generalized geometrically convex functions and inequalities ·...

Noor et al. Journal of Inequalities and Applications (2017) 2017:202 DOI 10.1186/s13660-017-1477-x

R E S E A R C H Open Access

Generalized geometrically convexfunctions and inequalitiesMuhammad Aslam Noor1,2*, Khalida Inayat Noor2 and Farhat Safdar2

*Correspondence:[email protected] of Mathematics, KingSaud University, Riyadh, SaudiArabia2Department of Mathematics,COMSATS Institute of InformationTechnology, Park Road, Islamabad,Pakistan

AbstractIn this paper, we introduce and study a new class of generalized functions, calledgeneralized geometrically convex functions. We establish several basic inequalitiesrelated to generalized geometrically convex functions. We also derive several newinequalities of the Hermite-Hadamard type for generalized geometrically convexfunctions. Several special cases are discussed, which can be deduced from our mainresults.

MSC: 26D15; 26D10; 90C23

Keywords: generalized convex functions; generalized geometrically convexfunctions; Hermite-Hadamard’s type inequalities; Hölder’s inequality

1 IntroductionDuring the last few decades, the theory of convex analysis has turned into one of the mostinteresting and useful fields to study a wide class of problems arising in pure and appliedsciences. Innovative techniques and calculations have yielded different directions for thestudy of convex analysis. In recent years, various inequalities for convex functions andtheir variant forms have been developed using novel techniques; see [–].

The theory of convex functions is closely related to theory of inequalities. It is wellknown that a function is convex, if and only if it satisfies an integral inequality, which isknown as the Hermite-Hadamard inequality; see [, ]. Such types of integral inequalitiesare useful in finding the upper and lower bounds. For recent developments and applica-tions, see [–].

The convex sets and convex functions have been extended and generalized in differentdirections using innovative ideas to study different problems in a general and unified framework; see [–]. One of the most recent significant generalizations of convex functionsis the ϕ-convex function, introduced by Gordji et al. []. These functions are non-convexfunctions. For recent developments, see [, –] and the references therein.

The main purpose of this paper is to introduce a new class of generalized convex func-tions, which are called generalized geometrically convex functions. We establish somenew results by using the basic inequalities. We derive new Hermite-Hadamard integralinequalities for the generalized geometrically convex functions. Several special cases areconsidered. Our results are a significant and important refinement of well-known resultsfor inequalities.

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2 PreliminariesLet I be an interval in real line R. Let f : I ⊂ R+ = (,∞) → R be a differentiable functionon the interior I of I and let f : I = [x, y] → R and η(·, ·) : R × R → R be a continuousbifunction. Throughout this paper, we will use the following notation:

R = (–∞, +∞), R+ = (,∞) and R– = (–∞, ).

Definition . ([]) Let I be an interval in real line R. A function f : I = [x, y] →R is saidto be generalized convex with respect to an arbitrary bifunction η(·, ·) : R×R →R, if

f(tx + ( – t)y

) ≤ f (y) + tη(f (x), f (y)

), ∀x, y ∈ I, t ∈ [, ].

If η(x, y) = x – y, then the generalized convex function reduces to a convex function.Every convex function is a generalized convex function, but the converse is not true; see,

for example, Examples . and ..

Example . ([]) For a convex function f , we may find another function η other thanthe function η(x, y) = x–y such that f is generalized convex. Consider f (x) = x and η(x, y) =x + y. Then we have

f(tx + ( – t)y

)=

(tx + ( – t)y

)

≤ tx + y + t( – t)xy

≤ tx + y + t( – t)(x + y)

≤ y + t(x + x + y)

= y + t(x + y)

= f (y) + tη(f (x), f (y)

), ∀x, y ∈R, t ∈ (, ).

Also the fact that x ≤ y + (x + y) and y ≤ y, for all x, y ∈ R shows the correctness ofthe inequality for t = and t = , respectively. This means that f is a generalized convexfunction. Note that the function f (x) = x is generalized convex with respect to all η(x, y) =ax + by with a ≥ , b ≥ – and x, y ∈R.

Example . ([]) Consider f : R →R as

f (x) =

⎧⎨

⎩–x if x ≥ ,

x if x < ,

and define a bifunction η = –x – y for all x, y ∈ R– = (–∞, ). Then f is η-convex, but the

converse is not true.

Definition . ([]) Let I ⊂R+. The set I is said to be a geometrically convex set, if

xty–t ∈ I, ∀x, y ∈ I, t ∈ [, ].


Definition . ([]) A function f : I ⊂ R+ = (,∞) →R is said to be geometrically convexon I , if

f(y–txt) ≤ ( – t)f (y) + t

(f (x)

), ∀x, y ∈ I, t ∈ [, ],

where (y–txt) and ( – t)f (y) + t(f (x)) are the weighted geometric mean of two positivenumbers x and y and the weighted arithmetic mean of f (x) and f (y), respectively.

We now introduce a new class of generalized convex functions on the geometricallyconvex set with respect to an arbitrary bifunction η(·, ·), which is called the generalizedgeometrically convex function.

Definition . A function f : I ⊂R+ = (,∞) →R is said to be generalized geometricallywith respect to a bifunction η(·, ·) : R×R →R, if

f(y–txt) ≤ ( – t)f (y) + t

(f (y)

)+ η

(f (x), f (y)

), ∀x, y ∈ I, t ∈ [, ]. (.)

If η(x, y) = x – y, then the generalized geometrically convex functions reduce to geomet-rically convex functions given in Definition ..

If t = in (.), then

f (√

xy) ≤ f (y) +η(f (x), f (y)

), ∀x, y ∈ I, t ∈ [, ], (.)

which is called a generalized Jensen geometrically convex function.We will use the following notations throughout this paper:() arithmetic mean:

A(a, b) =a + b

∀a, b ∈R+, a = b,

() logarithmic mean:

L(a, b) =b – a

log b – log a∀a, b ∈R+, a = b,

() generalized logarithmic mean:

Ł(a, b) =

⎧⎪⎪⎨

⎪⎪⎩

[ bp+–ap+

(p+)(b–a) ]p if p = –, ,

L(a, b) if p = –,e ( bb

aa )

b–a if p = .

3 Main resultsIn this section, we derive some new Hermite-Hadamard type inequalities for generalizedgeometrically convex functions. We denote I = [a, b], unless otherwise specified.

Theorem . Let f , g : I = [a, b] → (,∞) be generalized geometrically convex functionson I and a, b ∈ I with a < b. Then

f (√

ab) –

(ln b – ln a)

∫ b

a

x

[η

(f(

abx

), f (x)

)]dx


≤∫ b

a

x

f (x) dx ≤ f (a) + f (b)

+

(η(f (a), f (b)

)+ η

(f (b), f (a)

)).

Proof Let f be a generalized geometrically convex function on I . Then, ∀a, b ∈ I and t ∈[, ], we have

f(atb–t) ≤ f (b) + tη

(f (a), f (b)

). (.)

Integrating (.) over t on [, ], we have

∫

f(atb–t)dt ≤

∫

[f (b) + tη

(f (a), f (b)

)]dt

=[

f (b) +η(f (a), f (b)

)].

Thus

ln b – ln a

∫ b

a

x[f (x)

]dx ≤

[f (b) +

η(f (a), f (b)

)].

Using (.) and taking x = (atb–t) and y = (a–tbt), we have

f (√

ab) ≤[

f((

a–tbt)) +η(f((

atb–t)), f((

a–tbt)))]

.


f (√

ab) =

ln b – ln a

∫ b

a

x

[f (x) +

η

(f(

abx

), f (x)

)]dx.

Thus

f (√

ab) –

(ln b – ln a)

∫ b

a

xη

(f(

abx

), f (x)

)dx ≤

ln b – ln a

∫ b

a

x[f (x)

]dx.

Since f is a generalized geometrically convex function,

f(atb–t) ≤ f (b) + tη

(f (a), f (b)

), (.)

f(a–tbt) ≤ f (a) + tη

(f (b), f (a)

). (.)

Adding (.) and (.), we have

f(atb–t) + f

(a–tbt) ≤ f (a) + f (b) + tη

(f (b), f (a)

)+ tη

(f (a), f (b)

). (.)


ln b – ln a

∫ b

a

x[f (x)

]dx ≤ f (a) + f (b) +

(η(f (a), f (b)

)+ η

(f (b), f (a)

)).

This completes the proof. �


Corollary . If η(b, a) = b – a in Theorem ., then

f (√

ab) ≤∫ b

a

x

f (x) dx ≤ f (a) + f (b)

.


ln b – ln a

∫ b

a

x[f (x)g(x)

]dx ≤

M(a, b) +

N(a, b),

where

M(a, b) =[[

f (b) + η(f (a), f (b)

)][g(b) + η

(g(a), g(b)

)]+ f (b)g(b)

], (.)

N(a, b) =[f (b)

[g(b) + η

(g(a), g(b)

)]+ g(b)

[f (b) + η

(f (a), f (b)

)]]. (.)

Proof Let f , g be generalized geometrically convex functions on I . Then, ∀a, b ∈ I andt ∈ [, ], we have

f(atb–t) ≤ ( – t)f (b) + t

[f (b) + η

(f (a), f (b)

)], (.)

g(atb–t) ≤ ( – t)g(b) + t

[g(b) + η

(g(a), g(b)

)]. (.)

From (.) and (.), we have

f(atb–t)g

(atb–t)

≤ ( – t)f (b)g(b) + t( – t)[f (b)

[g(b) + η

(g(a), g(b)

)]+ g(b)

[f (b) + η

(f (a), f (b)

)]]

+ t[[f (b) + η(f (a), f (b)

)][g(b) + η

(g(a), g(b)

)]]. (.)

Integrating both sides of (.) over t on [, ], we have

∫

f(atb–t)g

(atb–t)dt

≤ f (b)g(b)∫

( – t) dt

+[f (b)

[g(b) + η

(g(a), g(b)

)]+ g(b)

[f (b) + η

(f (a), f (b)

)]] ∫

t( – t) dt

+[[

f (b) + η(f (a), f (b)

)][g(b) + η

(g(a), g(b)

)]] ∫

t dt

=

f (b)g(b) +

[f (b)

[g(b) + η

(g(a), g(b)

)]+ g(b)

[f (b) + η

(f (a), f (b)

)]]

+[[

f (b) + η(f (a), f (b)

)][g(b) + η

(g(a), g(b)

)]]

=

M(a, b) +

N(a, b).


Thus

ln b – ln a

∫ b

a

x[f (x)g(x)

]dx ≤

M(a, b) +

N(a, b).



ln b – ln a

∫ b

a

x

[f (x)g

(abx

)]dx ≤

M(a, b) +

N(a, b),

where M(a, b) and N(a, b) are defined by (.) and (.).


f(atb–t) ≤ ( – t)f (b) + t

[f (b) + η

(f (a), f (b)

)], (.)

g(a–tbt) ≤ ( – t)g(a) + t

[g(a) + η

(g(b), g(a)

)]. (.)


f(atb–t)g

(a–tbt)

≤ ( – t)f (b)g(a) + t( – t)[f (b)

[g(a) + η

(g(b), g(a)

)]

+ g(a)[f (b) + η

(f (a), f (b)

)]]

+ t[[f (b) + η(f (a), f (b)

)][g(a) + η

(g(b), g(a)

)]]. (.)

Integrating both sides of (.) over t on [, ], we have

∫

f(atb–t)g

(a–tbt)dt

≤ f (b)g(a)∫

( – t) dt

+[f (b)

[g(a) + η

(g(b), g(a)

)]+ g(a)

[f (b) + η

(f (a), f (b)

)]] ∫

t( – t) dt

+[[

f (b) + η(f (a), f (b)

)][g(a) + η

(g(b), g(a)

)]] ∫

t dt

=

f (b)g(a) +

[f (b)

[g(a) + η

(g(b), g(a)

)]+ g(a)

[f (b) + η

(f (a), f (b)

)]]

+[[

f (b) + η(f (a), f (b)

)][g(a) + η

(g(b), g(a)

)]]

=

N(a, b) +

M(a, b).


Thus

ln b – ln a

∫ b

a

x

[f (x)g

(abx

)]dx ≤

M(a, b) +

N(a, b).


Corollary . If f = g and η(x, y) = x – y in Theorem ., then

ln b – ln a

∫ b

a

x

[f (x)f

(abx

)]dx ≤

f (b)f (a) +

[f (b) + f (a)

].


ln b – ln a

∫ b

a

x

[f (x)g

(abx

)]dx

≤

{[f (b) + g(a)

]

+[[

g(a) + η(g(b), g(a)

)] +[f (b) + η

(f (a), f (b)

)]]

+[f (b)

[f (b) + η

(f (a), f (b)

)]+ g(a)

[g(a) + η

(g(b), g(a)

)]]}.


f(atb–t) ≤ ( – t)f (b) + t

[f (b) + η

(f (a), f (b)

)], (.)

g(a–tbt) ≤ ( – t)g(a) + t

[g(a) + η

(g(b), g(a)

)]. (.)

Now,

ln b – ln a

∫ b

a

x

[f (x)g

(abx

)]dx

=∫

(f(atb–t)g

(a–tbt))dt

≤

∫

[[f(atb–t)] +

[g(a–tbt)]]dt

≤

{∫

[( – t)f (b) + t

[f (b) + η

(f (a), f (b)

)]]

+[( – t)g(a) + t

[g(a) + η

(g(b), g(a)

)]]}

dt

=

{[f (b) + g(a)

] ∫

( – t) dt

+[[

g(a) + η(g(b), g(a)

)] +[f (b) + η

(f (a), f (b)

)]]∫

t dt

+[[

f (b)][

f (b) + η(f (a), f (b)

)]+

[g(a)

][g(a) + η

(g(b), g(a)

)]] ∫

t( – t) dt

}


=

{[f (b) + g(a)

]+

[[g(a) + η

(g(b), g(a)

)] +[f (b) + η

(f (a), f (b)

)]]

+[f (b)

[f (b) + η

(f (a), f (b)

)]+ g(a)

[g(a) + η

(g(b), g(a)

)]]},

which is the required result. �

Corollary . If f = g and η(b, a) = b – a in Theorem ., then

ln b – ln a

∫ b

a

x

[f (x)f

(abx

)]dx

≤

{[f (b) + f (a)

]+

[f (b)f (a) + f (a)f (b)

]}.


ln b – ln a

∫ b

a

x

[f (x)g

(abx

)]dx

=

{[

f (b) + g(a)] +

[[f (b) + η

(f (a), f (b)

)][g(a) + η

(g(b), g(a)

)]]

+(f (b) + g(a)

)[f (b) + η

(f (a), f (b)

)][g(a) + η

(g(b), g(a)

)]}.


f(atb–t) ≤ ( – t)f (b) + t

[f (b) + η

(f (a), f (b)

)], (.)

g(a–tbt) ≤ ( – t)g(a) + t

[g(a) + η

(g(b), g(a)

)]. (.)

Now,

ln b – ln a

∫ b

a

x

[f (x)g

(abx

)]dx

=∫

(f(atb–t)g

(a–tbt))dt

≤

∫

[[f(atb–t)] +

[g(a–tbt)]] dt

≤

{∫

[( – t)f (b) + t

[f (b) + η

(f (a), f (b)

)]

+[( – t)g(a) + t

[g(a) + η

(g(b), g(a)

)]]}

dt

=

{∫

[( – t)

[f (b) + g(a)

]+ t

[f (b) + η

(f (a), f (b)

)][g(a) + η

(g(b), g(a)

)]]}

dt

=

{∫

[( – t)[f (b) + g(a)

] + t[[f (b) + η(f (a), f (b)

)][g(a) + η

(g(b), g(a)

)]]

+ t( – t)(f (b) + g(a)

)[f (b) + η

(f (a), f (b)

)][g(a) + η

(g(b), g(a)

)]}dt


=

{[

f (b) + g(a)] +

[[f (b) + η

(f (a), f (b)

)][g(a) + η

(g(b), g(a)

)]]

+(f (b) + g(a)

)[f (b) + η

(f (a), f (b)

)][g(a) + η

(g(b), g(a)

)]},



ln b – ln a

∫ b

a

x

[f (x)f

(abx

)]dx

≤

{[f (b) + f (a)

] +[f (a)f (b)

] +[f (a)f (b)

(f (b) + f (a)

)]}.


f (√

ab)g(√

ab)

≤ ln b – ln a

∫ b

a

x[f (x)g(x)

]dx

+

(ln b – ln a)

∫ b

a

x[f (x)

][η

(g(

abx

), g(x)

)]

+[g(x)

][η

(f(

abx

), f (x)

)]dx

+

(ln b – ln a)

∫ b

a

x

[η(f

(abx

), f (x)

][η

(g(

abx

), g(x)

)]dx.


f (√

ab) ≤ f(a–tbt) +

η(f(atb–t), f

(a–tbt)), (.)

g(√

ab) ≤ g(a–tbt) +

η(g(atb–t), g

(a–tbt)). (.)


f (√

ab)g(√

ab)

≤[

f(a–tbt) +

η(f(atb–t), f

(a–tbt))

][g(a–tbt) +

η(g(atb–t), g

(a–tbt))

]

= f(a–tbt)[g

(a–tbt)] +

[[

f(a–tbt)][η

(g(atb–t), g

(a–tbt))]

+[g(a–tbt)][η

(f(atb–t), f

(a–tbt))]]

+

[[η(g(atb–t), g

(a–tbt))][η

(f(atb–t), f

(a–tbt))]]. (.)



∫

f (

√ab)g(

√ab) dt

≤∫

f(a–tbt)[g

(a–tbt)]dt

+

∫

[[f(a–tbt)][η

(g(atb–t), g

(a–tbt))]

+[g(a–tbt)][η

(f(atb–t), f

(a–tbt))]]dt

+

∫

[[η(g(atb–t), g

(a–tbt))][η

(f(atb–t), f

(a–tbt))]]dt.

Thus

f (√

ab)g(√

ab)

≤ ln b – ln a

∫ b

a

x[f (x)g(x)

]dx

+

(ln b – ln a)

∫ b

a

x[f (x)

][η

(g(

abx

), g(x)

)]

+[g(x)

][η

(f(

abx

), f (x)

)]dx

+

(ln b – ln a)

∫ b

a

x

[η

(f(

abx

), f (x)

)][η

(g(

abx

), g(x)

)]dx.



f (√

ab)f (√

ab) ≤ ln b – ln a

∫ b

a

x[f (x)f (x)

]dx.

Theorem . Let f , g : I = [a, b] → (,∞) be increasing and generalized geometrically con-vex functions on I and a, b ∈ I with a < b. Then

ln b – ln a

∫ b

a

x[f (x)

][g(b) +

η(g(a), g(b)

)]dx

+

ln b – ln a

∫ b

a

x[g(x)

][

f (b) +η(f (a), f (b)

)]

dx

≤ ln b – ln a

∫ b

a

x[f (x)

][g(x)

]dx

+[

f (b) +η(f (a), f (b)

)][g(b) +

η(g(a), g(b)

)].

Proof Let f and g be generalized geometrically convex functions on I . Then ∀a, b ∈ I andt ∈ [, ], we have

f(atb–t) ≤ [

f (b) + tη(f (a), f (b)

)], (.)


and

g(atb–t) ≤ [

g(b) + tη(g(a), g(b)

)]. (.)

Using the inequality

(a – b)(c – d) ≥ ∀a, b, c, d ∈R, a < b, c < d, (.)

we have

[f(atb–t)][g(b) + tη

(g(a), g(b)

)]+

[g(atb–t)][f (b) + tη

(f (a), f (b)

)]

≤ f(atb–t)g

(atb–t) +

[f (b) + tη

(f (a), f (b)

)][g(b) + tη

(g(a), g(b)

)].

Integrating the above inequality over t on [, ], we have

∫

[f(atb–t)]dt

∫

[g(b) + tη

(g(a), g(b)

)]dt

+∫

[g(atb–t)]dt

∫

[f (b) + tη

(f (a), f (b)

)]dt

≤∫

f(atb–t)dt

∫

g(atb–t)dt

+∫

[f (b) + tη

(f (a), f (b)

)]dt

∫

[g(b) + tη

(g(a), g(b)

)]dt.

Now after simple integration, we have

ln b – ln a

∫ b

a

x[f (x)

][

g(b) +η(g(a), g(b)

)]

dx

+

ln b – ln a

∫ b

a

x[g(x)

][

f (b) +η(f (a), f (b)

)]

dx

≤ ln b – ln a

∫ b

a

x[f (x)

][g(x)

]dx

+[

f (b) +η(f (a), f (b)

)][g(b) +

η(g(a), g(b)

)].



ln b – ln a

∫ b

a

x[f (x)

][ f (b) + f (a)

]dx

≤ ln b – ln a

∫ b

a

x[f (x)

][f (x)

]dx +

[f (b) + f (a)

].


4 Integral inequalitiesIn this section, we will use the following result to obtain our main results.

Lemma . ([]) Let f : I ⊂ R+ = (,∞) → R be a differentiable function on the interiorI of I , where a, b ∈ I with a < b and f ′ ∈ L[a, b]. Then

bf (b) – af (a) –∫ b

af (x) dx

=(ln b – ln a)

[∫

b+ta–tf ′(b

+t a

–t

)dt +

∫

b–ta+t f ′(b

–t a

+t

)dt

].

Theorem . Let f : I ⊂ R+ = (,∞) → R be a differentiable function on the interior I

of I , where a, b ∈ I with a < b and f ′ ∈ L[a, b]. If |f ′| is a generalized geometrically convexfunction on [a, b] for q ≥ , then

∣∣∣∣bf (b) – af (a) –

∫ b

af (x) dx

∣∣∣∣

≤ (b – a)– q

()q +

{b[(

b – a – L(a, b))∣∣f ′(b)

∣∣q

+(L(a, b) – a

)∣∣f ′(b) + η(f ′(a), f ′(b)

)∣∣q] q

+ a[(

b – a + L(a, b))∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q +(b – L(a, b)

)∣∣f ′(b)∣∣q]

q}

.

Proof Using Lemma . and Hölder’s inequality, we have

∣∣∣∣bf (b) – af (a) –

∫ b

af (x) dx

∣∣∣∣

=ab(ln b – ln a)

[∫

(ba

)t∣∣f ′(b

+t a

–t

)∣∣dt +∫

(ab

)t∣∣f ′(b

–t a

+t

)∣∣dt]

≤ ab(ln b – ln a)

{[∫

(ba

)t

dt]–

q[∫

(ba

)t∣∣f ′(b+t a

–t

)∣∣q dt]

q

+[∫

(ab

)t

dt]–

q[∫

(ab

)t∣∣f ′(b

–t a

+t

)∣∣q dt]

q}

. (.)

Now consider

I =∫

(ba

)t∣∣f ′(b

+t a

–t

)∣∣q dt

≤∫

∣∣f ′(b)∣∣q

( + t

)+

∣∣f ′(b) + η(f ′(a), f ′(b)

)∣∣q(

– t

)dt

=∣∣f ′(b)

∣∣q∫

(ab

)t( + t

)dt +

∣∣f ′(b) + η(f ′(a), f ′(b)

)∣∣q∫

(ba

)t( – t

)dt

=∣∣f ′(b)

∣∣q b – a – L(a, b)

a(ln b – ln a)+

∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q L(a, b) – aa(ln b – ln a)

. (.)


Similarly I becomes

∫

(ab

)t∣∣f ′(b–t a

+t

)∣∣q dt

≤ ∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q b – a + L(a, b)b(ln b – ln a)

+∣∣f ′(b)

∣∣q b – L(a, b)

b(ln b – ln a). (.)

Also

[∫

(ba

)t

dt]–

q=

(b – a

a(ln b – ln a)

)– q

. (.)

Combining (.), (.), (.) and (.), we have

∣∣∣∣bf (b) – af (a) –

∫ b

af (x) dx

∣∣∣∣

≤ (b – a)– q

()q +

{b[(

b – a – L(a, b))∣∣f ′(b)

∣∣q

+(L(a, b) – a

)∣∣f ′(b) + η(f ′(a), f ′(b)

)∣∣q] q

+ a[(

b – a + L(a, b))∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q +(b – L(a, b)

)∣∣f ′(b)∣∣q]

q}

,


If η(b, a) = b – a, then Theorem . reduces to the following result.

Corollary . ([]) Let f : I ⊂R+ = (,∞) →R be a differentiable function on I (interiorof I), where a, b ∈ I with a < b and f ′ ∈ L[a, b]. If |f ′| is a generalized geometrically convexfunction on [a, b] for q ≥ , then

∣∣∣∣bf (b) – af (a) –

∫ b

af (x) dx

∣∣∣∣

≤ (b – a)– q

()q +

{b[(

b – a – L(a, b))∣∣f ′(b)

∣∣q +(L(a, b) – a

)∣∣f ′(a)∣∣q]

q

+ a[(

b – a + L(a, b))∣∣f ′(a)

∣∣q +

(b – L(a, b)

)∣∣f ′(b)∣∣q]

q}

.

Theorem . Let f : I ⊂ R+ = (,∞) → R be a differentiable function on I (interior ofI), where a, b ∈ I with a < b and f ′ ∈ L[a, b]. If |f ′| is a generalized geometrically convexfunction on [a, b] for q > , then

∣∣∣∣bf (b) – af (a) –

∫ b

af (x) dx

∣∣∣∣

≤ (ln b – ln a)

()+ q

[L(a

qq– , b

qq–

]– q{

b[A

(∣∣f ′(b)

∣∣q,

∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q)] q

+ a[A

(∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q),∣∣f ′(b)

∣∣q]

q}

.



∣∣∣∣bf (b) – af (a) –

b – a

∫ b

af (x) dx

∣∣∣∣

=ab(ln b – ln a)

[∫

(ba

)t∣∣f ′(b+t a

–t

)∣∣dt +∫

(ab

)t∣∣f ′(b–t a

+t

)∣∣dt]


{[∫

(ba

) qtq–

dt]–

q[∫

∣∣f ′(b+t a

–t

)∣∣q dt]

q

+[∫

(ab

) qtq–

dt]–

q[∫

∣∣f ′(b

–t a

+t

)∣∣q dt]

q}

. (.)

Now we consider

I =∫

∣∣f ′(b

+t a

–t

)∣∣q dt

≤∫

∣∣f ′(b)∣∣q

( + t

)+

∣∣f ′(b) + η(f ′(a), f ′(b)

)∣∣q(

– t

)dt

=∣∣f ′(b)

∣∣q

∫

( + t

)dt +

∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q∫

( – t

)dt

=

∣∣f ′(b)∣∣q +

∣∣f ′(b) + η(f ′(a), f ′(b)

)∣∣q. (.)

Similarly I becomes

∫

∣∣f ′(b–t a

+t

)∣∣q dt ≤

∣∣f ′(b) + η(f ′(a), f ′(b)

)∣∣q +

∣∣f ′(b)∣∣q. (.)

Also

∫

(ba

) qtq–

dt]– q =

(b

qq– – a

qq–

( qq– )a

qq– (ln b – ln a)

)– q

. (.)


∣∣∣∣bf (b) – af (a) –

∫ b

af (x) dx

∣∣∣∣

≤ (ln b – ln a)

()+ q

[L(a

qq– , b

qq–

)]– q{

b[A

(∣∣f ′(b)

∣∣q,∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q)] q

+ a[A

(∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q),∣∣f ′(b)

∣∣q] q}

,



Corollary . ([]) Let f : I ⊂R+ = (,∞) → R be a differentiable function on I (interiorof I), where a, b ∈ I with a < b and f ′ ∈ L[a, b]. If |f ′| is a generalized geometrically convex


function on [a, b] for q > , then

∣∣∣∣bf (b) – af (a) –

∫ b

af (x) dx

∣∣∣∣

≤ (ln b – ln a)

()+ q

[L(a

qq– , b

qq–

)]– q{

b[A

(∣∣f ′(b)

∣∣q,

∣∣f ′(a)

∣∣q)]

q

+ a[A

(∣∣f ′(a)

∣∣q),∣∣f ′(b)

∣∣q] q}

.

Theorem . Let f : I ⊂ R+ = (,∞) → R be a differentiable function on I (interior ofI), where a, b ∈ I with a < b and f ′ ∈ L[a, b]. If |f ′| is a generalized geometrically convexfunction on [a, b] for q ≥ , then

∣∣∣∣bf (b) – af (a) –

∫ b

af (x) dx

∣∣∣∣

≤ (ln b – ln a)– q

(q)q

{b[(

L(aq, bq) – aq)∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q

+(bq – aq – L

(aq, bq))∣∣f ′(b)

∣∣q]

q

+ a[∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q(bq – aq + L(aq, bq))

+(bq – L

(aq, bq))∣∣f ′(b)

∣∣]q}

.


∣∣∣∣bf (b) – af (a) –

∫ b

af (x) dx

∣∣∣∣

=ab(ln b – ln a)

[∫

(ba

)t∣∣f ′(b+t a

–t

)∣∣dt +∫

(ab

)t∣∣f ′(b–t a

+t

)∣∣dt]


{[∫

dt

]– q[∫

(ba

)qt∣∣f ′(b

+t a

–t

)∣∣q dt]

q

+[∫

dt

]– q[∫

(ab

)qt∣∣f ′(b–t a

+t

)∣∣q dt]

q}

. (.)

Now consider

I =∫

(ba

)qt∣∣f ′(b+t a

–t

)∣∣q dt

≤∫

(ba

)qt[∣∣f ′(b)∣∣t(

+ t

)+

∣∣f ′(b) + η(f ′(a), f ′(b)

)∣∣q(

– t

)]dt

=∣∣f ′(b)

∣∣q∫

(ba

)qt( + t

)dt +

∣∣f ′(b) + η(f ′(a), f ′(b)

)∣∣q∫

(ba

)qt( – t

)dt

=∣∣f ′(b)

∣∣q bq – aq – L(aq, bq)

qaq(ln b – ln a)+

∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q L(aq, bq) – aq

qaq(ln b – ln a). (.)


Similarly I becomes

∫

(ab

)qt∣∣f ′(b–t a

+t

)∣∣q dt

≤ ∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q bq – aq + L(aq, bq)qbq(ln b – ln a)

+∣∣f ′(b)

∣∣q bq – L(aq, bq)

qbq(ln b – ln a). (.)

Combining (.), (.) and (.), we have

∣∣∣∣bf (b) – af (a) –

∫ b

af (x) dx

∣∣∣∣

≤ (ln b – ln a)– q

(q)q

{b[(

L(aq, bq) – aq)∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q

+(bq – aq – L

(aq, bq))∣∣f ′(b)

∣∣q] q

+ a[∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q(bq – aq + L(aq, bq)) +

(bq – L

(aq, bq))∣∣f ′(b)

∣∣]

q}

,



Corollary . ([]) Let f : I ⊂ R+ = (,∞) →R be a differentiable function on I (interiorof I), where a, b ∈ I with a < b and f ′ ∈ L[a, b]. If |f ′| is a generalized geometrically convexfunction on [a, b] for q ≥ , then

∣∣∣∣bf (b) – af (a) –

∫ b

af (x) dx

∣∣∣∣

≤ (ln b – ln a)– q

(q)q

{b[(

L(aq, bq) – aq)∣∣f ′(a)

∣∣q

+(bq – aq – L

(aq, bq))∣∣f ′(b)

∣∣q]

q + a[∣∣f ′(a)

∣∣q(bq – aq + L

(aq, bq))

+(bq – L

(aq, bq))∣∣f ′(b)

∣∣]q}

.

Theorem . Let f : I ⊂ R+ = (,∞) → R be a differentiable function on I (interior ofI), where a, b ∈ I with a < b and f ′ ∈ L[a, b]. If |f ′| is a generalized geometrically convexfunction on [a, b] for q > and q > p > , then

∣∣∣∣bf (b) – af (a) –

∫ b

af (x) dx

∣∣∣∣

≤ (ln b – ln a)

(p)q

[L(a

q–pq– , b

q–pq–

)]– q{

a– pq b

[(L(ap, bp) – ap)∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q

+(bp – ap – L

(ap, bp))∣∣f ′(b)

∣∣q]

q}

+{

ab– pq[(

bp – ap + L(ap, bp))∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q

+(bq – L

(ap, bp))∣∣f ′(b)

∣∣q]

q}

.



∣∣∣∣bf (b) – af (a) –

∫ b

af (x) dx

∣∣∣∣

=ab(ln b – ln a)

[∫

(ba

)t∣∣f ′(b

+t a

–t

)∣∣dt +∫

(ab

)t∣∣f ′(b

–t a

+t

)∣∣dt]


{[∫

(ba

) (q–p)tq–

dt]–

q[∫

(ba

)pt∣∣f ′(b+t a

–t

)∣∣q dt]

q

+[∫

(ab

) (q–p)tq–

dt]–

q[∫

(ab

)pt∣∣f ′(b

–t a

+t

)∣∣q dt]

q}

. (.)

Now consider

I =∫

(ba

)pt∣∣f ′(b

+t a

–t

)∣∣q dt

≤∫

(ba

)pt[∣∣f ′(b)∣∣t(

+ t

)+

∣∣f ′(b) + η(f ′(a), f ′(b)

)∣∣q(

– t

)]dt

=∣∣f ′(b)

∣∣q∫

(ab

)pt( + t

)dt +

∣∣f ′(b) + η(f ′(a), f ′(b)

)∣∣q∫

(ba

)pt( – t

)dt

=∣∣f ′(b)

∣∣q bp – ap – L(ap, bp)

pap(ln b – ln a)+

∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q L(ap, bp) – ap

pap(ln b – ln a). (.)

Similarly I becomes

[∫

(ab

)pt∣∣f ′(b–t a

+t

)∣∣q dt]

≤ ∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q bp – ap + L(ap, bp)pbp(ln b – ln a)

+∣∣f ′(b)

∣∣q bp – L(ap, bp)

pbp(ln b – ln a). (.)

Also

[∫

(ba

) (q–p)tq–

dt]–

q=

(b

q–pq– – a

q–pq–

( q–pq– )a

q–pq– (ln b – ln a)

)– q

. (.)


∣∣∣∣bf (b) – af (a) –

∫ b

af (x) dx

∣∣∣∣

≤ (ln b – ln a)

(p)q

[L(a

q–pq– , b

q–pq–

)]– q{

a– pq b

[(L(ap, bp) – ap)∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q

+ (bp – ap – L(ap, bp)∣∣f ′(b)

∣∣q] q}

+{

ab– pq[(

bp – ap + L(ap, bp))∣∣f ′(b) + η

(f ′(a), f ′(b)

)∣∣q

+(bq – L

(ap, bp))∣∣f ′(b)

∣∣q] q}

,




Corollary . ([]) Let f : I ⊂ R+ = (,∞) → R be a differentiable function on the inte-rior I of I , where a, b ∈ I with a < b and f ′ ∈ L[a, b]. If |f ′| is a generalized geometricallyconvex function on [a, b] for q > and q > p > , then

∣∣∣∣bf (b) – af (a) –

∫ b

af (x) dx

∣∣∣∣

≤ (ln b – ln a)

(p)q

[L(a

q–pq– , b

q–pq–

)]– q{

a– pq b

[(L(ap, bp) – ap)∣∣f ′(a)

∣∣q

+(bp – ap – L

(ap, bp))∣∣f ′(b)

∣∣q] q}

+{

ab– pq[(

bp – ap + L(ap, bp))∣∣f ′(a)

∣∣q

+(bq – L

(ap, bp))∣∣f ′(b)

∣∣q]

q}

.

5 ConclusionIn this paper, we have introduced and investigated a new class of generalized functions,called generalized geometrically convex functions. Some basic inequalities related to gen-eralized geometrically convex functions have been derived. Several new inequalities of theHermite-Hadamard type for generalized geometrically convex functions have been estab-lished. Several special cases are discussed, which can be deduced from our main results.The techniques and ideas of this paper may stimulate further research in this dynamicfield. It is an interesting problem to consider these estimates in the setting of fractionalcalculus of meromorphic functions; see [, , , , ] and the references therein.

AcknowledgementsThe authors would like to thank the Rector of the COMSATS Institute of Information Technology, Pakistan, for providingexcellent research and academic environments. The authors are pleased to acknowledge the support of theDistinguished Scientist Fellowship Program (DSFP), from King Saud University, Riyadh, Saudi Arabia. The authors aregrateful to the referees for their valuable and constructive comments.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsMAN, KIN and FS worked jointly. All the authors read and approved the final manuscript.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 11 July 2017 Accepted: 17 August 2017

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