# Generalized Hermite–Hadamard-Type Integral Inequalities forh-Godunova–Levin Functions

Authors: Rana Safdar Ali; Shahid Mubeen; Sabila Ali; Gauhar Rahman; Jihad Younis; Asad Ali
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The main objective of this article is to establish generalized fractional Hermite–Hadamard and related type integral inequalities forh-Godunova–Levin convexity and h-Godunova–Levin preinvexity with extended Wright generalized Bessel function acting as kernel. Moreover, Hermite–Hadamard-type and trapezoid-type inequalities for several known convexities including Godunova–Levin function, classical convex, s-Godunova–Levin function, P-function, and s-convex function are deduced as corollaries. These obtained results are analyzed in the form of generalization of fractional inequalities.


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## Body

## 1. Introduction

The convexity, preinvexity, and their generalizations have been widely discussed by researchers due to its immense uses in different fields [1–11].Many inequalities have been extensively analyzed and reported in research fields as a result of convexity and its generalizations in engineering and sciences [12–25]. Among them, a highly worked inequality is Hermite–Hadamard inequality is defined as(1)Θx+y2≤1y−x∫xyΘzdz≤Θx+Θy2,for convex function [26–29]; Θ:J⟶R, x,y∈J⊆R, x<y, J⊆R which is playing a significant role in immense applications of inequalities and is widely used by researchers [30,31].In recent years, the concept of convexity has been extended tos-Godunova–Levin type of convexity by Dragomir [32]. Moreover, s-Godunova–Levin-type convexity has been studied in [33]. The h-convexity was introduced by Varos̆anec in [34]. Ohud Almutari introduced h-Godunova–Levin convexity and h-Godunova–Levin preinvexity [35] by combining the concepts of Dragomir and Varos̆anec. In this study, we have considered h-Godunova–Levin convex and h-Godunova–Levin preinvex function to obtain generalized fractional version of Hermite–Hadamard-type inequality and trapezoid-type inequalities related to Hermite–Hadamard inequality.Definition 1 (see [27, 36]).
A functionΘ:J⟶ℝ is called convex if the following inequality holds:(2)Θδu+1−δv≤δΘu+1−δΘv,for δ∈0,1, ∀u,v∈JDefinition 2 (see [4]).
An invex setJ⊆ℝ, with respect to a real bifunction Θ:J×J⟶ℝ, is defined for u,v∈J, λ∈0,1 as follows:(3)v+λΘu,v∈J.Definition 3 (see [4]).
The preinvex functionΘ:J⟶ℝ is defined for x,y∈J and λ∈0,1 as follows:(4)Θy+λζx,y≤λΘx+1−λΘy,where J is an invex set with respect to ζ.Definition 4 (see [37]).
A positive valued functionΘ:J⊆ℝ⟶ℝ is said to be a Godunova–Levin if(5)Θδu+1−δv≤Θuδ+Θv1−δ,for all u,v∈J, δ∈0,1Definition 5 (see [35]).
Supposeh:0,1⟶ℝ. A nonnegative function Θ:J⟶ℝ is said to be h-Godunova–Levin, for all u,v∈J and δ∈0,1, if(6)Θδu+1−δv≤Θuhδ+Θvh1−δ.Definition 6 (see [35]).
A functionΘ:J⟶ℝ is said to be h-Godunova–Levin preinvex with respect to ζ if, for all u,v∈J, ϕ∈0,1,(7)Θu+ϕζv,u≤Θuh1−ϕ+Θvhϕ,holds.Definition 7 (see [38]).
Pochammer’s symbol is defined forδ∈ℕ as(8)γδ=1,forδ=0,γ≠0,γγ+1⋯γ+δ−1,forδ≥1,,γδ=Γγ+δΓγ,γmδ=Γγ+mδΓγ,for γ∈ℂ and m≥0, where Γ being the gamma function.Definition 8 (see [38]).
The integral representation of the gamma function is defined as(9)Γδ=∫0∞zδ−1e−zdz,for ℜt>0.Definition 9 (see [39–41]).
The classical beta function is defined forℜm>0 and ℜn>0 as(10)Bm,n=∫01δm−11−δn−1dδ=ΓmΓnΓm+n.Definition 10 (see [42–44]).
Extended beta functions are defined forℜm>0, ℜn>0, and ℜp>0 as follows:(11)Bpm,n=∫01zm−11−zn−1exp−pz1−zdz.Definition 11 (see [45]).
Ali et al. defined and investigated the generalized Bessel–Maitland function (eight parameters) with a new fractional integral operator and discussed its properties and relations with Mittag–Leffler functions. The function of generalized Bessel–Maitland is as follows:(12)Jϕ,ξ,m,σψ,θ,δ,ϑy=∑p=0∞θξpϑσp−ypΓϕp+ψ+1δmp,where ϕ,ψ,θ,δ,ϑ∈ℂ, ℜϕ>0, ℜν≥−1, ℜθ>0, ℜδ>0, ℜϑ>0, ξ,m,σ≥0, and m,ξ>ℜϕ+σ.Definition 12 (see [46]).
The extended generalized Bessel–Maitland function is defined forμ,ν,η,ρ,γ,c∈ℂ, ℜμ>0, ℜν≥−1, ℜη>0, ℜρ>0, ℜγ>0, ξ,m,σ≥0, and m,ξ>ℜμ+σ as follows:(13)Jν,η,ρ,γμ,ξ,m,σ,cω;p=∑n=0∞βpη+ξn,c−ηcξnγσnβη,c−ηΓμn+ν+1ρmn−ωn.
Generalized fractional integral operators are widely discussed, and many researchers have contributed to the field [47, 48]. Ali et al. defined a new generalized fractional operator as follows.Definition 13 (see [46]).
The generalized fractional integral operators, with extended generalized Bessel–Maitland function as kernel, are defined, forμ,ν,η,ρ,γ,c∈ℂ, ℜμ>0, ℜν≥−1, ℜη>0, ℜρ>0, ℜγ>0, ξ,m,σ≥0, and m,ξ>ℜμ+σ, as follows:(14)Tν,η,ρ,γ;p+μ,ξ,m,σ,cfx,r=∫pxx−tνJν,η,ρ,γμ,ξ,m,σ,cωx−tμ;rftdt,(15)Tν,η,ρ,γ;q−μ,ξ,m,σ,cfx,r=∫xqt−xνJν,η,ρ,γμ,ξ,m,σ,cωt−xμ;rftdt.
In the paper, we obtain Hermite–Hadamard- and trapezoid-type inequalities using the generalized fractional integral operator with extended generalized Bessel–Maitland function as its nonsingular kernel.
The structure of the paper is as follows.
In Section2, we present Hermite–Hadamard inequalities for h-Godunova–Levin convex function using the generalized fractional operator. Section 3 is devoted to trapezoid-type inequalities related to Hermite–Hadamard inequality for h-Godunova–Levin preinvex function using the generalized fractional operator.
In our work, we have frequently used the given notations:(16)ℑv,v′u+ω,Θ=Tv′,η,ρ,γ;u+μ,ξ,m,σ,cΘv,p,(17)ℑu,v′v−ω,Θ=Tv′,η,ρ,γ;v−μ,ξ,m,σ,cΘu,p.


## 2. Hermite–Hadamard Inequalities viah-Godunova–Levin Convex Function

In this section, we establish Hermite–Hadamard inequalities forh-Godunova–Levin convex function using the generalized fractional operator as follows.Theorem 1.
LetΘ:u,v⟶ℝ be a h-Godunova–Levin convex function, where 0<u<v and Θ∈L1u,v with h:0,1⟶ℝ is a positive function and hδ≠0; then, for the generalized fractional integral defined in (33), we have(18)h1/22Θu+v2ℑu,v′v−ω′,1≤12ℑu,v′v−ω′,Θ+ℑv,v′u+ω′,Θ≤Θu+Θv2∫011hδ+1h1−δδv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;pdδ,ω′=ωv−uμ.Proof.
By theh-Godunova–Levin convexity of Θ on the interval u,v, let x,y∈u,v, and we have(19)Θξx+1−ξy≤Θxhξ+Θyh1−ξ,where if we take(20)x=δu+1−δv,y=1−δu+δv,(21)ξ=12,leads to(22)Θu+v2≤1h1/2Θδu+1−δv+Θ1−δu+δv.
Multiplying both sides byδv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;p and integrating the resulting inequality on 0,1 with respect to δ, we have(23)Θu+v2∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;pdδ≤1h1/2∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;pΘδu+1−δvdδ+∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;pΘ1−δu+δvdδ,Θu+v2∑n=0∞βpη+ξn,c−ηcξnγσnβη,c−ηΓμn+v′+1ρmn−ωn∫01δv′+μndδ≤1h1/2∑n=0∞βpη+ξn,c−ηcξnγσnβη,c−ηΓμn+v′+1ρmn−ωn×∫01δv′+μnΘδu+1−δvdδ+∫01δv′+μnΘ1−δu+δvdδ.
Solving the integrals involved in inequality (23), we obtain(24)h1/22Θu+v2ℑu,v′v−ω′,1≤12ℑv,v′u+ω′;Θ+ℑu,v′v−ω′;Θ.
For the second part of inequality, again usingh-Godunova–Levin convexity of Θ, we have(25)Θδu+1−δv≤Θuhδ+Θvh1−δ,(26)Θ1−δu+δ≤Θuh1−δ+Θvhδ.
Addition of these inequalities gives(27)Θδu+1−δv+Θ1−δu+δv≤Θu+Θv1hδ+1h1−δ.
Multiplying both sides byδv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;p and integrating the resulting inequality on 0,1 with respect to δ, we obtain(28)∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;pΘδu+1−δvdδ+∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;pΘ1−δu+δvdδ≤Θu+Θv∫011hδ+1h1−δδv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;pdδ.
Solving the integrals involved leads to(29)12ℑv,v′u+ω′;Θ+ℑu,v′v−ω′;Θ≤Θu+Θv2∫011hδ+1h1−δδv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;pdδ.
Combining (24) and (29), we reach to inequality.Corollary 1.
Choosinghδ=δs in Theorem 1, we obtain Hermite–Hadamard-type inequality for s-Godunova–Levin function:(30)1/2s2Θu+v2ℑu,v′v−ω′,1≤12ℑu,v′v−ω′,Θ+ℑv,v′u+ω′,Θ≤Θu+Θv2∫011δs+11−δsδv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;pdδ.Corollary 2.
Choosinghδ=1 in Theorem 1, we obtain Hermite–Hadamard-type inequality for p function:(31)12Θu+v2ℑu,v′v−ω′,1≤12ℑu,v′v−ω′,Θ+ℑv,v′u+ω′,Θ≤Θu+Θvℑv,v′u+ω′,1.Corollary 3.
Choosinghδ=1/δ in Theorem 1, we obtain Hermite–Hadamard-type inequality for convex function:(32)Θu+v2ℑu,v′v−ω′,1≤12ℑu,v′v−ω′,Θ+ℑv,v′u+ω′,Θ≤Θu+Θv2ℑv,v′u+ω′,1.Corollary 4.
Choosinghδ=δ in Theorem 1, we obtain Hermite–Hadamard-type inequality for Godunova–Levin function:(33)14Θu+v2ℑu,v′v−ω′,1≤12ℑu,v′v−ω′,Θ+ℑv,v′u+ω′,Θ≤Θu+Θv2∫01δv′−11−δJv′,η,ρ,γμ,ξ,m,σ,cωδμ;pdδ.Corollary 5.
Choosinghδ=1/δs in Theorem 1, we obtain Hermite–Hadamard-type inequality for s-convex function:(34)2s−1Θu+v2ℑu,v′v−ω′,1≤12ℑu,v′v−ω′,Θ+ℑv,v′u+ω′,Θ≤Θu+Θv2∫01δs+1−δsδv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;pdδ.


## 3. Trapezoid-Type Inequalities Related to Hermite–Hadamard Inequalities forh-Godunova–Levin Preinvex Function

In this section, Wright generalized that the Bessel function is restricted to a real valued function. The trapezoid-type inequalities related to Hermite–Hadamard inequalities using fractional integral with Wright generalized Bessel function in its kernel can be obtained with the help of the following lemma.Lemma 1.
Consider a functionΘ:J=u,u+ζv,u⟶ℝ with u,v∈ℝ; let Θ∈L1u,u+ζv,u be a differentiable function, where J=u,u+ζv,u is taken to be an open invex set with respect to ζ:J×J⟶ℝ with ζv,u>0, for u,v∈J; then, for the generalized fractional integral defined in (33), we have(35)Θu+Θu+ζv,u2Jv′,η,ρ,γμ,ξ,m,σ,cω;p−12ζv,uv′ℑu+ζv,u,v′−1u+ω′;Θ+ℑu,v′−1u+η1v,u−ω′;Θ=ζv,u2I,where I=∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;pΘ′u+δζv,udδ+∫01−1−δv′Jv′,η,ρ,γμ,ξ,m,σ,cω1−δμ;pΘ′u+δζv,udδ, ω′=ω/ζv,uμ.Proof.
Consider(36)I=∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;pΘ′u+δζv,udδ+∫01−1−δv′Jv′,η,ρ,γμ,ξ,m,σ,cω1−δμ;pΘ′u+δζv,udδ.
Let(37)I=I1+I2.
First, we considerI1:(38)I1=∑n=0∞βpη+ξn,c−ηcξnγσnβη,c−ηΓμn+v′+1ρmn−ωn∫01δv′+μnΘ′u+δζv,udδ.
Integrating by parts, we have(39)I1=∑n=0∞βpη+ξn,c−ηcξnγσnβη,c−ηΓμn+v′+1ρmn−ωnδv′+μnΘu+δζv,uζv,u|01−v′+μnζv,u∫01δv′+μn−1Θu+δζv,udδ,I1=∑n=0∞βpη+ξn,c−ηcξnγσnβη,c−ηΓμn+v′+1ρmn−ωn×Θu+ζv,uζv,u−v′+μnζv,u∫01δv′+μn−1Θu+δζv,udδ,I1=Θu+ζv,uζv,uJv′,η,ρ,γμ,ξ,m,σ,cω;p−1ζv,uv′+1ℑu,v′−1u+ζv,u−ω′;Θ.
Continuing in the same manner, we obtain(40)I2=Θuζv,uJv′,η,ρ,γμ,ξ,m,σ,cω;p−1ζv,uv′+1ℑu+ζv,u,v′−1u+ω′,Θ,I=Θu+Θu+ζv,uζv,uJv′,η,ρ,γμ,ξ,m,σ,cω;p−1ζv,uv′+1×ℑu,v′−1u+ζv,u−ω′,Θ+ℑu+ζv,u,v′−1u+ω′,Θ.
Multiplying byζv,u/2, we get the required result.
By Lemma1, we present the following theorem.Theorem 2.
Consider a functionΘ:J=u,u+ζv,u⟶0,∞ with J∈ℝ, and let it be a differentiable function on J. Also, suppose that Θ′ is a h-Godunova–Levin preinvex function on J; then, for the generalized fractional integral defined in (33) with the restricted Wright generalized Bessel function to a real valued function, we have(41)Θu+Θu+ζv,u2Jv′,η,ρ,γμ,ξ,m,σ,cω;p−12ζv,uv′×ℑu+ζv,u,v′−1u+ω′,Θ+ℑu,v′−1u+ζv,u−ω′,Θ≤ζv,u2Θ′u+Θ′v∫01∑n=0∞βpη+ξn,c−ηcξnγσnβΓμn+v′+1ρmn−ωn×δv′+μn−1−δv′+μnhδdδ.Proof.
(42)|Θu+Θu+ζv,u2Jv′,η,ρ,γμ,ξ,m,σ,cω;p−12ζv,uv′×ℑu+ζv,u,v′−1u+ω′,Θ+ℑu,v′−1u+ζv,u−ω′,Θ|=ζv,u2I≤ζv,u2∑n=0∞βpη+ξn,c−ηcξnγσnβη,c−ηΓμn+v′+1ρmn−ωn∫01δv′+μn−1−δv′+μnΘ′u+δζv,udδ≤ζv,u2∑n=0∞βpη+ξn,c−ηcξnγσnβη,c−ηΓμn+v′+1ρmn−ωn×∫01δv′+μn−1−δv′+μnΘ′uhδ+Θ′vh1−δdδ≤ζv,u2∑n=0∞βpη+ξn,c−ηcξnγσnβη,c−ηΓμn+v′+1ρmn−ωn×Θ′u∫01δv′+μn−1−δv′+μn1hδdδ+Θ′v∫01δv′+μn−1−δv′+μn1h1−δdδ=ζv,u2Θ′u+Θ′v∫01∑n=0∞βpη+ξn,c−ηcξnγσnβη,c−ηΓμn+v′+1ρmn−ωn×δv′+μn−1−δv′+μnhδdδ.Corollary 6.
Takingζv,u=v−u in Theorem 2, we obtain the following inequality:(43)|Θu+Θv2Jv′,η,ρ,γμ,ξ,m,σ,cω;p−12v−uv′×ℑv,v′−1u+ω′,Θ+ℑu,v′−1v−ω′,Θ|≤v−u2Θ′u+Θ′v∫01∑n=0∞βpη+ξn,c−ηcξnγσnβη,c−ηΓμn+v′+1ρmn−ωn×δv′+μn−1−δv′+μnhδdδ.Theorem 3.
Suppose thatΘ:J=u,u+ζv,u⟶0,∞ with J∈ℝ, and let it be a differentiable function on J. Also, suppose that Θ′q is a h-Godunova–Levin preinvex function on J with p>1 and q=p/p−1; then, for the generalized fractional integral defined in (33) with the restricted Wright generalized Bessel function to a real valued function, we have(44)|Θu+Θu+ζv,u2Jv′,η,ρ,γμ,ξ,m,σ,cω;p−12ζv,uv′×ℑu+ζv,u,v′−1u+ω′,Θ+ℑu,v′−1u+ζv,u−ω′,Θ|≤ζv,u2Θ′uq+Θ′vq1/q∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;p−1−δv′Jv′,η,ρ,γμ,ξ,m,σ,cω1−δμ;ppdδ1/p∫011hδdδ1/q.Proof.
Using Lemma1, we have(45)|Θu+Θu+ζv,u2Jv′,η,ρ,γμ,ξ,m,σ,cω;p−12ζv,uv′×ℑu+ζv,u,v′−1u+ω′,Θ+ℑu,v′−1u+ζv,u−ω′,Θ|=ζv,u2I≤ζv,u2∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;p−1−δv′Jv′,η,ρ,γμ,ξ,m,σ,cω1−δμ;pΘ′u+δζv,udδ.
Using Hölder’s integral inequality, we have(46)≤ζv,u2∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;p−1−δv′Jv′,η,ρ,γμ,ξ,m,σ,cω1−δμ;ppdδ1/p∫01Θ′u+δζv,uqdδ1/q,where 1/p+1/q=1.
Now, sinceΘ′q is an h-Godunova–Levin preinvex, we obtain(47)∫01Θ′u+δζv,uqdδ≤∫01Θ′uqhδ+Θ′vqh1−δdδ≤Θ′uq+Θ′vq∫011hδdδ.
Using (47) in (46) leads to the result.Theorem 4.
With the assumptions of Theorem3, we get the following inequality related to Hermite–Hadamard inequality:(48)Θu+Θu+ζv,u2Jv′,η,ρ,γμ,ξ,m,σ,cω;p−12ζv,uv′ℑu+ζv,u,v′−1u+ω′,Θ+ℑu,v′−1u+ζv,u−ω′,Θ≤ζv,u21/qΘ′uq+Θ′vq1/qJv′+1,η,ρ,γμ,ξ,m,σ,cω;p−12v′Jv′+1,η,ρ,γμ,ξ,m,σ,cω12μ;p1−1/q∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;p−1−δv′Jv′,η,ρ,γμ,ξ,m,σ,cω1−δμ;phδdδ1/q,where v′,μ∈ℝ+.Proof.
From Lemma1, we have(49)Θu+Θu+ζv,u2Jv′,η,ρ,γμ,ξ,m,σ,cω;p−12ζv,uv′ℑu+ζv,u,v′−1u+ω′,Θ+ℑu,v′−1u+ζv,u−ω′,Θ=ζv,u2I≤ζv,u2∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;p−1−δv′Jv′,η,ρ,γμ,ξ,m,σ,cω1−δμ;pΘ′u+δζv,udδ.
Applying power-mean inequality, we obtain(50)Θu+Θu+ζv,u2Jv′,η,ρ,γμ,ξ,m,σ,cω;p−12ζv,uv′ℑu+ζv,u,v′−1u+ω′,Θ+ℑu,v′−1u+ζv,u−ω′,Θ≤ζv,u2∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;p−1−δv′Jv′,η,ρ,γμ,ξ,m,σ,cω1−δμ;pdδ1−1/q∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;p−1−δv′Jv′,η,ρ,γμ,ξ,m,σ,cω1−δμ;p×Θ′u+δζv,uqdδ1/q.
SinceΘ′q is an h-Godunova–Levin preinvex, we obtain(51)∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;p−1−δv′Jv′,η,ρ,γμ,ξ,m,σ,cω1−δμ;pΘ′u+δζv,uqdδ≤∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;p−1−δv′Jv′,η,ρ,γμ,ξ,m,σ,cω1−δμ;phδΘ′uq+δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;p−1−δv′Jv′,η,ρ,γμ,ξ,m,σ,cω1−δμ;ph1−δΘ′vqdδ=Θ′uq+Θ′vq∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;p−1−δv′Jv′,η,ρ,γμ,ξ,m,σ,cω1−δμ;phδ.
Now, consider(52)∫01δv′Jv′,η,ρ,γμ,ξ,m,σ,cωδμ;p−1−δv′Jv′,η,ρ,γμ,ξ,m,σ,cω1−δμ;p=∑n=0∞βpη+ξn,c−ηcξnγσnβη,c−ηΓμn+v′+1ρmn−ωn∫01δv′+μn−1−δv′+μndδ=∑n=0∞βpη+ξn,c−ηcξnγσnβη,c−ηΓμn+v′+1ρmn−ωn×∫01/21−δv′+μn−δv′+μndδ+∫1/21δv′+μn−1−δv′+μn=2Jv′+1,η,ρ,γμ,ξ,m,σ,cω;p−12v′Jv′+1,η,ρ,γμ,ξ,m,σ,cω12μ;p.


## 4. Conclusion

In the present paper, the advanced approach of the generalized fractional version of Hermite–Hadamard-type and trapezoid-type integral inequalities for a recently introduced function,h-Godunova–Levin convex, and h-Godunova–Levin preinvex have been established by using fractional integral operator with Wright generalized Bessel function as its kernel. Convexities and its different forms have remarkable uses in many fields and is extensively worked by researchers. Since h-Godunova–Levin convex function is generalization of several known convexities, so the results have been also deduced for them in the form of corollaries.



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