# On Generalized Strongly Convex Functions and Unified Integral Operators

Authors: Timing Yu; Ghulam Farid; Kahkashan Mahreen; Chahn Yong Jung; Soo Hak Shim
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we define a strongly exponentiallyα,h−m-convex function that generates several kinds of strongly convex and convex functions. The left and right unified integral operators of these functions satisfy some integral inequalities which are directly related to many unified and fractional integral inequalities. From the results of this paper, one can obtain various fractional integral operator inequalities that already exist in the literature.


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## Body

## 1. Introduction

Mathematical inequalities are very important and useful in the study of various kinds of problems in mathematics, operation research, economics, physics, and engineering and in many other subjects of diverse fields. Integral inequalities provide support in the establishment of solutions to differential equations and solutions of boundary/initial value problems and their uniqueness. In recent years, mathematical inequalities for different kinds of fractional integrals are in the focus of researchers, and a lot of papers have been published on this subject. For instance, the classical inequalities such as the Chebyshev inequality, Minkowski inequality, Hadamard inequality, and Ostrowski inequality have been published for different kinds of fractional derivative/integral operators.The existence of convex functions’ revolutionary enhances the theory of mathematical inequalities. The properties and presentations of convex functions in different forms (geometric and analytical) create the room to invent other kinds of notions and functions. There are many types of convex functions that have been formulated from the analytical presentation of convex functions such as logarithmic convex, geometric convex, superquadratic, strongly convex,h-convex, m-convex, α,m-convex, h,m-convex, and α,h−m-convex functions. For different kinds of convex functions and their utilizations, one can see [1–6] and references therein. For applications in fractional integral inequalities, we refer the readers to [7–23] and references therein.The aim of this paper is to study different kinds of convexities and fractional integral operators simultaneously in the form of unified integral inequalities. We will define a new function called strongly exponentiallyα,h−m-convex function that unifies several kinds of convex functions in a single inequality (20). Also, left- and right-sided unified integral operators given in Definition 2 are utilized for this new class of convex functions to obtain the required results.A huge amount of recent work in the theory of inequalities for real functions consists of the fractional integral inequalities and their applications. The Hadamard inequality is the most popular among classical inequalities which is studied extensively by researchers in near past decades for fractional integral operators. We will find an integral inequality of Hadamard type by applying operators (1) and (2) of the newly defined function. This will behave as a generalization as well as refinement of a lot of Hadamard and fractional Hadamard inequalities for convex functions of almost all kinds. Also, we will provide the upper bounds and bounds of several kinds of already known fractional integral operators connected with unified integral operators (1) and (2). Ultimately, the results of this paper work for all kinds of convexities and fractional integral operators that occur in the literature. Next, we give the results from the literature for fractional integral operators. A generalized fractional integral operator containing an extended generalized Mittag-Leffler function is defined as follows.Definition 1.
(see [7]). Let ω,μ,α,l,γ,c∈C, ℜμ,ℜα,ℜl>0, and ℜc>ℜγ>0 with p≥0, δ>0, and 0<k≤δ+ℜμ. Let f∈L1a,b and x∈a,b. Then, the generalized fractional integral operators εμ,α,l,ω,a+γ,δ,k,cf and εμ,α,l,ω,b−γ,δ,k,cf are defined by(1)εμ,α,l,ω,a+γ,δ,k,cfx,ω;p=∫axx−tα−1Eμ,α,lγ,δ,k,cωx−tμ;pftdt,(2)εμ,α,l,ω,b−γ,δ,k,cfx,ω;p=∫xbt−xα−1Eμ,α,lγ,δ,k,cωt−xμ;pftdt,where(3)Eμ,α,lγ,δ,k,ct;p=∑n=0∞βpγ+nk,c−γβγ,c−γcnkΓμn+αtnlnδ, is the extended generalized Mittag-Leffler function.Mittag-Leffler function, introduced in 1903, is a generalization of the exponential function just by a single parameter with a convergence condition. Further generalization by another parameter was given by Wiman and further extended by Prabhakar and then by other authors; to see its importance, we suggest the reader to [24, 25]. Next, we give recently established results in [26] for the fractional integral operators containing the extended generalized Mittag-Leffler function defined in [7] by utilizing α,h−m-convex functions.Theorem 1.
Letf:a,b⟶ℝ be a real-valued function. If f is positive and α,h−m-convex, α,m∈0,12, m≠0, then for α′,β≥1, the following fractional integral inequality for generalized fractional integral operators holds:(4)εμ,α′,l,ω,a+γ,δ,k,cfx;p+εμ,β,l,ω,b−γ,δ,k,cfx;p≤x−aJα′−1,a+x;pfa∫01hzαdz+mfxm∫01h1−zαdz+b−xJβ−1,b−x;pfb∫01hzαdz+mfxm∫01h1−zαdz.Theorem 2.
Letf:a,b⟶ℝ be a real-valued function. If f is differentiable and f′ is α,h−m-convex, α,m∈0,12, m≠0, then for α′,β≥1, the following fractional integral inequality for generalized fractional integral operators holds:(5)|εμ,α′+1,l,ω,a+γ,δ,k,cfx;p+εμ,β+1,l,ω,b−γ,δ,k,cfx;p−Jα′−1,a+x;pfa+Jβ−1,b−x;pfb|≤x−aJα′−1,a+x;pf′a∫01hzαdz+mf′xm∫01h1−zαdz+b−xJβ−1,b−x;pf′b∫01hzαdz+mf′xm∫01h1−zαdz.Theorem 3.
Letf:a,b⟶ℝ, a>b, be a real-valued function. If f is positive and α,h−m-convex, α,m∈0,12, m≠0, and fa+mb−x/m=fx, then for α′,β≥1, the following fractional integral inequality holds:(6)1h12α+mh1−12αfa+mb2Jβ+1,b−a;p+Jα′+1,a+b;p≤εμ,β+1,l,ω,b−γ,δ,k,cfa;p+εμ,α′+1,l,ω,a+γ,δ,k,cfb;p≤Jβ−1,b−a;p+Jα′−1,a+b;pb−a2×fb∫01hzαdz+mfam∫01h1−zαdz.Next, we give the definition of the unified integral operator based on a kernel which depends on a strictly increasing function along with two variables.Definition 2.
(see [16]). Let f,g:a,b⟶ℝ, 0<a<b, be the functions such that f is positive and f∈L1a,b and g is differentiable and strictly increasing. Also, let ϕ/x be an increasing function on a,∞ and α,l,γ,c∈ℂ, p,μ,δ≥0, and 0<k≤δ+μ. Then, for x∈a,b, the left and right integral operators are defined by(7)gFμ,α,l,a+ϕ,γ,δ,k,cfx,ω;p=∫axKxyEμ,α′,lγ,δ,k,c,g;ϕfydgy,(8)Fgμ,α,l,b−ϕ,γ,δ,k,cfx,ω;p=∫xbKyxEμ,α′,lγ,δ,k,c,g;ϕfydgy,where the involved kernel is defined by(9)KxyEμ,α′,lγ,δ,k,c,g;ϕ=ϕgx−gygx−gyEμ,α′,lγ,δ,k,cωgx−gyμ;p.
The following property of the kernel given in (9) is used frequently in the whole paper.
P: let g and ϕ/I be increasing functions. Then, for m<t<n, m,n∈a,b, the kernel KmnEμ,α′,lγ,δ,k,c,g;ϕ satisfies the following inequality:(10)KtmEμ,α′,lγ,δ,k,c,g;ϕg′t≤KnmEμ,α′,lγ,δ,k,c,g;ϕg′t.
The reverse of inequality (10) holds when g and ϕ/I are of opposite monotonicity. For further properties, see [27].For suitable settings of functionsϕ and g and certain values of parameters included in the Mittag-Leffler function, it unifies many fractional integral operators studied in [10, 18, 20, 25, 28–37]; see [38], Remarks 6 and 7. The following results are proved for unified integral operators of strongly convex functions in [39].Theorem 4.
Letf:a,b⟶ℝ be a positive, integrable, and strongly convex function with m∈0,1. Let g:a,b⟶ℝ be a differentiable and strictly increasing function; also, let ϕ/x be an increasing function on a,b. If α,β,l,γ,c∈ℝ+, p,μ,δ≥0, and 0<k≤δ+μ, then for x∈a,b, we have(11)Fgμ,α,l,a+ϕ,γ,δ,k,cfx,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,cfx,ω;p≤KxaEμ,α,lγ,δ,k,c,g;ϕgx−gafx+fa−λx−a2Ia,x,Idg−a+xIa,x,g+KbxEμ,β,lγ,δ,k,c,g;ϕgb−gxfb+fx−λb−x2Ix,b,Idg−x+bIx,b,g.Theorem 5.
With the assumptions of Theorem4, in addition, if fx=fa+b−x, then we have(12)fa+b2Fgμ,α,l,b−ϕ,γ,δ,k,c1a,ω;p+λ4Fgμ,α,l,b−ϕ,γ,δ,k,ca+b−2x2a,ω;p+fa+b2Fgμ,β,l,a+ϕ,γ,δ,k,c1b,ω;p+λ4Fgμ,β,l,a+ϕ,γ,δ,k,ca+b−2x2b,ω;p≤Fgμ,α,l,b−ϕ,γ,δ,k,cfa,ω;p+Fgμ,β,l,a+ϕ,γ,δ,k,cfb,ω;p≤KbaEμ,α,lγ,δ,k,c,g;ϕ+KbaEμ,β,lγ,δ,k,c,g;ϕgb−gafb+fa−b−aλ2Ia,b,Idg−a+bIa,b,g.Theorem 6.
Along with the assumptions of Theorem4, if the strong convexity of f is replaced with the strong convexity of f′, then for x∈a,b, we have(13)Fgμ,α,l,a+ϕ,γ,δ,k,cf∗gx,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,cf∗gx,ω;p≤KxaEμ,α,lγ,δ,k,c,g;ϕ×gx−gaf′x+f′a−λx−a2Ia,x,Idg−a+xIa,x,g+KbxEμ,β,lγ,δ,k,c,g;ϕgb−gxf′b+f′x−λb−x2Ix,b,Idg−x+bIx,b,g,where Fgμ,α′,l,a+ϕ,γ,δ,k,cf∗gx,ω;pandFgμ,β,l,b−ϕ,γ,δ,k,cf∗gx,ω;p are the same as defined in Theorem 9.The unified integral operators are also analyzed forα,m-convex, h−m-convex, s,m-convex, and α,h−m-convex functions in [14, 27, 38, 40]. Here, we will unify bounds of unified and several kinds of fractional integral operators for different classes of convex, exponentially convex, strongly convex, and strongly exponentially convex functions in the results of this paper. Next, we give the definition of the h−m-convex function as follows.Definition 3.
(see [4]). Let J⊆ℝ be an interval containing 0,1, and let h:J⟶ℝ be a nonnegative function. A function f:0,b⟶ℝ is called an h−m-convex function if f is nonnegative and for all x,y∈0,b, m∈0,1, and t∈0,1, one has(14)ftx+m1−ty≤htfx+mh1−tfy.By suitably defining the functionh and giving particular values to parameter m, the above definition produces many known functions.Remark 1.
(i)
Ifm=1, then the h-convex function can be obtained(ii)
Ifht=t, then the m-convex function can be obtained(iii)
Ifht=t and m=1, then the convex function can be obtained(iv)
Ifht=1 and m=1, the p-function can be obtained(v)
Ifht=ts and m=1, then the s-convex function can be obtained(vi)
Ifht=1/t and m=1, then the Godunova–Levin function can be obtained(vii)
Ifht=1/ts and m=1, then the s-Godunova–Levin function of the second kind can be obtainedThe notion of theα,m-convex function is defined as follows.Definition 4.
(see [3]). A function f:0,b⟶ℝ,b>0, is said to be α,m-convex, where α,m∈0,12, if(15)ftx+m1−ty≤tαfx+m1−tαfyholds for all x,y∈0,bandt∈0,1.Remark 2.
(i)
If we considerα,m=1,m, then (15) gives the definition of the m-convex function(ii)
If we considerα,m=1,1, then (15) gives the definition of the convex function(iii)
If we considerα,m=1,0, then (15) gives the definition of the star-shaped functionThe notion of theα,h−m-convex function is defined as follows.Definition 5.
(see [19]). Let J⊆ℝ be an interval containing 0,1, and let h:J⟶ℝ be a nonnegative function. A function f:0,b⟶ℝ is called an α,h−m-convex function if f is nonnegative and for all x,y∈0,b,t∈0,1, and α,m∈0,12, one has(16)ftx+m1−ty≤htαfx+mh1−tαfy.Definition 6.
(see [5]). Let I be a nonempty convex subset of the normed space X,.. A real-valued function f is said to be strongly convex, with modulus λ≥0, on I if for each x,y∈I and t∈0,1, we have(17)ftx+1−ty≤tfx+1−tfy−λt1−tb−a2.Definition 7.
(see [41]). A function f:0,b⟶ℝ is said to be an s,m-convex function, where s,m∈0,12 and b>0, if for every x,y∈0,b and t∈0,1, we have(18)ftx+m1−ty≤tsfx+m1−tsfy.The notion of the stronglys,m-convex function is defined as follows.Definition 8.
(see [42]). A function f:0,+∞⟶ℝ is said to be a strongly s,m-convex function, with modulus λ≥0, for s,m∈0,12 if(19)fta+m1−tb≤tsfa+m1−tsfb−λmt1−tb−a2holds for all a,b∈0,+∞ and t∈0,1.The rest of the paper is organized as follows: in Section2, first, we define the strongly exponentially α,h−m-convex function, and then upper bounds of unified integral operators of this function are obtained. Using the condition of symmetry and strongly exponentially α,h−m-convexity of f′, further bounds are obtained in the form of a modulus inequality. These results hold for all fractional integral operators comprised in [38], Remarks 6 and 7, deducible from unified integral operators (7) and (8) and for convex functions of various kinds.


## 2. Main Results

First, we give the definition of a new function called the strongly exponentiallyα,h−m-convex function.Definition 9.
LetJ⊆ℝ be an interval containing 0,1, and let h:J⟶ℝ be a nonnegative function. A function f:0,b⟶ℝ is called a strongly exponentially α,h−m-convex function, with modulus λ≥0, if f is nonnegative and for all x,y∈0,b,t∈0,1, η∈ℝ, and α,m∈0,12, one has(20)ftx+m1−ty≤htαfxeηx+mh1−tαfyeηy−mλex+yηhtαh1−tαy−x2.The definition of the stronglyα,h−m-convex function is obtained as follows.Definition 10.
LetJ⊆ℝ be an interval containing 0,1, and let h:J⟶ℝ be a nonnegative function. A function f:0,b⟶ℝ is called a strongly α,h−m-convex function, with modulus λ≥0, if f is nonnegative and for all x,y∈0,b,t∈0,1, and α,m∈0,12, one has(21)ftx+m1−ty≤htαfx+mh1−tαfy−mλhtαh1−tαy−x2.Many known and new convex, exponentially convex, strongly convex, and strongly exponentially convex functions can be obtained from the above definitions. The following theorem provides an upper bound for unified integral operators of strongly exponentiallyα,h−m-convex functions.Theorem 7.
Letf∈L1a,b,0≤a<mb, be a positive strongly exponentially α,h−m-convex function, m∈0,1. Let g be a differentiable and strictly increasing function; also, let ϕ/x be an increasing function and hxhy≤hx+y. If α′,β,l,γ,c∈ℝ+, p,μ≥0, δ≥0, and 0<k≤δ+μ, then for x∈a,b, we have(22)Fgμ,α′,l,a+ϕ,γ,δ,k,cfx,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,cfx,ω;p≤KxaEμ,α′,lγ,δ,k,c,g;ϕx−afaXxazα,h;g′eηa+mfx/mXxa1−zα,h;g′eηx/m−λx−ma2h1gx−gamx−aeηa+x/m+KbxEμ,β,lγ,δ,k,c,g;ϕb−xfbXxbzα,h;g′eηb+mfx/mXxb1−zα,h;g′eηx/m−λbm−x2h1gb−gxmb−xeηb+x/m,where Xxazα,h;g′=∫01hzαg′x−zx−adz and Xxa1−zα,h;g′=∫01h1−zαg′x−zx−adz.Proof.
By usingP, the following inequalities hold:(23)KxtEμ,α′,lγ,δ,k,c,g;ϕg′t≤KxaEμ,α′,lγ,δ,k,c,g;ϕg′t,a<t<x,(24)KtxEμ,α′,lγ,δ,k,c,g;ϕgt≤KbxEμ,α′,lγ,δ,k,c,g;ϕg′t,x<t<b.
Using strongly exponentiallyα,h−m-convexity of f, we have(25)ft≤hx−tx−aαfaeηa+mh1−x−tx−aαfx/meηx/m−λx−am2meηa+x/mhx−tx−aαh1−x−tx−aα,(26)ft≤ht−xb−xαfbeηb+mh1−t−xb−xαfx/meηx/m−λbm−x2meηb+x/mht−xb−xαh1−t−xb−xα.
From (23) and (25), the following integral inequality holds true:(27)∫axKxtEμ,α′,lγ,δ,k,c,g;ϕftdgt≤faKxaEμ,α′,lγ,δ,k,c,g;ϕ∫axhx−tx−aαdgt+mfx/meηx/mKxaEμ,α′,lγ,δ,k,c,g;ϕ∫axh1−x−tx−aαdgt−λbm−x2mKxaEμ,α′,lγ,δ,k,c,g;ϕ∫axh1−x−tx−aαhx−tx−aαdgt.
By using (7) of Definition 2 on the left-hand side and by setting z=x−t/x−a on the right-hand side, the following inequality is obtained:(28)Fgμ,α′,l,a+ϕ,γ,δ,k,cfx,ω;p≤KxaEμ,α′,lγ,δ,k,c,g;ϕx−a×faeηa∫01hzαg′x−zx−adz+mfx/meηx/m∫01h1−zαg′x−zx−adz−λx−am2meηa+x/m∫01h1−zαhzαg′x−zx−adz.
The above inequality can be written as follows:(29)Fgμ,α′,l,a+ϕ,γ,δ,k,cfx,ω;p≤KxaEμ,α′,lγ,δ,k,c,g;ϕx−a×faeηaXxazα,h;g′+mffx/meηx/mXxa1−zα,h;g′−λx−ma2h1gx−gamx−aeηa+x/m.
On the contrary, multiplying (24) and (26) and adopting the same pattern as we did for (23) and (25), the following inequality holds true:(30)Fgμ,β,l,b−ϕ,γ,δ,k,cfx,ω;p≤KbxEμ,β,lγ,δ,k,c,g;ϕb−x×fbeηb∫01hzαg′x−zx−bdz+mfx/meηx/m∫01h1−zαg′x−zx−bdz−λbm−x2meηb+x/m∫01h1−zαhzαg′x−zx−bdz.
The above inequality can be written as follows:(31)Fgμ,β,l,b−ϕ,γ,δ,k,cfx,ω;p≤KbxEμ,β,lγ,δ,k,c,g;ϕb−x×fbeηbXxbzα,h;g′+mfx/meηx/mXxb1−zα,h;g′−λbm−x2h1gb−gxmb−xeηb+x/m.
By adding (29) and (31), (22) can be obtained.Corollary 1.
If we takeη=0 and m=α=1, then (22) holds for strongly h-convex functions.(32)Fgμ,α′,l,a+ϕ,γ,δ,k,cfx,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,cfx,ω;p≤KxaEμ,α′,lγ,δ,k,c,g;ϕx−afaYxaz,h;g′+fxYxay,h;g′−λx−ah1gx−ga+KbxEμ,β,lγ,δ,k,c,g;ϕb−xfbYxbz,h;g′+fxYxby,h;g′−λb−xh1gb−gx.The following remark summarizes the connection of Theorem7 with already known results.Remark 3.
(i)
If we considerη=0 and α=1 in (22), then Theorem 1 of [40] can be obtained(ii)
If we considerη=0, α′=β, and ht=t in (22), then Theorem 1 of [14] can be obtained(iii)
If we considerη=0, λ=0, α′=β, m=α=1, and ht=t in (22), then Theorem 8 of [38] can be obtained(iv)
If we considerη=0, λ=0, ϕt=Γα′tα′/k/kΓkα′, ht=gt=t, α=m=1, and p=ω=0 in (22), then Theorem 1 of [13] can be obtained(v)
If we considerη=0, λ=0, and α′=β in the result of (iv), then Corollary 1 of [13] can be obtained(vi)
If we considerη=0, λ=0, k=1, and x=a or x=b in the result of (v), then Corollary 2 of [13] can be obtained(vii)
If we considerη=0, λ=0, k=1, and x=a+b/2 in the result of (v), then Corollary 3 of [13] can be obtained(viii)
If we considerη=0, λ=0, ϕt=tα′, α=1, and gt=t in (22), then Theorem 1 of [11] can be obtained(ix)
If we considerη=0, λ=0, ϕt=tα′, α=m=1, and ht=gt=t in (22), then Corollary 1 of [11] can be obtained(x)
If we considerη=0, λ=0, ϕt=Γα′tα′, α=1, p=ω=0, and gt=t in (22), then Theorem 1 of [12] can be obtained(xi)
If we considerη=0, λ=0, and α′=β in the result of (x), then Corollary 1 of [12] can be obtained(xii)
If we considerη=0, λ=0, α′=β, ϕt=Γα′tα′, p=ω=0, gt=t, and α=1 and using Remark (2) of [12] in (22), then Corollary 2 of [12] can be obtained(xiii)
If we considerη=0, λ=0, ϕt=Γα′tα′, α=1, p=ω=0,ht=1,ht=tp, respectively, and gt=t in (22), then Theorem 3 of [12] can be obtained(xiv)
If we considerη=0, λ=0, ϕt=Γα′tα′, p=ω=0, m=α=1, and gt=ht=t in (22), then Theorem 1 of [15] can be obtained(xv)
If we considerη=0, λ=0, and α′=β in the result of (xiv), then Corollary 1 of [15] can be obtained(xvi)
If we considerη=0, λ=0, ϕt=Γα′tα′, p=ω=0, m=α=1, and ht=t in (22), then Theorem 1 of [17] can be obtained(xvii)
If we considerη=0, λ=0, α′=β, ϕt=Γα′tα′, p=ω=0, m=α=1, and ht=t in (22), then Corollary 1 of [17] can be obtained(xviii)
If we considerη=0, λ=0, ϕt=tα′, gt=t, ht=ts, and α=m=1 in (22), then Theorem 2.1 of [43] can be obtained(xix)
If we considerη=0, λ=0, α′=β, ϕt=tα′, gt=t, m=α=1, and ht=ts in (22), then Corollary 2.1 of [43] can be obtained(xx)
If we considerη=0, λ=0, p=ω=0, α=1, and ht=ts in (22), then Theorem 1 of [44] can be obtained(xxi)
If we considerη=0, λ=0, ϕt=tα′, ht=ts, α=1, and gt=t in (22), then Theorem 1 of [41] can be obtained(xxii)
If we considerη=0, λ=0, and α′=β in the result of (xxi), then Corollary 1 of [41] can be obtained(xxiii)
If we considerη=0, λ=0, ϕt=Γα′tα′/k/kΓkα′, ht=z, and p=ω=0 in (22), then Theorem 1 of [21] can be obtained(xxiv)
If we considerη=0, λ=0, and α′=β in the result of (xxiii), then Corollary 1 of [21] can be obtainedThe following lemma will be used for the proof of the upcoming theorem.Lemma 1.
Letf:a,b⟶ℝ,0≤a<mb, be a strongly exponentially α,h−m-convex function. If fx/ex=fa+mb−x/m/ea+mb−x/m,m≠0, then the following inequality holds:(33)fa+mb2≤h12α+mh2α−12αfxeηx−λmh12αh2α−12αa+mb−x−mx2eηa+mb+xm−1/m.Proof.
Asf is a strongly exponentially α,h−m-convex function, we have(34)fa+mb2≤h12αf1−ta+mtbeη1−ta+mtb+mh2α−12αfta+m1−tb/meηta+m1−tb/m−λmh12αh2α−12αt1+ma−mb+mb−ma2eη1−ta+mtb+ta+m1−tb/m.
Letx=a1−t+mtb. Then, we have a+mb−x=ta+m1−tb, and using the given condition on f, inequality (33) can be obtained. □The following result provides bounds in the form of Hadamard inequality for strongly exponentiallyα,h−m-convex functions.Theorem 8.
The conditions onf,g,h, and ϕ are the same as those in Theorem 7, and in addition, if fx/ex=fa+mb−x/m/ea+mb−x/m, then we have(35)Aηh1/2α+mh2α−1/2αfa+mb2Fgμ,α′,l,a+ϕ,γ,δ,k,c1b,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,c1a,ω;p+λmh12αh2α−12αFgμ,α′,l,a+ϕ,γ,δ,k,ca+mb−x−mx2b,ω;peηmb+a−b/m+Fgμ,β,l,b−ϕ,γ,δ,k,ca+mb−x−mx2a,ω;peηb≤Fgμ,α′,l,a+ϕ,γ,δ,k,cfb,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,cfa,ω;p≤b−aKbaEμ,α′,lγ,δ,k,c,g;ϕ+KbaEμ,β,lγ,δ,k,c,g;ϕfbXbazα,h;g′eηb+mfa/mXba1−zα,h;g′eηa/m−λa−bm2h1gb−gamb−aeηb+a/m,where Aη=eηa when η≥0 and Aη=eηb when η<0.Proof.
By usingP, the following inequalities hold:(36)KxaEμ,β,lγ,δ,k,c,g;ϕg′x≤KbaEμ,β,lγ,δ,k,c,g;ϕg′x,a<x<b,(37)KbxEμ,α′,lγ,δ,k,c,g;ϕg′x≤KbaEμ,α′,lγ,δ,k,c,g;ϕg′x,a<x<b.
Using the strongly exponentiallyα,h−m-convexity of f, we have(38)fx≤hx−ab−aαfbeηb+mh1−x−ab−aαfa/meηa/m−λa−bm2meηb+a/mhx−ab−aαh1−x−ab−aα.
Multiplying (36) and (38) and integrating the resulting inequality over a,b, we obtain(39)∫abKxaEμ,β,lγ,δ,k,c,g;ϕfxdgx≤fbeηbKbaEμ,β,lγ,δ,k,c,g;ϕ×∫abhx−ab−aαdgx+mfa/meηa/mKbaEμ,β,lγ,δ,k,c,g;ϕ∫abh1−x−ab−aαdgx−λa−bm2meηb+a/m∫abhx−ab−aαh1−x−ab−aαdgx.
Now, by using (7) of Definition 2 on the left-hand side and by setting z=x−a/b−a on the right-hand side, the following inequality is obtained:(40)Fgμ,β,l,b−ϕ,γ,δ,k,cfa,ω;p≤KbaEμ,β,lγ,δ,k,c,g;ϕb−a×fbeηb∫01hzαg′a+zb−adz+mfa/meηa/m∫01h1−zαg′a+zb−adz−λa−bm2meηb+a/m∫01hzαh1−zαg′a+zb−adz.
The above inequality can be written as(41)Fgμ,β,l,b−ϕ,γ,δ,k,cfa,ω;p≤KbaEμ,β,lγ,δ,k,c,g;ϕb−a×fbeηbXbazα,h;g′+mfa/meηa/mXba1−zα,h;g′−λa−bm2h1gb−gamb−aeηb+a/m.
Adopting the same pattern of simplification as we did for (36) and (38), the following inequality can be observed from (37) and (38):(42)Fgμ,α′,l,a+ϕ,γ,δ,k,cfb,ω;p≤KbaEμ,α′,lγ,δ,k,c,g;ϕb−a×fbeηbXbazα,h;g′+mfa/meηa/mXba1−zα,h;g′−λa−bm2h1gb−gamb−aeηb+a/m.
By adding (41) and (42), the following inequality can be obtained:(43)Fgμ,α′,l,a+ϕ,γ,δ,k,cfb,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,cfa,ω;p≤b−aKbaEμ,α′,lγ,δ,k,c,g;ϕ+KbaEμ,β,lγ,δ,k,c,g;ϕfbeηbXbazα,h;g′+mfa/meηa/mXba1−zα,h;g′−λa−bm2h1gb−gamb−aeηb+a/m.
Multiplying both sides of (33) by KxaEμ,β,lγ,δ,k,c,g;ϕdgx and integrating over a,b, we have(44)fa+mb2∫abKxaEμ,β,lγ,δ,k,c,g;ϕdgx≤h12α+mh2α−12α∫abKxaEμ,β,lγ,δ,k,c,g;ϕfxeηxdgx−λmh12αh2α−12α∫abKxaEμ,β,lγ,δ,k,c,g;ϕa+mb−x−mx2eηa+mb+xm−1/mdgx.
From Definition2, the following inequality is obtained:(45)Aηh1/2α+mh2α−1/2αfa+mb2Fgμ,β,l,b−ϕ,γ,δ,k,c1a,ω;p+λmeηbh12α×h2α−12αFgμ,β,l,b−ϕ,γ,δ,k,ca+mb−x−mx2a,ω;p≤Fgμ,β,l,b−ϕ,γ,δ,k,cfa,ω;p.
Similarly, multiplying both sides of (33) by KbxEμ,α″γ,δ,k,c,g;ϕdgx and integrating over a,b, we have(46)Aηh1/2α+mh2α−1/2αfa+mb2Fgμ,α′,l,a+ϕ,γ,δ,k,c1b,ω;p+λmeηmb+a−b/mh12α×h2α−12αFgμ,α′,l,a+ϕ,γ,δ,k,ca+mb−x−mx2b,ω;p≤Fgμ,α′,l,a+ϕ,γ,δ,k,cfb,ω;p.
From (43), (45), and (46), inequality (35) can be obtained.Corollary 2.
If we takeη=0 and m=α=1, then (35) holds for strongly h-convex functions.(47)1h1/21+mfa+b2Fgμ,α′,l,a+ϕ,γ,δ,k,c1b,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,c1a,ω;p+λh12h12Fgμ,α′,l,a+ϕ,γ,δ,k,ca+b−2x2b,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,ca+b−2x2a,ω;p≤Fgμ,α′,l,a+ϕ,γ,δ,k,cfb,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,cfa,ω;p≤b−aKbaEμ,α′,lγ,δ,k,c,g;ϕ+KbaEμ,β,lγ,δ,k,c,g;ϕfbYbaz,h;g′+faYbay,h;g′−λb−ah1gb−ga.
The following remark summarizes the connection of Theorem8 with already known results.Remark 4.
(i)
If we considerη=0, λ=0, α′=β, m=α=1, and ht=t in (35), then Theorem 22 of [38] can be obtained(ii)
If we considerη=0, λ=0, ϕt=Γα′tα′/k+1, ht=gt=t, α=m=1, and p=ω=0 in (35), then Theorem 3 of [13] can be obtained(iii)
If we considerη=0, λ=0, and α′=β in the result of (ii), then Corollary 6 of [13] can be obtained(iv)
If we considerη=0, λ=0, ϕt=Γα′tα′+1, p=ω=0, m=α=1, and gt=ht=t in (35), then Theorem 3 of [15] can be obtained(vi)
If we considerη=0, λ=0, α′=β, and ϕt=Γα′tα′ in the result of (v), then Corollary 6 of [15] can be obtained(vii)
If we considerη=0, λ=0, ϕt=Γα′tα′+1, p=ω=0, m=α=1, and ht=t in (35), then Theorem 3 of [17] can be obtained(viii)
If we considerη=0, λ=0, and α′=β in the result of (vii), then Corollary 3 of [17] can be obtained(ix)
If we considerη=0, λ=0, ϕt=tα′+1, gt=t, ht=ts, and α=m=1 in (35), then Theorem 2.4 of [43] can be obtained(x)
If we considerη=0, λ=0, and α′=β in the result of (ix), then Corollary 2.6 of [43] can be obtained
The forthcoming result is established for a differentiable functionf by utilizing the strongly exponentially α,h−m-convexity of f′.Theorem 9.
Letf:a,b⟶ℝ be a differentiable function. f′ is strongly exponentially α,h−m-convex. Let g:a,b⟶ℝ be a differentiable and strictly increasing function; also, let ϕ/x be an increasing function on a,b and hxhy≤hx+y. If α′,β,l,γ,c∈ℝ+, p,μ≥0, δ≥0, and 0<k≤δ+μ, then for x∈a,b, we have(48)Fgμ,α′,l,a+ϕ,γ,δ,k,cf∗gx,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,cf∗gx,ω;p≤KxaEμ,α′,lγ,δ,k,c,g;ϕ×x−af′aeηaXxazα,h;g′+mf′x/meηx/mXxa1−zα,h;g′−λx−ma2h1gx−gamx−a+KbxEμ,β,lγ,δ,k,c,g;ϕb−x×f′beηbXxbzα,h;g′+mf′x/meηx/mXxb1−zα,h;g′−λmb−x2h1gb−gxmb−x,where(49)Fgμ,α′,l,a+ϕ,γ,δ,k,cf∗gx,ω;p≔∫axKxtEμ,α′,lγ,δ,k,c,g;ϕf′tdgt,Fgμ,β,l,b−ϕ,γ,δ,k,cf∗gx,ω;p≔∫xbKtxEμ,α′,lγ,δ,k,c,g;ϕf′tdgt.Proof.
Using the strongly exponentiallyα,h−m-convexity of f′, we have(50)f′t≤hx−tx−aαf′aeηa+mh1−x−tx−aαf′x/meηx/m−λx−am2meηa+x/mhx−tx−aαh1−x−tx−aα,from which we can write the following inequality:(51)−hx−tx−aαf′aeηa+mh1−x−tx−aαf′x/meηx/m−λx−am2meηa+x/mhx−tx−aαh1−x−tx−aα≤f′t≤hx−tx−aαf′aeηa+mh1−x−tx−aαf′x/meηx/m−λx−am2meηa+x/mhx−tx−aαh1−x−tx−aα.
Let us consider the other inequality of (51):(52)f′t≤hx−tx−aαf′aeηa+mh1−x−tx−aαf′x/meηx/m−λx−am2meηa+x/mhx−tx−aαh1−x−tx−aα.
Multiplying (23) and (52) and integrating over a,x, we obtain(53)∫axKxtEμ,α′,lγ,δ,k,c,g;ϕf′tdgt≤f′aeηaKxaEμ,α′,lγ,δ,k,c,g;ϕ×∫axhx−tx−aαdgt+mf′x/meηx/mKxaEμ,α′,lγ,δ,k,c,g;ϕ×∫axh1−x−tx−aαdgt−λx−ma2meηa+x/mKxaEμ,α′,lγ,δ,k,c,g;ϕ×∫axh1−x−tx−aαhx−tx−aαdgt,from which we get(54)Fgμ,α′,l,a+ϕ,γ,δ,k,cfx,ω;p≤KxaEμ,α′,lγ,δ,k,c,g;ϕx−a×f′aeηaXxazα,h;g′+mf′x/meηx/mXxa1−zα,h;g′−λx−ma2h1gx−gamx−aeηa+x/m.
If we consider the first inequality of (51) and adopt the same pattern as we did for the second inequality, then we have(55)Fgμ,α′,l,a+ϕ,γ,δ,k,cgfx,ω;p≥−KxaEμ,α′,lγ,δ,k,c,g;ϕx−a×f′aeηaXxazα,h;g′+mf′x/meηx/mXxa1−zα,h;g′−λx−ma2h1gx−gamx−aeηa+x/m.
From (54) and (55), the following inequality is obtained:(56)Fgμ,α′,l,a+ϕ,γ,δ,k,cfx,ω;p≤KxaEμ,α′,lγ,δ,k,c,g;ϕx−a×f′aeηaXxazα,h;g′+mf′x/meηx/mXxa1−zα,h;g′−λx−ma2h1gx−gamx−aeηa+x/m.
Now, using the strongly exponentiallyα,h−m-convexity of f′, we have(57)f′t≤ht−xb−xαf′beηb+mh1−t−xb−xαf′x/meηx/m−λbm−x2meηb+x/mht−xb−xαh1−t−xb−xα.
On the same pattern as we did for (23) and (50), one can obtain the following inequality from (24) and (58):(58)Fgμ,β,l,b−ϕ,γ,δ,k,cfx,ω;p≤KbxEμ,β,lγ,δ,k,c,g;ϕb−x×f′beηbXxbzα,h;g′+mf′x/meηx/mXxb1−zα,h;g′−λbm−x2h1gb−gxmb−xeηb+x/m.
By adding (56) and (58), inequality (48) can be achieved.Corollary 3.
If we takeη=0 and m=α=1, then (48) holds for strongly h-convex functions.(59)Fgμ,α′,l,a+ϕ,γ,δ,k,cf∗gx,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,cf∗gx,ω;p≤KxaEμ,α′,lγ,δ,k,c,g;ϕx−af′aYxaz,h;g′+f′xYxay,h;g′−λx−ah1gx−ga+KbxEμ,β,lγ,δ,k,c,g;ϕb−xf′bYxbz,h;g′+f′xYxby,h;g′−λb−xh1gb−gx.The following remark summarizes the connection of Theorem9 with already known results.Remark 5.
(i)
If we considerη=0, λ=0, and α=1 in (48), then Theorem 3 of [40] can be obtained(ii)
If we considerη=0, λ=0, α′=β, and ht=t in (48), then Theorem 3 of [14] can be obtained(iii)
If we considerη=0, λ=0, α′=β, m=α=1, and ht=t in (48), then Theorem 25 of [38] can be obtained(iv)
If we considerη=0, λ=0, ϕt=Γα′tα′/k+1, ht=gt=t, α=m=1, and p=ω=0 in (48), then Theorem 2 of [13] can be obtained(v)
If we considerη=0, λ=0, and α′=β in the result of (iv), then Corollary 4 of [13] can be obtained(vi)
If we considerη=0, λ=0, ϕt=tα′, α=1, and gt=t in (48), then Theorem 2 of [11] can be obtained(vii)
If we considerη=0, λ=0, m=1, and ht=t in the result of (vi), then Corollary 2 of [11] can be obtained(viii)
If we considerη=0, λ=0, ϕt=Γα′tα′+1, α=1, p=ω=0, and gt=t in (48), then Theorem 2 of [12] can be obtained(ix)
If we considerη=0, λ=0, and α′=β in the result of (viii), then Corollary 4 of [12] can be obtained(x)
If we considerη=0, λ=0, ϕt=Γα′tα′+1, p=ω=0, m=α=1, and gt=ht=t in (48), then Theorem 2 of [15] can be obtained(xi)
If we considerη=0, λ=0, α′=β, and ϕt=Γα′tα′ in the result of (x), then Corollary 4 of [15] can be obtained(xii)
If we considerη=0, λ=0, ϕt=Γα′tα′+1, p=ω=0, m=α=1, and ht=t in (48), then Theorem 2 of [17] can be obtained(xiii)
If we considerη=0, λ=0, and α′=β in the result of (xii), then Corollary 2 of [17] can be obtained(xiv)
If we considerη=0, λ=0, ϕt=tα′+1, gt=t, ht=ts, and α=m=1 in (48), then Theorem 2.3 of [43] can be obtained(xv)
If we considerη=0, λ=0, and α′=β in the result of (xiv), then Corollary 2.5 of [43] can be obtained(xvi)
If we considerη=0, λ=0, p=ω=0, α=1, and ht=ts in (48), then Theorem 2 of [44] can be obtained(xvii)
If we considerη=0, λ=0, ϕt=Γα′tα′, ht=ts, α=1, and gt=t in (48), then Theorem 3 of [41] can be obtained(xviii)
If we considerη=0, λ=0, and α′=β in the result of (xvii), then Corollary 5 of [41] can be obtained(xix)
If we considerη=0, λ=0, ϕt=Γα′tα′/k+1, ht=t, and p=ω=0 in (48), then Theorem 2 of [21] can be obtained(xx)
If we considerη=0, λ=0, and α′=β in the result of (xix), then Corollary (2) of [21] can be obtained


## 3. Concluding Remarks

Recently defined functions of different kinds related to the notion of convexity are combined in a single definition (Definition9). From this definition, the reader can obtain at once convex, strongly convex, and exponentially convex functions of various kinds defined by different authors in near past decades in [3–6, 19, 42, 45–51]. Therefore, this paper provides new as well as results of several published papers in compact forms. The reader simultaneously can obtain inequalities for fractional integrals of many types for various kinds of convex functions by specific settings of involved functions and parameters. The remarks elaborate connections of the proved results with well-known inequalities that already exist in the literature.



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