# Hermite–Hadamard-Type Inequalities for the Generalized Geometrically Strongly Modifiedh-Convex Functions

Authors: Xishan Yu; Muhammad Shoaib Saleem; Shumaila Waheed; Ilyas Khan
Journal: Journal of Mathematics
 (2021)

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## Abstract

Convexity theory becomes a hot area of research due to its applications in pure and applied mathematics, especially in optimization theory. The aim of this paper is to introduce a broader class of convex functions by unifying geometrically strong convex function withh convex functions. This new class of functions is called as generalized geometrically strongly modified h-convex functions. We established Hermite–Hadamard-type inequalities for the generalized geometrically strongly modified h-convex functions. Our results can be considered as generalization and extension of literature.


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## Body

## 1. Introduction

The modern analysis involves the applications of convexity, and no mathematician, working in applied mathematics, especially, in nonlinear programming and optimization theory, can ignore the significant role of convex sets and convex functions. It can be viewed as one of the most natural and simple notation in mathematics. The convexity of sets and the convexity of functions have been the object of many studies during the last century. Early contributions to convex analysis were made by Hölder [1] and Minkowski [2]. Convexity also plays a fundamental role in mathematical economics and statistics [3–5], and the importance of convex functions is well known in statistics, management sciences, and engineering. Since the classical convexity is not enough to attain certain goals in applied mathematics, the classical convexity has been generalized in many ways, see [6, 7].The theory of inequalities produced important contributions in convex analysis, for example, the Hermite–Hadamard, Schur, and Fejér inequalities are very important and produce many applications [8, 9]. The Hermite–Hadamard inequalities give us an estimate of the mean value of a convex function and are defined as follows.The Hermite–Hadamard inequality for a convex functionϕ:L=r,s⊂ℝ⟶ℝ is(1)ϕr+s2≤1s−r∫rsϕxdx≤ϕr+ϕs2.There are many generalizations of the Hermite–Hadamard inequality, see [10–12]. Different notions of convex functions and different variants of fractional intergrades have been used in literature to establish Hermite–Hadamard inequalities, see, for example, Hermite–Hadamard-type inequalities for harmonically convex functions [6], Hermite–Hadamard inequality for Caputo-Fabrizio fractional integrals [13], Hermite–Hadamard-type inequalities for h-convexity [14], and Hermite–Hadamard inequalities for s-convex and s-concave functions [15]. For more details about inequality theory, we refer the readers to [16–20] and references therein.The other interesting inequalities are Chebyshev-type inequalities [15, 21, 22], Trapezium-type inequalities [23], generalized fractional Hilfer integral inequalities [24], etc. [25]. Also, many different extensions and versions appeared in a number of papers, see, for example, [26–30].In this paper, we introduced a broad class of convex functions and established Hermite–Hadamard-type inequalities in the setting of newly introduced class of functions.The paper is organized as follows. In Section2, we will give some basic definitions and basic algebraic properties for this class of functions. Sections 3 and 4 are devoted to Hermite–Hadamard and some other inequalities for the generalized geometrically strongly modified h-convex functions.


## 2. Definitions and Basic Results

We start with some basic definitions related to our work. Throughout this paper,J is an interval in ℝ and η:N×N⟶M⊂ℝ is a bi-function.Definition 1.
(convex function, see [31]). Let ϕ:J⟶ℝ be an extended real-valued function defined on a convex set J⊂ℝn. Then, the function ϕ is convex on J if(2)ϕαr+1−αs≤αϕr+1−αϕsholds, for all r,s∈J and α∈0,1.Definition 2.
(h-convex function, see [14]). Let ϕ,h:J⊂ℝ⟶ℝ be nonnegative functions. Then, ϕ is h-convex function, if(3)ϕαr+1−αs≤hαϕr+h1−αϕsholds, for all r,s∈J and α∈0,1.Definition 3.
(modifiedh-convex function, see [32]). Let ϕ,h:J⊂ℝ⟶ℝ be nonnegative functions. Then, ϕ is called modified h-convex function if(4)ϕαr+1−αs≤hαϕr+1−hαϕsholds, for all r,s∈J and α∈0,1.Definition 4.
(generalized convex function, see [33]). A function ϕ:J=r,s⟶ℝ is said to be generalized convex function with respect to a bi-function η.,.:ℝ×ℝ⟶ℝ if(5)ϕαr+1−αs≤ϕs+αηϕr,ϕs,∀r,s∈J,α∈0,1,holds.Remark 1.
If we takeηr,s=r−s in Definition 1, then the generalized convex function reduces to a convex function.Definition 5.
(generalized modifiedh-convex function). Let ϕ,h:J⊂ℝ⟶ℝ be nonnegative functions. Then, ϕ is the generalized modified h-convex function if(6)ϕαr+1−αs≤ϕs+hαηϕr,ϕsholds, for all r,s∈J and α∈0,1.Definition 6.
(generalized strongly modifiedh-convex function, see [34]). Let ϕ,h:J⊂ℝ⟶ℝ be nonnegative functions. Then, ϕ is the generalized strongly modified h-convex function if(7)ϕαr+1−αs≤ϕs+hαηϕr,ϕs−μα1−αr−s2holds, for all r,s∈J and α∈0,1.Definition 7.
(additive). A functionη is said to be additive if(8)ηr1,s1+ηr2,s2=ηr1+r2,s1+s2,∀,r1,r2,s1,s2∈ℝ.Definition 8.
(nonnegatively homogeneous). A functionη is said to be nonnegatively homogeneous if(9)ηλr1,λr2=ληr1,r2,∀,r1,r2∈ℝ.Definition 9.
(supermultiplicative function, see [35]). A function ϕ:J⊂ℝ⟶ℝ+ is said to be supermultiplicative function if(10)ϕr,s≥ϕrϕs,∀,r,s∈J,α∈0,1.Definition 10.
(logarithmic mean). Letr,s∈ℝ such that r,s≠0 and r≠s; then, logarithmic mean for real numbers is defined as(11)Lr,s=r−slnr−lns.Definition 11.
(geometrically convex function, see [14]). A function ϕ:J⊂ℝ+=0,∞⟶ℝ is said to be a geometrically convex function or geometric arithmetic convexity with respect to a bi-function η.,.:ℝ×ℝ⟶ℝ if(12)ϕrαs1−α≤αϕr+1−αϕs,∀r,s∈0,1.Definition 12.
(generalized geometrically convex function, see [24]). A function ϕ:J⊂ℝ+=0,∞⟶ℝ is said to be generalized geometrically convex function with respect to a bi-function η.,.:ℝ×ℝ⟶ℝ if(13)ϕrαs1−α≤1−αϕs+αϕs+ηϕr,ϕs,∀r,s∈0,1.Definition 13.
(generalized geometrically strongly modifiedh-convex functions). A function ϕ:J⊂ℝ+=0,∞⟶ℝ is said to be generalized geometrically strongly modified h-convex function with respect to a bi-function η.,.:ℝ×ℝ⟶ℝ if(14)ϕrαs1−α≤1−hαϕs+hαϕs+ηϕr,ϕs−μα1−αr−s2.Lemma 1.
Forr,s∈0,∞ and m,α∈0,1, if r<s and s≥1, then(15)rαsm1−α≤αr+1−αs.Remark 2.
The abovementioned definitions can be related with each other as follows:(1)
Forηr,s=r−s, the generalized convex function reduces to the convex function(2)
Forhα=α, the generalized modified h−convex function reduces to the generalized convex function(3)
Forhα=α and ηr,s=r−s, the generalized modified h−convex function reduces to the convex function(4)
Forηr,s=r−s, the generalized modified h−convex function reduces to the modified h−convex function(5)
Forμ=0,ηr,s=r−s and hα=α, the generalized geometrically strongly modified h−convex functions reduces to the convex function(6)
Forhα=α and μ=0, the generalized geometrically strongly modified h−convex functions reduces to the generalized geometrically convex functionThe following proposition is about the addition of two functions.Proposition 1.
Letη be additive μ∗≥0 and 2μ=μ∗; then, the sum of two generalized geometrically strongly modified h-convex functions is also generalized geometrically strongly modified h-convex functions.proof.
Letϕ and ψ be two generalized geometrically strongly modified h-convex functions; then, by definition, we have(16)ϕrαs1−α≤1−hαϕs+hαϕs+ηϕr,ϕs−μα1−αr−s2,ψrαs1−α≤1−hαψs+hαψs+ηψr,ψs−μα1−αr−s2.
So,(17)ϕrαs1−α+ψrαs1−α≤1−hαϕ+ψs+hαϕs+ηϕr,ϕs+ψs+ηψr,ψs−2μα1−αr−s2.
Now, by additive property ofη and taking 2μ=μ∗, we obtain(18)ϕrαs1−α+ψrαs1−α≤1−hαϕ+ψs+hαϕ+ψs+ηϕr+ψr,ϕs+ψs−μ∗α1−αr−s2.
This completes the proof.
The next result is about the scalar multiplication of a function.Proposition 2.
Letϕ be generalized geometrically strongly modified h-convex function with λ≥0 and η is nonnegatively homogeneous. Then, λϕ is also a generalized geometrically strongly modified h-convex function.Proof.
Sinceϕ is the generalized geometrically strongly modified h-convex function, so, by definition, we have(19)ϕrαs1−α≤1−hαϕs+hαϕs+ηϕr,ϕs−μα1−αr−s2.
So,(20)λϕrαs1−α≤λ1−hαϕs+hαϕs+ηϕr,ϕs−μα1−αr−s2≤1−hαλϕs+λhαϕs+ηϕr,ϕs−λμα1−αr−s2≤1−hαλϕs+hαλϕs+ληϕr,ϕs−λμα1−αr−s2.
Asη is nonnegatively homogeneous and by taking λμ=μ∗, we obtain(21)λϕrαs1−α≤1−hαλϕs+hαλϕs+ηλϕr,λϕs−μ∗α1−αr−s2.
This completes the proof.Proposition 3.
Letϕ and ψ be two generalized geometrically strongly modified h-convex functions, where η is additive and nonnegatively homogeneous. Then, rϕ+sψ is also the generalized geometrically strongly modified h-convex function ∀r,s∈ℝ.Proposition 4.
Letψ be a supermultiplicative function and ϕ be a generalized geometrically strongly modified h-convex function. If ψr−ψs=r−s, then ϕ°ψ is also a generalized geometrically strongly modified h-convex function.Proof.
Letψ be a supermultiplicative function and ϕ is a generalized geometrically strongly modified h-convex function. Also, since ψr−ψs=r−s, so ∀,r,s∈J and α∈0,1, and we obtain(22)ϕ°ψrαs1−α≥ϕψrαψs1−α≤1−hαϕ°ψs+hαϕ°ψs+ηϕ°ψr,ϕ°ψs−μα1−αψr−ψs2=1−hαϕ°ψs+hαϕ°ψs+ηϕ°ψr,ϕ°ψs−μα1−αr−s2.
This shows thatϕ°ψ is a generalized geometrically strongly modified h-convex function.Proposition 5.
Let us considerϕi:J⊂ℝ⟶ℝ such that ϕi are generalized geometrically strongly modified h-convex functions, where ∑i=1mλi=1 and η is additive and nonnegatively homogeneous. Then, the linear combination ψ:ℝ⟶ℝ is also a generalized geometrically strongly modified h-convex function.Proof.
Lettingfx=∑i=1mλiϕix and setting r=rαs1−α, we obtain(23)frαs1−α=∑i=1mλiϕirαs1−α≤1−hα∑i=1mλiϕis+hα∑i=1mλiϕis+ηϕir,ϕis−μα1−αr−s2=∑i=1mλiϕis+hαη∑i=1mλiϕir,∑i=1mλiϕis−μα1−αr−s2=fs+hαηfr,fs−μα1−αr−s2.
This completes the proof.Proposition 6.
Ifϕ is a generalized geometrically strongly modified h-convex function, then ϕ is also a generalized geometrically modified h-convex function.Proof.
We have(24)ϕrαs1−α≤1−hαϕs+hαϕs+ηϕr,ϕs−μα1−αr−s2≤1−hαϕs+hαϕs+ηϕr,ϕs,∀r,s∈J⊂ℝ. This completes the proof.Corollary 1.
Ifϕ is the generalized geometrically strongly convex function and α≤hα, then ϕ is also a generalized geometrically strongly modified h-convex function.


## 3. Hermite–Hadamard-Type Inequality

In this section, we derive some new Hermite–Hadamard-type inequalities for generalized geometrically strongly modifiedh-convex functions. Throughout this section, consider J=r,s.Theorem 1.
Letϕ,ψ:J=r,s⟶0,∞ be two generalized geometrically strongly modified h-convex functions and r,s∈J with r<s; then, the following inequality holds:(25)1lns−lnr∫rs1xϕxψxdx≤ϕsψs∫011−hα2dα+M1r,s∫01hα1−hαdα+M2r,s∫01hα2dα−μr−s2ηϕr+ψr,ϕs+ψs∫01α1−αhαdα−16μr−s2ϕs+ψs−15μr−s2,where M1r,s=ϕsψs+ηψr,ψs+ψsϕs+ηϕr,ϕs and M2r,s=ϕs+ηϕr,ϕsψs+ηψr,ψs.Proof.
Sinceϕ,ψ are generalized geometrically strongly modified h-convex functions, then ∀,r,s∈J and α∈0,1, and we have(26)ϕrαs1−α≤1−hαϕs+hαϕs+ηϕr,ϕs−μα1−αr−s2,ψrαs1−α≤1−hαψs+hαψs+ηψr,ψs−μα1−αr−s2,which implies that(27)ϕrαs1−αψrαs1−α≤1−hα2ϕsψs+hα1−hαϕsψs+ηψr,ψs+ψsϕs+ηϕr,ϕs+hα2ϕs+ηϕr,ϕs×ψs+ηψr,ψs−μα1−αr−s2ϕs+ψs1−hα+hαϕs+ηϕr,ϕs+ψs+ηψr,ψs+μ2α21−α2r−s4.
Integrate both sides overα on 0,1 and by the additive property of η, i.e., ηr1,s1+ηr2,s2=ηr1+r2,s1+s2, and we have(28)∫01ϕrαs1−αψrαs1−αdα=1lns−lnr∫rs1xϕxψxdx≤ϕsψs∫011−hα2dα+ϕsψs+ηψr,ψs+ψsϕs+ηϕr,ϕs∫01hα1−hαdα+ϕs+ηϕr,ϕsψs+ηψr,ψs∫01hα2dα−μr−s2×∫01α1−α1−hαϕs+ψs+hαϕs+ψs+ηϕr+ψr,ϕs+ψsdα+μ2r−s4∫01α21−α2dα≤ϕsψs∫011−hα2dα+ϕsψs+ηψr,ψs+ψsϕs+ηϕr,ϕs∫01hα1−hαdα+ϕs+ηϕr,ϕsψs+ηψrψs∫01hα2dα−μr−s2ηϕr+ψr,ϕs+ψs∫01α1−αhαdα−16μr−s2ϕs+ψs−15μr−s2≤ϕsψs∫011−hα2dα+M1r,s∫01hα1−hαdα+M2r,s∫01hα2dα−μr−s2ηϕr+ψr,ϕs+ψs×∫01α1−αhαdα−16μr−s2ϕs+ψs−15μr−s2.
This completes the proof.Theorem 2.
Letϕ,ψ:J=r,s⟶0,∞ be generalized geometrically strongly modified h-convex functions and r,s∈J with r<s; then, the following inequality holds:(29)1lns−lnr∫rs1xϕxψrsxdx≤ϕsψr∫011−hα2dα+M1r,s∫01hα1−hαdα+M2r,s∫01hα2dα−μr−s2ηϕr+ψs,ϕs+ψr∫01α1−αhαdα−16μr−s2ϕs+ψr+15μr−s2,where M1r,s=ϕsψr+ηψb,ψr+ψrϕs+ηϕr,ϕs and M2r,s=ϕs+ηϕr,ϕsψr+ηψs,ψr.Proof.
Sinceϕ,ψ are generalized geometrically strongly modified h-convex functions, so ∀r,s∈J and α∈0,1, and we have(30)ϕrαs1−α≤1−hαϕs+hαϕs+ηϕr,ϕs−μα1−αr−s2,ψr1−αsα≤1−hαψr+hαψr+ηψs,ψr−μα1−αr−s2,which implies that(31)ϕrαs1−αψr1−αsα≤ϕsψr1−hα2+ϕsψr+ηψs,ψr+ψr×ϕs+ηϕr,ϕshα1−hα+ϕs+ηϕr,ϕs×ψr+ηψs,ψrhα2−μr−s2ϕs+ψrα1−α×1−hα−μr−s2ϕs+ηϕr,ϕs+ψr+ηψs,ψr×α1−αhα+μ2r−s4α21−α2.
Integrate both sides of (31) over α on [0,1] and by the additive property of η, that is, ηr1,s1+ηr2,s2=ηr1+r2,s1+s2, and we have(32)∫01ϕrαs1−αψr1−αsαdα=1lns−lnr∫rs1xϕxψrsxdx≤ϕsψr∫011−hα2dα+ϕsψr+ηψs,ψr+ψrϕs+ηϕr,ϕs∫01hα1−hαdα+ϕs+ηϕr,ϕsψr+ηψs,ψr∫01hα2dα−μr−s2×∫01α1−α1−hαϕs+ψr+htϕs+ψr+ηϕr+ψs,ϕs+ψrdα+μ2a−b4∫01α21−α2dα≤ϕsψr∫011−hα2dα+ϕsψr+ηψs,ψr+ψrϕs+ηϕr,ϕs∫01hα1−hαdα+ϕs+ηϕr,ψsψr+ηψs,ψr∫01hα2dα−μr−s2ηϕr+ψs,ϕs+ψr∫01α1−αhαdα−16μr−s2ϕs+ψr−15μr−s2≤ϕsψr∫011−hα2dα+M1r,s∫01hα1−hαdt+M2r,s∫01hα2dα−μr−s2ηϕr+ψs,ϕs+ψr×∫01α1−αhαdα−16μr−s2ϕs+ψr+15μr−s2.
This completes the proof.Remark 3.
Takinghα=α and μ=0 in (25) and (31), we obtained Hermite–Hadamard-type inequalities for generalized geometrically convex functions [24].Corollary 2.
Takingϕ=ψ and ηr,s=r−s in (25), we obtain(33)1lns−lnr∫rs1xϕxϕrsxdx≤ϕsϕr∫011−hα2dα+ϕ2s+ϕ2r∫01hα1−hαdα+ϕrϕs∫01hα2dα−16μr−s2ϕs+ϕr−15μr−s2.Theorem 3.
Letϕ,ψ:r,s⟶0,∞ be generalized geometrically strongly modified h-convex functions and r,s∈J with r<s; then, the following inequality holds:(34)1lns−lnr∫rs1xϕxψrsxdx≤12ϕ2s+ψ2r∫011−hα2dα+ψr+ηψs,ψr2+ϕs+ηϕr,ϕs2×∫01hα2dα+ϕsϕs+ηϕr,ϕs+ψrψr+ηψs,ψr∫012hα1−hαdα+ϕs+ηϕr,ϕs+ψr+ηψs,ψr2μr−s2∫01α1−αhαdα+ϕs+ψr2μr−s2∫01α1−α1−hαdα+μ2r−s415.Proof.
Sinceϕ,ψ:r,s⟶0,∞ are generalized geometrically strongly modified h-convex functions, so ∀r,s∈J and α∈0,1, and we have(35)ϕrαs1−α≤1−hαϕs+hαϕs+ηϕr,ϕs−μα1−αr−s2,ψr1−αsα≤1−hαψr+hαψr+ηψs,ψr−μα1−αr−s2,which gives(36)1lns−lnr∫rs1xϕxψrsxdx=∫01ϕrαs1−αψr1−αsαdα≤12∫01ϕrαs1−α2+ψr1−αsα2dα≤12∫011−hαϕs+hαϕs+ηϕr,ϕs−μα1−αr−s22+1−hαψr+hαψr+ηψs,ψr−μα1−αr−s22dα=12ϕ2s+ψ2r∫011−hα2dα+ψr+ηψs,ψr2+ϕs+ηϕr,ϕs2∫01hα2dα+ϕsϕs+ηϕr,ϕs+ψrψr+ηψs,ψr×∫012hα1−hαdα+ϕs+ηϕr,ϕs+ψr+ηψs,ψr2μr−s2∫01α1−αhαdα+ϕs+ψr2μr−s2∫01α1−α1−hαα+2μ2r−s4∫01α21−α2dα≤12ϕ2s+ψ2r∫011−hα2dα+ψr+ηψs,ψr2+ϕs+ηϕr,ϕs2∫01hα2dα+ϕsϕs+ηϕr,ϕs+ψrψr+ηψs,ψr∫012hα1−hαdα+ϕs+ηϕr,ϕs+ψr+ηψs,ψr2μr−s2∫01α1−αhαdα+ϕs+ψr2μr−s2∫01α1−α1−hαdα+μ2r−s415.
This completes the proof.Remark 4.
Takinghα=α and μ=0 in (34) we obtained Hermite–Hadamard-type inequality for the generalized geometrically convex functions [24].Corollary 3.
Takingg1=g2 and ηx,y=x−y in Theorem 3, we obtain(37)1lns−lnr∫rs1xϕxϕrsxdx≤12ϕ2s+ϕ2r∫011−2hα2+2hα2dα+ϕsϕr+ϕrϕs∫012hα1−hαdt+2μr−s2ϕr+ϕs×∫01α1−αhαdα+∫01α1−α1−hαdα+μ2r−s415.Lemma 2.
Letϕ:J⊂ℝ⟶ℝ be a differentiable function on the interior J° of J, where r,s∈J with r<s and ϕ″∈L1r,s. Then, we have(38)ϕr+ϕs2−1s−r∫rsϕxdx=s−r22∫01α1−αϕ″αr+1−αsdα.Theorem 4.
Letϕ:J⊂ℝ+⟶ℝ+ be a twice differentiable function on J°, r,s∈J with r<s and ϕ″∈L1r,s. If ϕ″q is a generalized geometrically strongly modified h−convex function, monotonically decreasing on r,s for q≥1 and α∈0,1, then the following inequality holds:(39)ϕr+ϕs2−1s−r∫rsϕxdx≤s−r22161−1/q16ϕ″bq+∫01α1−αhαηϕ″r,ϕ″sqdα−μr−s2∫01α1−αα1−αqdα1/q.Proof.
From Lemma2 and using the well-known power-mean inequality, we have(40)ϕr+ϕs2−1s−r∫rsϕxdx=s−r22∫01α1−αϕ″αr+1−αsdα≤s−r22∫01α1−α1−1/q∫01α1−αϕ″αr+1−αsqdα1/q=s−r22161−1/q∫01α1−αϕ″αr+1−αsqdα1/q.
Sinceϕ″q is generalized geometrically strongly modified h-convex function and monotonically decreasing on r,s, we obtain(41)rαs1−α≤αr+1−αs,ϕ″αr+1−αsq≤ϕ″rαs1−αq.
Therefore, we have(42)ϕr+ϕs2−1s−r∫rsϕxdx≤s−r22161−1/q∫01α1−αϕ″rαs1−αqdα1/q≤s−r22161−1/q∫01α1−α|1−hαϕ″s+hαϕ″s+ηϕ″r,ϕ″s−μr−s2α1−α|qdα1/q≤s−r22161−1/q16ϕ″sq+∫01α1−αhαηϕ″r,ϕ″sqdα−μr−s2∫01α1−αα1−αqdα1/q.
This completes the proof.Remark 5.
Takinghα=α, μ=0, and ηr,s=r−s in (39), we obtained Hermite–Hadamard inequality for the geometric arithmetic convex function.


## 4. Some Other Inequalities

Lemma 3.
Letϕ:J⊂ℝ+=0,∞⟶ℝ be a differentiable function on the interior J° of J, where r,s∈J with r<s and ϕ′∈Lr,s. Then, we have(43)sϕs−rϕr−∫rsϕxdx=lns−lnr2∫01s1+αr1−αϕ′s1+α/2r1−α/2dα+∫01s1+αr1−αϕ′s1−α/2r1+α/2dα.Theorem 5.
Letϕ:J⊂ℝ+=0,∞⟶ℝ be a differentiable function on the interior J° of J, where r,s∈J with r<s and ϕ′∈Lr,s. If ϕ′ is generalized geometrically strongly modified h-convex function for q≥1, then the following inequality holds:(44)sϕs−rϕr−∫rsϕxdx≤s−r1−1/q2sϕ′sq∫01srα1−h1−α2dα+ϕ′s+ηϕ′r,ϕ′sq∫01srαh1−α2dα1/q+rϕ′sq∫01rsα1−h1+α2dα+ϕ′s+ηϕ′r,ϕ′sq∫01rsαh1+α2dα1/q−μr−s24Lr,ss2+r2rs1/q.Proof.
Using Lemma3 and Holder’s inequality, we have(45)sϕs−rϕr−∫rsϕxdx=rslns−lnr2∫01srαϕ′s1+α/2r1−α/2dα+∫01rsαϕ′s1−α/2r1+α/2dα≤rslns−lnr2∫01srαdα1−1/q∫01srαϕ′s1+α/2r1−α/2dα1/q+∫01rsαdα1−1/q∫01rsαϕ′s1−α/2r1+α/2dα1/q.
Now, consider(46)I1=∫01srαϕ′s1+α/2r1−α/2qdα≤∫01srαϕ′sq1−h1−α2+ϕ′s+ηϕ′r,ϕ′sqh1−α2−μr−s21−α21+α2dα=ϕ′sq∫01srα1−h1−α2dα+ϕ′s+ηϕ′r,ϕ′sq∫01srαh1−α2dα−μr−s2∫01srα1−α21+α2dα=ϕ′sq∫01srα1−h1−α2dα+ϕ′s+ηϕ′r,ϕ′sq×∫01srαh1−α2dα−μr−s2∫01srα1−α21+α2dα.
Similarly,I2 becomes(47)∫01rsαϕ′s1−α/2r1+α/2qdα≤ϕ′sq∫01rsα1−h1+α2dα+ϕ′s+ηϕ′r,ϕ′sq×∫01rsαh1+α2dα−μr−s2∫01rsα1+α21−α2dα.
Also,(48)∫01srαdα1−1/q=s−rrlns−lnr1−1/q.
Combining (45)–(48), we have(49)sϕs−rϕr−∫rsϕxdx≤s−r1−1/q2sϕ′sq∫01srα1−h1−α2dα+ϕ′s+ηϕ′r,ϕ′sq∫01srαh1−α2dα−sμr−s24s−rrlns−lnr1/q+rϕ′sq∫01rsα1−h1+α2dα+ϕ′s+ηϕ′r,ϕ′sq∫01rsαh1+α2dα−rμr−s24r−sslnr−lns1/q≤s−r1−1/q2sϕ′sq∫01srα1−h1−α2dα+ϕ′s+ηϕ′r,ϕ′sq∫01srαh1−α2dt1/q+rϕ′sq∫01rsα1−h1+α2dα+ϕ′s+ηϕ′r,ϕ′sq∫01rsαh1+α2dα1/q−μr−s24Lr,ss2+r2rs1/q.
This completes the proof.
If we takeηr,s=r−s, then (45) reduces to the following result.Corollary 4.
Letϕ:J⊂ℝ+=0,∞⟶ℝ be a differentiable function on the interior J° of J, where r,s∈J with r<s and ϕ′∈Lr,s. If ϕ′ is a generalized geometrically strongly modified h-convex function for q≥1, then the following inequality holds:(50)sϕs−rϕr−∫rsϕxdx≤s−r1−1/q2sϕ′sq∫01srα1−h1−α2dα+ϕ′rq∫01srαh1−α2dα1/q+rϕ′sq∫01rsα1−h1+α2dα+ϕ′rq∫01rsαh1+α2dα1/q−μr−s24Lr,ss2+r2rs1/q.Theorem 6.
Letϕ:J⊂ℝ+=0,∞⟶ℝ be a differentiable function on the interior J° of J, where r,s∈J with r<s and ϕ′∈Lr,s. If ϕ′ is generalized geometrically strongly modified h-convex function for q>1, then the following inequality holds:(51)sϕs−rϕr−∫rsϕxdx≤lns−lnr2Lrq/q−1,sq/q−11−1/qsϕ′sq∫011−h1−α2dα+ϕ′s+ηϕ′r,ϕ′sq∫01h1−α2dα−μr−s261/q+rϕ′sq∫01−h1+α2dα+ϕ′s+ηϕ′r,ϕ′sq×∫01h1+α2dα−μr−s261/q.Proof.
Using Lemma3 and Holder’s inequality, we have(52)sϕs−rϕr−∫rsϕxdx=rslns−lnr2∫01srαϕ′s1+α/2r1−α/2dα+∫01rsαϕ′s1−α/2r1+α/2dα≤rslns−lnr2∫01srqα/q−1dα1−1/q∫01ϕ′s1+α/2r1−α/2qdα1/q+∫01rsqα/q−1dα1−1/q∫01ϕ′s1−α/2r1+α/2qdα1/q.
Now, consider(53)I1=∫01ϕ′s1+α/2r1−α/2qdα≤∫01ϕ′sq1−h1−α2+ϕ′s+ηϕ′r,ϕ′sq×h1−α2−μr−s21−α21+α2dα=ϕ′sq∫011−h1−α2dα+|ϕ′s+ηϕ′r,ϕ′s|q∫01h1−α2dα−μr−s26.
Similarly,I2 becomes(54)I2=∫01ϕ′s1−α/2r1+α/2qdα=ϕ′sq∫011−h1+α2dα+|ϕ′s+ηϕ′r,ϕ′s|q∫01h1+α2dα−μr−s26.
Also,(55)∫01srqα/q−1dα1−1/q=sq/q−1−rq/q−1q/q−1rq/q−1lns−lnr1−1/q.
Combining (52)–(55), we have(56)sϕs−rϕr−∫rsϕxdx≤lns−lnr2Lrq/q−1,sq/q−11−1/qsϕ′sq∫011−h1−α2dα+ϕ′s+ηϕ′r,ϕ′sq∫01h1−α2dα−μr−s261/q+rϕ′sq∫01−h1+α2dα+|ϕ′s+ηϕ′r,ϕ′s|q∫01h1+α2dα−μr−s261/q.
This completes the proof.Remark 6.
Takinghα=α and μ=0 in (45) and (47), we obtained fractional integral inequalities for generalized geometrically convex function, see [24].Ifηr,s=r−s, then (47) reduces to the following result:Corollary 8.
Letϕ:J⊂ℝ+=0,∞⟶ℝ be a differentiable function on the interior J° of J, where r,s∈I with r<s and ϕ′∈Lr,s. If ϕ′ is generalized geometrically strongly modified h-convex function for q>1, then the following inequality holds:(57)sϕs−rϕr−∫rsϕxdx≤lns−lnr2Lrq/q−1,sq/q−11−1/qsϕ′sq∫011−h1−α2dα+ϕ′rq∫01h1−α2dα−μr−s261/q+rϕ′sq×∫01−h1+α2dα+ϕ′sq∫01h1+α2dα−μr−s261/q.


## 5. Conclusions

The theory of the convex function is applicable in almost every field of mathematics specially in approximation theory. The inequality theory got application in diverse areas of pure and applied mathematics. In this paper, we introduced a broader class of convex functions by unifying geometrically strong convex function withηh convex function and developed the Hermite–Hadamard-type and fractional integral inequalities for this class of function.



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