# Refinements of Some Integral Inequalities forφ-Convex Functions

Authors: Moquddsa Zahra; Yu-Ming Chu; Ghulam Farid
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we are interested to deal with unified integral operators for stronglyφ-convex function. We will present refinements of bounds of these unified integral operators and use them to get associated results for fractional integral operators. Several known results are connected with particular assumptions.


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## Body

## 1. Introduction and Preliminaries

Convex functions play an important role in the formation of new definitions of related functions which help to give the generalization of classical results. Therefore, in recent years, many generalizations of convex functions are defined and utilized to study the Hadamard and other well-known inequalities (see [1–9]). In this paper, we deal with the strongly φ-convex functions to study the bounds of unified integral operators. The obtained results are compared with already known results.First, we give some definitions of functions which are necessary for the findings of this paper.Definition 1 (see [7]).
A functionf:I⟶ℝ is said to be convex on I if(1)fςu+1−ςv≤ςfu+1−ςfv,holds for all u,v∈I and ς∈0,1, where I⊆ℝ is an interval. Reverse of inequality (1) defines f as concave function.Definition 2 (see [10]).
A functionf:I⟶ℝ is said to be strongly convex with modulus λ>0 if(2)fςu+1−ςv≤ςfu+1−ςfv−λς1−ςv−u2,holds for all u,v∈I and ς∈0,1.Definition 3 (see [3]).
A functionf:I⟶ℝ is said to be φ-convex on I if(3)fςu+1−ςv≤fv+ςφfu,fv,holds for all u,v∈I and ς∈0,1, where φ is a bifunction.Definition 4 (see [2]).
A functionf:I⟶ℝ is said to be strongly φ-convex on I if(4)fςu+1−ςv≤fv+ςφfu,fv−λς1−ςv−u2,holds for all u,v∈I and ς∈0,1,λ≥0, where φ is a bifunction.
It is to be noted that forφx,y=x−y, strongly φ-convex function reduces to strongly convex function. Farid in [11] defined the unified integral operators (5) and (6) and has proved the continuity and the boundedness of these integral operators. The aim of this paper is the study of integral inequalities for strongly φ-convex functions via unified integral operators. Next, we give definition of the unified integral operators.Definition 5.
Letf,g:u,v⟶ℝ where 0<u<v be the function such that f is positive and integrable over u,v and g is differentiable and strictly increasing. Also, let Ψ/x be an increasing function on u,∞ and α,ξ,γ,ζ∈ℂ, p,μ ,δ≥0 and 0<k≤δ+μ. Then, for x∈u,v, the left and right integral operators are defined as follows:(5)Fgμ,α,ξ,u+Ψ,γ,δ,k,ζfx,η;p=∫uxJxyEμ,α,ξγ,δ,k,ζ,g;Ψg′yfydy,(6)Fgμ,α,ξ,v−Ψ,γ,δ,k,ζfx,η;p=∫xvJyxEμ,α,ξγ,δ,k,ζ,g;Ψg′yfydy,where(7)JxyEμ,α,ξγ,δ,k,ζ,g;Ψ=Ψgx−gygx−gyEμ,α,ξγ,δ,k,ζηgx−gyμ;p.
By choosing specific functionsΨandg and fixing parameters involved in the Mittag-Leffler function Eμ,α,ξγ,δ,k,ζηgx−gyμ;p, various known fractional integrals can be reproduced (see [5], Remarks 6 and 7). In [4], by using unified integral operators, we have obtained integral inequalities for φ-convex functions. In the following, we give these inequalities in the form of Theorems 1–3.Theorem 1.
Letf:u,v⟶ℝ be a positive φ-convex function and g:u,v⟶ℝ be differentiable and strictly increasing function. Also, let Ψ/x be an increasing function on u,v, η,α,ξ,γ,ζ∈ℂ, p,μ,ν,δ≥0, 0<k≤δ+μ, and 0<k≤δ+ν. Then, for x∈u,v, we have(8)Fgμ,α,ξ,u+Ψ,γ,δ,k,ζfx,η;p+Fgν,α,ξ,v−Ψ,γ,δ,k,ζfx,η;p≤Eμ,α,ξγ,δ,k,ζηgx−guμ;pΨgx−gufx+JxuEμ,α,ξγ,δ,k,ζ,g;Ψφfu,fxIu,x;g−gu+Eν,α,ξγ,δ,k,ζηgv−gxν;pΨgv−gxfv+JvxEν,α,ξγ,δ,k,ζ,g;Ψφfx,fvIx,v;g−gx.Theorem 2.
Along with the assumptions of Theorem1, if fu+v−x=fx and φx,y=x+y, then the following result holds:(9)12fu+v2Fgμ,α,ξ,v−Ψ,γ,δ,k,ζ1u,η;p+Fgμ,α,ξ,u+Ψ,γ,δ,k,ζ1v,η;p≤Fgμ,α,ξ,v−Ψ,γ,δ,k,ζfu,η;p+Fgμ,α,ξ,u+Ψ,γ,δ,k,ζfv,η;p≤2Ψgv−guEμ,α,ξγ,δ,k,ζηgv−guμ;pfv+2fu+fvJvuEμ,α,ξγ,δ,k,ζ,g;ΨIu,v;g−gu.
Also, the following result holds for the convolution of functionsf and g.Theorem 3.
Letf,g:u,v⟶ℝ be two differentiable functions such that f′ is φ-convex and g be strictly increasing for 0<u<v. Also, Ψ/x be an increasing function on u,v and α,ξ,γ,ζ∈ℂ, p,μ,ν,δ≥0 and 0<k≤δ+μ and 0<k≤δ+ν. Then, for x∈u,v, we have(10)Fgμ,α,ξ,u+Ψ,γ,δ,k,ζf∗gx,η;p+Fgν,α,ξ,v−Ψ,γ,δ,k,ζf∗gx,η;p≤Eμ,α,ξγ,δ,k,ζηgx−guμ;pΨgx−guf′x+JxuEμ,α,ξγ,δ,k,ζ,g;Ψφf′u,f′xIu,x;g−gu+Eν,α,ξγ,δ,k,ζηgv−gxν;pΨgv−gxf′v+JvxEν,α,ξγ,δ,k,ζ,g;Ψφf′x,f′vIx,v;g−gx.
Although we follow the same method which was adopted to prove the results of [4], here we will get refinements of these results by using strongly φ-convex functions. In Section 2, we give the refinements of bounds of unified integral operators given in Definition 5. In Section 3, we will present refinements of bounds of fractional integral operators.


## 2. Main Results

Throughout this section, we have adopted the following notations:(11)Iu,v;g≔1v−u∫uvgtdt,Sμ,ν,u+,v−=Fgμ,α,ξ,u+Ψ,γ,δ,k,ζfx,η;p+Fgν,α,ξ,v−Ψ,γ,δ,k,ζfx,η;p.Theorem 4.
Iff is positive strongly φ-convex function with modulus λ≥0, along with other assumptions of Theorem 1, then we have(12)Sμ,ν,u+,v−≤Eμ,α,ξγ,δ,k,ζηgx−guμ;pΨgx−gufx+JxuEμ,α,ξγ,δ,k,ζ,g;Ψφfu,fxIu,x;g−gu−λx−u2Iu,x;Idg−x+uIu,x;g+Eν,α,ξγ,δ,k,ζηgv−gxν;pΨgv−gxfv+JvxEν,α,ξγ,δ,k,ζ,g;Ψφfx,fvIx,v;g−gx−λv−x2Ix,v;Idg−v+xIx,v;g,where Id is the identity function.Proof.
For the kernel defined in (7) and the strongly φ-convexity of the function f on u,x, the following inequalities hold, respectively:(13)JxςEμ,α,ξγ,δ,k,ζ,g;Ψg′ς≤JxuEμ,α,ξγ,δ,k,ζ,g;Ψg′ς,ς∈u,x,x∈u,v,(14)fς≤fx+x−ςx−uφfu,fx−λx−ςx−uς−ux−uu−x2.
The aforementioned inequalities are used to obtain the following integral inequality:(15)∫uxJxςEμ,α,ξγ,δ,k,ζ,g;Ψg′ςfςdς≤fxJxuEμ,α,ξγ,δ,k,ζ,g;Ψ∫uxg′ςdς+φfu,fxx−uJxuEμ,α,ξγ,δ,k,ζ,g;Ψ∫uxx−ςg′ςdς−λJxuEμ,α,ξγ,δ,k,ζ,g;Ψ∫uxx−ςς−ug′ςdς.
In view of Definition5 and applying integration by parts, from inequality (15), we get the following upper bound of the right-sided unified integral operator:(16)Fgμ,α,ξ,u+Ψ,γ,δ,k,ζfx,η;p≤Eμ,α,ξγ,δ,k,ζηgx−guμ;pΨgx−gufx+JxuEμ,α,ξγ,δ,k,ζ,g;Ψφfu,fxIu,x;g−gu−λx−u2Iu,x;Idg−x+uIu,x;g.
Again for the kernel defined in (7) and the strongly φ-convexity of the function f on x,v, the following inequalities hold, respectively:(17)JςxEν,α,ξγ,δ,k,ζ,g;Ψg′ς≤JvxEν,α,ξγ,δ,k,ζ,g;Ψg′ς,(18)fς≤fv+v−ςv−xφfx,fv−λv−ςv−xς−xv−xx−v2.
The aforementioned inequalities (17) and (18) are used to obtain the following integral inequality:(19)∫xvJςxEν,α,ξγ,δ,k,ζ,g;Ψg′ςfςdς≤JxvEν,α,ξγ,δ,k,ζ,g;Ψfv∫xvg′ςdς+φfx,fvv−x∫xvv−ςg′ςdς−λJxvEν,α,ξγ,δ,k,ζ,g;Ψ∫xvv−ςς−xg′ςdς.
In view of Definition5 and applying integration by parts, from inequality (19), we get the following upper bound of the left-sided unified integral operator:(20)Fgν,α,ξ,v−Ψ,γ,δ,k,ζfx,η;p≤Eν,α,ξγ,δ,k,ζηgv−gxν;pΨgv−gxfv+JvxEν,α,ξγ,δ,k,ζ,g;Ψφfx,fvIx,v;g−gx−λv−x2Ix,v;Idg−v+xIx,v;g.
Inequality (12) will be obtained by combining (16) and (20).Corollary 1.
By settingμ=ν in (12), we get(21)Sμ,μ,u+,v−≤Eμ,α,ξγ,δ,k,ζηgx−guμ;pΨgx−gufx+JxuEμ,α,ξγ,δ,k,ζ,g;Ψφfu,fxIu,x;g−gu−λx−u2Iu,x;Idg−x+uIu,x;g+Eμ,α,ξγ,δ,k,ζηgv−gxμ;pΨgv−gxfv+JvxEμ,α,ξγ,δ,k,ζ,g;Ψφfx,fvIx,v;g−gx−λv−x2Ix,v;Idg−v+xIx,v;g.Remark 1.
Forλ=0 in (12), we get inequality (8) of Theorem 1; if 2Iu,x;Idg>x+uIu,x;g and 2Ix,v;Idg>v+xIx,v;g, then we will get the refinement of (8).Forφx,y=x−y in (21), we get the result for strongly convex function.Forφx,y=x−y and λ=0 in (21), we get the result of Theorem 8 in [5].
We will use the following lemma for our next result.Lemma 1.
Letf be strongly φ-convex function with modulus λ≥0. If fx=fu+v−x, then(22)fu+v2≤fx+12φfx,fx−λu−v24,holds for x∈u,v.Proof.
Stronglyφ-convexity of f implies (23)fu+v2≤fx−uv−uu+v−xv−uv+12φfx−uv−uv+v−xv−uu,fx−uv−uu+v−xv−uv−λu−v24=fu+v−x+12φfx,fu+v−x−λu−v24.
Using the conditionfx=fu+v−x in the above inequality, we get (22).Remark 2.
Forλ=0, Lemma 1 reduces to Lemma 1 of [4]. For λ>0, we get its refinement.Forφx,y=x−y and λ=0, Lemma 1 reduces to Lemma 21 of [5].Theorem 5.
Letfu+v−x=fx and φx,y=x+y in addition with the assumptions of Theorem 4. Then, the following inequality holds:(24)12fu+v2+λu−v24Fgμ,α,ξ,v−Ψ,γ,δ,k,ζ1u,η;p+Fgμ,α,ξ,u+Ψ,γ,δ,k,ζ1v,η;p≤Sμ,ν,u+,v−≤2Ψgv−guEμ,α,ξγ,δ,k,ζηgv−guμ;pfv+2fu+fvJvuEμ,α,ξγ,δ,k,ζ,g;ΨIu,v;g−gu−2λJvuEμ,α,ξγ,δ,k,ζ,g;Ψv−u2Iu,v;Idg−u+vIu,v;g.Proof.
For the kernel defined in equation (7) and the strongly φ-convexity of the function f on u,v, the following inequalities hold, respectively:(25)JxuEμ,α,ξγ,δ,k,ζ,g;Ψg′x≤JvuEμ,α,ξγ,δ,k,ζ,g;Ψg′x,x∈u,v,(26)fx≤fv+v−xv−uφfu,fv−λv−xx−u.
The aforementioned inequalities are used to obtain the following integral inequality:(27)∫uvJxuEμ,α,ξγ,δ,k,ζ,g;Ψfxg′xdx≤fvJvuEμ,α,ξγ,δ,k,ζ,g;Ψ∫uvg′xdx+φfu,fvv−uJvuEμ,α,ξγ,δ,k,ζ,g;Ψ∫uvv−xg′xdx−λJvuEμ,α,ξγ,δ,k,ζ,g;Ψ∫uvv−xx−ug′xdx.
In view of Definition5, applying integration by parts, and using φx,y=x+y, from inequality (27), we get the following upper bound of the left-sided unified integral operator:(28)Fgμ,α,ξ,v−Ψ,γ,δ,k,ζfu,η;p≤Eμ,α,ξγ,δ,k,ζηgv−guμ;pΨgv−gufv+JvuEμ,α,ξγ,δ,k,ζ,g;Ψfu+fvIu,v;g−gu−λJvuEμ,α,ξγ,δ,k,ζ,g;Ψv−u2Iu,v,Idg−v+uIu,v,g.
Also, the following inequality holds:(29)JvxEμ,α,ξγ,δ,k,ζ,g;Ψg′x≤JvuEμ,α,ξγ,δ,k,ζ,g;Ψg′x,x∈u,v.
The aforementioned inequalities (26) and (29) are used to obtain the following integral inequality:(30)∫uvJvxEμ,α,ξγ,δ,k,ζ,g;Ψg′xfxdx≤JvuEμ,α,ξγ,δ,k,ζ,g;Ψfv∫uvg′xdx+φfu,fvv−u∫uvg′xv−xdx−λJvuEμ,α,ξγ,δ,k,ζ,g;Ψ∫uvv−xx−ug′xdx.
In view of Definition5 and applying integration by parts, from inequality (30), we get the following upper bound of the right-sided unified integral operator:(31)Fgμ,α,ξ,u+Ψ,γ,δ,k,ζfv,η;p≤Eμ,α,ξγ,δ,k,ζηgv−guμ;pΨgv−gufv+JvuEμ,α,ξγ,δ,k,ζ,g;Ψfu+fvIu,v;g−gu−λJvuEμ,α,ξγ,δ,k,ζ,g;Ψv−u2Iu,v,Idg−v+uIu,v,g.
Now, using Lemma1, we can write(32)∫uvfu+v2JxuEμ,α,ξγ,δ,k,ζ,g;Ψg′xdx≤∫uvJxuEμ,α,ξγ,δ,k,ζ,g;Ψg′xfxdx+12∫uvJxuEμ,α,ξγ,δ,k,ζg;Ψg′xφfx,fxdx−λu−v24∫uvJxuEμ,α,ξγ,δ,k,ζ,g;Ψg′xdx.
In view of Definition5 and φx,y=x+y, from (32), we get the following upper bound of the left-sided unified integral operator:(33)fu+v2Fgμ,α,ξ,v−Ψ,γ,δ,k,ζ1u,η;p≤2Fgμ,α,ξ,v−Ψ,γ,δ,k,ζfu,η;p−λu−v24Fgμ,α,ξ,v−Ψ,γ,δ,k,ζ1u,η;p.
Also, from Lemma1, we can write(34)∫uvfu+v2JvxEμ,α,ξγ,δ,k,ζ,g;Ψg′xdx≤∫uvJvxEμ,α,ξγ,δ,k,ζ,g;Ψg′xfxdx+12∫uvJvxEμ,α,ξγ,δ,k,ζ,g;Ψg′xφfx,fxdx−λu−v24∫uvJvxEμ,α,ξγ,δ,k,ζ,g;Ψg′xdx.
In view of Definition5 and φx,y=x+y, from (34), we get the following upper bound of the right-sided unified integral operator:(35)fu+v2Fgμ,α,ξ,u+Ψ,γ,δ,k,ζ1v,η;p≤2Fgμ,α,ξ,u+Ψ,γ,δ,k,ζfv,η;p−λu−v24Fgμ,α,ξ,u+Ψ,γ,δ,k,ζ1v,η;p.
Inequality (24) will be obtained by using (28), (31), (33), and (35).Remark 3.
Forλ=0 in (24), we get (9) of Theorem 2; if 2Iu,v;Idg>u+vIu,v;g, then we will get refinement of (9).Forφx,y=x−y in (24), we get the result for strongly convex function.Forφx,y=x−y and λ=0 in (24), we get the result of Theorem 22 in [5].Theorem 6.
Iff′ is strongly φ-convex with modulus λ≥0 along with other assumptions of Theorem 3, then the inequality(36)Fgμ,α,ξ,u+Ψ,γ,δ,k,ζf∗gx,η;p+Fgν,α,ξ,v−Ψ,γ,δ,k,ζf∗gx,η;p≤Eμ,α,ξγ,δ,k,ζηgx−guμ;pΨgx−guf′x+JxuEμ,α,ξγ,δ,k,ζ,g;Ψφf′u,f′xIu,x;g−gu−λx−u2Iu,x;Idg−x+uIu,x;g+Eν,α,ξγ,δ,k,ζηgv−gxν;pΨgv−gxf′v+JvxEν,α,ξγ,δ,k,ζ,g;Ψφf′x,f′vIx,v;g−gx−λv−x2Ix,v;Idg−v+xIx,v;gholds for x∈u,v, where(37)Fgμ,α,ξ,u+Ψ,γ,δ,k,ζf∗gx,η;p=∫uxJxςEμ,α,ξγ,δ,k,ζ,g;Ψg′ςf′ςdς,(38)Fgν,α,ξ,v−Ψ,γ,δ,k,ζf∗gx,η;p=∫xvJςxEν,α,ξγ,δ,k,ζ,g;Ψg′ςf′ςdς,and Id is the identity function.Proof.
Using stronglyφ-convexity of f′ over u,x gives(39)f′ς≤f′x+x−ςx−uφf′u,f′x−λx−ςς−u,ς∈u,x.
Using absolute value property, we can write(40)−f′x+x−ςx−uφf′u,f′x−λx−ςς−u≤f′ς≤f′x+x−ςx−uφf′u,f′x−λx−ςς−u.
The aforementioned inequality (13) and second inequality of (40) are used to obtain the following integral inequality:(41)∫uxJxςEμ,α,ξγ,δ,k,ζ,g;Ψg′ςf′ςdς≤JxuEμ,α,ξγ,δ,k,ζ,g;Ψf′x∫uxg′ςdς+φf′u,f′xx−u∫uxx−ςg′ςdς−λ∫uxx−ςς−ug′ςdς.
In view of (37) and applying integration by parts, from inequality (41), we get the following upper bound:(42)Fgμ,α,ξ,u+Ψ,γ,δ,k,ζf∗gx,η;p≤Eμ,α,ξγ,δ,k,ζηgx−guμ;pΨgx−guf′x+JxuEμ,α,ξγ,δ,k,ζ,g;Ψφf′u,f′xIu,x;g−gu−λx−u2Iu,x;Idg−x+uIu,x;g.
Also, inequality (13) and the first inequality of (40) are used to obtain the following integral inequality:(43)Fgμ,α,ξ,u+Ψ,γ,δ,k,ζf∗gx,η;p≥−Eμ,α,ξγ,δ,k,ζηgx−guμ;pΨgx−guf′x+JxuEμ,α,ξγ,δ,k,ζ,g;Ψφf′u,f′xIu,x;g−gu−λx−u2Iu,x;Idg−x+uIu,x;g.
Now, usingφ-convexity of f′ over x,v, we have(44)f′ς≤f′v+v−ςv−xφf′x,f′v−λv−ςς−x,ς∈x,v.
Inequalities (17), (38), and (44) are used to obtain the following upper bounds:(45)Fgν,α,ξ,v−Ψ,γ,δ,k,ζf∗gx,η;p≤Eν,α,ξγ,δ,k,ζηgv−gxν;pΨgv−gxf′v+JvxEν,α,ξγ,δ,k,ζ,g;Ψφf′x,f′vIx,v;g−gx−λv−x2Ix,v;Idg−v+xIx,v;g,(46)Fgν,α,ξ,v−Ψ,γ,δ,k,ζf∗gx,η;p≥−Eν,α,ξγ,δ,k,ζηgv−gxν;pΨgv−gxf′v+JvxEν,α,ξγ,δ,k,ζ,g;Ψφf′x,f′vIx,v;g−gx−λv−x2Ix,v;Idg−v+xIx,v;g.
Inequality (36) will be obtained by using (42)–(46).Corollary 2.
By settingμ=ν in (36), we get the following inequality:(47)Fgμ,α,ξ,u+Ψ,γ,δ,k,ζf∗gx,η;p+Fgμ,α,ξ,v−Ψ,γ,δ,k,ζf∗gx,η;p≤Eμ,α,ξγ,δ,k,ζηgx−guμ;pΨgx−guf′x+JxuEμ,α,ξγ,δ,k,ζ,g;Ψφf′u,f′xIu,x;g−gu−λx−u2Iu,x;Idg−x+uIu,x;g+Eμ,α,ξγ,δ,k,ζηgv−gxμ;pΨgv−gxf′v+JvxEμ,α,ξγ,δ,k,ζ,g;Ψφf′x,f′vIx,v;g−gx−λv−x2Ix,v;Idg−v+xIx,v;g.Remark 4.
Forλ=0 in (36), we get inequality (10) of Theorem 3; if 2Iu,x;Idg>x+uIu,x;g and 2Ix,v;Idg>v+xIx,v;g, then we will get the refinement of (10).Forφx,y=x−y in (47), we get the result for strongly convex function.Forφx,y=x−y and λ=0 in (47), we get the result of Theorem 25 in [5].


## 3. Results for Fractional Integral Operators

In this section, we give the bounds of some of the fractional integral operators which will be deduced from the results ofSection 2. Throughout this section, we assume that p=η=0.Proposition 1.
Under the assumptions of Theorem4, the following result holds:(48)ΓαIgαu+fx+Igαv−fx≤gx−guαfx+gv−gxαfv+gx−guα−1φfu,fxIu,x;g−gu−λx−u2Iu,x;Idg−x+uIu,x;g+gv−gxα−1φfx,fvIx,v;g−gx−λv−x2Ix,v;Idg−v+xIx,v;g.Proof.
ForΨς=ςα,whereα>0, Theorem 4 gives (48).Proposition 2.
Under the assumptions of Theorem4, the following inequality holds:(49)ΓαIu+Ψfx+Iv−Ψfx≤Ψx−ufx+Ψx−u2φfu,fx+Ψv−xfv+Ψv−x2φfx,fv−Ψx−uλx−u26−Ψv−xλv−x26.Proof.
Forg as identity function, Theorem 4 gives (49).Corollary 3.
ForΨς=ςα/k/kΓkα, (5) and (6) reduce to the fractional integral operators given in [5]. Further, the following bound for α≥k is also satisfied:(50)Igαu+kfx+Igαv−kfx≤1kΓkαgx−guα/kfx+gv−gxα/kfv+gx−guα/k−1φfu,fxIu,x;g−gu−λx−u2Iu,x;Idg−x+uIu,x;g+gv−gxα/k−1φfx,fvIx,v;g−gx−λv−x2Ix,v;Idg−v+xIx,v;g.Corollary 4.
ForΨς=ςα, where α≥1, and g as identity function, (5) and (6) give fractional integrals defined in [12]. Further, the following bound is also satisfied:(51)ΓαIαu+fx+Iαv−fx≤x−uαfx+v−xαfv+x−uα2φfu,fx+v−xα2φfx,fv−λx−uα+26−λv−xα+26.Corollary 5.
UsingΨς=ςα/k/kΓkα and g as identity functions, (5) and (6) reduce to the fractional integral operators given in [13]. Further, the following bound is also satisfied:(52)kΓkαIαu+kfx+Iαv−kfx≤x−uα/kfx+v−xα/kfv+x−uα/k2φfu,fx+v−xα/k2φfx,fv−λx−uα/k+26−λv−xα/k+26.Corollary 6.
ForΨς=ςα,whereα>0, and gx=xρ/ρ,whereρ>0, (5) and (6) reduce to the fractional integral operators given in [14]. Further, the following bound is also satisfied:(53)Iρu+αfx+Iρv−αfx≤1ραΓαxρ−uραfx+vρ−xραfv+xρ−uρα−1φfu,fxxρ+1−uρ+1x−uρ+1−uρ−λ2xρ+2−uρ+2ρ+2−x+uxρ+1−uρ+1ρ+1+vρ−xρα−1φfx,fvvρ+1−xρ+1v−xρ+1−xρ−λ2vρ+2−xρ+2ρ+2−v+xvρ+1−xρ+1ρ+1.Corollary 7.
ForΨς=ςα,whereα>0, and gx=xs+1/s+1,wheres>0, (5) and (6) give the following fractional integral operators:(54)Fu+ςα/Γα,gfx=Isu+αfx=s+11−αΓα∫uxxs+1−ςs+1α−1ςsfςdς,Fv−ςα/Γα,gfx=Isv−αfx=s+11−αΓα∫xvςs+1−xs+1α−1ςsfςdς.
Further, the following bound is also satisfied:(55)s+1αΓαISu+αfx+ISv−αfx≤xs+1−us+1αfx+vs+1−xs+1αfv+xs+1−us+1α−1φfu,fxxs+2−us+2x−us+2−us+1−λ2xs+3−us+3s+3−x+uxs+2−us+2s+2+vs+1−xs+1α−1φfx,fvvs+2−xs+2v−xs+2−xs+1−λ2vs+3−xs+3s+3−v+xvs+2−xs+2s+2.Corollary 8.
ForΨς=ςα/k/kΓkα and gx=xs+1/s+1,wheres>0, (5) and (6) reduce to the fractional integral operators given in [15]. Further, the following bound is also satisfied:(56)Iksu+αfx+Iksv−αfx≤1s+1α/kkΓkαxs+1−us+1α/kfx+vs+1−xs+1α/kfv+xs+1−us+1α/k−1φfu,fxxs+2−us+2x−us+2−us+1−λ2xs+3−us+3s+3−x+uxs+2−us+2s+2+vs+1−xs+1α/k−1φfx,fvvs+2−xs+2v−xs+2−xs+1−λ2vs+3−xs+3s+3−v+xvs+2−xs+2s+2.Corollary 9.
ForΨς=ςα,whereα>0, gx=xβ+s/β+s,whereβands>0, (5) and (6) reduce to the fractional integral operators given in [16]. Further, the following bound is also satisfied:(57)Iβsu+αfx+Iβsv−αfx≤xβ+s−uβ+sαfx+vβ+s−xβ+sαfv+xβ+s−uβ+sα−1φfu,fxxβ+s+1−uβ+s+1x−uβ+s+1−uβ+s−λ2xβ+s+2−uβ+s+2β+s+2−x+uxβ+s+1−uβ+s+1β+s+1+vβ+s−xβ+sα−1φfx,fvvβ+s+1−xβ+s+1v−xβ+s+1−xβ+s−λ2vβ+s+2−xβ+s+2β+s+2−v+xvβ+s+1−xβ+s+1β+s+1.Corollary 10.
UsingΨς=ςα and gx=x−uρ/ρ in (5) and gx=−v−xρ/ρ in (6), where ρ>0, fractional integral operators given in [17] are obtained. Further, the following bound is also satisfied:(58)Iρu+αfx+Iρv−αfx≤1ραΓαx−uραfx+φfu,fxx−uραρ+1−λρρ+1ρ+2x−uρα+1+v−xραfv+φfx,fvv−xραρ+1−λρρ+1ρ+2v−xρα+1.Corollary 11.
ForΨς=ςα/k/kΓkα,whereα>k, and gx=x−uρ/ρ in (5) and gx=−v−xρ/ρ in (6), where ρ>0, fractional integral operators given in [18] are obtained. Further, the following bound is also satisfied:(59)Ikρu+αfx+Ikρv−αfx≤1ρα/kkΓkαx−uρα/kfx+φfu,fxx−uρα/kρ+1−λρρ+1ρ+2x−uρα/k+2+v−xρα/kfv+φfx,fvv−xρα/kρ+1−λρρ+1ρ+2v−xρα/k+2.Remark 5.
Forλ=0, all the results of Section 3 reduce to the results of Section 3 in [4]; if λ>0, then all the results of Section 3 give the refinements of the results of Section 3 in [4].Forφx,y=x−y and λ=0, all the results of Section 3 reduce to the propositions and corollaries of [5].
Further, various bounds can be obtained by applying Theorems5 and 6 which we leave for the reader.


## 4. Concluding Remarks

The paper presents bounds of unified integral operators (5) and (6) for strongly φ-convex functions. These bounds are refinements of bounds obtained for unified integral operators for φ-convex functions in [4]. The results for fractional integral operators have been deduced which provide bounds for Riemann–Liouville and other well-known fractional integral operators.



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