# Fractional Version of Hermite-Hadamard and Fejér Type Inequalities for a Generalized Class of Convex Functions

Authors: Lei Geng; Muhammad Shoaib Saleem; Kiran Naseem Aslam; Rahat Bano
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In the present paper, we deal with some fractional integral inequalities for strongly reciprocallyp,h-convex functions. We established fractional version of Hermite-Hadamard and Fejér type inequalities for strongly reciprocally p,h-convex functions. Our results extend and generalize many exiting results of literate.


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## Body

## 1. Introduction

The convex functions nowadays are widely used in many branches of mathematics like optimization theory, functional analysis, and modeling theory [1, 2]. The interesting geometry of convex functions makes its study distinct from other functions. From geometric point of view, a function ψx is convex provided that the line segment connecting any two points of its graph lies on or above the graph of function. The new inequalities in analysis are always appreciable [3, 4].Since the classical convexity is not enough to attain certain goals in applied mathematics, so the classical convexity has been generalized in many directions. For recent generalizations, one can see [5, 6].For the class of convex functions, various inequalities have been developed [7, 8], but the most famous inequality is Hermite-Hadamard’s inequality. It is stated as follows:Letψ:M⊆ℝ⟶ℝ be a convex function, and let d1,d2∈M with d1d2; then, the following double inequality holds:
(1)ψd1+d22≤1d2−d1∫d1d2ψxdx≤ψd1+ψd22.In [9], Fejér gave the weighted version of Hermite-Hadamard inequality (1) as follows:Letψ:d1,d2⊆ℝ⟶ℝ be a convex function and w:d1,d2⟶ℝ a non-negative, integrable, and symmetric function about d1+d2/2; then, the following inequality holds:
(2)ψd1+d22∫d1d2wxdx≤∫d1d2ψxwxdx≤ψd1+ψd22∫d1d2wxdx.The aim of present paper is to establish fractional version of Hermite-Hadamard and Fejér type inequalities for a more generalized class of functions. The present paper is organized as follows: The §2 is concerned with some preliminary material. In Section3, we give some basic results for strongly reciprocally (p,h)-convex functions, and in Sections 4 and 5, we develop Hermite-Hadamard and Fejér type inequalities, respectively, for strongly reciprocally (p,h)-convex function. Moreover, the last section is devoted for fractional integral inequalities.


## 2. Preliminaries

In this section, we give a brief review of some preliminaries.Definition 1 (p-convex set see [10, 11]).
A setM=d1,d2⊆ℝ\0 is p-convex set, if
(3)jxp+1−jyp1/p∈M,for all x,y∈M, j∈0,1, where p=2u+1 or p=a/b,a=2v+1,b=2w+1 and u,v,w∈N.Definition 2 (p-convex function see [10]).
LetM=d1,d2⊆ℝ\0 be a p-convex set. A function ψ:M=d1,d2⟶ℝ is called p-convex function, if
(4)ψjxp+1−jyp1/p≤jψx+1−jψy∈M,
holds for allx,y∈M and j∈0,1.Definition 3 (Strongly convex function see [4]).
A functionψ:M=d1,d2⟶ℝ is called strongly convex function with modulus μ on M, where μ≥0, if
(5)ψjx+1−jy≤jψx+1−jψy−μj1−jy−x2,holds for all x,y∈M and j∈0,1.Definition 4 (Stronglyp-convex function see [12]).
A functionψ:M=d1,d2⟶ℝ is called strongly p-convex function, if
(6)ψjxp+1−jyp1/p≤jψx+1−jψy−μj1−jyp−xp2,
holds for allx,y∈M and j∈0,1.Definition 5 (Harmonic convex set see [11, 13]).
A setM=d1,d2⊆ℝ\0 is said to be harmonic convex set, if
(7)xyjx+1−jy∈M,
for allx,y∈M and j∈0,1.Definition 6 (Harmonic convex function see [11, 14]).
LetM=d1,d2⊆ℝ\0 be the harmonic convex set. A function ψ:M=d1,d2⊆ℝ\0⟶ℝ is harmonic convex function, if
(8)ψxyjx+1−jy1−jψx+jψy,
holds for allx,y∈M and j∈0,1.Definition 7 (p-harmonic convex set see [11, 15]).
A setM=d1,d2⊆ℝ\0 is p-harmonic convex set, if
(9)xpypjxp+1−jyp1/p∈M,
for allx,y∈M and j∈0,1.Definition 8 (p-harmonic convex function see [11, 15]).
LetM=d1,d2⊆ℝ\0 be p-harmonic convex set. A function ψ:M=d1,d2⊆ℝ\0⟶ℝ is p-harmonic convex, if
(10)ψxpypjxp+1−jyp1/p≤1−jψx+jψy,
holds for allx,y∈M and j∈0,1.Definition 9 (Strongly reciprocally convex function see [16]).
LetM be an interval and let μ∈0,∞. A function ψ:M=d1,d2⊆ℝ\0⟶ℝ is said to be strongly reciprocally convex function with modulus μ on M, if
(11)ψxyjx+1−jy≤1−jψx+jψy−μj1−j1x−1y2,holds for all x,y∈M and j∈0,1.Definition 10 (Strongly reciprocallyp-convex function see [17]).
LetM be a p-harmonic convex set and let μ∈0,∞. A function ψ:M=d1,d2⊆ℝ\0⟶ℝ is said to be strongly reciprocally p-convex function with modulus μ on M, if
(12)ψxpypjxp+1−jyp1/p≤1−jψx+jψy−μj1−j1xp−1yp2,holds for all x,y∈M and j∈0,1.Definition 11 (h-convex function see [18]).
Choose the functionsψ,h:M=d1,d2⟶ℝ that are non-negative; then, ψ is called h-convex function, if
(13)ψjx+1−jy≤hjψx+h1−jψy,for all x,y∈M and j∈0,1.Now, we are ready to introduce a new class of convex functions by generalizing the concept of strongly reciprocallyp-convex functions, which we will call strongly reciprocally (p,h)-convex functions.Definition 12 (Strongly reciprocallyp,h-convex function).
LetM be a p-harmonic convex set and let μ∈0,∞. A function ψ:M=d1,d2⊆ℝ\0⟶ℝ is said to be strongly reciprocally p,h-convex function with modulus μ on M, if
(14)ψxpypjxp+1−jyp1/p≤h1−jψx+hjψy−μj1−j1xp−1yp2,hold for all x,y∈M and j∈0,1.
Throughout the paper, for convenience, we represent the class of strongly reciprocally (p, h) convex functions by SRp,h.Remark 13.
Insertinghj=j in Definition 12, we obtain Definition 10, and inserting hj=j and p=1, Definition 12 reduces to Definition 9.
Similarly, from Definition12, Definitions 6 and 8 can be obtained by inserting hj=j, μ=0, p=1, and hj=j with μ=0, respectively.


## 3. Basic Results

This section collects some basic and straight forward facts based on algebraic operations.The following proposition is concerned about the addition of two functions fromSRp,h.Proposition 14.
Letψ,σ:M⟶ℝ be two strongly reciprocally (p,h)-convex functions with modulus μ on M; then, ψ+σ:M⟶ℝ is also strongly reciprocally (p,h)-convex function with modulus μ∗ on M, where μ∗=2μ.Proof.
We will start by definition of strongly reciprocally (p,h)-convexity of ψ and σ:
(15)ψ+σxpypjxp+1−jyp1/p=ψxpypjxp+1−jyp1/p+σxpypjxp+1−jyp1/p≤hjψx+h1−jψy−μj1−j1yp−1xp2+hjσx+h1−jσy−μj1−j1yp−1xp2,
which in turns implies that(16)ψ+σxpypjxp+1−jyp1/p≤hjψ+σx+h1−jψ+σy−2μj1−j1yp−1xp2=hjψ+σx+h1−jψ+σy−μ∗j1−j1yp−1xp2,
whereμ∗=2μ and μ≥0. This completes the proof.Our next result is concerned with the scalar multiplication of strongly reciprocally (p,h)-convex function.Proposition 15.
Letψ:M⟶ℝ be a strongly reciprocally (p,h)-convex function; then, for any λ≥0, λψ:M⟶ℝ is also strongly reciprocally (p,h)-convex function with modulus ν∗ on M, where ν∗=λμ.Proof.
Letλ≥0, ψ∈SRp,h, we obtain
(17)λψxpypjxp+1−jyp1/p+λψxpypjxp+1−jyp1/p≤λhjψx+h1−jψy−μj1−j1yp−1xp2=hjλψx+h1−jλψy−λμj1−j1yp−1xp2=hjλψx+h1−jλψy−ν∗j1−j1yp−1xp2,
whereν∗=λμ and μ≥0. This completes the proof.Proposition 16.
Letψi:M⟶ℝ, where 1≤i≤n be in SRp,h with modulus μ; then, for λi≥0 where 1≤i≤n, the function ψ:M⟶ℝ, where ψ=∑i=1nλiψi is also in SRp,h with modulus γ≥0, where γ=∑i=1nλiμ.Proof.
LetM is a p-harmonic convex set. Then, ∀x,y∈M and j∈0,1, we have
(18)ψxpypjxp+1−jyp1/p=∑i=1nλiψixpypjxp+1−jyp1/p≤∑i=1nλihjψix+h1−jψiy−μj1−j1yp−1xp2=hj∑i=1nλiψix+h1−j∑i=1nλiψiy−∑i=1nλiμj1−j1yp−1xp2=hjfx+h1−jψy−γj1−j1yp−1xp2,
whereγ=∑i=1nλiμ. This completes the proof.Proposition 17.
Letψi:M⟶ℝ, where 1≤i≤n be strongly reciprocally (p,h)-convex functions with modulus μ; then ψ=maxψi,i=1,2,⋯,n, is also strongly reciprocally (p,h)-convex functions with modulus μ.Proof.
LetM is p-harmonic convex set. Then ∀x,y∈M and j∈0,1, we have
(19)ψxpypjxp+1−jyp1/p=maxψixpypjxp+1−jyp1/p,i=1,2,3,⋯,n=ψcxpypjxp+1−jyp1/p≤hjψcx+h1−jψcy−μj1−j1yp−1xp2=hjmaxψix+h1−jmaxψiy−μj1−j1yp−1xp2=hjψx+h1−jψy−μj1−j1yp−1xp2.
This completes the proof.Our next intension is to develop Hermite-Hadamard’s inequality for this generalization.


## 4. Hermite-Hadamard Type Inequality

Theorem 18.
LetM⊂ℝ\0 be an interval. If ψ:M⟶ℝ be a strongly reciprocally (p,h)-convex function with modulus μ≥0 and ψ∈Ld1,d2, then for h1/2≠0, we have
(20)12h1/2ψ2d1pd2pd1p+d2p1/p+μ12d2p−d1pd1pd2p2≤pd1pd2pd2p−d1p∫d1d2ψxx1+pdx≤∫01h1−jψd1+hjψd2dj−μ6d2p−d1pd1pd2p2.Proof.
Since,ψ∈SRp,h, and allowing j=1/2 yields
(21)ψ2xpypxp+yp1/p≤h12ψx+h12ψy−μ12121xp−1yp2.
Letx=d1pd2p/jd1p+1−jd2p1/p and y=d1pd2p/jd2p+1−jd1p1/p and integrating (21) w.r.t j over 0,1, we obtain
(22)ψ2d1pd2pd1p+d2p1/p≤h12ψd1pd2pjd1p+1−jd2p1/p+h12ψd1pd2pjd2p+1−jd1p1/p−μ4d2p−d1pd1pd2p21−2j2∫01ψ2d1pd2pd1p+d2p1/pdj≤∫01h12ψd1pd2pjd1p+1−jd2p1/pdj+∫01h12ψd1pd2pjd2p+1−jd1p1/pdj−μ4d2p−d1pd1pd2p2∫011−2j2djψ2d1pd2pd1p+d2p1/p≤2h12pd1pd2pd2p−d1p∫d1d2ψxx1+pdx−μ12d2p−d1pd1pd2p2,
and(23)ψ2d1pd2pd1p+d2p1/p+μ12d2p−d1pd1pd2p2≤2h12pd1pd2pd2p−d1p∫d1d2ψxx1+pdx12h1/2ψ2d1pd2pd1p+d2p1/p+μ12d2p−d1pd1pd2p2≤pd1pd1pd2p−d1p∫d1d2ψxx1+pdx,
which gives one side of (20).
For the right side of (20), since ψ∈SRp,h, so by setting x=d1 and y=d2, we obtain the following result:
(24)ψd1pd2pjd1p+1−jd2p1/p≤h1−jψd1+hjψd2−μj1−j1d1p−1d2p2.
Integrating (24) w.r.t j over 0,1, we obtain
(25)∫01ψd1pd2pjd1p+1−jd2p1/pdj≤∫01h1−jψd1dj+∫01hjψd2dj−μd2p−d1pd1pd2p2∫01j1−jdj,
and(26)pd1pd2pd2p−d1p∫d1d2ψxx1+pdx≤∫01h1−jψd1+hjψd2dj−μ6d2p−d1pd1pd2p2,
which is right side of (20), which completes the proof.Remark 19.
(1)
Insertinghj=j and p=1 in Theorem 18, we obtained Hermite-Hadamard inequality for strongly reciprocally convex function [16] (Theorem 3.1)(2)
Insertionhj=j, p=1, and μ=0 in Theorem 18 yields Hermite-Hadamard inequality for harmonic convex functions [14] (Theorem 2.4)
Now, we develop Fejér type inequality for this new class of convex functions.


## 5. Fejér Type Inequality

Theorem 20.
LetM⊂ℝ\0 be an interval. If ψ:M⟶ℝ be a strongly reciprocally (p,h)-convex function with modulus μ≥0, then for h1/2≠0, we have
(27)12h1/2ψ2d1pd2pd1p+d2p1/p∫d1d2wxx1+pdx+μ2d1pd2p2∫d1d22d1pd2p−d1p+d2pxp2wxx1+3pdx≤∫d1d2ψxwxx1+pdx≤ψd1+ψd2∫d1d2hd1pd2p−xpxpd2p−d1pwxx1+pdx−μd1pd2p∫d1d2xp−d1pd2p−xpwxx1+3pdx,
holds ford1,d2∈M with d1≤d2 and ψ∈Ld1,d2, where w:M⟶ℝ is a non-negative integrable function that satisfies
(28)wd1pd2pxp1/p=wd1pd2pd1p+d2p−xp1/p.Proof.
Sinceψ∈SRp,h, and allowing j=1/2 yields
(29)ψ2xpypxp+yp1/p≤h12ψx+h12ψy−μ12121xp−1yp2.
Insertingx=d1pd2p/jd1p+1−jd2p1/p and y=d1pd2p/jd2p+1−jd1p1/p in (29), we obtain
(30)ψ2d1pd2pd1p+d2p1/p≤h12ψd1pd2pjd1p+1−jd2p1/p+h12ψd1pd2pjd2p+1−jd1p1/p−μ4jd1p+1−jd2pd1pd2p−jd2p+1−jd1pd1pd2p2.
Since,w is a non-negative symmetric and integrable function, so
(31)ψ2d1pd2pd1p+d2p1/pwd1pd2pjd1p+1−jd2p1/p≤h12ψd1pd2pjd1p+1−jd2p1/pwd1pd2pjd1p+1−jd2p1/p+h12ψd1pd2pjd2p+1−jd1p1/pwd1pd2pjd1p+1−jd2p1/p−μ4jd1p+1−jd2pd1pd2p−jd2p+1−jd1pd1pd2p2wd1pd2pjd1p+1−jd2p1/p.
Integrating inequality (31) w.r.t j over 0,1, we obtain
(32)∫01ψ2d1pd2pd1p+d2p1/pwd1pd2pjd1p+1−jd2p1/pdj≤∫01h12ψd1pd2pjd1p+1−jd2p1/pwd1pd2pjd1p+1−jd2p1/pdj+∫01h12ψd1pd2pjd2p+1−jd1p1/pwd1pd2pjd1p+1−jd2p1/pdj−μ4∫01jd1p+1−jd2pd1pd2p−jd2p+1−jd1pd1pd2p2wd1pd2pjd1p+1−jd2p1/pdj,
and(33)ψ2d1pd2pd1p+d2p1/p∫a1b1wxx1+pdx+μ2d1pd2p2∫d1d22d1pd2p−d1p+d2pxp2wxx1+3pdx≤2h12∫a1b1ψxwxx1+pdx,
so,(34)12h1/2ψ2d1pd2pd1p+d2p1/p∫d1d2wxx1+pdx+μ2d1pd2p2∫d1d22d1pd2p−d1p+d2pxp2wxx1+3pdx≤∫d1d2ψxwxx1+pdx,
which is left side of (27).Finally, for the right side of (27), since ψ∈SRp,h and by setting x=d1 and y=d2, we obtain the following result:
(35)ψd1pd2pjd1p+1−jd2p1/p≤h1−jψd1+hjψd2−μj1−j1d1p−1d2p2.Sincew is a non-negative symmetric and integrable function, so
(36)ψd1pd2pjd1p+1−jd2p1/pwd1pd2pjd1p+1−jd2p1/p≤h1−jψd1wd1pd2pjd1p+1−jd2p1/p+hjψd2wd1pd2pjd1p+1−jd2p1/p−μj1−j1d1p−1d2p2wd1pd2pjd1p+1−jd2p1/p.Integrating inequality (36) w.r.t j over 0,1, we obtain
(37)∫01ψd1pd2pjd1p+1−jd2p1/pwd1pd2pjd1p+1−jd2p1/pdj≤∫01h1−jψd1wd1pd2pjd1p+1−jd2p1/pdj+∫01hjψd2wd1pd2pjd1p+1−jd2p1/pdj−μ∫01j1−j1d1p−1d2p2wd1pd2pjd1p+1−jd2p1/pdj,and(38)∫d1d2ψxwxx1+pdx≤ψd1+ψd2∫a1b1hd1pd2p−xpxpd2p−d1pwxx1+pdx−μd1pd2p∫d1d2xp−d1pd2p−xpwxx1+3pdx,which is the right side of inequality (27); this completes the proof.Remark 21.
(1)
Insertinghj=j and p=1 in Theorem 20, we obtained Fejér type inequality for strongly reciprocally convex function [16] (Theorem 3.7)(2)
Insertionhj=j, in Theorem 20, yields Fejér type inequality for strongly reciprocally p-convex functions [17] (Theorem 3.5)


## 6. Fractional Integral Inequalities

Fractional integral inequalities are important to study means [19–22]. This section is devoted for some fractional integral inequalities for functions whose derivatives are in SRp,h. A source for results of the desired type is the following lemma.Lemma 22 see ([23], Lemma 2.1).
Letψ:M=d1,d2⊆ℝ is differentiable defined on the interior M of M. If ψ′∈Ld1,d2 and λ∈0,1, then
(39)1−λψ2d1pd2pd1p+d2p1/p+λψd1+ψd22−pd1pd2pd2p−d1p∫d1d2ψxx1+pdx=d2p−d1p2pd1pd2p∫01/22j−λd1pd2pjd1p+1−jd2p1+1/pψ′a1pb1pja1p+1−jb1p1/pdj+∫1/212j−2+λd1pd2pjd1p+1−jd2p1+1/pψ′d1pd2pjd1p+1−jd2p1/pdj.
As a first application of Lemma22, we prove the following result.Theorem 23.
LetM=d1,d2⊂ℝ\0 be a p-harmonic convex set, and let ψ:M=d1,d2⊆ℝ\0⟶ℝ be differentiable defined on the interior M of M. If M′∈Ld1,d2 and ψ′q is strongly reciprocally (p,h)-convex function on M, q≥1, and λ∈0,1, then
(40)1−λψ2d1pd2pd1p+d2p1/p+λψd1+ψd22−pd1pd2pd2p−d1p∫d1d2ψxx1+pdx≤d2p−d1p2pd1pd2pk1p,d1,d21−1/qk15p,d1,d2ψ′d1q+k17p,d1,d2ψ′d2q+k7p,d1,d2μ1/q+k2p,d2,d11−1/qk18p,d2,d1ψ′d1q+k16p,d2,d1ψ′d2q+k8p,d2,d1μ1/q,
where(41)k1p,d1,d2=∫01/22j−λd1pd2pjd1p+1−jd2p1+1/pdj,(42)k2p,d2,d1=∫1/212j−2+λd1pd2pjd1p+1−jd2p1+1/pdj,(43)k15p,d1,d2=∫01/2h1−j2j−λd1pd2pjd1p+1−jd2p1+1/pdj,(44)k16p,d2,d1=∫1/21hj2j−2+λd1pd2pjd1p+1−jd2p1+1/pdj,(45)k17p,d1,d2=∫01/2hj2j−λd1pd2pjd1p+1−jd2p1+1/pdj,(46)k18p,d2,d1=∫1/21h1−j2j−2+λd1pd2pjd1p+1−jd2p1+1/pdj,(47)k7p,d1,d2=∫01/2jj−12j−λd1pd2pjd1p+1−jd2p1+1/p1d2p−1d1p2dj,(48)k8p,d2,d1=∫1/21jj−12j−2+λd1pd2pjd1p+1−jd2p1+1/p1d2p−1d1p2dj.Proof.
Using Lemma22 and power mean inequality, we have
(49)1−λψ2d1pd2pd1p+d2p1/p+λψd1+ψd22−pd1pd2pd2p−d1p∫d1d2ψxx1+pdx≤d2p−d1p2pd1pd2p∫01/22j−λd1pd2pjd1p+1−jd2p1+1/p×ψ′d1pd2pjd1p+1−jd2p1/pdj+∫1/212j−2+λd1pd2pjd1p+1−jd2p1+1/p×ψ′d1pd2pjd1p+1−jd2p1/pdj.≤d2p−d1p2pd1pd2p∫01/22j−λd1pd2pjd1p+1−jd2p1+1/pdj1−1/q×∫01/22j−λd1pd2pjd1p+1−jd2p1+1/p×ψ′d1pd2pjd1p+1−jd2p1/pqdj1/q+∫1/212j−2+λd1pd2pjd1p+1−jd2p1+1/pdt1−1/q×∫1/212j−2+λd1pd2pjd1p+1−jd2p1+1/p×ψ′d1pd2pjd1p+1−jd2p1/pqdj1/q.
Sinceψ′xq∈SRp,h, so
(50)1−λψ2d1pd2pd1p+d2p1/p+λψd1+ψd22−pd1pd2pd2p−d1p∫d1d2ψxx1+pdx≤d2p−d1p2pd1pd2p∫01/22j−λd1pd2pjd1p+1−jd2p1+1/pdj1−1/q×∫01/22j−λd1pd2pjd1p+1−jd2p1+1/p×h1−jψ′d1q+hjψ′d2q−μj1−j1d2p−1d1p2dj1/q+∫1/212j−2+λd1pd2pjd1p+1−jd2p1+1/pdj1−1/q×∫1/212j−2+λd1pd2pjd1p+1−jd2p1+1/p×h1−jψ′d1q+hjψ′d2q−μj1−j1d2p−1d1p2dj1/q=d2p−d1p2pd1pd2pk1p,d1,d21−1/qk15p,d1,d2ψ′d1q+k17p,d1,d2ψ′d2q+k7p,d1,d2μ1/q+k2p,d2,d11−1/qk18p,d2,d1ψ′d1q+k16p,d2,d1ψ′d2q+k8p,d2,d1μ1/q.
This completes the proof.Remark 24.
Insertingμ=0 and hj=j in Theorem 23, we obtained [23] Theorem 2.2.
Forq=1, Theorem 23 reduces to the following result.Corollary 25.
LetM=d1,d2⊂ℝ\0 be a p-harmonic convex set, and let ψ:M=d1,d2⊆ℝ\0⟶ℝ be differentiable defined on the interior MofM. Ifψ′∈Ld1,d2 and ψ′q is in SRp,h on M and λ∈0,1, then
(51)1−λψ2d1pd2pd1p+d2p1/p+λψd1+ψd22−pd1pd2pd2p−d1p∫d1d2ψxx1+pdx≤d2p−d1p2pd1pd2pk15p,d1,d2+k18p,d2,d1ψ′d1+k16p,d2,d1+k17p,d1,d2ψ′d2+k7p,d1,d2+k8p,d2,d1μ,
wherek15,k16,k17,k18,k7, and k8 are given by (43) to (48).Remark 26.
Insertinghj=j and μ=0 in Corollary 25, we obtained [23] Corollary 2.3.
Using Lemma22, we can prove the following result.Theorem 27.
LetM=d1,d2⊂ℝ\0 be a p-harmonic convex set, and let ψ:M=d1,d2⊆ℝ\0⟶ℝ be differentiable defined on the interior M of M. If M′∈Ld1,d2 and ψ′q is in SRp,h on M, r,q>1, 1/r+1/q=1 and λ∈0,1, then
(52)1−λψ2d1pd2pd1p+d2p1/p+λψd1+ψd22−pd1pd2pd2p−d1p∫d1d2ψxx1+pdx≤d2p−d1p2pd1pd2p×λr+1+1−λr+12r+11/rk19q,p;d1,d2ψ′d1q+k21q,p;d1,d2ψ′d2q+k13q,p;d1,d2μ1/q+k22q,p;d2,d1ψ′d1q+k20q,p;d2,d1ψ′d2q+k14q,p;d2,d1μ1/q,
where(53)k19q,p;d1,d2=∫01/2h1−jd1pd2pjd1p+1−jd2pq+q/pdj,(54)k20q,p;d2,d1=∫1/21hjd1pd2pjd1p+1−jd2pq+q/pdj,(55)k21q,p;d1,d2=∫01/2hjd1pd2pjd1p+1−jd2pq+q/pdj,(56)k22q,p;d2,d1=∫1/21h1−jd1pd2pjd1p+1−jd2pq+q/pdj,(57)k13q,p;d1,d2=∫01/2jj−1d1pd2pjd1p+1−jd2pq+q/p1b1p−1a1p2dj,(58)k14q,p;d2,d1=∫1/21jj−12j−2+λd1pd2pjd1p+1−jd2pq+q/p1d2p−1d1p2dj.Proof.
Using Lemma22, we have
(59)1−λψ2d1pd2pd1p+d2p1/p+λψd1+ψd22−pd1pd2pd2p−d1p∫d1d2ψxx1+pdx≤d2p−d1p2pd1pd2p∫01/22j−λd1pd2pjd1p+1−jd2p1+1/p×ψ′d1pd2pjd1p+1−jd2p1/pdj+∫1/212j−2+λd1pd2pjd1p+1−jd2p1+1/p×ψ′d1pd2pjd1p+1−jd2p1/pdj.
Applying Holder’s integral inequality,(60)1−λψ2d1pd2pd1p+d2p1/p+λψd1+ψd22−pd1pd2pd2p−d1p∫d1d2ψxx1+pdx≤d2p−d1p2pd1pd2p∫01/22j−λrdj1/r∫01/2d1pd2pjd1p+1−jd2p1+1/p×ψ′d1pd2pjd1p+1−jd2p1/pqdj1/q+∫1/212j−2+λrdj1/r∫1/21d1pd2pjd1p+1−jd2p1+1/p×ψ′d1pd2pjd1p+1−jd2p1/pqdj1/q=d2p−d1p2pd1pd2p∫01/22j−λrdj1/r∫01/2d1pd2pjd1p+1−jd2pq+q/p×ψ′d1pd2pjd1p+1−jd2p1/pqdj1/q+∫1/212j−2+λrdj1/r∫1/21d1pd2pjd1p+1−jd2pq+q/p×ψ′d1pd2pjd1p+1−jd2p1/pqdj1/q.
Sinceψ′xq∈SRp,h, so
(61)1−λψ2d1pd2pd1p+d2p1/p+λψd1+ψd22−pd1pd2pd2p−d1p∫d1d2ψxx1+pdx≤d2p−d1p2pd1pd2p∫01/22j−λrdj1/r×∫01/2d1pd2pjd1p+1−jd2pq+q/p×h1−jψ′d1q+hjψ′d2q−μj1−j1d2p−1d1p2dj1/q+∫1/212j−2+λrdj1/r∫1/21d1pd2pjd1p+1−jd2pq+q/p×h1−jψ′d1q+hjψ′d2q−μj1−j1d2p−1d1p2dj1/q≤d2p−d1p2pd1pd2p∫01/22j−λrdj1/r×k19q,p;d1,d2ψ′d1q+k21q,p;d1,d2ψ′d2q×+k13q,p;d1,d2μ1/q+∫01/22j−2+λrdj1/r×k22q,p;d2,d1ψ′d1q+k20q,p;d2,d1ψ′d2q+k14q,p;d2,d1μ1/q≤d2p−d1p2pd1pd2p×λr+1+1−λr+12r+11/r×k19q,p;d1,d2ψ′d1q+k21q,p;d1,d2ψ′d2q+k13q,p;d1,d2μ1/q+k22q,p;d2,d1ψ′d1q+k20q,p;d2,d1ψ′d2q+k14q,p;d2,d1μ1/q.
Which is the required result.Remark 28.
Insertinghj=j and μ=0 in Theorem 27, we obtained [23] Theorem 2.5.
Forλ=0, Theorem 27 reduces to the following result.Corollary 29.
LetM=d1,d2⊂ℝ\0 be a p-harmonic convex set, and let ψ:M=d1,d2⊆ℝ\0⟶ℝ be differentiable defined on the interior M of M. If ψ′∈Ld1,d2 and ψ′q is in SRp,h on M, r,q>1, 1/r+1/q=1 and λ∈0,1, then
(62)ψ2d1pd2pd1p+d2p1/p−pd1pd2pd2p−d1p∫d1d2ψxx1+pdx≤d2p−d1p2pd1pd2p×12r+11/rk19q,p;d1,d2ψ′d1q+k21q,p;d1,d2ψ′d2q+k13q,p;d1,d2μ1/q+k22q,p;d2,d1ψ′d1q+k20q,p;d2,d1ψ′d2q+k14q,p;d2,d1μ1/q,
wherek19,k20,k21,k22,k13, and k14 are given by (53)-(58).Remark 30.
Insertinghj=j and μ=0 in Corollary 29, we obtained [23] Corollary 3.5.
Forλ=1, Theorem 27 reduces to the following result.Corollary 31.
LetM=d1,d2⊂ℝ\0 be a p-harmonic convex set, and let ψ:M=d1,d2⊆ℝ\0⟶ℝ be differentiable defined on the interior M of M. If M′∈La1,b1 and f′q is in SRp,h on M, r,q>1, 1/r+1/q=1 and λ∈0,1 then,
(63)ψd1+ψd22−pd1pd2pd2p−d1p∫d1d2ψxx1+pdx≤d2p−d1p2pd1pd2p×12r+11/rk19q,p;d1,d2ψ′d1q+k21q,p;d1,d2ψ′d2q+k13q,p;d1,d2μ1/q+k22q,p;d2,d1ψ′d1q+k20q,p;d2,d1ψ′d2q+k14q,p;d2,d1μ1/q,
wherek19,k20,k21,k22,k13, and k14 are given by (53)-(58).Remark 32.
Insertinghj=j and μ=0 in Corollary 31, we obtained [23] Corollary 3.6.
Forλ=1/3, Theorem 27 reduces to the following result.Corollary 33.
LetM=d1,d2⊂ℝ\0 be a p-harmonic convex set, and let ψ:M=d1,d2⊆ℝ\0⟶ℝ be differentiable defined on the interior M of M. If ψ′∈Ld1,d2 and ψ′q is in SRp,h on M, r,q>1, 1/r+1/q=1 and λ∈0,1 then,
(64)16ψd1+4ψ2d1pd2pd1p+d2p1/p+ψd2−pd1pd2pd2p−d1p∫d1d2ψxx1+pdx≤d2p−d1p2pd1pd2p×1+2r+16.3rr+11/rk19q,p;d1,d2ψ′d1q+k21q,p;d1,d2ψ′d1q+k13q,p;d1,d2μ1/q+k22q,p;d2,d1ψ′d1q+k20q,p;d2,d1ψ′d2q+k14q,p;d2,d1μ1/q,
wherek19,k20,k21,k22,k13, and k14 are given by (53)-(58).Remark 34.
Insertinghj=j and μ=0 in Corollary 33, we obtained [23] Corollary 3.7.
Forλ=1/2, Theorem 27) reduces to the following result.Corollary 35.
LetM=d1,d2⊂ℝ\0 be a p-harmonic convex set, and let ψ:M=d1,d2⊆ℝ\0⟶ℝ be differentiable defined on the interior M of M. If ψ′∈Ld1,d2 and ψ′q is in SRp,h on M, r,q>1, 1/r+1/q=1 and λ∈0,1 then,
(65)14ψd1+2ψ2d1pd2pd1p+d2p1/p+ψd2−pd1pd2pd2p−d1p∫d1d2ψxx1+pdx≤d2p−d1p2pd1pd2p×24.2rr+11/rk19q,p;d1,d2ψ′d1q+k21q,p;d1,d2ψ′d2q+k13q,p;d1,d2μ1/q+k22q,p;d2,d1ψ′d1q+k20q,p;d2,d1ψ′d2q+k14q,p;d2,d1μ1/q
wherek19,k20,k21,k22,k13, and k14 are given by (53)–(58).Remark 36.
Insertinghj=j and μ=0 in Corollary 35, we obtained [23] Corollary 3.8.


## 7. Conclusion

Convexity plays very important role in pure and applied mathematics. In many problems, the classical definition of convexity is not enough, so the definition of convex functions is generalized in various directions. In this paper, we developed a new generalization called strongly reciprocally (p,h)-convex function. We have also developed Hermite-Hadamard and Fejér type inequality for this generalization. Moreover, we conclude that the results proved in this article are the generalization of results in [14, 16, 17, 23].



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