# Some Inequalities of Generalized p-Convex Functions concerning Raina’s Fractional Integral Operators

Authors: Changyue Chen; Muhammad Shoaib Sallem; Muhammad Sajid Zahoor
Journal: Journal of Mathematics
 (2021)

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## Abstract

Convex functions play an important role in pure and applied mathematics specially in optimization theory. In this paper, we will deal with well-known class of convex functions named as generalized p-convex functions. We develop Hermite–Hadamard-type inequalities for this class of convex function via Raina’s fractional integral operator.


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## Body

## 1. Introduction

The subject of fractional calculus got rapid development in the last few decades. As a matter of fact, fractional calculus give more accuracy to model applied problems in engineering and other sciences then classical calculus. In order to model recent complicated problems, scientists are using fractional inequalities and fractional equations. For more on this, we refer the books [1, 2]. The models with fractional calculus have been applied successfully in ecology, aerodynamics, physics, biochemistry, environmental science, and many other branches. For more about fractional calculus and models, we refer [3–5].Fractional integral inequalities are considered one of the important tools to study the behavior and properties of solutions of various fractional problems [6–14]. There are many interesting generalization of fractional derivatives as per need of practical problems or some theoretical approach, for example, Raina’s fractional integral operator, Caputo-Fabrizio fractional integral, and extended Caputo-Fabrizio fractional integral. For recent work on it, we refer [15–20].Convex functions also play an important role in pure and applied mathematics specially in optimization theory. Classical convexity does not fulfil needs of modern mathematics; therefore, several generalizations of convex functions are presented in literature.s-convex function [21], M-convex functions [22], and h-convex function [23] are some examples of generalized convex functions. It is always interesting to study properties of some generalized convex function in the setting of fractional integral operators. This paper is an effort in this direction. In this paper, we study the p-convex functions and present some of its properties in the setting of Raina’s fractional integral operators.The paper is organized as follows. In Section2, we present some basic definition and properties of Raina’s fractional integral operator. Section 3 is devoted for Hermite–Hadamard type inequalities for generalized p-convex functions in terms of Raina’s fractional integral operators.


## 2. Preliminaries

Here, we present some basic definitions and known results.Definition 1 (convex function).
A functionϕ:I⟶R is said to be convex function if the following inequality holds:(1)ϕϑx+1−ϑy≤ϑϕx+1−ϑϕy,for ∀x,y∈I and ϑ∈0,1.One of the novel generalization of convexity isη-convexity introduced by M. R. Delavar and S. S. Dragomir in [24].Definition 2.
A functionϕ:I⟶R is said to be generalized convex function with respect to η:A×A⟶B for appropriate A,B⊆R if(2)ϕϑx+1−ϑy≤ϕy+ϑηϕx,ϕy,for ∀x,y∈I and ϑ∈0,1In [25], Zhang and Wan gave definition of p-convex function as follows.Definition 3.
LetI be a p-convex set. A function f:I⟶ℝ is said to be p-convex function if(3)ϕϑxp+1−ϑyp1/p≤ϑϕx+1−ϑϕy,holds, for all x,y∈I and ϑ∈0,1.In [26], the authors gave the definition of the generalized p-convex function as follows.Definition 4.
A functionϕ:I⟶R is said to be generalized p-convex function with respect to η:A×A⟶B for appropriate A,B⊆R if(4)ϕϑxp+1−ϑyp1/p≤ϕy+ϑηϕx,ϕy,for ∀x,y∈I,p>0 and ϑ∈0,1.For some important properties and results about generalizedp-convexity, see [26]. Moreover, in [26], the following Hermite–Hadamard type inequality for p-convex functions can be found.Theorem 1.
Letϕ:I⟶ℝ be generalized p-convex function for ξ1,ξ2∈I with condition ξ1<ξ2; then, we obtain the inequality(5)ϕξ1p+ξ2p21/p−p2ξ2p−ξ1p∫ξ1ξ2xp−1ηϕξ1p+ξ2p−xp1/p,ϕxdx≤pξ2p−ξ1p∫ξ1ξ2xp−1ϕxdx≤ϕξ1+ϕξ22+14ηϕξ1,ϕξ2+ηϕξ2,ϕξ1.In [27], the author introduced a class of functions defined formally by(6)ℱρ,λσz=ℱρ,λσ0,σ1,…z=∑k=0∞σkΓρk+λzk,where ρ,λ>0, z<ℝ (ℝ is the set of real numbers), and σ=σ1,…,σk,… is a bounded sequence of positive real numbers.Using (6), in [28], the authors defined the following left-sided and right-sided fractional integral operators, respectively:(7)Jρ,λ,ξ1+;wσϕz=∫ξ1zz−ϑλ−1ℱρ,λσzwz−ϑρϕϑdϑ,z>ξ1,Jρ,λ,b−;wσϕz=∫ξ1zϑ−zλ−1ℱρ,λσzwϑ−zρϕϑdϑ,z<ξ2,where ρ,λ>0,ω∈ℝ, and ϕ is such that the integral on the right side exits.It is easy to verify thatJρ,λ,ξ1+;wσϕz and Jρ,λ,ξ2−;wσϕz are bounded integral operators on Lpξ1,ξ2,1≤p≤∞ if(8)M:=ℱρ,λ+1σwξ2−ξ1ρ≤∞.In fact, forϕ∈Lpξ1,ξ2, we have(9)Jρ,λ,ξ1+;wσϕp≤Mϕp,Jρ,λ,ξ2−;wσϕp≤Mϕp,where(10)ϕp=∫ξ1ξ2ϕzpdz1/p.The importance of these operators stems indeed from their generality. Many useful fractional integral operators can be obtained by specializing the coefficientσk. Let ϕ∈Lξ1,ξ2. The right-hand side and left-hand side Riemann–Liouville fractional integral of order α>0 with ξ2>ξ1>0 are defined by(11)Jξ1+αϕz=1Γα∫ξ1zz−kα−1ϕkdk,x>ξ1,Jξ2−αϕz=1Γα∫zξ2k−zα−1ϕkdk,z<ξ2,respectively, where Γα is the Gamma function defined as Γα=∫0∞e−kkα−1dk.Lemma 1 (see [29, 30]).
Letλ,ρ>0,ω∈ℝ, and σ be a sequence of nonnegative real numbers. Let ϕ:ξ1,ξ2⟶ℝ be a differentiable mapping on ξ1,ξ2 with ξ1<ξ2 and λ>0. If ϕ′∈Lξ1;ξ2, the following equality for the fractional integral operator holds:(12)ϕξ1+ϕξ22−12ξ2−ξ1λℱρ,λ+1σwξ2−ξ1ρJρ,λ,ξ2−;wσϕξ2+Jρ,λ,ξ1+;wσϕξ1=ξ2−ξ12ξ2−ξ1λℱρ,λ+1σwξ2−ξ1ρ∫011−ϑλℱρ,λ+1σwξ2−ξ1ρ1−ϑρϕ′ϑξ1+1−ϑξ2dϑ−∫01ϑλℱρ,λ+1σwξ2−ξ1ρϑρϕ′ϑξ1+1−ϑξ2dϑ.


## 3. Main Results

In this section, we establish new Hermite–Hadamard type inequalities for generalizedp-convex functions in terms of Raina’s fractional integral operators.Theorem 2.
Letϕ:I⟶ℝ be generalized p-convex function and provided η (.,.) is bounded from above on ϕI×ϕI and ϕ∈Lξ1,ξ2 with ξ1<ξ2 and p>0. Then, following fractional integral inequality holds:(13)ϕξ1p+ξ2p21/p−Nη≤pJρ,λ,ξ2−;wσϕξ2+Jρ,λ,ξ1+;wσϕξ12ξ2p−ξ1pλℱρ,λ+1σwξ2p−ξ1pρ,≤ϕξ1+ϕξ22+ℱρ,λ+2σ1wξ2p−ξ1pρℱρ,λ+1σwξ2p−ξ1pρMη,where σ1k=σkkρ+λ, for all k=0,1,2,… and NηandMη are bounds of ϕ.Proof. From inequality (6), we have(14)ϕxp+yp21/p−Nη2≤ϕx+ϕy2+Nη2,where Nη are bounds of ϕ. Substitute xP=ϑξ1p+1−ϑξ2p and yp=1−ϑξ1p+ϑξ2p; then, (6) can be written as(15)2ϕξ1p+ξ2p21/p−Nη≤ϕϑξ1p+1−ϑξ2p1/p+ϕ1−ϑξ1p+ϑξ2p1/p+Nη.Multiplying both sides byϑλ−1ℱρ,λσwξ2p−ξ1pρϑρ, we obtain(16)2ϕξ1p+ξ2p21/p−Nηϑλ−1ℱρ,λσwξ2p−ξ1pρϑρ≤ϑλ−1ℱρ,λσwξ2p−ξ1pρϑρϕϑξ1p+1−ϑξ2p1/p+ϑλ−1ℱρ,λσwξ2p−ξ1pρϑρϕ1−ϑξ1p+ϑξ2p1/p+ϑλ−1ℱρ,λσwξ2p−ξ1pρϑρNη.Integrate overϑ∈0,1, we obtain(17)2ϕξ1p+ξ2p21/p−Nηℱρ,λ+1σwξ2p−ξ1pρ≤∫01ϑλ−1ℱρ,λσwξ2p−ξ1pρϑρϕϑξ1p+1−ϑξ2p1/pdϑ+∫01ϑλ−1ℱρ,λσwξ2p−ξ1pρϑρϕ1−ϑξ1p+ϑξ2p1/pdϑ+∫01ϑλ−1ℱρ,λσwξ2p−ξ1pρϑρNηdϑ.With the convenient change of the variable, we can observe that(18)∫01ϑλ−1ℱρ,λσwξ2p−ξ1pρϑρϕϑξ1p+1−ϑξ2p1/pdϑ=−pξ2p−ξ1p∫ξ2ξ1ξ2p−xpξ2p−ξ1pλ−1ℱρ,λσwξ2p−ξ1pρξ2p−xpξ2p−ξ1pρϕxxp−1dx=pξ2p−ξ1pλ∫ξ1ξ2ξ2p−xpλ−1ℱρ,λσwξ2p−ξ1pρξ2p−xpξ2p−ξ1pρϕxxp−1dx=pξ2p−ξ1pλJρ,λ,ξ1+;wσϕξ2.Similarly, the second integral can be written as(19)∫01ϑλ−1ℱρ,λσwξ2p−ξ1pρϑρϕ1−ϑξ1p+ϑξ2p1/pdϑ=pξ2p−ξ1pλJρ,λ,ξ2−;wσϕξ1.Now, equation (17) becomes(20)2ϕξ1p+ξ2p21/p−Nηℱρ,λ+1σwξ2p−ξ1pρ≤pξ2p−ξ1pλJρ,λ,ξ1+;wσϕξ2+Jρ,λ,ξ2−;wσϕξ1+ℱρ,λ+1σwξ2p−ξ1pρNηϕξ1p+ξ2p21/p−Nη≤p2ξ2p−ξ1pλℱρ,λ+1σwξ2p−ξ1pρJρ,λ,ξ1+;wσϕξ2+Jρ,λ,ξ2−;wσϕξ1,which is the left-hand side of inequality (13). To prove right-hand side of (13), using the Definition 4 of generalized p-convex function,(21)ϕϑξ1p+1−ϑξ2p1/p≤ϕξ2+ϑηϕξ1,ϕξ2,ϕϑξ2p+1−ϑξ1p1/p≤ϕξ1+ϑηϕξ2,ϕξ1.Multiplying both inequalities byϑλ−1ℱρ,λσwξ2p−ξ1pρϑρ and then adding, we obtain(22)ϑλ−1ℱρ,λσwξ2p−ξ1pρϑρϕϑξ1p+1−ϑξ2p1/p+ϑλ−1ℱρ,λσwξ2p−ξ1pρϑρϕϑξ2p+1−ϑξ1p1/p≤ϑλ−1ℱρ,λσwξ2p−ξ1pρϑρϕξ2+ϑηϕξ1,ϕξ2+ϑλ−1ℱρ,λσwξ2p−ξ1pρϑρϕξ1+ϑηϕξ2,ϕξ1.Integrate overϑ∈0,1, we obtain(23)pJρ,λ,ξ1+;wσϕξ2+Jρ,λ,ξ2−;wσϕξ1ξ2p−ξ1pλ≤ϕa+ϕξ2ℱρ,λ+1σ1wξ2p−ξ1pρ+ℱρ,λ+2σwξ2p−ξ1pρηϕξ1,ϕξ2+ηϕξ2,ϕξ1pJρ,λ,ξ1+;wσϕξ2+Jρ,λ,ξ2−;wσϕξ12ξ2p−ξ1pλℱρ,λ+1σ1wξ2p−ξ1pρ≤ϕξ1+ϕξ22+ℱρ,λ+2σwξ2p−ξ1pρℱρ,λ+1σ1wξ2p−ξ1pρMη,where σ1k=σkkρ+λ. This completes the proof.Corollary 1.
Letϕ:I⟶ℝ be generalized p-convex function and provided η (.,.) is bounded from above on ϕI×ϕI and ϕ∈La,ξ2 with a<ξ2 and p>0. Then, the following inequality holds:(24)ϕξ1p+ξ2p21/p−Nη≤pΓα+1Iξ1+αϕξ2+Iξ2−αϕξ12ξ2p−ξ1pα,≤ϕξ1+ϕξ22+αα+1Mη,where NηandMη are bounds of ϕ.Proof. By takingλ=α, σ=1,0,0,…, w=0, and p=1, we obtain(25)ℱρ,λ+1σwξ2p−ξ1pρ=∑k=0∞σkΓρk+λ+1=1Γα+1,ℱρ,λ+2σwξ2p−ξ1pρ=αΓα+2,Jρ,λ,ξ1+;wσϕx=Iξ1+αϕx,Jρ,λ,ξ2−;wσϕx=Iξ2−αϕx.Making the substitution in (13), we obtain (26).Remark 1.
In Corollary1, if we take ηx,y=x−y, λ=α, σ=1,0,0,…, w=0, and p=1, then we get Theorem 1.4 of [29, 31].Theorem 3.
Letϕ:ξ1,ξ2⟶ℝ be a differentiable function on ξ1,ξ2 with ξ1<ξ2. If ϕ′ is a generalized p-convex function on ξ1,ξ2, then the following inequality for fractional integral operator holds:(26)ϕξ1+ϕξ22−pJρ,λ,ξ2−;wσϕξ2+Jρ,λ,ξ1+;wσϕξ12ξ2p−ξ1pλℱρ,λ+1σwξ2p−ξ1pρ≤ξ2p−ξ1pℱρ,λ+2σ1wξ2p−ξ1pρ2ℱρ,λ+1σwξ2p−ξ1pρ2ϕ′ξ2+ηϕ′ξ1,ϕ′ξ2,where σ1k=σk=1−1/2kρ+λ.Proof. Using Lemma1 and definition of generalized p-convexity of ϕ′, we have(27)ϕξ1+ϕξ22−pJρ,λ,ξ2−;wσϕξ2+Jρ,λ,ξ1+;wσϕξ12ξ2p−ξ1pλℱρ,λ+1σwξ2p−ξ1pρ=ξ2p−ξ1p2ℱρ,λ+1σwξ2p−ξ1pρ|∑k=0∞σkΓρk+λ+1wkξ2p−ξ1pkρ∫011−ϑkρ+λϕ′ϑξ1p+1−ϑξ2p1/2dϑ−∑k=0∞σkΓρk+λ+1wkξ2p−ξ1pkρ∫01ϑkρ+λϕ′ϑξ1p+1−ϑξ2p1/pdϑ|=ξ2p−ξ1p2ℱρ,λ+1σwξ2p−ξ1pρ∑k=0∞σkΓρk+λ+1wkξ2p−ξ1pkρ×∫011−ϑkρ+λ−ϑkρ+λϕ′ϑξ1p+1−ϑξ2p≤ξ2p−ξ1p2ℱρ,λ+1σwξ2p−ξ1pρ∑k=0∞σkΓρk+λ+1wkξ2p−ξ1pkρ×∫01/21−ϑkρ+λ−ϑkρ+λϕ′ξ2+ϑηϕ′ξ1,ϕ′ξ2dϑ+∫1/21ϑkρ+λ−1−ϑkρ+λϕ′ξ2+ϑηϕ′ξ1,ϕ′ξ2dϑ.It is easy to verify that(28)∫01/21−ϑkρ+λ−ϑkρ+λdϑ=1−1/2kρ+λkρ+λ+1,∫01/21−ϑkρ+λ−ϑkρ+λϑdϑ=1/2kρ+λ+1−1kρ+λ+2,∫1/21ϑkρ+λ+1−ϑ1−ϑkρ+λdϑ=1kρ+λ+2−12kρ+λ+1.Equation (27) becomes(29)ϕξ1+ϕξ22−pJρ,λ,ξ2−;wσϕξ2+Jρ,λ,ξ1+;wσϕξ12ξ2p−ξ1pλℱρ,λ+1σwξ2p−ξ1pρ≤ξ2p−ξ1p2ℱρ,λ+1σwξ2p−ξ1pρ∑k=0∞σkΓρk+λ+1wkξ2p−ξ1pkρ×2ϕ′ξ21−1/2kρ+λkρ+λ+1+ηϕ′ξ1,ϕ′ξ21−1/2kρ+λkρ+λ+1.Finally, we can write it as(30)ϕξ1+ϕξ22−pJρ,λ,ξ2−;wσϕξ2+Jρ,λ,ξ1+;wσϕξ12ξ2p−ξ1pλℱρ,λ+1σwξ2p−ξ1pρ≤,ξ2p−ξ1pℱρ,λ+2σ1wξ2p−ξ1pρ2ℱρ,λ+1σwξ2p−ξ1pρ2ϕ′ξ2+ηϕ′ξ1,ϕ′ξ2,where σ1k=σk=1−1/2kρ+λ, which is our required result.Corollary 2.
Letϕ:ξ1,ξ2⟶ℝ be a differentiable function on ξ1,ξ2 with ξ1<ξ2. If ϕ′ is a generalized p-convex function on ξ1,ξ2; then, the following inequality for fractional integral operator holds:(31)ϕξ1+ϕξ22−pΓα+1Iξ2−αϕξ1+Iξ1+αϕξ22ξ2p−ξ1pα≤ξ2p−ξ1p2α+11−12α2ϕ′ξ2+ηϕ′ξ1,ϕ′ξ2.Proof. By takingλ=α, σ=1,0,0,…, w=0, and p=1, we obtain(32)ℱρ,λ+1σwξ2p−ξ1pρ=1Γα+1,ℱρ,λ+2σwξ2p−ξ1pρ=1−1/2αΓα+2,Jρ,λ,ξ1+;wσϕx=Iξ1+αϕx,Jρ,λ,b−;wσϕx=Iξ2−αϕx.Making the substitution in (35), we obtain (31).Remark 2.
In Corollary2, if we take ηx,y=x−y, λ=α, σ=1,0,0,…, w=0, and p=1, then we obtain Theorem 1.5 in [29, 31].Theorem 4.
Letϕ:ξ1,ξ2⟶ℝ be a differentiable function on ξ1,ξ2 with ξ1<ξ2. If ϕ′q1 is a generalized p-convex function on ξ1,ξ2 with q1=p1/p1+1 for some fixed p1>0, then the following inequality for fractional integral operator holds:(33)ϕξ1+ϕξ22−pJρ,λ,ξ2−;wσϕξ2+Jρ,λ,ξ1+;wσϕξ12ξ2p−ξ1pλℱρ,λ+1σwξ2p−ξ1pρ≤ξ2p−ξ1p2ℱρ,λ+1σwξ2p−ξ1pρ∑k=0∞σ1kΓkρ1+λ1+1wkξ2p−ξ1p1/pkρ1×ϕ′ξ2q12+ϑηϕ′ξ1q1,ϕ′ξ2q18dϑ1/q1+ϕ′ξ2q12+ϑηϕ′ξ1q1,ϕ′ξ2q18dϑ1/q1,where ρ1=ρp, λ1=λp, and(34)σ1k=σk1−1/2kρ+λp1+1kρ+λ+1p1+11/p1,for all k=0,1,2….Proof. Using Lemma1, definition of generalized p-convexity of ϕ, and Hölder inequality, we have(35)ϕξ1+ϕξ22−pJρ,λ,ξ2−;wσϕξ2+Jρ,λ,ξ1+;wσϕξ12ξ2p−ξ1pλℱρ,λ+1σwξ2p−ξ1pρ≤ξ2p−ξ1p2ℱρ,λ+1σwξ2p−ξ1pρ∑k=0∞σkΓρk+λ+1wkξ2p−ξ1pkρ×∫01/21−ϑkρ+λ−ϑkρ+λp1dϑ1/p1∫01/2ϕ′ξ2q1+ϑηϕ′ξ1q1,ϕ′ξ2q1dϑ1/q1+∫121ϑkρ+λ−1−ϑkρ+λp1dϑ1/p1∫1/21ϕ′ξ2q1+ϑηϕ′ξ1q1,ϕ′ξ2q1dϑ1/q1≤ξ2p−ξ1p2ℱρ,λ+1σwξ2p−ξ1pρ∑k=0∞σkΓρk+λ+1wkξ2p−ξ1pkρ×∫01/21−ϑkρ+λp1−ϑkρ+λp1dϑ1/p1∫01/2ϕ′ξ2q1+ϑηϕ′ξ1q1,ϕ′ξ2q1dϑ1/q1+∫1/21ϑkρ+λp1−1−ϑkρ+λp1dϑ1/p1∫1/21ϕ′ξ2q1+ϑηϕ′aq1,ϕ′ξ2q1dϑ1/q1≤ξ2p−ξ1p2ℱρ,λ+1σwξ2p−ξ1pρ∑k=0∞σkΓρk+λ+1wkξ2p−ξ1pkρ1−1/2kρ+λp1+1kρ+λ+1p1+11/p1×ϕ′ξ2q12+ϑηϕ′ξ1q1,ϕ′ξ2q18dϑ1/q1+ϕ′ξ2q12+ϑηϕ′ξ1q1,ϕ′ξ2q18dϑ1/q1≤ξ2p−ξ1p2ℱρ,λ+1σwξ2p−ξ1pρ∑k=0∞σ1kΓkρ1+λ1+1wkξ2p−ξ1p1/pkρ1×ϕ′ξ2q12+ϑηϕ′ξ1q1,ϕ′ξ2q18dϑ1/q1+ϕ′ξ2q12+ϑηϕ′ξ1q1,ϕ′ξ2q18dϑ1/q1,which completes the proof.



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