# Hermite–Hadamard and Jensen-Type Inequalities via Riemann Integral Operator for a Generalized Class of Godunova–Levin Functions

Authors: Xiaoju Zhang; Khurram Shabbir; Waqar Afzal; He Xiao; Dong Lin
Journal: Journal of Mathematics
 (2022)

---

## Abstract

The generalization of Godunova–Levin interval-valued functions has been drastically studied in last few decades, as it has a remarkable applications in both pure and applied mathematics. The goal of this study is to introduce the notion of h-Godunova–Levin interval-valued functions. We establish Hermite–Hadamard and Jensen-type inequalities via Riemann integral operator.


---

## Body

## 1. Introduction

In last few years, the inequality theory gained the attention of many researchers working in analysis and other branches of mathematics [1–3]. Most of the real world problems may be viewed as integral equations. So, the generalization of integral inequalities is always appreciable and is more closer to applied problems[4, 5].In the renowned celebrated book by Moore, the interactive analysis of numerical data starts the introduction to interval analysis in numerical analysis, see [6]. A tremendous number of applications have been developed over the past 50 years in areas including computer graphics [7], aeroelasticity [8], interval differential equations [9], and neural network optimization [10]. Numerous various integral inequalities have been investigated recently by different authors in the context of interval-valued functions (see [11, 12]).It is well known that the convexity of functions plays an extremely important role in mathematics and other scientific fields such as economics, probability theory, and optimal control theory; moreover, several inequalities have been recorded in the literature (see [13–17]). In equality, the following inequality is called a classical Hermite–Hadamard inequality:(1)φv+w2≤1v−w∫vwφμdμ≤φv+φw2,where φ:S⊆R⟶R is a convex on interval S and v,w∈S with v<w. In the context of different generalizations and extensions of this inequality (see [16, 18]), the notion of h-convex was originally developed by Varoşanec in 2007 (see [19]). Several authors have developed more sophisticated Hermite–Hadamard inequalities that include h-convex functions (see [20, 21]). Furthermore, Costa presented an inequality of the Jensen type for fuzzy interval-valued functions (see [22]). As well, Zhao et al. provide a new Hermite–Hadamard inequality for h-convex functions in the context of interval-valued functions (see [23]).The following inequality was proved in 2019 by Almutairi and Kiliman using theh-Godunova-Levin function (see [24]).Theorem 1.
Letφ:v,w⟶R. Ifφis h-Godunova–Levin convex function andh1/2≠0. Then,(2)h1/22φv+w2≤1w−v∫vwφμdμ≤φv+φw∫01dxhx.Motivated by Costa [22], Zhao et al. [23], Dragomir [25], and Almutairi and Kiliman [24], we introduce and explore the notion of h-Godunova–Levin interval-valued functions. Our new concept allowed us to develop fractional version of Hermite–Hadamard and Jensen-type inequalities via Riemann integral operator.


## 2. Preliminaries

In this study, we will review some fundamental definitions, properties, and notations. Let us sayI is the collection of all intervals of R, v∈I is defined as follows:v=v¯,v¯=x∈R|v¯≤x≤v¯, v¯,v¯∈R, where real interval v is closed and bounded subset of R. The interval v is said to be degenerated when v¯ and v¯ both are equal. We call v is positive when v¯>0 or negative when v¯<0. We denote the space of all intervals by RI of R and the set of all negative and positive intervals are represented as RI− and RI+, respectively. The inclusion “⊆” is defined as(3)v⊆w⇔v¯,v¯⊆w¯,w¯⇔w¯≤v¯,v¯≤w¯.For any random real numberμ and v, the interval μv is given as(4)μ⋅v¯,v¯=μv¯,μv¯,ifμ>0,0,ifμ=0,μv¯,μv¯,ifμ<0.Forv=v¯,v¯ and w=w¯,w¯, algebraic operations are defined as(5)v+w=v¯+w¯,v¯+w¯,v−w=v¯−w¯,v¯−w¯,v−w=v¯−w¯,v¯−w¯,v⋅w=minvw¯,v¯w¯,v¯w¯,vw¯,maxvw¯,v¯w¯,vw¯,v¯w¯,vw=minv¯w¯,v¯w¯,v¯w¯,v¯w¯,maxv¯w¯,v¯w¯,v¯w¯,v¯w¯,where(6)0∉v¯,w¯.For intervalsv¯,w¯,v¯,w¯ the Hausdorff–Pompeiu distance is defined as(7)dv¯,v¯,w¯,w¯=maxv¯−w¯,v¯−w¯.It is well known that the entireRI,d is complete metric space.Definition 1 (see [26]).
Letφ:v,w⟶RIbe such thatφs=φ¯s,φ¯s, for eachs∈v,wandφ¯,φ¯are Riemann integrable over intervalv,w. Then, we say that our function f is Riemann integrable over intervalv,wand denotes(8)∫vwφsds=∫vwφ¯sds,∫vwφ¯sds.Definition 2 (see [27]).
A functionφ:S⟶Ris known as convex function; if∀v,w∈Sandμ∈0,1, we have(9)φμv+1−μw≤μφv+1−μφw.Definition 3 (see [28]).
A nonnegative functionφ:S⟶Ris known as Godunova–Levin function; if∀v,w∈Sandμ∈0,1, we have(10)φμv+1−μw≤φvμ+φw1−μ.Definition 4 (see [19]).
Leth:0,1⊆M⟶Rbe a nonnegative function. We sayφ:M⟶Ris known as h-convex function, orφ∈SXh,M; if∀v,w∈Mandμ∈0,1, we have(11)φμv+1−μw≤hμφv+h1−μφw.
If the above inequality is inverted, it is referred to ash-concave or φ∈SVh,M.Definition 5 (see [24]).
Leth:0,1⊆M⟶Rbe a nonnegative function. We sayφ:M⟶Ris known as h-Godunova–Levin convex function, orφ∈SGX1/h,M; if∀v,w∈Mandμ∈0,1, we have(12)φμv+1−μw≤φvhμ+φwh1−μ.
If the above inequality is inverted, it is referred to as h-Godunova–Levin concave orφ∈SGV1/h,M.Remark 1.
(i)
Ifhμ=1, then Definition 2.5 reduces to P-function [18](ii)
Ifhμ=1/hμ, then Definition 2.5 reduces to h-convex function [19](iii)
Ifhμ=μ, then Definition 2.5 reduces to Godunova–Levin function [28](iv)
Ifhμ=1/μs, then Definition 2.5 reduces to s-convex function [29](v)
Ifhμ=μs, then Definition 2.5 reduces to s-Godunova–Levin function [30]


## 3. Main Results

Now, we are ready to introduce the notion of interval-valuedh-Godunova–Levin convex functions.Definition 6.
Leth:0,1⊆M⟶Rbe a nonnegative function. We sayφ:M⟶Ris the interval h-Godunova–Levin convex function, orφ∈SGX1/h,M,RI+; if, for allv,w∈Mandμ∈0,1, we have(13)φvhμ+φwh1−μ⊆φμv+1−μw.
If the above inequality is inverted, it is referred to intervalh-Godunova–Levin concave or φ∈SGV1/h,M,RI+. The space of all interval-valued h-Godunova–Levin convex and h-Godunova–Levin concave functions are denoted by φ∈SGX1/h,M,RI+ and φ∈SGV1/h,M,RI+, respectively.Proposition 1.
Letφ:v,w⟶RI+be h-Godunova–Levin convex interval-valued function such thatφμ=φ¯μ,φ¯μ. Then, ifφ∈SGX1/h,v,w,RI+if and only ifφ¯∈SGX1/h,v,w,RI+, φ¯∈SGV1/h,v,w,RI+.Proof.
Letφ be h-Godunova–Levin convex interval-valued function, and suppose that x,y∈v,w, μ∈0,1; then,(14)φxhμ+φyh1−μ⊆φμx+1−μy,that is,(15)φ¯xhμ+φ¯yh1−μ,φ¯xhμ+φ¯xh1−μ⊆φ¯μx+1−μy,φ¯μx+1−μy.
It follows that we have(16)φ¯xhμ+φ¯yh1−μ≥φ¯μx+1−μyand(17)φ¯xhμ+φ¯yh1−μ≤φ¯μx+1−μy.
This shows thatφ¯∈SGX1/h,v,w,RI+ and φ¯∈SGV1/h,v,w,RI+. Conversely, suppose that if φ¯∈SGX1/h,v,w,RI+ and φ¯∈SGV1/h,v,w,RI+, then, from above definition and set inclusion, we have φ∈SGX1/h,v,w,RI+ This completes the proof.Proposition 2.
Letφ:v,w⟶RI+be h-Godunova–Levin concave interval-valued function such thatφμ=φ¯μ,φ¯μ. Then, ifφ∈SGV1/h,v,w,RI+if and only ifφ¯∈SGV1/h,p,q,RI+, φ¯∈SGX1/h,p,q,RI+.Proof.
This can be similar to Proposition1.
In the next theorem, we Hermite–Hadamard-type inequality forh-Godunova–Levin interval-valued functions.Theorem 2.
Letφ:v,w⟶RI+,h:0,1⟶R+andh1/2≠0; ifφ∈SGX1/h,v,w,RI+andφ∈IRv,w, then we have(18)h1/22φv+w2⊇1w−v∫vwφμdμ⊇φv+φw∫01dxhx.Proof.
By supposition, we have(19)φxv+1−xwh1/2+φ1−xv+xwh1/2⊆φv+w2.
Integrating the above inequality w.r.t “x” over 0,1, we obtain(20)∫01φ¯xm+1−xndx+∫01φ¯1−xm+xndx≥h12∫01φ¯v+w2dx,(21)∫01φ¯xv+1−xwdx+∫01φ¯1−xv+xwdx≤h12∫01φ¯v+w2dx.
It follows that we have(22)2w−v∫vwφ¯μdμ≥h1/2∫01φ¯v+w2dx=h12φ¯v+w2.
Similarly,(23)2w−v∫vwφ¯μdμ≤h12∫01φ¯v+w2dx=h12φ¯v+w2.
This implies that(24)h1/22φ¯v+w2,φ¯v+w2⊇1w−v∫vwφμdμ.
As a result of applying the intervalh-Godunova–Levin convex function, we have(25)φvhx+φwh1−x⊆φxv+1−xw.
Integrate w.r.t “x” over 0,1; we have(26)φv∫01dxhx+φw∫01dxh1−x⊆∫01φxv+1−xwdx.
Accordingly,(27)φv+φw∫01dxhx⊆1w−v∫vwφμdμ.
Now, combining (24) and (26), we get the required result:(28)h1/22φv+w2⊇1w−v∫vwφμdμ⊇φv+φw∫01dxhx.Remark 2.
(i)
Ifhx=1, then Theorem2has the following result for intervalPfunctions:(29)12φv+w2⊇1w−v∫vwφμdκ⊇φv+φw.(ii)
Ifhx=1/x, then Theorem2has the following result for interval convex functions:(30)φv+w2⊇1w−v∫vwφμdκ⊇φv+φw2.(iii)
Ifhx=1/xs, then Theorem2has the following result for interval s-convex function:(31)2s−1φv+w2⊇1w−v∫vwφμdμ⊇φv+φws+1.(iv)
Ifφ¯=φ¯, then Theorem2has the following result of Ohud Almutairi and Adem Kiliman (see [24], Theorem1).Example 1.
Suppose thath:0,1⟶RI+is defined ashx=1/xforx∈0,1,v,w=−1,1, andφ:v,w⟶RI+is defined asφμ=μ2,4−eμ, where(32)h1/22φv+w2=φ0=0,3,1w−v∫vwφμdμ=12∫−11μ2dμ,∫−114−eμdμ=12∫−11μ2dμ,∫−114−eμdμ=13,4−e−1/e2φv+φw∫01dxhx=1,4−e+1/e2.
As a result,(33)0,3⊇13,4−e−1/e2⊇1,4−e+1/e2.
This proves the above theorem.Corollary 1.
Letφ:v,w⟶RI+,h:0,1⟶R+andh1/2≠0ifφ∈SGV1/h,v,w,RI+andφ∈IRv,w; then, we have(34)h1/22φv+w2⊆1w−v∫vwφμdμ⊆φv+φw∫01dxhx.Theorem 3.
Letφ:v,w⟶RI+,h0,1⟶R+andh1/2≠0; ifφ∈SGX1/h,I,v,w,R+andφ∈IRv,w, then we have(35)h1/224φv+w2⊇Δ1⊇1w−v∫vwφμdμ⊇Δ2⊇φv+φw12+1h1/2∫01dxhx,where(36)Δ1=h1/24φ3v+w4+φ3w+v4,Δ2=φv+w2+φv+φw2∫01dxhx.Proof.
We break the intervalv,w into v,v+w/2 and v+w/2,w; then, for interval v,v+w/2v,v+w/2,(37)φxv+1−xv+w/2h1/2+φ1−xv+xv+w/2h1/2⊆φxv+1−xv+w/2+xv+w/2+1−xv2=φv+v+w/22=φ3v+w2.
Integrate the above inequality over0,1 w.r.t x:(38)1h1/2∫01φxv+1−xv+w2dx+∫01φxv+w2+1−xvdx⊆φ3v+w2.
Then, the above inequality becomes as(39)1h1/22v−w∫v+w/2vφμdμ+2w−v∫vv+w/2φμdμ⊆φ3v+w2.
Now, by using the property of integral, the above inequality becomes as(40)1h1/22w−v∫vv+w/2φμdμ+2w−v∫vv+w/2φμdμ⊆φ3v+w2,4h1/21w−v∫vv+w/2φμdμ⊆φ3v+w2,(41)1w−v∫vv+w/2φμdμ⊆h1/24φ3v+w2.
Similarly, for intervalv+w/2,w, we have(42)1w−v∫v+w/2wφμdμ⊆h1/24φ3w+v2.
Adding inequality (41) and (42), we obtain(43)Δ1=h1/24φ3v+w4+φ3w+v4⊇1w−v∫vwφμdμ=122w−v∫vv+w/2φμdμ+2w−v∫v+w/2wφμdμ⊇12φv+φv+w2∫01dxhx+12φw+φv+w2∫01dxhx=12φv+φw+2φv+w2∫01dxhx=φv+φw2+φv+w2∫01dxhx=Δ2.
Now,(44)h1/224φv+w2=h1/224φ123v+w4+123w+v4⊇h1/224φ3w+v/4h1/2+φ3w+v/4h1/2=h1/224h1/2φ3v+w4+φ3v+w4=h1/24φ3v+w4+φ3v+w4=Δ1⊇h1/241h1/2φv+φv+w2+1h1/2φw+φv+w2=h1/241h1/2φv+φw+2φv+w2=14φv+φw+2φv+w2=12φv+φw2+φv+w2⊇φv+φw2+φv+w2∫01dxhx=Δ2⊇φv+φw2+φvh1/2+φwh1/2∫01dxhx=φv+φw2+1h1/2φv+φw∫01dxhx=φv+φw12+1h1/2∫01dxhx.Example 2.
Leth:0,1⟶RI+be defined ash1x=1/x, and forx∈0,1,v,w=−1,1,φ,ϕ:v,w⟶RI+is defined asφμ=μ2,4−eμ, where(45)h1/224φv+w2⊇Δ1⊇1w−v∫vwφμdμ⊇Δ2⊇φv+φw12+1h1/2∫01dxhx.
Consider and put values:(46)h1/224φv+w2=φ0=0,3,Δ1=h1/24φ3v+w4+φ3v+w4,Δ1=12φ−12,φ12=14,4−e+1/e2,1w−v∫vwφμdμ=12∫−11μ2dμ,∫−114−eμdμ=12∫−11μ2dμ,∫−114−eμdμ=13,4−e−1/e2,Δ2=φv+φw2+φv+w2∫01dxhx,Δ2=121,4−e+1/e2+0,3,Δ2=12,72−e+1/e4φv+φw12+1h1/2∫01dxhx=1,4−e+1/e2.
Thus, we obtain(47)0,3⊇14,4−e+1/e2⊇13,4−e−1/e2⊇12,72−e+1/e4⊇1,4−e+1/e2.
This proves the above theorem.Corollary 2.
Letφ:v,w⟶RI+,h:0,1⟶R+andh1/2≠0; ifφ∈SGV1/h,I,v,w,R+andφ∈IRv,w, then we have(48)h1/224φv+w2⊆Δ1⊆1w−v∫vwφμdμ⊆Δ2⊆φv+φw12+1h1/2∫01dxhx.Theorem 4.
Letφ:v,w⟶RI+,h:0,1⟶R+be a continuous function. Ifφ∈SGX1/h1,p,q,RI+,g∈SGX1/h2,v,w,RI+, andφ,ϕ∈IRv,w, then we have(49)1w−v∫vwφμϕμdμ⊇Mv,w∫011h1xh2xdx+Nv,w∫011h1xh21−xdx,where(50)Mv,w=φvϕv+φwϕw,Nv,w=φvϕw+φwϕv.Proof.
We assume thatφ∈SGX1/h1,v,w,RI+ and ϕ∈SGX1/h2,v,w,RI+; then, we have(51)φvh1x+φwh11−x⊆φvx+1−xw,ϕvh2x+ϕwh21−x⊆ϕvx+1−xw.
Then,(52)φvx+1−xwϕvx+1−xw⊇φvϕvh1xh2x+φvϕwh1xh21−x+φwϕvh11−xh2x+φwϕwh11−xh21−x.
Integrating both sides over0,1, w.r.t “x,” we have(53)∫01φvx+1−xwϕvx+1−xwdx=∫01φ¯vx+1−xwϕ¯vx+1−xwdx,∫01φ¯vx+1−xwϕ¯vx+1−xwdx=1w−v∫vwφ¯μϕ¯μdμ,1w−v∫vwφ¯μϕ¯μdμ=1w−v∫vwφμϕμdμ⊇∫01φvϕv+φwϕwh1xh2xdx+∫01φvϕw+φwϕvh1xh21−xdx.
It follows that(54)1w−v∫vwφμϕμdμ⊇Mv,w∫011h1xh2xdx+Nv,w∫011h1xh21−xdx.
Theorem is proved.Example 3.
Suppose thath:0,1⟶RI+defined ash1x=1/xandh2x=1forx∈0,1,v,w=0,1, andφ,ϕ:v,w⟶RI+is defined asφμ=μ2,4−eμandϕμ=μ,3−μ2.
Then,(55)1w−v∫vwφμϕμdμ=∫01μ3dμ,∫014−eμ3−μ2dμ=∫01μ3dμ,∫014−eμ3−μ2dμ=14,353−2e,Mv,w∫011h1xh2xdx=M0,1∫01xdx=12,172−e,Nv,w∫011h1xh21−xdx=N0,1∫01xdx=0,92−3e4.
It follows that(56)14,353−2e⊇12,172−e+0,92−3e4=12,8−7e4.
Consequently, the above theorem is verified.Corollary 3.
Letφ:v,w⟶RI+,h:0,1⟶R+be a continuous function. Ifφ∈SGV1/h1,v,w,RI+,ϕ∈SGV1/h2,v,w,RI+, andφ,ϕ∈IRv,w, then we have(57)1w−v∫vwφμϕμdμ⊆Mv,w∫011h1xh2xdx+Nv,w∫011h1xh21−xdx.Theorem 5.
Letφ,ϕ:v,w⟶RI+,h:0,1⟶R+be a continuous function. Ifφ∈SGX1/h1,v,w,RI+,ϕ∈SGX1/h2,v,w,RI+andφ,ϕ∈IRv,w, then we have(58)h11/2h21/22φv+w2ϕv+w2⊇1w−v∫vwφμϕμdμ+Mv,w∫011h1xh2xdx+Nv,w∫011h1xh21−xdx.Proof.
By hypothesis, one has(59)φv+w2⊇φvx+1−xwh11/2+φv1−x+xwh11/2,ϕv+w2⊇ϕvx+1−xwh21/2+ϕv1−x+xwh21/2.
Then,(60)φv+w2ϕv+w2⊇1h1/2h1/2φvx+1−xwϕvx+1−xw+φv1−x+xwϕv1−x+xw+1h1/2h1/2φvx+1−xwϕv1−x+xw+φv1−x+xwϕvx+1−xw⊇1h1/2h1/2φvx+1−xwϕvx+1−xw+φv1−x+xwϕv1−x+xw+1h1/2h1/2φvh1x+φwh11−xϕwh21−x+ϕwh2x+φvh11−x+φwh1xϕvh2x+ϕwh21−x=1h1/2h1/2φvx+1−xwϕvx+1−xw+φv1−x+wxϕv1−x+wx+1h1/2h1/21h1xh21−x+1h11−xh2xMv,w+1h1xh2x+1h11−xh21−xNv,w.
Integrating above inequality over0,1, w.r.t x, we obtain(61)∫01φv+w2ϕv+w2dx=∫01φ¯v+w2ϕ¯v+w2dx,∫01φ¯v+w2ϕ¯v+w2dx=φv+w2ϕv+w2dx⊇2h1/2h1/21w−v∫vwφμϕμdμ+2h1/2h1/2Mv,w∫011h1xh21−xdx+Nv,w∫011h1xh2xdx.
Multiply both sides byh1/2h1/2/2; from the above equation, we obtain(62)h11/2h21/22φv+w2ϕv+w2⊇1w−v∫vwφμϕμdμ+Mv,w∫011h1xh21−xdx+Nv,w∫011h1xh2xdx.
This completes the proof.Example 4.
Suppose thath:0,1⟶RI+ is defined ash1x=1/x,h2x=2forx∈0,1,v,w=0,1, andφ:v,w⟶RI+is defined asφμ=μ2,4−eμandϕμ=μ,3−μ2.
Then,(63)h11/2h21/22φv+w2ϕv+w2=2φ12ϕ12=14,44−11e2,1w−v∫vwφμϕμdμ=∫01μ3dμ,∫014−eμ3−μ2dμ=14,353−2e,Mv,w∫011h1xh21−xdx=M0,12∫01xdx=14,174−e2,Nv,w∫011h1xh2xdx=N0,12∫01xdx=0,94−3e8.
It follows that(64)18,11−11e4⊇14,353−2e+14,174−e2+0,94−3e8=12,436−69e24.
This proves the above theorem.Corollary 4.
Letφ,ϕ:v,w⟶RI+,h:0,1⟶R+be a continuous function. Ifφ∈SGV1/h1,v,w,RI+,ϕ∈SGX1/h2,v,w,RI+, andφ,ϕ∈IRv,w, then we have(65)h11/2h21/22φv+w2ϕv+w2⊆1w−v∫vwφμϕμdμ+Mv,w∫011h1xh2xdx+Nv,w∫011h1xh21−xdx.Theorem 6.
(Jensen-type inequality). Lets1,s2,s3…sk∈R+withk≥2. Ifhis super multiplicative nonnegative function and ifφ∈SGX1/h,v,w,RI+z1,z2,z3,…,zk∈I, then the inequality become as(66)φ1Sk∑i=1ksizi⊇∑i=1kφzihsi/Sk,where Sk=∑i=1k.Proof.
Whenk=2, the above inequality is trivially true, i.e., reduce to official definition of h−Goduova−Levin interval-valued function. Now, we suppose that inequality is true for k−1; then, consider(67)φ1Sk∑i=1ksizi=φskSkzk+∑i=1k−1siSkzi=φskSkzk+Sk−1Sk∑i=1k−1siSk−1zi⊇φzkhsk/Sk+φ∑k−1i=1si/Sk−1zihSk−1/Sk⊇φzkhsk/Sk+∑i=1k−1φzihsi/Sk−11hSk−1/Sk⊇φzkhsk/Sk+∑i=1k−1φzihsi/Sk⊇∑i=1kφzihsi/Sk.
Thus, the result is proved by using mathematical induction.


## 4. Conclusions

We introduced the notion of h-Godunova–Levin interval-valued functions and established Hermite–Hadamard and Jensen-type inequalities for the introduced class of functions. Our results are extension of many existing results. It is interesting for the researchers to establish fractional version of established inequalities for theh-Godunova–Levin interval-valued functions.



---

*Source: 3830324-2022-08-31.xml*