# Quantum Inequalities of Hermite–Hadamard Type forr-Convex Functions

Authors: Xuexiao You; Hasan Kara; Hüseyin Budak; Humaira Kalsoom
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this present study, we first establish Hermite–Hadamard type inequalities forr-convex functions via qκ2-definite integrals. Then, we prove some quantum inequalities of Hermite–Hadamard type for product of two r-convex functions. Finally, by using these established inequalities and the results given by (Brahim et al. 2015), we prove several quantum Hermite–Hadamard type inequalities for coordinated r-convex functions and for the product of two coordinated r-convex functions.


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## Body

## 1. Introduction

Quantum calculus research is an unlimited analysis of calculus and is known asq-calculus. We get the initial mathematical formulas in q-calculus as q reaches 1−. The commencement of the analysis of q-calculus was initiated by Euler (1707–1783). The aforementioned results lead to an intensive investigation on q-calculus in the twentieth century. The concept of q-calculus is used in many areas in mathematics and physics such as theory, orthogonal polynomials, integration, basic hypergeometric functions, mechanical theory, and quantum and relativity theory. For more information about q-calculus, one can refer to [1–10].Mathematically, convexity is very simple and natural which plays a very important role in various fields of pure and applied science, such as in the field of practicality, engineering science, and management science. In the recent past, the classical concept of convexity has been extended and generalized in different directions. Another factor that makes the theory of the most popular convex works is its relationship to the concept of inequality. Many inequalities can be achieved using the definition of convex functions. One of the widely studied inequalities involving convex works is the Hermite–Hadamard inequality, which is the first basic result of convex design with natural geometric descriptions and multiple uses and has attracted great interest in elementary mathematics. Many mathematicians have devoted their efforts to generalization, refinement, modelling, and multiplication of various fields of work such as the use of convex mappings (see, e.g., [11], p.137, and [12]).The classical Hermite–Hadamard inequality states that ifϜ:I⟶ℝ is a convex function on the interval I of real numbers and κ1,κ2∈I with κ1<κ2, then(1)Ϝκ1+κ22≤1κ2−κ1∫κ1κ2Ϝϰdϰ≤Ϝκ1+Ϝκ22.The inequality holds in the reversed direction ifϜ is concave. We see that the Hermite–Hadamard inequality can be regarded as a refinement of the concept of integration and is easily followed by Jensen’s inequality. The Hermite–Hadamard inequality of convex works has received renewed attention in recent years and has been studied in significant and practical variations.In [13], Pachpatte proved the following inequalities for products of convex functions.Theorem 1.
LetϜ and G be real-valued, nonnegative, and convex functions on κ1,κ2. Then, we have(2)1κ2−κ1∫κ1κ2ϜϰGϰdϰ≤13Aκ1,κ2+16ℬκ1,κ2,(3)2Ϝκ1+κ22Gκ1+κ22≤1κ2−κ1∫κ1κ2ϜϰGϰdϰ+16Aκ1,κ2+13ℬκ1,κ2,where Aκ1,κ2=Ϝκ1Gκ1+Ϝκ2Gκ2 and ℬκ1,κ2=Ϝκ1Gκ2+Ϝκ2Gκ1.A positive function is calledr-convex on κ1,κ2, if for all ϰ,y∈κ1,κ2 and ξ∈0,1,(4)Ϝξϰ+1−ξy≤ξϜϰr+1−ξϜyr1/r,ifr≠0,ϜϰξϜy1−ξ,ifr=0.It is obvious ifr=1, then the inequality classical convex functions. It should be noted that if Ϝ is r-convex function, then Ϝ is convex function. We have that 0-convex functions are simply log-convex functions and 1-convex functions are ordinary convex functions [14].In [15], the definition of r-convex functions on coordinates is given, such thatDefinition 1.
A functionϜ:Δ:=κ1,κ2×κ3,κ4⟶ℝ+ will be called r-convex on Δ for all ξ,λ∈0,1 and ϰ,y,u,v∈Δ, if the following inequality holds:(5)Ϝξϰ+1−ξy,λu+1−λv≤ξλϜrϰ,u+ξ1−λϜrϰ,v+1−ξλϜry,u+1−ξ1−λϜry,v1/r,ifr≠0,Ϝξλϰ,uϜξ1−λϰ,vϜ1−ξλy,uϜ1−ξ1−λy,v,ifr=0.It is simply to see that if we chooser=0, we have coordinated log-convex functions and if we choose r=1, we have coordinated convex functions. In [15], Ekinci et al. also prove several Hermite–Hadamard type inequalities for coordinated r-convex functions. In literature, many studies have been done on r-convex functions. For some of them, one can see [16–23].


## 2. Preliminaries ofq-Calculus and Some Inequalities

In this section, we present some required definitions and related inequalities aboutq-calculus. For more information about q-calculus, one can refer to [1–10, 24, 25]. Also, here and further, we use the following notation (see [5]):(6)nq=1−qn1−q=1+q+q2+⋯+qn−1,q∈0,1.In [4], Jackson gave the q-Jackson integral from 0 to κ2 for 0<q<1 as follows:(7)∫0κ2Ϝϰdqϰ=1−qκ2∑n=0∞qnϜκ2qn,provided the sum converges absolutely.Jackson in [4] gave the q-Jackson integral in a generic interval κ1,κ2 as(8)∫κ1κ2Ϝϰdqϰ=∫0κ2Ϝϰdqϰ−∫0κ1Ϝϰdqϰ.Definition 2 (see [9]).
For a continuous functionϜ:κ1,κ2⟶ℝ, then q-derivative of Ϝ at ϰ∈κ1,κ2 for 0<q<1 is characterized by the expression(9)Dqκ1Ϝϰ=Ϝϰ−Ϝqϰ+1−qκ11−qϰ−κ1,ϰ≠κ1.
SinceϜ:κ1,κ2⟶ℝ is a continuous function, thus we haveDqκ1Ϝκ1=limϰ⟶κ1Dqκ1Ϝϰ. The function Ϝ is said to be q-differentiable on κ1,κ2 if Dqκ1Ϝξ exists for all ϰ∈κ1,κ2. If κ1=0 in (9), then Dq0Ϝϰ=DqϜϰ, where DqϜϰ is familiar q-derivative of Ϝ at ϰ∈κ1,κ2 defined by the expression (see [5])(10)DqϜϰ=Ϝϰ−Ϝqϰ1−qϰ,ϰ≠0.Definition 3 (see [9]).
LetϜ:κ1,κ2⟶ℝ be a continuous function. Then, the qκ1-definite integral on κ1,κ2 and 0<q<1 are defined as(11)∫κ1κ2Ϝϰdqκ1ϰ=1−qκ2−κ1∑n=0∞qnϜqnκ2+1−qnκ1=κ2−κ1∫01Ϝ1−ξκ1+ξκ2dqξ.In [26], Alp et al. proved the following qκ1-Hermite–Hadamard inequality for convex functions in the setting of quantum calculus.Theorem 2.
IfϜ:κ1,κ2⟶ℝ is a convex differentiable function on κ1,κ2 and 0<q<1. Then, q-Hermite–Hadamard inequalities are as follows:(12)Ϝqκ1+κ21+q≤1κ2−κ1∫κ1κ2Ϝϰκ1dqϰ≤qϜκ1+Ϝκ21+q.On the other hand, Bermudo et al. gave the following new definition and related Hermite–Hadamard type inequalities.Definition 4 (see [27]).
LetϜ:κ1,κ2⟶ℝ be a continuous function. Then, the qκ2-definite integral on κ1,κ2 for 0<q<1 is defined as(13)∫κ1κ2Ϝϰκ2dqϰ=1−qκ2−κ1∑n=0∞qnϜqnκ1+1−qnκ2=κ2−κ1∫01Ϝξκ1+1−ξκ2dqξ.Theorem 3 (see [27]).
IfϜ:κ1,κ2⟶ℝ is a convex differentiable function on κ1,κ2 and 0<q<1. Then, q-Hermite–Hadamard inequalities are as follows:(14)Ϝκ1+qκ21+q≤1κ2−κ1∫κ1κ2Ϝϰκ2dqϰ≤Ϝκ1+qϜκ21+q.From Theorem2 and Theorem 3, one can get the following inequalities.Corollary 1 (see [27]).
For any convex functionϜ:κ1,κ2⟶ℝ and 0<q<1, we have(15)Ϝqκ1+κ21+q+Ϝκ1+qκ21+q≤1κ2−κ1∫κ1κ2Ϝϰκ1dqϰ+∫κ1κ2Ϝϰκ2dqϰ≤Ϝκ1+Ϝκ2,(16)Ϝκ1+κ22≤12κ2−κ1∫κ1κ2Ϝϰκ1dqϰ+∫κ1κ2Ϝϰκ2dqϰ≤Ϝκ1+Ϝκ22.Brahim et al. prove the following lemma and theorem forr-convex functions.Lemma 1 (see [28]).
Forp≥1 and 0<q<1, the following inequality is valid:(17)∫011−ξpdqξ≤qp+1q.Theorem 4 (see [28]).
LetϜ:κ1,κ2⟶ℝ+ be r1-convex on κ1,κ2. Then, the following inequality holds for 0<r1≤1 and 0<q<1:(18)1κ2−κ1∫κ1κ2Ϝϰκ1dqϰ≤11/r1+1qqϜκ1r1+Ϝκ2r11/r1.Theorem 5 (see [28]).
LetϜ,G:κ1,κ2⟶ℝ+ be r1-convex and r2-convex functions, respectively, on κ1,κ2. Then, the following inequality holds for 0<r1,r2≤2 and 0<q<1:(19)2κ2−κ1∫κ1κ2ϜϰGϰκ1dqϰ≤12/r1+1qq1/2Ϝκ1r1+Ϝκ2r12/r1+12/r2+1qq1/2Gκ1r2+Gκ2r22/r2.Theorem 6 (see [28]).
LetϜ,G:κ1,κ2⟶ℝ+ be r1-convex and r2-convex functions, respectively, on κ1,κ2 and 0<q<1. Then, the following inequality holds if r1>1 and 1/r1+1/r2=1:(20)1κ2−κ1∫κ1κ2ϜϰGϰκ1dqϰ≤qϜκ1r1+Ϝκ2r12q1/r1qGκ1r2+Gκ1r22q1/r2.In [29], Latif defined the qκ1κ3-integral and related properties for two variable functions as follows.Definition 5.
Suppose thatϜ:Δ⟶ℝ is continuous function and 0<q1,q2<1. Then, the definite qκ1κ3-integral on Δ is defined by(21)∫κ1ϰ∫κ3yϜξ,sκ3dq2sκ1dq1ξ=1−q11−q2ϰ−κ1y−κ3×∑n=0∞∑m=0∞q1nq2mϜq1nϰ+1−q1nκ1,q2my+1−q2mκ3,for ϰ,y∈Δ.In [29], Latif et al. also proved a q-Hermite–Hadamard inequality for coordinated convex functions.By Definitions4 and 5, Budak et al. defined the following qκ1κ4, qκ3κ2 and qκ2κ4 integrals.Definition 6 (see [30]).
Suppose thatϜ:Δ⟶ℝ is a continuous function and 0<q1,q2<1. Then, the following qκ1κ4, qκ3κ2, and qκ2κ4 integrals on Δ are defined by(22)∫κ1ϰ∫yκ4Ϝξ,sκ4dq2sκ1dq1ξ=1−q11−q2ϰ−κ1κ4−y×∑n=0∞∑m=0∞q1nq2mϜq2mϰ+1−q2mκ1,q1ny+1−q1nκ4,∫ϰκ2∫κ3yϜξ,sκ3dq2sκ2dq1ξ=1−q11−q2κ2−ϰy−κ3×∑n=0∞∑m=0∞q1nq2mϜq2mϰ+1−q2mκ2,q1ny+1−q1nκ3,(23)∫ϰκ2∫yκ4Ϝξ,sκ4dq2sκ2dq1ξ=1−q11−q2κ2−ϰκ4−y×∑n=0∞∑m=0∞q1nq2mϜq2mϰ+1−q2mκ2,q1ny+1−q1nκ4,respectively, for ϰ,y∈Δ.Budak et al. also proved some quantum Hermite–Hadamard type inequalities for coordinated convex functions. For other similar quantum inequalities, please see [31,32].In this paper, we first prove the new variant of results of Brahim et al. forqκ2-integrals. We also obtain quantum versions of the inequalities in [15].


## 3. Quantum Hermite–Hadamard Type Inequalities forr-Convex Functions

In this section, we obtain some quantum inequalities of Hermite–Hadamard type forr-convex functions and for product of two r-convex functions.Theorem 7.
LetϜ:κ1,κ2⟶ℝ+ be a r1-convex function on κ1,κ2. Then, the following inequality holds for 0<r1≤1:(24)1κ2−κ1∫κ1κ2Ϝϰκ2dqϰ≤11/r1+1qϜκ1r1+qϜκ2r11/r1,where 0<q<1.Proof.
According to definitionr1-convex, for all ξ∈0,1, we have(25)Ϝξκ1+1−ξκ2≤ξϜκ1r1+1−ξϜκ2r11/r1.
By integrating the inequality on0,1, we obtain(26)∫01Ϝξκ1+1−ξκ2dqξ≤∫01ξϜκ1r1+1−ξϜκ2r11/r1dqξ.
From Definition4, we get(27)∫κ1κ2Ϝϰκ2dqϰ≤∫01ξϜκ1r1+1−ξϜκ2r11/r1dqξ.
Using Minkowski’s inequality for right side of inequality (26), (28)∫01ξϜκ1r1+1−ξϜκ2r11/r1dqξ≤∫01ξ1/r1dqξr1Ϝκ1r1+∫011−ξ1/r1dqξr1Ϝκ2r11/r1.
By Lemma1, we have(29)∫01ξ1/r1dqξr1=11/r1+1qr1,(30)∫011−ξ1/r1dqξr1≤q1/r1+1qr1.
Thus, by substituting (29) and (30) in (28), we obtain(31)∫κ1κ2Ϝϰκ2dqϰ≤11/r1+1qr1Ϝκ1r1+q1/r1+1qr1Ϝκ2r11/r1≤11/r1+1qϜκ1r1+qr1Ϝκ2r11/r1.
The proof is completed.Remark 1.
If we take the limitq⟶1− in Theorem 7, then Theorem 7 reduces to Theorem 2.1 in [33].Remark 2.
If we chooser1=1 in Theorem 7, then inequality (24) reduces to the second inequality in (14).Theorem 8.
LetϜ,G:κ1,κ2⟶ℝ+ be r1-convex and r2-convex functions, respectively, on κ1,κ2. Then, the following inequality holds for 0<r1,r2≤2:(32)2κ2−κ1∫κ1κ2ϜϰGϰκ2dqϰ≤12/r1+1qϜκ1r1+q1/2Ϝκ2r12/r1+12/r2+1qGκ1r2+q1/2Gκ2r22/r2,where 0<q<1.Proof.
By the assumptions thatϜ is an r1-convex function and G is an r2-convex function, we can write(33)Ϝξκ1+1−ξκ2≤ξϜκ1r1+1−ξϜκ2r11/r1,(34)Gξκ1+1−ξκ2≤ξGκ1r2+1−ξGκ2r21/r2,for all ξ∈0,1 and r1,r2>0.
Then,(35)Ϝξκ1+1−ξκ2Gξκ1+1−ξκ2≤ξϜκ1r1+1−ξϜκ2r11/r1ξGκ1r2+1−ξGκ2r21/r2.
Integrating both sides with respect toξ on 0,1 and from Definition 4, we obtain(36)1κ2−κ1∫κ1κ2ϜϰGϰκ2dqϰ≤∫01ξϜκ1r1+1−ξϜκ2r11/r1ξGκ1r2+1−ξGκ2r21/r2dqξ.
Using Cauchy’s inequality for right side of inequality (36), we obtain(37)∫01ξϜκ1r1+1−ξϜκ2r11/r1ξGκ1r2+1−ξGκ2r21/r2dqξ≤12∫01ξϜκ1r1+1−ξϜκ2r12/r1dqξ+12∫01ξGκ1r2+1−ξGκ2r22/r2dqξ.
By using Minkowski’s inequality, we have(38)∫01ξϜκ1r1+1−ξϜκ2r12/r1dqξ≤∫01ξ2/r1dqξr1/2Ϝκ1r1+∫011−ξ2/r1dqξr1/2Ϝκ2r12/r1=12/r1+1qr1/2Ϝκ1r1++q2/r1+1qr1/2Ϝκ2r12/r1.
Similarly, we have(39)∫01ξGκ1r2+1−ξGκ2r22/r2dqξ≤12/r2+1qr2/2Gκ1r2+q2/r2+1qr2/2Gκ2r22/r2.
Thus, from the inequalities (36)–(39), we obtain the desired result.Remark 3.
If we take the limitq⟶1− in Theorem 8, then Theorem 8 reduces to Theorem 2.3 in [33].Corollary 2.
If we chooser1=r2=2 in Theorem 8, then we have the inequality(40)2κ2−κ1∫κ1κ2ϜϰGϰκ2dqϰ≤Ϝκ12+qϜκ222q+Gκ12+qGκ222q.Particularly, ifϜϰ=Gϰ for all ϰ∈κ1,κ2, then we get(41)1κ2−κ1∫κ1κ2Ϝϰ2κ2dqϰ≤Ϝκ12+qϜκ222q.Theorem 9.
LetϜ,G:κ1,κ2⟶ℝ+ be r1-convex and r2-convex functions, respectively, on κ1,κ2. Then, we get the following inequality:(42)1κ2−κ1∫κ1κ2ϜϰGϰκ2dqϰ≤Ϝκ1r1+qϜκ2r12q1/r1Gκ1r2+qGκ2r22q1/r2,where 0<q<1 and 1/r1+1/r2=1 with r1>1.Proof.
From (36), we have(43)1κ2−κ1∫κ1κ2ϜϰGϰκ2dqϰ≤∫01ξϜκ1r1+1−ξϜκ2r11/r1ξGκ1r2+1−ξGκ2r21/r2dqξ.
Using Hölder inequality for quantum integrals, we have(44)1κ2−κ1∫κ1κ2ϜϰGϰκ2dqϰ≤∫01ξϜκ1r1+1−ξϜκ2r1dqξ1/r1∫01ξGκ1r2+1−ξGκ2r2dqξ1/r2=Ϝκ1r1+qϜκ2r12q1/r1Gκ1r2+qGκ2r22q1/r2.
This completes the proof.Remark 4.
If we take the limitq⟶1− in Theorem 9, then Theorem 9 reduces to Theorem 2.6 in [33].Corollary 3.
If we chooser1=r2=2 in Theorem 9, then we have the inequality(45)1κ2−κ1∫κ1κ2ϜϰGϰκ2dqϰ≤Ϝκ12+qϜκ222qGκ12+qGκ222q.Particularly, ifϜϰ=Gϰ for all ϰ∈κ1,κ2, then we get(46)1κ2−κ1∫κ1κ2Ϝϰ2dqκ2ϰ≤Ϝκ12+qϜκ222q.


## 4. Quantum Hermite–Hadamard Type Inequalities for Coordinatedr -Convex Functions

In this section, we present several Hermite–Hadamard type inequalities for coordinatedr-convex functions via qκ2κ4, qκ1κ4, qκ3κ2, and qκ1κ3 integrals. We also prove some quantum inequalities of Hermite–Hadamard type for the product of two coordinated r-convex functions.where 0<r1≤1and 0<q1,q2<1.Theorem 10.
Suppose thatϜ:Δ⟶ℝ+ is a positive coordinated r1-convex function on Δ. Then, one has the inequality(47)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yκ2dq1ϰκ4dq2y≤11/r1+1q212κ2−κ1∫κ1κ2Ϝϰ,κ3r1+q2Ϝϰ,κ4r11/r1dq1κ2ϰ+11/r1+1q112κ4−κ3∫κ3κ4Ϝκ1,yr1+q1Ϝκ2,yr11/r1dq2κ4ϰ,Proof.
SinceϜ:Δ⟶ℝ+ is a coordinated r1-convex function, then the partial mappings,(48)Ϝϰ:κ3,κ4⟶ℝ+,Ϝϰv=Ϝϰ,v,(49)Ϝy:κ1,κ2⟶ℝ+,Ϝyu=Ϝu,y,are r1-convex. By inequality (24), we can write(50)1κ4−κ3∫κ3κ4Ϝϰyκ4dq2y≤11/r1+1q2Ϝϰκ3r1+q2Ϝϰκ4r11/r1,i.e.,(51)1κ4−κ3∫κ3κ4Ϝϰ,yκ4dq2y≤11/r1+1q2Ϝϰ,κ3r1+q2Ϝϰ,κ4r11/r1.
Dividing both sides of the inequalityκ2−κ1 and qκ2-integrating with respect to ϰ on κ1,κ2, we get(52)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,ydq1κ2ϰκ4dq2y≤11/r1+1q21κ2−κ1∫κ1κ2Ϝϰ,κ3r1+q2Ϝϰ,κ4r11/r1dq1κ2ϰ.
By a similar argument for the mapping(53)Ϝy:κ1,κ2⟶ℝ+,Ϝyu=Ϝu,y,we have(54)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yκ2dq1ϰdq2κ4y≤11/r1+1q11κ4−κ3∫κ3κ4Ϝκ1,yr1+q1Ϝκ2,yr11/r1dq2κ4y.
By adding inequalities (52) and (54), we can obtain inequality (47).Remark 5.
If we take the limitq1⟶1−andq2⟶1− in Theorem 10, then Theorem 10 reduces to Theorem 5 in [15].Remark 6.
If we chooser1=1 in Theorem 10, then inequality (47) reduces to the third inequality of Theorem 3.6 in [30].Theorem 11.
Suppose thatϜ:Δ⟶ℝ+ is a positive coordinated r1-convex function on Δ. Then, one has the inequality(55)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yκ1dq1ϰκ3dq2y≤11/r1+1q212κ2−κ1∫κ1κ2q2Ϝϰ,κ3r1+Ϝϰ,κ4r11/r1dq1κ1ϰ+11/r1+1q112κ4−κ3∫κ3κ4q1Ϝκ1,yr1+Ϝκ2,yr11/r1dq2κ3y,where 0<r1≤1and 0<q1,q2<1.Proof.
The proof is similar to the proof of Theorem10 by using Theorem 4.Theorem 12.
Suppose thatϜ:Δ⟶ℝ+ is a positive coordinated convex function on Δ. Then, one has the inequality(56)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yκ1dq1ϰκ4dq2y≤11/r1+1q212κ2−κ1∫κ1κ2q2Ϝϰ,κ3r1+Ϝϰ,κ4r11/r1dq1κ1ϰ+11/r1+1q112κ4−κ3∫κ3κ4Ϝκ1,yr1+q1Ϝκ2,yr11/r1dq2κ4y,where 0<r1≤1 and 0<q1,q2<1.Proof.
The proof is similar to the proof of Theorem10 by using Theorems 4 and 7.Theorem 13.
Suppose thatϜ:Δ⟶ℝ+ is a positive coordinated r1-convex function on Δ. Then, one has the inequality(57)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝχ,yκ2dq1χκ3dq2y≤11/r1+1q212κ2−κ1∫κ1κ2Ϝχ,κ3r1+q2Ϝχ,κ4r11/r1dq1κ2χ+11/r1+1q112κ4−κ3∫κ3κ4q1Ϝκ1,yr1+Ϝκ2,yr1κ31/r1dq2,where 0<r1≤1 and 0<q1,q2<1.Proof.
The proof is similar to the proof of Theorem10 by using Theorems 4 and 7.Theorem 14.
Suppose thatϜ,G:Δ⟶ℝ+ is a coordinated r1-convex function and coordinated r2-convex function, respectively, on Δ. Then, we have the inequality(58)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yGϰ,ydq2κ4ydq1κ2ϰ≤142/r1+1q21κ2−κ1∫κ1κ2Ϝϰ,κ3r1+q21/2Ϝϰ,κ4r12/r1dq1ϰ+142/r2+1q21κ2−κ1∫κ1κ2Gϰ,κ3r2+q21/2Gϰ,κ4r22/r2dq1κ2ϰ+142/r1+1q11κ4−κ3∫κ3κ4Ϝκ1,yr1+q11/2Ϝκ2,yr12/r1dq2κ4y+142/r2+1q11κ4−κ3∫κ3κ4Gκ1,yr2+q11/2Gκ2,yr22/r2dq2κ4y,where 0<r1, r2≤2, and 0<q1,q2<1.Proof.
SinceϜ:Δ⟶ℝ+ is a coordinated r1-convex on Δ, then the partial mappings,(59)Ϝϰ:κ3,κ4⟶ℝ+,Ϝϰv=Ϝϰ,v,Ϝy:κ1,κ2⟶ℝ+,Ϝyu=Ϝu,y,are r1-convex on Δ. On the other hand, if G is a coordinated r2-convex function, then the partial mappings,(60)Gϰ:κ3,κ4⟶ℝ+,Gϰv=Gϰ,v,Gy:κ1,κ2⟶ℝ+,Gyu=Gu,y,are r2-convex on Δ. From inequality (32), we get(61)2κ4−κ3∫κ3κ4ϜϰyGϰyκ4dq2y≤12/r1+1q2Ϝϰκ3r1+q21/2Ϝϰκ4r12/r1+12/r2+1q2Gϰκ3r2+q21/2Gϰκ4r22/r2,i.e.,(62)2κ4−κ3∫κ3κ4Ϝϰ,yGϰ,yκ4dq2y≤12/r1+1q2Ϝϰ,κ3r1+q21/2Ϝϰ,κ4r12/r1+12/r2+1q2Gϰ,κ3r2+q21/2Gϰ,κ4r22/r2.
Dividing both sides of the inequalityκ2−κ1 and qκ2-integrating with respect to ϰ on κ1,κ2, we have(63)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yGϰ,yκ4dq2yκ2dq1ϰ≤1212/r1+1q21κ2−κ1∫κ1κ2Ϝϰ,κ3r1+q21/2Ϝϰ,κ4r12/r1dq1κ2ϰ+1212/r2+1q21κ2−κ1∫κ1κ2Gϰ,κ3r2+q21/2Gϰ,κ4r22/r2dq1κ2ϰ.
By a similar argument, we obtain(64)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yGϰ,ydq2κ4ydq1κ2ϰ≤1212/r1+1q11κ4−κ3∫κ3κ4Ϝκ1,yr1+q11/2Ϝκ2,yr12/r1dq2κ4y+1212/r2+1q11κ4−κ3∫κ3κ4Gκ1,yr2+q11/2Gκ2,yr22/r2dq2κ4y.
By adding inequalities (63) and (64), we obtain the required result.Remark 7.
If we take the limitq1⟶1− and q2⟶1− in Theorem 14, then Theorem 14 reduces to Theorem 6 in [15].Corollary 4.
If we chooser1=r2=2 in Theorem 14, then we have the inequality(65)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yGϰ,ydq2κ4ydq1κ2ϰ≤14κ2−κ1∫κ1κ2Ϝϰ,κ32+q2Ϝϰ,κ422q2dq1κ2ϰ+14κ2−κ1∫κ1κ2Gϰ,κ32+q2Gϰ,κ422q2dq1κ2ϰ+14κ4−κ3∫κ3κ4Ϝκ1,y2+q1Ϝκ2,y22q1dq2κ4y+14κ4−κ3∫κ3κ4Gκ1,y2+q1Gκ2,y22q1dq2κ4y.
Particularly, ifϜϰ,y=Gϰ,y for all ϰ,y∈Δ, then we get(66)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,y2dq2κ4ydq1κ2ϰ≤12κ2−κ1∫κ1κ2Ϝϰ,κ32+q2Ϝϰ,κ422q2dq1κ2ϰ+12κ4−κ3∫κ3κ4Ϝκ1,y2+q1Ϝκ2,y22q1dq2κ4y.Theorem 15.
Suppose thatϜ,G:Δ⟶ℝ+ is a coordinated r1-convex function and coordinated r2-convex function, respectively, on Δ. Then, we have the inequality(67)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yGϰ,ydq2κ3ydq1κ2ϰ≤142/r1+1q21κ2−κ1∫κ1κ2Ϝϰ,κ3r1+q21/2Ϝϰ,κ4r12/r1dq1κ2ϰ+142/r2+1q21κ2−κ1∫κ1κ2Gϰ,κ3r2+q21/2Gϰ,κ4r22/r2dq1κ2ϰ+142/r1+1q11κ4−κ3∫κ3κ4q11/2Ϝκ1,yr1+Ϝκ2,yr12/r1dq2κ3y+142/r2+1q11κ4−κ3∫κ3κ4q11/2Gκ1,yr2+Gκ2,yr22/r2dq2κ3y,where 0<r1, r2≤2, and 0<q1,q2<1.Proof.
The proof is similar to the proof of Theorem14 by using Theorems 5 and 8.Theorem 16.
Suppose thatϜ,G:Δ⟶ℝ+ is a coordinated r1-convex function and coordinated r2-convex function, respectively, on Δ. Then, we have the inequality(68)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yGϰ,ydq2κ3ydq1k1ϰ≤142/r1+1q21κ2−κ1∫κ1κ2q21/2Ϝϰ,κ3r1+Ϝϰ,κ4r12/r1dq1k1ϰ+142/r2+1q21κ2−κ1∫κ1κ2q21/2Gϰ,κ3r2+Gϰ,κ4r22/r2dq1k1ϰ+142/r1+1q11κ4−κ3∫κ3κ4q11/2Ϝκ1,yr1+Ϝκ2,yr12/r1dq2κ3y+142/r2+1q11κ4−κ3∫κ3κ4q11/2Gκ1,yr2+Gκ2,yr22/r2dq2κ3y,where 0<r1, r2≤2, and 0<q1,q2<1.Proof.
The proof is similar to the proof of Theorem14 by using Theorem 5.Theorem 17.
Suppose thatϜ,G:Δ⟶ℝ+ is a coordinated r1-convex function and coordinated r2-convex function, respectively, on Δ. Then, we have the inequality(69)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yGϰ,ydq2κ4ydq1k1ϰ≤142/r1+1q21κ2−κ1∫κ1κ2q21/2Ϝϰ,κ3r1+Ϝϰ,κ4r12/r1dq1k1ϰ+142/r2+1q21κ2−κ1∫κ1κ2q21/2Gϰ,κ3r2+Gϰ,κ4r22/r2dq1k1ϰ+142/r1+1q11κ4−κ3∫κ3κ4Ϝκ1,yr1+q11/2Ϝκ2,yr12/r1dq2κ4y+142/r2+1q11κ4−κ3∫κ3κ4Gκ1,yr2+q11/2Gκ2,yr22/r2dq2κ4y,where 0>r1, r2≤2, and 0<q1,q2<1.Proof.
The proof is similar to the proof of Theorem14 by using Theorems 5 and 8.Theorem 18.
Suppose thatϜ,G:Δ⟶ℝ+ is a coordinated r1-convex function and coordinated r2 -convex function, respectively, on Δ. Then, we have the inequality(70)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yGϰ,ydq2κ4dq1κ2ϰ≤121κ2−κ1∫κ1κ2Ϝϰ,κ3r1+q2Ϝϰ,κ4r12q21/r1dq1κ2ϰ×1κ2−κ1∫κ1κ2Gϰ,κ3r2+q2Gϰ,κ4r22q21/r2dq1κ2ϰ+121κ4−κ3∫κ3κ4Ϝκ1,yr1+q1Ϝκ2,yr12q11/r1dq2κ4×1κ4−κ3∫κ3κ4Gκ1,yr2+q1Gκ2,yr22q11/r2dq2κ4.where 0<q1,q2<1, and 1/r1+1/r2=1 with r1>1.Proof.
By applying inequality (42) for the partial mapping Ϝϰ and Gϰ, we can write(71)1κ4−κ3∫κ3κ4Ϝϰ,yGϰ,ydq2κ4y≤Ϝϰ,κ3r1+q2Ϝϰ,κ4r12q21/r1Gϰ,κ3r2+q2Gϰ,κ4r22q21/r2.
By usingqκ2-integral, we obtain(72)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yGϰ,ydq2κ4ydq1κ2ϰ≤1κ2−κ1∫κ1κ2Ϝϰ,κ3r1+q2Ϝϰ,κ4r12q21/r1dq1κ2ϰ×1κ2−κ1∫κ1κ2Gϰ,κ3r2+q2Gϰ,κ4r22q2dq1κ2ϰ.
Similarly, by applying inequality (42) for the partial mapping Ϝy and Gy, we can write(73)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yGϰ,ydq2κ4ydq1κ2ϰ≤1κ4−κ3∫κ3κ4Ϝκ1,yr1+q1Ϝκ2,yr12q11/r1dq2κ4y×1κ4−κ3∫κ3κ4Gκ1,yr2+q1Gκ2,yr22q11/r2dq2κ4y.
By adding inequalities (72) and (73), we obtain the desired result (70).Remark 8.
If we take the limitq1⟶1− and q2⟶1− in Theorem 18, then Theorem 18 reduces to Theorem 7 in [15].Corollary 5.
If we chooser1=r2=2 in Theorem 18, then we have the inequality(74)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yGϰ,ydq2κ4ydq1κ2ϰ≤121κ2−κ1∫κ1κ2Ϝϰ,κ32+q2Ϝϰ,κ422q2dq1κ2ϰ×1κ2−κ1∫κ1κ2Gϰ,κ32+q2Gϰ,κ422q2dq1κ2ϰ+121κ4−κ3∫κ3κ4Ϝκ1,y2+q1Ϝκ2,y22q1dq2κ4y×1κ4−κ3∫κ3κ4Gκ1,y2+q1Gκ2,y22q1dq2κ4y.Theorem 19.
Suppose thatϜ,G:Δ⟶ℝ+ is a coordinated r1-convex function and coordinated r2-convex function, respectively, on Δ. Then, we have the inequality(75)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yGϰ,ydq2k3ydq1κ2ϰ≤121κ2−κ1∫κ1κ2Ϝϰ,κ3r1+q2Ϝϰ,κ4r12q21/r1dq1κ2ϰ×1κ2−κ1∫κ1κ2Gϰ,κ3r2+qGϰ,κ4r22q21/r2dq1κ2ϰ+121κ4−κ3∫κ3κ4q1Ϝκ1,yr1+Ϝκ2,yr12q11/r1dq2k3y×1κ4−κ3∫κ3κ4q1Gκ1,yr2+Gκ2,yr22q11/r2dq2k3y.where 0<q1,q2<1, and 1/r1+1/r2=1 with r1>1.Proof.
The proof is similar to the proof of Theorem18 by using Theorems 6 and 9.Theorem 20.
Suppose thatϜ,G:Δ⟶ℝ+ is a coordinated r1-convex function and coordinated r2-convex function, respectively, on Δ. Then, we have the inequality(76)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yGϰ,ydq2κ4ydq1κ1ϰ≤121κ2−κ1∫κ1κ2q2Ϝϰ,κ3r1+Ϝϰ,κ4r12q21/r1dq1κ1ϰ×1κ2−κ1∫κ1κ2q2Gϰ,κ3r2+Gϰ,κ4r22q21/r2dq1κ1ϰ+121κ4−κ3∫κ3κ4Ϝκ1,yr1+q1Ϝκ2,yr12q11/r1dq2κ4y×1κ4−κ3∫κ3κ4Gκ1,yr2+q1Gκ2,yr22q11/r2dq2κ4y,where 0<q1,q2<1, and 1/r1+1/r2=1 with r1>1.Proof.
The proof is similar to the proof of Theorem18 by using Theorems 6 and 9.Theorem 21.
Suppose thatϜ,G:Δ⟶ℝ+ is a coordinated r1-convex function and coordinated r2 -convex function, respectively, on Δ. Then, we have the inequality(77)1κ2−κ1κ4−κ3∫κ1κ2∫κ3κ4Ϝϰ,yGϰ,ydq2κ3ydq1κ1ϰ≤121κ2−κ1∫κ1κ2q2Ϝϰ,κ3r1+Ϝϰ,κ4r12q21/r1dq1κ1ϰ×1κ2−κ1∫κ1κ2q2Gϰ,κ3r2+Gϰ,κ4r22q21/r2dq1κ1ϰ+121κ4−κ3∫κ3κ4q1Ϝκ1,yr1+Ϝκ2,yr12q11/r1dq2κ3y×1κ4−κ3∫κ3κ4q1Gκ1,yr2+Gκ2,yr22q11/r2dq2κ3y.where 0<q1,q2<1, and 1/r1+1/r2=1 with r1>1.Proof.
The proof is similar to the proof of Theorem18 by using Theorem 6.


## 5. Conclusions

In this study, we present several quantum Hermite–Hadamard type inequalities forr-convex functions and coordinated r-convex functions. We also give some quantum inequalities for the product of two r-convex functions and for the product of two coordinated r-convex functions. In the future work, we can establish the similar quantum inequalities by using generalized r-convex functions.



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*Source: 6634614-2021-02-23.xml*