# Estimations of Upper Bounds forn-th Order Differentiable Functions Involving χ-Riemann–Liouville Integrals via γ-Preinvex Functions

Authors: Sadia Talib; Muhammad Uzair Awan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

A new fractional integral identity is obtained involvingn-th order differentiable functions and χ-Riemann–Liouville fractional integrals. Then, some associated estimates of upper bounds involving γ-preinvex functions are obtained. In order to relate some unrelated results, several special cases are discussed.


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## Body

## 1. Introduction and Preliminaries

A setC⊂ℝ is said to be convex, if(1)1−v♭1+v♭2∈C,∀♭1,♭2∈C,v∈0,1.A functionϜ:C↦ℝ is said to be convex, if(2)Ϝ1−v♭1+v♭2≤1−vϜ♭1+vϜ♭2,∀♭1,♭2∈C,v∈0,1.In recent years, several new generalizations of the classical concepts of convexity have been proposed in the literature. For example, Hanson [1] introduced the notion of differentiable invex functions, without calling them as invex, in connection with their special global optimum behaviour. It was Craven [2] who introduced the term invex for calling this class of functions, due to their property described as invariance by convexity. The concept of invex sets is defined as follows.A setCμ⊂ℝ is said to be invex with respect to bi-function μ⋅,⋅:Cμ×Cμ↦ℝ, if(3)♭1+vμ♭2,♭1∈Cμ,∀♭1,♭2∈Cμ,v∈0,1.Remark 1.
Ifμ♭2,♭1=♭2−♭1, then the concept of invex sets reduces to classical convexity. Thus, it is true that every convex set is also an invex set with respect to μ♭2,♭1=♭2−♭1, but the converse is not necessarily true. For more details, see [3].Weir and Mond [4] introd3uced the concept of preinvex functions as follows.A functionϜ:Cμ↦ℝ is said to be preinvex with respect to bi-function μ⋅,⋅:Cμ×Cμ↦ℝ, if(4)Ϝ♭1+vμ♭2,♭1≤1−vϜ♭1+vϜ♭2,∀♭1,♭2∈Cμ,v∈0,1.Note that if we takeμ♭2,♭1=♭2−♭1, then from the class of preinvex functions, we recapture the class of classical convex functions.Having inspiration from the research work of [5–7], Awan et al. [8] introduced the notion of γ-preinvex functions.Letγ:0,1↦0,∞ be a real function. A function Ϝ:Cμ↦ℝ is said to be γ-preinvex, if(5)Ϝ♭1+vμ♭2,♭1≤1−vγ1−vϜ♭1+vγvϜ♭2,∀♭1,♭2∈Cμ,v∈0,1.It has been observed that the class ofγ-preinvex functions generalizes several other classes of preinvexity and convexity. For example:(1)
If we takeγv=1, then we have the class of classical preinvex function.(2)
If we takeγv=v−1, then we have the definition of P-preinvex function (see [6]).(3)
If we takeγv=vs−1 where s∈0,1, then we have the class of s-preinvex functions of Breckner type (see [6]).(4)
If we takeγv=v−s−1, then we have the class of s-Godunova–Levin–Dragomir type of preinvex functions (see [9]).(5)
If we takeγv=1−v, then we have the definition of tgs-preinvex functions (see [8]).It is obvious that if we takeμ♭2,♭1=♭2−♭1 in the above discussed special cases, then we can recapture the classes of classical convexity.Theory of convexity has played significant role in the development of theory of inequalities. Many famously known results in theory of inequalities can easily be obtained using the convexity property of the functions. The Hermite–Hadamard inequality which provides us a necessary and sufficient condition for a function to be convex is one of the most studied results pertaining to convexity. It reads as follows.LetϜ:C=♭1,♭2↦ℝ be a convex function; then,(6)Ϝ♭1+♭22≤1♭2−♭1∫♭1♭2Ϝxdx≤Ϝ♭1+Ϝ♭22.In recent years, several new extensions and generalizations for the Hermite–Hadamard inequality have been obtained in the literature. For example, Noor [10] obtained a new refinement of the Hermite–Hadamard inequality using the class of preinvex functions. Awan et al. [8] obtained its new version by utilizing the class of γ-preinvex functions. The authors have also discussed various applications for some special means. Sarikaya et al. [11] introduced a new dimension by introducing the fractional analogue of the Hermite–Hadamard inequality. The idea of Sarikaya and his co-authors has attracted many inequalities experts and consequently a variety of new fractional analogues of classical inequalities have been obtained in the literature using different variants of classical concepts of fractional calculus and also by different generalizations of classical convexity. For example, Hwang et al. [12] obtained different refinements and extensions of the Hermite–Hadamard inequality via fractional integrals. Turhan et al. [13] obtained Hermite–Hadamard type of inequalities via n-times differentiable convex functions involving Riemann–Liouville fractional integrals. Wu et al. [14] obtained fractional analogues of k-th order differentiable functions involving Riemann–Liouville integrals via higher order strongly h-preinvex functions. Zhang et al. [15] obtained new k-fractional integral inequalities containing multiple parameters via generalized s,m-preinvexity. The aim of this paper is to derive a new integral identity involving n-times differentiable functions and χ-Riemann–Liouville fractional integrals. Some associated estimates of upper bounds involving γ-preinvex functions are also obtained. In order to relate some unrelated results, several special cases are discussed. This shows that our results are more generalized and quite unifying. In order to show the significance of our obtained results, we also present applications to special means of real numbers. We hope that the ideas of this paper will inspire interested readers working in this field.Before we proceed further, let us recall some basic preliminaries from fractional calculus. These preliminaries will be helpful during the study of this paper.Definition 1.
(see [16]). LetϜ∈L1♭1,♭2. The Riemann–Liouville fractional integralsJ♭1+αϜandJ♭2−αϜof orderα>0with♭1≥0are defined by(7)J♭1+αϜx=1Γα∫♭1xx−vα−1Ϝvdv,x>♭1,J♭2−αϜx=1Γα∫x♭2v−xα−1Ϝvdv,x<♭2,respectively, and Γα is the gamma function. Also, we define J♭1+0Ϝx=J♭2−0Ϝx=Ϝx.Mobeen and Habibullah extended the notion of Riemann–Liouville fractional integrals and introduced the concept ofχ-Riemann–Liouville fractional integrals.Definition 2.
(see [17]). LetϜ∈L1♭1,♭2. Theχ-Riemann–Liouville fractional integralsJ♭1+αχϜandJ♭2−αχϜof orderα,χ>0with♭1≥0are given as follows:(8)J♭1+αχϜx=1χΓχα∫♭1xx−vα/χ−1Ϝvdv,x>♭1,J♭2−αχϜx=1χΓχα∫x♭2v−xα/χ−1Ϝvdv,x<♭2,respectively, where(9)Γχx=∫0∞vx−1e−vχ/χdv,is the χ-gamma function (see also [18]).
Note that whenχ⟶1,χ-Riemann–Liouville fractional integral reduces to classical Riemann–Liouville fractional integral [16].
χ-beta function is defined as(10)Bχx,y=1χ∫01vx/χ−11−vy/χ−1dv=ΓχxΓχyΓχx+y,x>0,y>0.


## 2. Auxiliary Result

We now derive a key lemma which will be helpful in obtaining the coming results of the paper.Lemma 1.
Letn≥1andϜ:♭1,♭1+μ♭2,♭1⟶ℝben-times differentiable function. Ifμ♭2,♭1>0andϜn∈L♭1,♭1+μ♭2,♭1, then(11)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+χI♭1+μ♭2,♭1−αϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12=μ♭2,♭12Cα,χ,n−1∫01−1n−11−vα/χ+n−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dv,where(12)Cα,χ,n=μn♭2,♭1χnα+χα+2χ,…,α+nk.Proof.
We prove this result by using the mathematical induction principle. The case forn=1 is obvious. Suppose Lemma 1 holds for n−1, that is,(13)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+χI♭1+μ♭2,♭1−αϜ♭1+∑j=1n−2Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12=μ♭2,♭12Cα,χ,n−2∫01−1n−21−vα/χ+n−2−vα/χ+n−2Ϝn−1♭1+1−vμ♭2,♭1dv.
Now, we prove (11) for n. Integrating by parts, we have(14)μ♭2,♭12Cα,χ,n−1∫01−1n−11−vα/χ+n−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dv=Cα,χ,n−1Ϝn−1♭1+−1n−1Ϝn−1♭1+μ♭2,♭12+μ♭2,♭12Cα,χ,n−2∫01−1n−21−vα/χ+n−2−vα/χ+n−2Ϝn−1♭1+1−vμ♭2,♭1dv=Cα,χ,n−1Ϝn−1♭1+−1n−1Ϝn−1♭1+μ♭2,♭12+Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+χI♭1+μ♭2,♭1−αϜ♭1+∑j=1n−2Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12=Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+χI♭1+μ♭2,♭1−αϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12.
This completes the proof.Remark 2.
(i)
If we taken=1 in (11), then we have the following identity:(15)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+χI♭1+μ♭2,♭1−αϜ♭1=μ♭2,♭12∫011−vα/χ−vα/χϜ′♭1+1−vμ♭2,♭1dv.(ii)
If we taken=2 to (11), then(16)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+χI♭1+μ♭2,♭1−αϜ♭1+μ♭2,♭1χ2α+χϜ′♭1−Ϝ′♭1+μ♭2,♭12=μ2♭2,♭1χ2α+χ∫01−1−vα/χ+1−vα/χ+1Ϝ″♭1+1−vμ♭2,♭1dv=μ♭2,♭1χ2α+χϜ′♭1+μ♭2,♭1−Ϝ′♭12+μ2♭2,♭12α+χ∫01−1−vα/χ+1−vα/χ+1Ϝ″♭1+1−vμ♭2,♭1dv.On the other hand, one can easily see that(17)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1=μ2♭2,♭1χ2α+χ∫011−1−vα/χ+1−vα/χ+1Ϝ″♭1+1−vμ♭2,♭1dv=μ2♭2,♭1χ2α+χ∫01Ϝ″♭1+1−vμ♭2,♭1dv+μ2♭2,♭1χ2α+χ∫01−1−vα/χ+1−vα/χ+1Ϝ″♭1+1−vμ♭2,♭1dv=μ♭2,♭1χ2α+χ∫♭1♭1+μ♭2,♭1Ϝ″sds+μ2♭2,♭1χ2α+χ∫01−1−vα/χ+1−vα/χ+1Ϝ″♭1+1−vμ♭2,♭1dv=μ2♭2,♭1χ2α+χϜ′♭1+μ♭2,♭1−Ϝ′♭1+μ2♭2,♭1χ2α+χ∫01−1−vα/χ+1−vα/χ+1Ϝ″♭1+1−vμ♭2,♭1dv.One can easily see that (16) and (17) are identical.(iii)
If we takeχ=1 to (11), then we have(18)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γα+12μα♭2,♭1I♭1+αϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αϜ♭1+∑j=1n−1Cα,jϜj♭1+−1jϜj♭1+μ♭2,♭12=μ♭2,♭12Cα,n−1∫01−1n−11−vα+n−1−vα+n−1Ϝn♭1+1−vμ♭2,♭1dv,where(19)Cα,n=μn♭2,♭1α+1α+2,…,α+n.


## 3. Main Results

We now derive our main results of the paper.Theorem 1.
Letn≥1andϜ:♭1,♭1+μ♭2,♭1⟶ℝben-times differentiable function withμ♭2,♭1>0. IfϜn∈L♭1,♭1+μ♭2,♭1andϜnis aγ-preinvex function, then(20)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭12Cα,χ,n−1Ϝn♭1∫01vγv1−vα/χ+n−1+vα/χ+n−1dv+|Ϝn♭2|∫011−vγ1−v1−vαχ+n−1+vαχ+n−1dv,nis even,Ϝn♭1∫01/2vγv1−vα/χ+n−1−vα/χ+n−1dv+Ϝn♭2∫01/21−vγ1−v1−vα/χ+n−1−vα/χ+n−1dv+Ϝn♭1∫1/21vγvvα/χ+n−1−1−vα/χ+n−1dv+Ϝn♭2∫1/211−vγ1−vvα/χ+n−1−1−vα/χ+n−1dv,nis odd.Proof.
Using Lemma1 and the γ-preinvexity of Ϝn, we have(21)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭12Cα,χ,n−1∫01−1n−11−vα/χ+n−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dv=μ♭2,♭12Cα,χ,n−1∫01−1−vα/χ+n−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dv,nis even,∫011−vα/χ+n−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dvnis odd,≤μ♭2,♭12Cα,χ,n−1∫01−1−vα/χ+n−1−vα/χ+n−1vγvϜn♭1+1−vγ1−vϜn♭2dv,nis even,∫011−vα/χ+n−1−vα/χ+n−1vγvϜn♭1+1−vγ1−vϜn♭2dv,nis odd,=μ♭2,♭12Cα,χ,n−1∫011−vα/χ+n−1+vα/χ+n−1vγvϜn♭1+1−vγ1−vϜn♭2dv,nis even,∫011−vα/χ+n−1−vα/χ+n−1vγvϜn♭1+1−vγ1−vϜn♭2dv,nis odd,=μ♭2,♭12Cα,χ,n−1∫011−vα/χ+n−1+vα/χ+n−1vγvϜn♭1+1−vγ1−vϜn♭2dv,nis even,∫01/21−vα/χ+n−1−vα/χ+n−1vγvϜn♭1+1−vγ1−vϜn♭2dv+∫1/21vα/χ+n−1−1−vα/χ+n−1vγvϜn♭1+1−vγ1−vϜn♭2dv,nis odd,=μ♭2,♭12Cα,χ,n−1Ϝn♭1∫01vγv1−vα/χ+n−1+vα/χ+n−1dv+Ϝn♭2∫011−vγ1−v1−vα/χ+n−1+vα/χ+n−1dv,nis even,Ϝn♭1∫01/2vγv1−vα/χ+n−1−vα/χ+n−1dv+Ϝn♭2∫01/21−vγ1−v1−vα/χ+n−1−vα/χ+n−1dv+Ϝn♭1∫1/21vγvvα/χ+n−1−1−vα/χ+n−1dv+Ϝn♭2∫1/211−vγ1−vvα/χ+n−1−1−vα/χ+n−1dv,nis odd.
This completes the proof.We now discuss some special cases of Theorem1.(i)
Ifγv=1, then Theorem 1 reduces to the following result for the class of preinvex function.Corollary 1.
Under the assumptions of Theorem 1, ifϜnis classical preinvex function on♭1,♭1+μ♭2,♭1with respect toμ⋅,⋅, then(22)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤Cα,χ,n2Ϝn♭1+Ϝn♭2,nis even,1−12α/χ+n−1Ϝn♭1+Ϝn♭2,nis odd.
(ii)
Ifγv=v−1, then Theorem 1 reduces to the following result for the class of P-preinvex function.Corollary 2.
Under the assumptions of Theorem 1, ifϜnisP-preinvex function on♭1,♭1+μ♭2,♭1with respect toμ⋅,⋅, then(23)|Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤Cα,χ,nϜn♭1+Ϝn♭2,nis even,1−12α/χ+n−1Ϝn♭1+Ϝn♭2,nis odd.
(iii)
Ifγv=vs−1, then Theorem 1 reduces to the following result for the class of s-preinvex functions of Breckner type.Corollary 3.
Under the assumptions of Theorem 1, ifϜniss-preinvex function on♭1,♭1+μ♭2,♭1with respect toμ⋅,⋅, then(24)|Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭12Cα,χ,n−1Lα,n,χϜn♭1+Mα,n,χϜn♭2,nis even,Nα,n,χNα,n,χϜn♭1+Rα,n,χϜn♭2,nis odd,where(25)Lα,n,χ=χα+nχ+sχ+χℬχsk+χ,α+nk,(26)Mα,n,χ=χα+nχ+sχ+χℬχα+nχ,sχ+χ,(27)Nα,n,χ=χα+nχ+sχ−χ2α/χ+n+s−1α+nχ+sχ+2χ1/2ℬχsχ+χ,α+nχ−χℬχsχ+χ,α+nχ,(28)Rα,n,χ=χα+nχ+sχ−χ2α/χ+n+s−1α+nχ+sχ−2χ1/2ℬχsχ+χ,α+nχ+χℬχsχ+χ,α+nχ.(iv)
Ifγv=v−s−1, then Theorem 1 reduces to the following result in the class of s-Godunova–Levin preinvex function.Corollary 4.
Under the assumptions of Theorem 1, ifϜniss-Godunova–Levin preinvex function on♭1,♭1+μ♭2,♭1with respect toμ⋅,⋅, then(29)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭12Cα,χ,n−1L⋆α,n,χϜn♭1+M⋆α,n,χϜn♭2,nis even,N⋆α,n,χϜn♭1+R⋆α,n,χϜn♭2,nis odd,where(30)L⋆α,n,χ=χα+nχ−sχ+χℬχα+nχ,χ−sχ,(31)M⋆α,n,χ=χα+nχ−sχ+χℬχχ−sχ,α+nχ,(32)N⋆α,n,χ=χα+nχ−sχ−χ2α/χ+n−s−1α+nχ−sχ+2χ1/2ℬχχ−sχ,α+nχ−χℬχχ−sχ,α+nχ,(33)R⋆α,n,χ=χα+nχ−sχ−χ2α/χ+n−s−1α+nχ−sχ−2χ1/2ℬχχ−sχ,α+nχ+χℬχχ−sχ,α+nχ.(v)
Ifγv=1−v, then Theorem 1 reduces to the following result for the class of tgs-preinvex function.Corollary 5.
Under the assumptions of Theorem 1, ifϜnistgs-preinvex function on♭1,♭1+μ♭2,♭1with respect toμ⋅,⋅, then(34)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭1χ2Cα,χ,n−12χ+nχ+1χ+nχ+22Ϝn♭1+Ϝn♭2,nis even,1−α+nχ+3χ2α/χ+n+1χϜn♭1+Ϝn♭2,nis odd.Theorem 2.
Letn≥1andϜ:♭1,♭1+μ♭2,♭1⟶ℝben-times differentiable function withμ♭2,♭1>0. IfϜn∈L♭1,♭1+μ♭2,♭1andϜnqis aγ-preinvex function, then(35)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭12Cα,χ,n−1Ϝn♭1q∫01vγvdv+Ϝn♭2q∫011−vγ1−vdv1/q,nis even,1pα/χ+n−1+11−12pα/χ+n−11/p×Ϝn♭1q∫01/2vγvdv+Ϝn♭2q∫01/21−vγ1−vdv1/q+Ϝn♭1q∫1/21vγvdv+Ϝn♭2q∫1/211−vγ1−vdv1/q,nis odd.Proof.
Using Lemma1, Hölder’s integral inequality, and γ-preinvexity of Ϝnq, we have(36)|Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭12Cα,χ,n−1∫01−1n−11−vα/χ+n−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dv≤μ♭2,♭12Cα,χ,n−1∫01−1−vα/χ+n−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dv,nis even,∫011−vα/χ+n−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dv,nis odd,=μ♭2,♭12Cα,χ,n−1∫011−vα/χ+n−1+vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dv,nis even,∫011−vα/χ+n−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dv,nis odd,=μ♭2,♭12Cα,χ,n−1∫011−vα/χ+n−1+vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dv,nis even,∫01/21−vα/χ+n−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dv+∫1/21vα/χ+n−1−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dv,nis odd,≤μ♭2,♭12Cα,χ,n−1∫011−vα/χ+n−1+vα/χ+n−1pdv1/p∫01Ϝn♭1+1−vμ♭2,♭1qdv1/q,nis even,∫01/21−vα/χ+n−1−vα/χ+n−1pdv1/p∫01/2Ϝn♭1+1−vμ♭2,♭1qdv1/q+∫1/21vα/χ+n−1−1−vα/χ+n−1pdv1/p∫1/21Ϝn♭1+1−vμ♭2,♭1qdv1/q,nis odd,≤μ♭2,♭12Cα,χ,n−1∫011−vα/χ+n−1+vα/χ+n−1pdv1/p×∫01vγvϜn♭1q+1−vγ1−vϜn♭2qdv1/q,nis even,∫01/21−vα/χ+n−1−vα/χ+n−1pdv1/p×∫01/2vγvϜn♭1q+1−vγ1−vϜn♭2qdv1/q+∫1/21vα/χ+n−1−1−vα/χ+n−1pdv1/p×∫1/21vγvϜn♭1q+1−vγ1−vϜn♭2qdv1/q,n is odd,≤μ♭2,♭12Cα,χ,n−1∫011−vα/χ+n−1+vα/χ+n−1pdv1/p×∫01vγvϜn♭1q+1−vγ1−vϜn♭2qdv1/q,nis even,∫01/21−vpα/χ+n−1−vpα/χ+n−1dv1/p×∫01/2vγvϜn♭1q+1−vγ1−vϜn♭2qdv1/q+∫1/21vα/χ+n−1−1−vpα/χ+n−1dv1/p×∫1/21vγvϜn♭1q+1−vγ1−vϜn♭2qdv1/q,n is odd,=μ♭2,♭12Cα,χ,n−1Ϝn♭1q∫01vγvdv+Ϝn♭2q∫011−vγ1−vdv1/q,nis even,1pα/χ+n−1+11−12pα/χ+n−11/p×Ϝn♭1q∫01/2vγvdv+Ϝn♭2q∫01/21−vγ1−vdv1/q+Ϝn♭1q∫1/21vγvdv+Ϝn♭2q∫1/211−vγ1−vdv1/q,n is odd.
This completes the proof.We now discuss some special cases of Theorem2.(i)
Ifγv=1, then Theorem 2 reduces to the following result for the class of classical preinvex function.Corollary 6.
Under the assumptions of Theorem 2, ifϜnqis classical preinvex function on♭1,♭1+μ♭2,♭1with respect toμ⋅,⋅, then(37)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭12Cα,χ,n−1Ϝn♭1q+Ϝn♭2q21/q,nis even,1pα/χ+n−1+11−12pα/χ+n−11/p×Ϝn♭1q+3Ϝn♭2q81/q+3Ϝn♭1q+Ϝn♭2q81/q,nis odd.(ii)
Ifγv=v−1, then Theorem 2 reduces to the following result for the class of P-preinvex function.Corollary 7.
Under the assumptions of Theorem 2, ifϜnqisP-preinvex function on♭1,♭1+μ♭2,♭1with respect toμ⋅,⋅, then(38)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭12Cα,χ,n−1Ϝn♭1q+Ϝn♭2q1/q,nis even,21pα/χ+n−1+11−12pα/χ+n−11/p×Ϝn♭1q+Ϝn♭2q21/q,nis odd.(iii)
Ifγv=vs−1, then Theorem 2 reduces to the following result for the class of s-preinvex function.Corollary 8.
Under the assumptions of Theorem 2, ifϜnqiss-preinvex function on♭1,♭1+μ♭2,♭1with respect toμ⋅,⋅, then(39)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭12Cα,χ,n−1Ϝn♭1q+Ϝn♭2qs+11/q,nis even,1pα/χ+n−1+11−12pα/χ+n−11/p×1s+11/q12s+1Ϝn♭1q+1−12s+1Ϝn♭2q1/q+1−12s+1Ϝn♭1q+12s+1Ϝn♭2q1/q,nis odd.(iv)
Ifγv=v−s−1, then Theorem 2 reduces to the following result for the class of s-Godunova–Levin preinvex function.Corollary 9.
Under the assumptions of Theorem 2, ifϜnqiss-Godunova–Levin preinvex function on♭1,♭1+μ♭2,♭1with respect toμ⋅,⋅, then(40)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭12Cα,χ,n−1Ϝn♭1q+Ϝn♭2q1−s1/q,nis even,1pα/χ+n−1+11−12pα/χ+n−11/p×11−s1/q121−sϜn♭1q+1−121−sϜn♭2q1/q+1−121−sϜn♭1q+121−sϜn♭2q1/q,nis odd.(v)
Ifγv=1−v, then Theorem 2 reduces to the following result for the class of tgs-preinvex function.Corollary 10.
Under the assumptions of Theorem 2, ifϜnqistgs-preinvex function on♭1,♭1+μ♭2,♭1with respect toμ⋅,⋅, then(41)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭12Cα,χ,n−1Ϝn♭1q+Ϝn♭2q61/q,nis even,21pα/χ+n−1+11−12pα/χ+n−11/p×Ϝn♭1q+Ϝn♭2q121/q,nis odd.Theorem 3.
Letn≥1andϜ:♭1,♭1+μ♭2,♭1⟶ℝben-times differentiable function withμ♭2,♭1>0. IfϜn∈L♭1,♭1+μ♭2,♭1andϜnqis aγ-preinvex function, then(42)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭12Cα,χ,n−12χα+nχ1−1/q×Ϝn♭1q∫01vγv1−vα/χ+n−1+vα/χ+n−1dv+Ϝn♭2q∫011−vγ1−v1−vα/χ+n−1+vα/χ+n−1dv1/q,nis even,2χα+nχ1−1/q1−12α/χ+n−11−1/qϜn♭1q∫01/2vγv1−vα/χ+n−1−vα/χ+n−1dv×+Ϝn♭2q∫01/21−vγ1−v1−vα/χ+n−1−vα/χ+n−1dv+Ϝn♭1q∫1/21vγvvα/χ+n−1−1−vα/χ+n−1dv+Ϝn♭2q∫111−vγ1−vvα/χ+n−1−1−vα/χ+n−1dv1/q,nis odd.Proof.
Using Lemma1, power mean integral inequality, and γ-preinvexity of Ϝnq, we have(43)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭12Cα,χ,n−1∫01−1n−11−vα/χ+n−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dv≤μ♭2,♭12Cα,χ,n−1∫01−1−vα/χ+n−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dv,nis even,∫011−vα/χ+n−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1dv,nis odd,≤μ♭2,♭12Cα,χ,n−1∫01−1−vα/χ+n−1−vα/χ+n−1dv1−1/q×∫01−1−vα/χ+n−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1qdv1/q,nis even,∫011−vα/χ+n−1−vα/χ+n−1dv1−1/q×∫011−vα/χ+n−1−vα/χ+n−1Ϝn♭1+1−vμ♭2,♭1qdv1/q,nis odd,≤μ♭2,♭12Cα,χ,n−12χα+nχ1−1/q×∫01−1−vα/χ+n−1−vα/χ+n−1×vγvϜn♭1q+1−vγ1−vϜn♭2qdv1/q,nis even,2χα+nχ1−1/q1−12α/χ+n−11−1/q×∫011−vα/χ+n−1−vα/χ+n−1×vγvϜn♭1q+1−vγ1−vϜn♭2qdv1/q,nis odd,=μ♭2,♭12Cα,χ,n−12χα+nχ1−1/q×Ϝn♭1q∫01vγv1−vα/χ+n−1+vα/χ+n−1dv+Ϝn♭2q∫011−vγ1−v1−vα/χ+n−1+vα/χ+n−1dv1/q,nis even,2χα+nχ1−1/q1−12α/χ+n−11−1/q×Ϝn♭1q∫01/2vγv1−vα/χ+n−1−vα/χ+n−1dv+Ϝn♭2q∫01/21−vγ1−v1−vα/χ+n−1−vα/χ+n−1dv+Ϝn♭1q∫1/21vγvvα/χ+n−1−1−vα/χ+n−1dv+Ϝn♭2q∫1/211−vγ1−vvα/χ+n−1−1−vα/χ+n−1dv1/q,nis odd.
This completes the proof.Now we will discuss some special cases of Theorem3.(i)
Ifγv=1, then Theorem 3 reduces to the following result for the class of classical preinvex function.Corollary 11.
Under the assumptions of Theorem 3, ifϜnqis classical preinvex function on♭1,♭1+μ♭2,♭1with respect toμ⋅,⋅, then(44)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤Cα,χ,nϜn♭1q+Ϝn♭2q21/q,nis even,1−12α/χ+n−1Ϝn♭1q+Ϝn♭2q21/q,nis odd.(ii)
Ifγv=v−1, then Theorem 3 reduces to the following result for the class of P-preinvex function.Corollary 12.
Under the assumptions of Theorem 3, ifϜnqisP-preinvex function on♭1,♭1+μ♭2,♭1with respect toμ⋅,⋅, then(45)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤Cα,χ,nϜn♭1q+Ϝn♭2q1/q,nis even,1−12α/χ+n−1Ϝn♭1q+Ϝn♭2q1/q,nis odd.(iii)
Ifγv=vs−1, then Theorem 3 reduces to the following result for the class of s-preinvex function.Corollary 13.
Under the assumptions of Theorem 3, ifϜnqiss-preinvex function on♭1,♭1+μ♭2,♭1with respect toμ⋅,⋅, then(46)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭12Cα,χ,n−12χα+nχ1−1/qLα,n,χϜn♭1q+Mα,n,χϜn♭2q1/q,nis even,1−12α/χ+n−11−1/qNα,n,χϜn♭1q+Rα,n,χϜn♭2q1/q,nis odd,whereLα,n,χ,Mα,n,χ,Nα,n,χ, and Rα,n,χ are given by (25), (26), (27), and (28), respectively.(iv)
Ifγv=v−s−1, then Theorem 3 reduces to the following result for the class of s-Godunova–Levin preinvex function.Corollary 14.
Under the assumptions of Theorem 3, ifϜnqiss-Godunova–Levin preinvex function on♭1,♭1+μ♭2,♭1with respect toμ⋅,⋅, then(47)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭12Cα,χ,n−12χα+nχ1−1/qL⋆α,n,χϜn♭1q+M⋆α,n,χϜn♭2q1/q,nis even,1−12α/χ+n−11−1/qN⋆α,n,χϜn♭1q+R⋆α,n,χϜn♭2q1/q,nis odd,whereL⋆α,n,χ,M⋆α,n,χ,N⋆α,n,χ, and R⋆α,n,χ are given by (30), (31), (32), and (33), respectively.(v)
Ifγv=1−v, then Theorem 3 reduces to the following result for the class of tgs-preinvex function.Corollary 15.
Under the assumptions of Theorem 3, ifϜnqistgs-preinvex function on♭1,♭1+μ♭2,♭1with respect toμ⋅,⋅, then(48)Ϝ♭1+Ϝ♭1+μ♭2,♭12−Γχα+χ2μα/χ♭2,♭1I♭1+αχϜ♭1+μ♭2,♭1+I♭1+μ♭2,♭1−αχϜ♭1+∑j=1n−1Cα,χ,jϜj♭1+−1jϜj♭1+μ♭2,♭12≤μ♭2,♭1χ2Cα,χ,n−122χα+nχ1−1/q2Ϝn♭1q+Ϝn♭2qχ+nχ+1χ+nχ+21/q,nis even,1−12α/χ+n−11−1/q1−α+nχ+3χ2α/χ+n+2χ1/qϜn♭1q+Ϝn♭2qχ+nχ+1χ+nχ+21/q,nis odd.


## 4. Applications

In this section, we present some applications to special means. We now recall some special means for two different positive real numbers♭1<♭2:(i)
The arithmetic mean:A♭1,♭2≔♭1+♭2/2.(ii)
The logarithmic mean:ℒ♭1,♭2≔♭2−♭1/ln♭2−ln♭1.(iii)
Thep-logarithmic mean: ℒp♭1,♭2≔♭2p+1−♭1p+1/p+1♭2−♭11/p,p∈ℝ\−1,0.Proposition 1.
For♭1,♭2∈ℝ+,0<♭1<♭2, and s∈0,1, the following inequality holds:(49)A♭1s,♭1s−ℒpp♭1,♭2+s♭2−♭12A♭1s−1,−♭2s−1≤♭2−♭12s1−ss2+3s+44s+1s+2s+3♭1s−1+♭1s−2.Proof.
Takingn=2,χ=α=1, and μ♭2,♭1=♭2−♭1 in inequality (24), we have(50)Ϝ♭1+Ϝ♭12−1♭2−♭1∫♭1♭2Ϝudu+♭2−♭12Ϝ′♭1−Ϝ′♭22≤♭2−♭12s2+3s+44s+1s+2s+3Ϝ″♭1+Ϝ″♭2.
Now for the functionϜ:0,1⟶0,1,Ϝx=xs, Ϝ″x=ss−1xs−2 is a s-convex function. This completes the proof.Proposition 2.
For♭1,♭2∈ℝ+,0<♭1<♭2, and s∈0,1, the following inequality holds:(51)A♭1s,♭1s−ℒpp♭1,♭2+s♭2−♭12A♭1s−1,−♭2s−1≤♭2−♭12sq1−sq4♭1s−2q+♭2s−2qs+11/q,where 1/p+1/q=1.Proof.
Takingn=2,χ=α=1, and μ♭2,♭1=♭2−♭1 in inequality (39), we have(52)Ϝ♭1+Ϝ♭12−1♭2−♭1∫♭1♭2Ϝudu+♭2−♭12Ϝ′♭1−Ϝ′♭22≤♭2−♭124Ϝ″♭1q+Ϝ″♭2qs+11/q.
Now for the functionϜ:0,1⟶0,1,Ϝx=xs, Ϝ″xq=sq1−sqxs−2q is a s-convex function. This completes the proof.Proposition 3.
For♭1,♭2∈ℝ+,0<♭1<♭2, ands∈0,1, the following inequality holds:(53)A♭1s,♭1s−ℒpp♭1,♭2+s♭2−♭12A♭1s−1,−♭2s−1≤♭2−♭12s−s24s2+3s+41/q231−1/q♭1s−2q+♭2s−2qs+1s+2s+31/q,where q≥1.Proof.
Takingn=2,χ=α=1, and μ♭2,♭1=♭2−♭1 in inequality (46), we have(54)Ϝ♭1+Ϝ♭12−1♭2−♭1∫♭1♭2Ϝudu+♭2−♭12Ϝ′♭1−Ϝ′♭22≤♭2−♭124s2+3s+41/q231−1/qϜ″♭1q+Ϝ″♭2qs+1s+2s+31/q.
Now for the functionϜ:0,1⟶0,1,Ϝx=xs, Ϝ″xq=sq1−sqxs−2q is a s-convex function. This completes the proof.


## 5. Conclusion

We have derived a new fractional integral identity by using theχ-fractional integral and n-times differentiable functions. Using this identity as an auxiliary result, we have obtained some new associated upper bounds involving γ-preinvex functions. Several new special cases are also discussed in detail which show that our results represent significant generalizations and are quite unifying. In order to show the significance of our theocratical results, we also presented applications to special means of our obtained results.



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