that two non-negative functions f and g are proportional if and only if there is anon-negative real constant k such that f = kg (or g = k f ).Now, using above theorem and properties of interval integration, we can prove the followinginterval version of Minkowski’s inequality:

Theorem 4 (Interval Minkowski’s inequality) If F,G : [a, b] → KC are two integrableinterval-valued functions, with F = [ f , f ], G = [g, g], f (x), g(x) ≥ 0 and p ≥ 1, then

(∫

[a,b](F(x) + G(x))p dx

) 1p ≤

(∫

[a,b]F(x)pdx

) 1p +

(∫

[a,b]G(x)pdx

) 1p

(21)

with equality if F and G are proportional, and if 0 < p < 1, then

(∫


) 1p ≥

(∫

[a,b]F(x)pdx

) 1p +

(∫

[a,b]G(x)pdx

) 1p

(22)

with equality if F and G are proportional.

Proof Due to (3), (7), (13), (14), Theorem 2, and Theorem 3 (19), we have(∫


) 1p

=(∫

[a,b]

[f (x) + g(x), f (x) + g(x)

]pdx

) 1p

=(∫

[a,b]

[(f (x) + g(x)

)p,(f (x) + g(x)

)p]dx

) 1p

=([∫

[a,b]

(f (x) + g(x)

)pdx,

∫

[a,b](f (x) + g(x)

)pdx

]) 1p

=[(∫

[a,b]

(f (x) + g(x)

)pdx

) 1p

,

(∫

[a,b](f (x) + g(x)

)pdx

) 1p]

≤[(∫

[a,b]f (x)pdx

) 1p +

(∫

[a,b]g(x)pdx

) 1p

,

(∫

[a,b]f (x)pdx

) 1p +

(∫

[a,b]g(x)pdx

) 1p]

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=[(∫

[a,b]f (x)pdx

) 1p

,

(∫

[a,b]f (x)pdx

) 1p]

+[(∫

[a,b]g(x)pdx

) 1p

,

(∫

[a,b]g(x)pdx

) 1p]

=[∫

[a,b]f (x)pdx,

∫

[a,b]f (x)pdx

] 1p +

[∫

[a,b]g(x)pdx,

∫

[a,b]g(x)pdx

] 1p

=(∫

[a,b]

[f (x)p, f (x)p

]dx

) 1p +

(∫

[a,b]

[g(x)p, g(x)p

]dx

) 1p

=(∫

[a,b]

[f (x), f (x)

]pdx

) 1p +

(∫

[a,b]

[g(x), g(x)

]pdx

) 1p

=(∫

[a,b]F(x)]pdx

) 1p +

(∫

[a,b]G(x)pdx

) 1p

Analogously, using (3), (7), (13), (14), Theorems 2 and 3 (20), we can prove the second partof our theorem for 0 < p < 1.

Finally, a straightforward calculation shows that equality is reached if F and G are pro-portional. This completes the proof. ��

4 Beckenbach’s inequality

The well-known Beckenbach’s inequality can be stated as follows (see Beckenbach andBellman 1992, pp. 27):

Theorem 5 (Beckenbach and Bellman 1992) If 0 < p < 1, and f (x), g(x) > 0, then∫( f (x) + g(x))p+1dx∫( f (x) + g(x))pdx

≤∫f (x)p+1dx∫f (x)pdx

+∫g(x)p+1dx∫g(x)pdx

. (23)

The aim of this section is to show a Beckenbach type inequality for interval-valued functions,and for this we will use an interval version of the Radon’s inequality. We recall that theclassical Radon’s inequality (published by Radon 1913), establishes that

Theorem 6 (Radon 1913) For every real numbers p > 0, xk ≥ 0, ak > 0, for 1 ≤ k ≤ n,the inequality

n∑

k=1

x p+1k

a pk

≥(∑n

k=1 xk)p+1

(∑nk=1 ak

)p (24)

holds.

Inequality (24) has been widely studied by many authors because of its utility in practicaland theoretical applications (see for example Mortici 2011; Zhao 2012). The next result isan extension of Radon’s inequality to the interval context.

Theorem 7 (Interval Radon’s inequality) Let [xk, xk], [ak, ak] ∈ KC , with xk ≥ 0, ak > 0,for all 1 ≤ k ≤ n. If p > 0, then the inequality

n∑

k=1

[xk, xk]p+1

[ak, ak]p≥

(∑nk=1[xk, xk]

)p+1

(∑nk=1[ak, ak]

)p (25)

holds.

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Proof Working on the left-side of (25), we have

n∑

k=1

[xk, xk]p+1

[ak, ak]p=

n∑

k=1

[x p+1k , x p+1

k ][a p

k , a pk ]

=n∑

k=1

[x p+1k

a pk

,x p+1k

a pk

]

=[

n∑

k=1

x p+1k

a pk

,

n∑

k=1

x p+1k

a pk

]

.

On the other hand, working on the right-side of (25), we obtain(∑n

k=1[xk, xk])p+1

(∑nk=1[ak, ak]

)p =([∑n

k=1 xk,∑n

k=1 xk])p+1

([∑nk=1 ak,

∑nk=1 ak]

)p

=[(∑n

k=1 xk)p+1

,(∑n

k=1 xk)p+1

]

[(∑nk=1 ak

)p,(∑n

k=1 ak)p]

=[(∑n

k=1 xk)p+1

(∑nk=1 ak

)p ,

(∑nk=1 xk

)p+1

(∑nk=1 ak

)p

]

.

Now, by Radon’s inequality (24) we have

n∑

k=1

x p+1k

a pk

≥(∑n

k=1 xk)p+1

(∑nk=1 ak

)p (26)

andn∑

k=1

x p+1k

a pk

≥(∑n

k=1 xk)p+1

(∑nk=1 ak

)p (27)

which implies that inequality (25) holds. ��Theorem 8 (Interval Beckenbach’s inequality) If F,G : [a, b] → KC are two integrableinterval-valued functions, with F = [ f , f ], G = [g, g], f (x), g(x) > 0 and 0 < p < 1,then

∫[a,b] (F(x) + G(x))p+1 dx∫[a,b] (F(x) + G(x))p dx

≤∫[a,b] F(x)p+1dx∫[a,b] F(x)pdx

+∫[a,b] G(x)p+1dx∫[a,b] G(x)pdx

. (28)

Proof Taking

I1 =(∫

[a,b]F(x)p+1dx

) 1p+1

, J1 =(∫

[a,b]F(x)pdx

) 1p

(29)

I2 =(∫

[a,b]G(x)p+1dx

) 1p+1

, J2 =(∫

[a,b]G(x)pdx

) 1p

(30)

and using the intervalar Radon inequality (25) we have

I p+11

J p1

+ I p+12

J p2

≥ (I1 + I2)p+1

(J1 + J2)p, (31)

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that is to say,∫[a,b] F(x)p+1dx∫[a,b] F(x)pdx

+∫[a,b] G(x)p+1dx∫[a,b] G(x)pdx

≥

((∫[a,b] F(x)p+1dx

) 1p+1 +

(∫[a,b] G(x)p+1dx

) 1p+1

)p+1

((∫[a,b] F(x)pdx

) 1p +

(∫[a,b] G(x)pdx

) 1p)p . (32)

Now, because 0 < p < 1 then 1 < p + 1 < 2 and , due to (21) and (22), we obtain

(∫

[a,b](F(x) + G(x))p+1 dx

) 1p+1 ≤

(∫

[a,b]F(x)p+1dx

) 1p+1 +

(∫

[a,b]G(x)p+1dx

) 1p+1

,

(33)

(∫


) 1p ≥

(∫

[a,b]F(x)pdx

) 1p +

(∫

[a,b]G(x)pdx

) 1p

. (34)

Finally, due (8), (9), (33) and (34) we obtain that

((∫[a,b] F(x)p+1dx

) 1p+1 +

(∫[a,b] G(x)p+1dx

) 1p+1

)p+1

((∫[a,b] F(x)pdx

) 1p +

(∫[a,b] G(x)pdx

) 1p)p ≥

∫[a,b] (F(x) + G(x))p+1 dx∫[a,b] (F(x) + G(x))p dx

,

(35)

and the proof is completed. ��

Example 2 Let p = 1/2 and let F,G : [0, 1] → KC two interval-valued functions definedby F(x) = [x, 2x] and G(x) = [x2, x], with x ∈ [0, 1]. Using (14), interval operations andproperties of the Aumann integral, a straightforward calculation shows that:

∫

[0,1]F(x)p+1dx =

[2

5, 2

322

5

],

∫

[0,1]F(x)pdx =

[2

3, 2

122

3

](36)

∫

[0,1]G(x)p+1dx =

[1

4,2

5

],

∫

[0,1]G(x)pdx =

[1

2,2

3

]. (37)

On the other hand,

(∫

[0,1](F + G)p+1dx

) 1p+1 =

[(3

128ln(

√2 + 3

2) + 3

128ln2 + 39

64

√2

) 23

,

(2

3

√3

) 23]

(38)

(∫

[0,1](F + G)pdx

) 1p =

[(3

4

√2 − 1

8ln2 − 1

8ln(

√2 + 3

2)

)2

,12

9

]

. (39)

123


Also,

(∫

[0,1]F(x)p+1dx

) 1p+1 =

[(2

5

) 23

, 2

(2

5

) 23]

,

(∫

[0,1]F(x)pdx

) 1p =

[4

9,8

9

](40)

and(∫

[0,1]G(x)p+1dx

) 1p+1 =

[(1

4

) 23

,

(2

5

) 23]

,

(∫

[0,1]G(x)pdx

) 1p =

[1

4,4

9

]. (41)

Thus, from (38), (40) and (41), because

(3

128ln

(√2 + 3

2

)+ 3

128ln2 + 39

64

√2

) 23 ≤

(2

5

) 23 +

(1

4

) 23

and(2

3

√3

) 23 ≤ 2

(2

5

) 23 +

(2

5

) 23

,

we obtain that(∫

[0,1](F + G)p+1dx

) 1p+1 ≤

(∫

[0,1]F p+1dx

) 1p+1 +

(∫

[0,1]Gp+1dx

) 1p+1

and, consequently, Minkowski’s inequality (21) is verified.Analogously, from (39)to (41), because

(3

4

√2 − 1

8ln2 − 1

8ln

(√2 + 3

2

))2

≥ 4

9+ 1

4

and

12

9≥ 8

9+ 4

9,

we obtain that(∫

[0,1](F + G)pdx

) 1p ≥

(∫

[0,1]F pdx

) 1p +

(∫

[0,1]Gpdx

) 1p

and, consequently, Minkowski’s inequality (22) is verified.Additionally,

∫[0,1] F

p+1dx∫[0,1] F pdx

+∫[0,1] G

p+1dx∫[0,1] Gpdx

=[

3

5√2,3√8

5

]

+[3

8,4

5

](42)

and∫[0,1] (F + G)p+1 dx∫[0,1] (F + G)p dx

=⎡

⎣3

128 ln(√

2 + 32

)+ 3

128 ln√2 + 39

64

√2

23

√3

,

65

√3

34

√2 − 1

8 ln2 − 18 ln

(√2 + 3

2

)

⎤

⎦ . (43)

123


Thus, from (42) and (43), because

3

5√2

+ 3

8≥

3128 ln

(√2 + 3

2

)+ 3

128 ln√2 + 39

64

√2

23

√3

and

3√8

5+ 4

5≥

65

√3

34

√2 − 1

8 ln2 − 18 ln

(√2 + 3

2

)

we obtain∫[0,1] F

p+1dx∫[0,1] F pdx

+∫[0,1] G

p+1dx∫[0,1] Gpdx

≥∫[0,1] (F + G)p+1 dx∫[0,1] (F + G)p dx

and, consequently, Beckenbach’s inequality (28) is verified.

5 Conclusion

In this paper, using the Kulisch–Miranker order on the space KC of non-empty compactand convex subsets of R, we have proved the Minkowski’s inequality (see Theorem ***4)for non-negative interval-valued functions, i.e., for interval-valued functions taken values inK+C = {[a, b] ∈ KC | 0 ≤ a ≤ b}.This fact shows that the functional ‖·‖p defined by

‖F‖p =(∫

[a,b]F(x)pdx

) 1p

is a seminorm (for p ≥ 1) on the convex and positive cone I([a, b],K+

C

)of non-negative

and integrable interval-functions, opening an interesting route toward the class of Lp-typeinterval spaces.

On the other hand, Radon and Beckenbach inequalities have important applications inconvex geometry on R

n through the concept of width-integral of convex bodies (see Zhao2012) and, in this context, in the near future we wish to extend these ideas for studying someproblems connected with convexity on the space Kn

C .

References

Agahi H, Mesiar R, Ouyang Y (2010) General Minkowski type inequalities for Sugeno integrals. Fuzzy SetsSyst. 161:708–715

Agahi H, Ouyang Y, Mesiar R, Pap E, Štrboja M (2011) Hölder and Minkowski type inequalities for pseudo-integral. Appl. Math. Comput. 217:8630–8639

Agahi H, Román-Flores H, Flores-Franulic A (2011) General Barnes–Godunova–Levin type inequalities forSugeno integral. Inf. Sci. 181:1072–1079

Agahi H, Mesiar R, Ouyang Y, Pap E, Strboja M (2012) General Chebyshev type inequalities for universalintegral. Inf. Sci. 187:171–178

Anastassiou GA (2011) Advanced Inequalities. World Scientific, New JerseyAubin JP, Cellina A (1984) Differential Inclusions. Springer, New YorkAubin JP, Franskowska H (1990) Set-Valued Analysis. Birkhäuser, BostonAubin JP, Franskowska H (2000) Introduction: set-valued analysis in control theory. Set Valued Anal. 8:1–9Aumann RJ (1965) Integrals of set-valued functions. J. Math. Anal. Appl. 12:1–12

123


Beckenbach, E., Bellman, R.: Inequalities. Springer, Berlin (1961) (New York, 1992)Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy number valued functions with appli-

cations to fuzzy differential equation. Fuzzy Sets Syst. 151:581–599Birsan T, Tiba D (2006) One hundred years since the introduction of the set distance by Dimitrie Pompeiu.

In: Ceragioli F, Dontchev A, Furuta H, Marti K, Pandolfi L (eds) IFIP International Federation forInformation Processing, vol. 199. System Modeling and Optimization. Springer, Boston, pp 35–39

Chalco-Cano Y, Flores-Franulic A, Román-Flores H (2012) Ostrowski type inequalities for inteval-valuedfunctions using generalized Hukuhara derivative. Comput. Appl. Math. 31:457–472

Chalco-Cano Y, Román-Flores H, Jiménez-Gamero MD (2011) Generalized derivative and π -derivative forset-valued functions. Inf. Sci. 181:2177–2188

Chalco-Cano Y, Rufián-Lizana A, Román-Flores H, Jiménez-GameroMD (2013) Calculus for interval-valuedfunctions using generalized Hukuhara derivative and applications. Fuzzy Sets Syst. 219:49–67

DiamondP,KloedenP (1994)Metric Space of FuzzySets: Theory andApplication.World Scientific, SingaporeFlores-Franulic A, Román-Flores H (2007) A Chebyshev type inequality for fuzzy integrals. Appl. Math.

Comput. 190:1178–1184Flores-Franulic A, Román-Flores H, Chalco-Cano Y (2009) Markov type inequalities for fuzzy integrals.

Appl. Math. Comput. 207:242247Hardy G, Littlewood J, Pólya G (1934) Inequalities. Cambridge University Press, CambridgeKulisch U, Miranker W (1981) Computer Arithmetic in Theory and Practice. Academic Press, New YorkMarkov S (1979) Calculus for interval functions of a real variable. Computing 22:325–377Mesiar R, Ouyang Y (2009) General Chebyshev type inequalities for Sugeno integrals. Fuzzy Sets Syst.

160:58–64Mitrinovic DS, Pecaric JE, Fink AM (1991) Inequalities Involving Functions and Their Integrals and Deriva-

tives. Kluwer Academic Publishers, BostonMoore RE (1966) Interval Analysis. Prince-Hall, Englewood CliffsMoore RE (1985) Computational Functional Analysis. Ellis Horwood Limited, EnglandMoore RE, Kearfott RB, Cloud MJ (2009) Introduction to Interval Analysis. SIAM, PhiladelphiaMortici C (2011) A new refinement of the Radon inequality. Math. Commun. 16:319–324Pap E (1995) Null-Additive Set Functions. Kluwer, DordrechtPuri M, Ralescu D (1986) Fuzzy random variables. J. Math. Anal. Appl. 114:409–422Radon J (1913) Über die absolut additiven Mengenfunktionen. Wien. Sitzungsber 122:1295–1438Ralescu D, Adams G (1980) The fuzzy integral. J. Math. Anal. Appl. 75:562–570Rokne, J.G.: Interval arithmetic and interval analysis: an introduction. In: Pedrycz, W. (ed.) Granular Com-

puting: An Emerging Paradigm. Physica, Heldelberg (2001)Román-Flores H, Chalco-Cano Y (2006) H-continuity of fuzzy measures and set defuzzification. Fuzzy Sets

Syst. 157:230–242Román-Flores H, Chalco-Cano Y (2007) Sugeno integral and geometric inequalities. Int. J. Uncertain. Fuzzi-

ness Knowl. Based Syst. 15:1–11Román-Flores H, Flores-Franulic A, Chalco-Cano Y (2007) The fuzzy integral for monotone functions. Appl.

Math. Comput. 185:492–498Román-Flores, H., Flores-Franulic, A., Chalco-Cano, Y.: (2007) A Jensen type inequality for fuzzy integrals.

Inf. Sci. 177:3192–3201Román-Flores H, Flores-Franulic A, Chalco-CanoY (2008) AHardy-type inequality for fuzzy integrals. Appl.

Math. Comput. 204:178–183Román-Flores H, Flores-Franulic A, Chalco-Cano Y (2013) Dan Ralescu. A two-dimensional Hardy type

inequality for fuzzy integrals. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 21:165–173Sugeno, M.: Theory of fuzzy integrals and its applications. Ph.D. thesis, Tokyo Institute of Technology (1974)Wang Z, Klir G (2009) Generalized Measure Theory. Springer, New YorkZhao C-J (2012) On Dresher’s inequalities for width-integrals. Appl. Math. Lett. 25:190–194

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