# New Estimates ofq1q2-Ostrowski-Type Inequalities within a Class of n-Polynomial Prevexity of Functions

Authors: Humaira Kalsoom; Muhammad Idrees; Dumitru Baleanu; Yu-Ming Chu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this article, we develop a novel framework to study for a new class of preinvex functions depending on arbitrary nonnegative function, which is calledn-polynomial preinvex functions. We use the n-polynomial preinvex functions to develop q1q2-analogues of the Ostrowski-type integral inequalities on coordinates. Different features and properties of excitement for quantum calculus have been examined through a systematic way. We are discussing about the suggestions and different results of the quantum inequalities of the Ostrowski-type by inferring a new identity for q1q2-differentiable function. However, the problem has been proven to utilize the obtained identity, we give q1q2-analogues of the Ostrowski-type integrals inequalities which are connected with the n-polynomial preinvex functions on coordinates. Our results are the generalizations of the results in earlier papers.


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## Body

## 1. Introduction

Calculus is an imperative study of the derivatives and integrals. The classical derivative was convoluted with the strength regulation kind kernel, and eventually, this gave upward thrust to new calculus referred to as the quantum calculus. In mathematics, quantum calculus (namedq-calculus) is the study of calculus without limits. The interest in this subject has exploded, and the q-calculus has in the last twenty years served as a bridge between mathematics and physics. The q-calculus has numerous applications in various fields of mathematics, for example, dynamical systems, number theory, combinatorics, special functions, fractals, and also for scientific problems in some applied areas such as computer science, quantum mechanics, and quantum physics. Jackson [1] defined the q-analogue of derivative and integral operator as well as provided some of their applications. It is imperative to mention that quantum integral inequalities are more practical and informative than their classical counterparts. It has been mainly due to the fact that quantum integral inequalities can describe the hereditary properties of the processes and phenomena under investigation. Historically, the subject of quantum calculus can be traced back to Euler and Jacobi, but in recent decades, it has experienced a rapid development. As a result, new generalizations of the classical concepts of quantum calculus have been initiated and reviewed in many literature. Tariboon and Ntouyas [2, 3] proposed the quantum calculus concepts on finite intervals and obtained several q-analogues of classical mathematical objects, which inspired many other researchers to study the subject in depth, and as a consequence, numerous novel results concerning quantum analogues of classical mathematical results have been launched. Noor et al. [4] obtained new q-analogues of inequality utilizing the first-order q-differentiable convex function.Inequality plays an irreplaceable role in the development of mathematics. Very recently, many new inequalities such as the Hermite-Hadamard-type inequality [5–9], Petrović-type inequality [10], Pólya-Szegö and Ćebyšev-type inequalities [11], Ostrowski-type inequality [12], reverse Minkowski inequality [13], Jensen-type inequality [14–16], Bessel function inequality [17], trigonometric and hyperbolic functions inequalities [18], fractional integral inequality [19–22], complete and generalized elliptic integrals inequalities [23–28], generalized convex function inequality [29–31], and mean values inequality [32–34] have been discovered by many researchers. While the concept of classical convexity has been brought into a streamline by mathematical inequalities [35–50]. In fact, convex function and its connection with mathematical inequalities have wide applications in the estimation of some parameters in scientific observations and calculations [51–65]. In recent years, the classical concept of convexity has been extended and generalized in different directions, one of the important generalization of convexity is the invexity, which was studied by Hanson [66]; this work has greatly expanded the role of invexity in optimization. In [67, 68], the authors introduced a class of functions, which is called preinvexity as a generalization of convex functions.Now, we recall the classical and well-known Hermite-Hadamard inequality [69], which can be stated as
(1)Φξ1+ξ22≤1ξ2−ξ1∫ξ1ξ2Φγdγ≤Φξ1+ϕξ22,for all ξ1,ξ2∈I if Φ:I⟶ℝ is a convex function.Ostrowski [70] established an integral inequality for continuous and differentiable function as follows.Theorem 1 (See [70]).
LetΦ:ξ1,ξ2⟶ℝ be continuous and differentiable on ξ1,ξ2 such that Φ′γ≤M for all γ∈ξ1,ξ2. Then, one has
(2)Φϱ−1ξ2−ξ1∫ξ1ξ2Φγdγ≤14+ϱ−ξ1+ξ2/22ξ2−ξ12ξ2−ξ1M,for all ϱ∈ξ1,ξ2 with the best possible constant 1/4.
The inequality (2) can be described in an identical kind as
(3)Φϱ−1ξ2−ξ1∫ξ1ξ2Φγdγ≤Mξ2−ξ1ϱ−ξ12+ϱ22.Latif et al. [71] generalized the Ostrowski inequality (2) to the coordinated convex function by establishing an identity as follows.Theorem 2 (See [71]).
Letξ1<ξ2,ξ3<ξ4 and Φ:ξ1,ξ2×ξ3,ξ4⟶ℝ be continuous and differentiable on ξ1,ξ2×ξ3,ξ4 such that ∂2Φ/∂z∂w∈Lξ1,ξ2×ξ3,ξ4.
Then the identity(4)Φϱ,ρ+1ξ2−ξ1ξ4−ξ3∫ξ1ξ2∫ξ3ξ4Φu,vdudv−y=ϱ−ξ12ρ−ξ32ξ2−ξ1ξ4−ξ3∫01∫01zw∂2∂z∂wΦzϱ+1−zξ1,wρ+1−wξ3dzdw−ϱ−ξ12ξ4−ρ2ξ2−ξ1ξ4−ξ3∫01∫01zw∂2∂z∂wΦzϱ+1−zξ1,wρ+1−wξ4dzdw−ξ2−ϱ2ρ−ξ32ξ2−ξ1ξ4−ξ3∫01∫01zw∂2∂z∂wΦzϱ+1−zξ2,wρ+1−wξ3dzdw+ξ2−ϱ2ξ4−ρ2ξ2−ξ1ξ4−ξ3∫01∫01zw∂2∂z∂wΦzϱ+1−zξ2,wρ+1−wξ4dzdw,holds for all ϱ,ρ∈ξ1,ξ2×ξ3,ξ4, where
(5)y=1ξ4−ξ3∫ξ3ξ4Φϱ,vdv+1ξ2−ξ1∫ξ1ξ2Φu,ρdu.
Noor et al. [4] presented the Ostrowski-type inequality for quantum calculus.Theorem 3 (See [4]).
Letq∈0,1,ξ1<ξ2 and Φξ1,ξ2⟶ℝ be continuous such that Dξ1qΦ is integrable on ξ1,ξ2. Then
(6)Φϱ−1ξ2−ξ1∫ξ1ξ2Φuξ1dqu=qϱ−ξ12ξ2−ξ1∫01zξ1DqΦ1−zξ1+zϱ0dqz+qξ2−ϱ2ξ2−ξ1∫01zξ1DqΦ1−zξ2+zϱ0dqz.
The following quantum integral version of the Hermite-Hadamard-type inequality for the coordinated convex function was proved by Alp and Sarıkaya [72].Theorem 4 (See [72]).
Letq1,q2∈0,1, ξ1<ξ2, ξ3<ξ4 and Φ:N=ξ1,ξ2×ξ3,ξ4⟶ℝ be a coordinated convex function on N. Then one has
(7)Φq1ξ1+ξ21+q1,q2ξ3+ξ41+q2≤121ξ2−ξ1∫ξ1ξ2Φz,q2ξ3+ξ41+q2dξ1q1z+1ξ3−ξ4∫ξ3ξ4Φq1ξ1+ξ21+q1,wdξ3q2w≤1ξ1−ξ2ξ3−ξ4∫ξ1ξ2∫ξ3ξ4Φz,wξ1dq1zξ3dq2w≤1411+q1ξ2−ξ1q2∫ξ1ξ2Φz,ξ3ξ1dq1z+∫ξ1ξ2Φz,ξ4ξ1dq1z+11+q2ξ3−ξ4q1∫ξ3ξ3Φξ1,wξ3dq2w+∫ξ3ξ4Φξ2,wξ3dq2w≤q1q2Φξ1,ξ3+q2Φξ2,ξ3+q1Φξ1,ξ4+Φξ2,ξ41+q11+q2.
Kalsoom et al. [73] found the quantum integral inequality for two parameters function on the finite rectangle.Next, we present the definitions ofq1q2-derivative and integral, and their two known results.Definition 5.
Letq1,q2∈0,1,ξ1<ξ2,ξ3<ξ4 and Φ:ξ1,ξ2×ξ3,ξ4⟶ℝ be a continuous function. Then, the partially q1-derivative, q2-derivative, and q1q2-derivative at z,w∈ξ1,ξ2×ξ3,ξ4 for the function Φ are defined by
(8)ξ1∂q1Φz,wξ1∂q1z=Φz,w−Φq1z+1−q1ξ1,w1−q1z−ξ1z≠ξ1,ξ3∂q2Φz,wξ3∂q2w=Φz,w−Φz,q2w+1−q2ξ31−q2w−ξ3w≠ξ3,ξ1,ξ3∂q1,q22Φz,wξ1∂q1zξ3∂q2w=11−q11−q2z−ξ1w−ξ3×Φq1z+1−q1ξ1,q2w+1−q2ξ3−Φq1z+1−q1ξ1,w−Φz,q2w+1−q2ξ3+Φz,wz≠ξ1,w≠ξ3,respectively. The function Φ is said to be partially q1-, q2-, and q1q2-differentiable on ξ1,ξ2×ξ3,ξ4 if ξ1∂q1Φz,w/ξ1∂q1z, ξ3∂q2Φz,w/ξ3∂q2w, and ξ1,ξ3∂q1,q22Φz,w/ξ1∂q1zξ3∂q2w exist for all z,w∈ξ1,ξ2×ξ3,ξ4.Definition 6.
Letq1,q2∈0,1,ξ1<ξ2,ξ3<ξ4 and Φ:ξ1,ξ2×ξ3,ξ4⟶ℝ be a continuous function. Then the q1q2-integral of the function Φ on ξ1,ξ2×ξ3,ξ4 is defined by
(9)∫ξ3t∫ξ1sΦz,wξ1dq1zξ3dq2w=1−q11−q2s−ξ1t−ξ3×∑m=0∞∑n=0∞q1nq2mΦq1ns+1−q1nξ1,q2mt+1−q2mξ3×∑m=0∞∑n=0∞q1nq2mΦq1ns+1−q1nξ1,q2mt+1−q2mξ3,for s,t∈ξ1,ξ2×ξ3,ξ4.Theorem 7.
Letξ1<ξ2,ξ3<ξ4 and Φ:ξ1,ξ2×ξ3,ξ4⟶ℝ be a continuous function. Then, we have the identities
(10)ξ1,ξ3∂q1,q22ξ1∂q1sξ3∂q2t∫ξ4t∫ξ1sΦz,wξ1dq1zξ3dq2w=Φs,t,∫ξ3t∫ξ1sξ1,ξ3∂q1,q22Φz,wξ1∂q1zξ3∂q2wξ1dq1zξ3dq2w=Φs,t,∫t1t∫s1sξ1,ξ3∂q1,q22Φz,wξ1∂q1zξ3∂q2wξ1dq1zξ3dq2w=Φs,t−Φs,t1−Φs1,t+Φs1,t1,for s1,t1∈ξ1,s×ξ4,t.Theorem 8.
Leta∈ℝ,ξ1<ξ2,<ξ3<ξ4 and Φ1,Φ2:ξ1,ξ2×ξ3,ξ4⟶R be continuous functions. Then, the identities
(11)∫ξ3t∫ξ1sΦ1z,w+Φ2z,wξ1dq1zξ4dq2w=∫ξ3t∫ξ1sΦ1z,wξ1dq1zξ3dq2w+∫ξ3t∫ξ1sΦ2z,wξ1dq1zξ3dq2w,∫ξ3t∫ξ1saΦz,wξ1dq1zξ3dq2w=a∫ξ3t∫ξ1sΦz,wξ1dq1zξ3dq2w,holds for s,t∈ξ1,ξ2×ξ3,ξ4.Very recently, Toplu et al. [74] improved the Hermite-Hadamard inequality (1) by investigating the n-polynomial convexity. The main purpose of the article is to introduce the notion of n-polynomial preinvex function, provide a new generalized quantum integral identity, establish new quantum analogues of Ostrowski-type inequalities for the n-polynomial preinvex function on coordinates, and generalize and unify the previous known results.


## 2. Discussions and Main Results

In the beginning of this section, we introduce the definition ofn-polynomial prevexity.Definition 9 (See [75]).
Letη⋅,⋅:ℝn×ℝn⟶ℝn be a continuous bi-function. Then, Ωη⊂ℝn is said to be invex if
(12)ξ1+γηξ2,ξ1∈Ωη,for all ξ1,ξ2∈Ωη and γ∈0,1.Definition 10 (See [67]).
The functionΦ:Ωη⊂ℝn⟶ℝ is said to be preinvex if
(13)Φξ1+γηξ2,ξ1≤1−γΦξ1+γΦξ2,for all ξ1,ξ2∈Ωη and γ∈0,1.Definition 11.
Letn∈ℕ. Then, the nonnegative function Φ:Ωη⊂ℝn⟶ℝ is said to be n-polynomial preinvex if
(14)Φξ1+γηξ2,ξ1≤1n∑p=1n1−1−γpΦξ1+1n∑p=1n1−1−γpΦξ2,for all ξ1,ξ2∈Ωη and γ∈0,1.
Note that ifn=1, then the definition of n-polynomial preinvex function reduce to the definition of preinvex function.
If we taken=2, then we have 2-polynomial preinvex function inequality
(15)Φξ1+γηξ2,ξ1≤3γ−γ22Φξ1+1−γ−γ22Φξ2.Proposition 12.
Letξ2>0 and Φα:ξ1,ξ1+ηξ2,ξ1⟶ℝ be an arbitrary family of n-polynomial preinvex functions and Φξ=supαΦαξ. If Jη=u∈ξ1,ξ1+ηξ2,ξ1:Φu<1 is nonempty, then Jη is an interval and Φ is an n-polynomial preinvex function on Jη.Proof.
Letγ∈0,1 and ξ1,ξ1+ηξ2,ξ1∈Jη. Then, we have
(16)Φξ1+γηξ2,ξ1=supαΦαξ1+γηξ2,ξ1≤supα1n∑p=1n1−1−γpΦαξ1+1n∑p=1n1−γpΦαξ2≤1n∑p=1n1−1−γpsupαΦαξ1+1n∑p=1n1−γpsupαΦαξ2≤1n∑p=1n1−1−γpΦξ1+1n∑p=1n1−γpΦξ2<∞.This completes the proof.In order to establish new quantum analogues of the Ostrowski-type inequalities on coordinates for then-polynomial preinvex function, we need a key lemma, which we present in this section.Lemma 13.
Letq1,q2∈0,1,ξ1<ξ2,ξ3<ξ4,N=ξ1,ξ2×ξ3,ξ4,N∘, be the interior of N, and Φ:N⟶ℝ be mixed partial q1q2-differentiable on N° such that ξ1,ξ3∂q1,q22/ξ1∂q1zξ3∂q2wΦ is continuous and integrable on ξ1,ξ1+η1ξ2,ξ1×ξ3,ξ3+η2ξ4,ξ3⊂N° for η1ξ2,ξ1,η2ξ4,ξ3>0. Then, we have the identity
(17)Ωq1q2ξ1,ξ2,ξ3,ξ4Φ=−q1q2η1ϱ,ξ12η2ρ,ξ32η1ξ2,ξ1η2ξ4,ξ3×∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ30dq1z0dq2w−q1q2η1ϱ,ξ12η2ξ4,ρ2η1ξ2,ξ1η2ξ4,ξ3×∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ4+wη2ρ,ξ40dq1z0dq2w−q1q2η1ξ2,ϱ2η2ρ,ξ32η1ξ2,ξ1η2ξ4,ξ3×∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ3+wη2ρ,ξ30dq1z0dq2w−q1q2η1ξ2,ϱ2η2ξ4,ρ2η1ξ2,ξ1η2ξ4,ξ3×∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ4+wη2ρ,ξ40dq1z0dq2w,where
(18)Ωq1q2ξ1,ξ2,ξ3,ξ4Φ=Φϱ,ρ+1η2ξ4,ξ3∫ξ3ξ3+η2ρ,ξ3Φϱ,v0dq2v+∫ξ4ξ4+η2ρ,ξ4Φϱ,v0dq2v+1η1ξ2,ξ1∫ξ1ξ1+η1ϱ,ξ1Φu,ρ0dq1u+∫ξ2ξ2+η1ϱ,ξ2Φu,ρ0dq1v−Q,Q=1η1ξ2,ξ1η2ξ4,ξ3∫ξ1ξ1+η1ϱ,ξ1∫ξ3ξ3+η2ρ,ξ3Φu,v0dq1u0dq2v+1η1ξ2,ξ1η2ξ4,ξ3∫ξ1ξ1+η1ϱ,ξ1∫ξ4ξ4+η2ρ,ξ4Φu,v0dq1u0dq2v+1η1ξ2,ξ1η2ξ4,ξ3∫ξ2ξ2+η1ϱ,ξ2∫ξ3ξ3+η2ρ,ξ3Φu,v0dq1u0dq2v+1η1ξ2,ξ1η2ξ4,ξ3∫ξ2ξ2+η1ϱ,ξ2∫ξ4ξ4+η2ρ,ξ4Φu,v0dq1u0dq2v.Proof.
Considering(19)−∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ30dq1z0dq2w−∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ4+wη2ρ,ξ40dq1z0dq2w−∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ3+wη2ρ,ξ30dq1z0dq2w−∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ4+wη2ρ,ξ40dq1z0dq2w,it follows from the definitions of partial q1q2-derivative and q1q2-integral that
(20)∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ30dq1z0dq2w=11−q11−q2η1ϱ,ξ1η2ρ,ξ3×∫01∫01Φξ1+zq1η1ϱ,ξ1,ξ3+wq2η2ρ,ξ20dq1z0dq2w−∫01∫01Φξ1+zq1η1ϱ,ξ1,ξ3+wη2ρ,ξ30dq1z0dq2w−∫01∫01Φξ1+zη1ϱ,ξ1,ξ3+wq2η2ρ,ξ30dq1z0dq2w+∫01∫01Φξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ30dq1z0dq2w=1η1ϱ,ξ1η2ρ,ξ3×∑n=0∞∑m=0∞q1nq2mΦξ1+q1n+1η1ϱ,ξ1,ξ3+q2m+1η2ρ,ξ3−∑n=0∞∑m=0∞q1nq2mΦξ1+q1n+1η1ϱ,ξ1,ξ3+q2mη2ρ,ξ3−∑n=0∞∑m=0∞q1nq2mΦξ1+q1nη1ϱ,ξ1,ξ3+q2m+1η2ρ,ξ3+∑n=0∞∑m=0∞q1nq2mΦξ1+q1n+1η1ϱ,ξ1,ξ3+q2mη2ρ,ξ3=1q1q2η1ϱ,ξ1η2ρ,ξ3∑n=1∞∑m=1∞q1nq2mΦξ1+q1nη1ϱ,ξ1,ξ3+q2mη2ρ,ξ3−1q1η1ϱ,ξ1η2ρ,ξ3∑n=1∞∑m=0∞q1nq2mΦξ1+q1nη1ϱ,ξ1,ξ3+q2mη2ρ,ξ3−1q2η1ϱ,ξ1η2ρ,ξ3∑n=0∞∑m=1∞q1nq2mΦξ1+q1nη1ϱ,ξ1,ξ3+q2mη2ρ,ξ3+1η1ϱ,ξ1η2ρ,ξ3∑n=0∞∑m=0∞q1nq2mΦξ1+q1nη1ϱ,ξ1,ξ3+q2mη2ρ,ξ3.
Note that(21)1q1q2η1ϱ,ξ1η2ρ,ξ3∑n=1∞∑m=1∞q1nq2mΦξ1+q1nη1ϱ,ξ1,ξ3+q2mη2ρ,ξ3=−Φϱ,ρq1q2η1ϱ,ξ1η2ρ,ξ3−1q1q2η1ϱ,ξ1η2ρ,ξ3∑n=0∞qnΦξ1+q1nη1ϱ,ξ1,ρ−1q1q2η1ϱ,ξ1η2ρ,ξ3∑m=0∞q2mϱ,ξ3+q2mη2ρ,ξ3−1q1q2η1ϱ,ξ1η2ρ,ξ3×∑n=0∞∑m=0∞q1nq2mΦξ1+q1nη1ϱ,ξ1,ξ3+q2mη2ρ,ξ3,(22)−1q1η1ϱ,ξ1η2ρ,ξ3∑n=1∞∑m=1∞q1nq2mΦξ1+q1nη1ϱ,ξ1,ξ3+q2mη2ρ,ξ3=1q1η1ϱ,ξ1η2ρ,ξ3∑m=0∞q2mΦϱ,ξ3+q2mη2ρ,ξ3−1q1η1ϱ,ξ1η2ρ,ξ3×∑n=0∞∑m=0∞q1nq2mΦξ1+q1nη1ϱ,ξ1,ξ3+q2mη2ρ,ξ3,(23)−1q2η1ϱ,ξ1η2ρ,ξ3∑n=0∞∑m=1∞q1nq2mΦξ1+q1nη1ϱ,ξ1,ξ3+q2mη2ρ,ξ3=1q2η1ϱ,ξ1η2ρ,ξ3∑n=0∞q1nΦξ1+q1nη1ϱ,ξ1,ρ−1q2η1ϱ,ξ1η2ρ,ξ3×∑n=0∞∑m=0∞q1nq2mΦξ1+q1nη1ϱ,ξ1,ξ3+q2mη2ρ,ξ3.
From (20)–(23) we get
(24)∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ30dq1z0dq2w=−Φϱ,ρq2q2η1ϱ,ξ1η2ρ,ξ3−1−q2η2ρ,ξ3q1q2η1ϱ,ξ1η2ρ,ξ32∑m=0∞q2mϱ,ξ3+q2mη2ρ,ξ3−1−q1η1ϱ,ξ1q1q2η1ϱ,ξ12η2ρ,ξ3∑n=0∞q2nΦξ1+q1mη1ϱ,ξ1,ρ+1−q11−q2η1ϱ,ξ1η2ρ,ξ3q1q2η1ϱ,ξ12η2ρ,ξ32∑n=0∞∑m=0∞q1nq2mΦξ1+q1nη1ϱ,ξ1,ξ3+q2mη2ρ,ξ3,(25)∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ2,ξ3+wη2ρ,ξ30dq1z0dq2w=−Φϱ,ρq1q2η1ϱ,ξ1η2ρ,ξ3−Φϱ,ρq1q2η1ϱ,ξ1η2ρ,ξ32∫ξ3ξ3+η2ρ,ξ3Φϱ,v0dq2v−1q1q2η1ϱ,ξ12η2ρ,ξ3∫ξ1ξ3+η1ϱ,ξ1Φu,ρ0dq1v+1q1q2η1ϱ,ξ12η2ρ,ξ32∫ξ1ξ2+η1ϱ,ξ2∫ξ3ξ3+η2ϱ,ξ3Φu,v0dq1u0dq2v.
Multiplying both sides of equality (25) by
(26)q1q2η1ϱ,ξ12η2ρ,ξ32η1ξ2,ξ1η2ξ4,ξ3,leads to
(27)q1q2η1ϱ,ξ12η1ρ,ξ32η1ξ2,ξ1η2ξ4,ξ3∫01∫01zw∂q1,q22ξ1∂q1zξ3∂q2w×Φξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ30dq1z0dq2w=−η1ϱ,ξ1η2ρ,ξ3η1ξ2,ξ1η2ξ4,ξ3Φϱ,ρ−η1ϱ,ξ1η1ξ2,ξ1η2ξ4,ξ3∫ξ3ξ3+η2ρ,ξ3Φϱ,v0dq2v−η2ρ,ξ3η1ξ2,ξ1η2ξ4,ξ3∫ξ1ξ1+η1ϱ,ξ1Φu,ρ0dq1u+1η1ξ2,ξ1η2ξ4,ξ3∫ξ3ξ3+η2ρ,ξ3Φu,v0dq1u0dq2v.
Similarly, we have(28)q1q2η1ϱ,ξ12η2ξ4,ρ2η1ξ2,ξ1η2ξ4,ξ3∫01∫01zw∂q1,q22∂ξ1q1zξ3∂q2w×Φξ1+zη1ϱ,ξ1,ξ4+zη2ρ,ξ4d0q1zd0q2w=−η1ϱ,ξ1η2ξ4,ρη1ξ2,ξ1η2ξ4,ξ3Φϱ,ρ−η1ϱ,ξ1η1ξ2,ξ1η2ξ4,ξ3∫ξ4ξ4+η2ρ,ξ4Φϱ,vd0q2v+η2ξ4,ρη1ξ2,ξ1η2ξ4,ξ3∫ξ1ξ1+η1ϱ,ξ1Φu,ρd0q1u+1η1ξ2,ξ1η2ξ4,ξ3∫ξ1ξ1+η1ϱ,ξ1∫ξ4ξ4+η2ρ,ξ4Φu,vd0q1ud0q2v,q1q2η1ξ2,ϱ2η2ρ,ξ32η1ξ2,ξ1η2ξ4,ξ3∫01∫01zw∂q1,q22∂ξ1q1zξ3∂q2w×Φξ2+zη1ϱ,ξ2+wη2ρ,ξ3d0q1z0dq2w=−η1ξ2,ϱη2ρ,ξ3η1ξ2,ξ1η2ξ4,ξ3Φϱ,ρ−η1ξ2,ϱη1ξ2,ξ1η2ξ4,ξ3∫ξ3ξ3+η2ρ,ξ3Φϱ,vd0q2v−η2ρ,ξ3η1ξ2,ξ1η2ξ4,ξ3∫ξ2ξ2+η1ρ,ξ2Φu,ρd0q1u+1η1ξ2,ξ1η2ξ4,ξ3∫ξ2ξ2+η1ϱ,ξ2∫ξ3ξ3+η2ρ,ξ3Φu,vd0q1u0dq2v,(29)q1q2η1ξ2,ϱ2η2ρ,ξ32η1ξ2,ξ1η2ξ4,ξ3∫01∫01zw∂q1,q22∂ξ1q1zξ3∂q2w×Φξ2+zη1ϱ,ξ2+wη2ρ,ξ3d0q1z0dq2w=−η1ξ2,ϱη2ρ,ξ3η1ξ2,ξ1η2ξ4,ξ3Φϱ,ρ−η1ξ2,ϱη1ξ2,ξ1η2ξ4,ξ3∫ξ3ξ3+η2ρ,ξ3Φϱ,vd0q2v−η2ρ,ξ3η1ξ2,ξ1η2ξ4,ξ3∫ξ2ξ2+η1ρ,ξ2Φu,ρd0q1u+1η1ξ2,ξ1η2ξ4,ξ3∫ξ2ξ2+η1ϱ,ξ2∫ξ3ξ3+η2ρ,ξ3Φu,vd0q1u0dq2v,(30)q1q2η1ξ2,ϱ2ξ4,ρ2η1ξ2,ξ1η2ξ4,ξ3∫01∫01zw∂q1,q22∂ξ1q1zξ3∂q2w×Φξ2+zη1ϱ,ξ2,ξ4+wη2ρ,ξ4d0q1z0dq2w=−η1ξ2,ϱη2ξ4,ρη1ξ2,ξ1η2ξ4,ξ3Φϱ,ρ−η1ξ2,ϱη1ξ2,ξ1η2ξ4,ξ3∫ξ4ξ4+η2ρ,ξ4Φϱ,vd0q2v−η2ξ4,ρη1ξ2,ξ1η2ξ4,ξ3∫ξ2ξ2+η1ϱ,ξ2Φu,ρd0q1u+1η1ξ2,ξ1η2ξ4,ξ3∫ξ2ξ2+η1ϱ,ξ2∫ξ4ξ4+η2ρ,ξ4Φu,vd0q1u0dq2v.
Therefore, Lemma13 follows from (19), (27), (28), (29), and (30).Theorem 14.
Letq1,q2∈0,1, ξ1<ξ2, ξ3<ξ4,Φ:N⊆ℝ2⟶ℝ be mixed partial q1q2 differentiable on N° (the interior of N) such that its mixed partial q1q2-derivatives is continuous and integrable on ξ1,ξ1+η1ξ2,ξ1×ξ3,ξ3+η2ξ4,ξ3⊂N∘ for η1ξ2,ξ1,η2ξ4,ξ3>0. If ξ1,ξ3∂q1,q22/ξ1∂q1zξ3∂q2wΦ is an n-polynomial preinvex function on the coordinates on ξ1,ξ1+η1ξ2,ξ1×ξ3,ξ3+η2ξ4,ξ3 and ξ1,ξ3∂q1,q22/ξ1∂q1zξ3∂q2wΦϱ,ρ≤M for ϱ,ρ∈N, then we have
(31)Φϱ,ρ+1η2ξ4,ξ1∫ξ3ξ3+η2ρ,ξ3Φϱ,vd0q2v+∫ξ4ξ3+η2ρ,ξ3Φϱ,vd0q2v+1η1ξ2,ξ1∫ξ1ξ1+η1ϱ,ξ1Φu,ρd0q1u+∫ξ2ξ2,η1ϱ,ξ3Φu,ρd0q1u−Q≤q1q2MAq1+Bq1Aq2+Bq2η1ϱ,ξ12+η1ξ2,ϱ2η1ξ2,ξ1×η2ρ,ξ32+η2ξ4,ρ2η2ξ4,ξ3,where Q is defined in Lemma 13.Proof.
It follows from (19) and the properties of the n-polynomial preinvexity function of the function ∂q1,q22/ξ1∂q1zξ3∂q2wΦ on coordinates that
(32)Φϱ,ρ+1η2ξ4ξ3∫ξ3ξ3+η2ρ,ξ3Φϱ,ρ0dq2v+∫ξ4ξ4+η2ρ,ξ4Φϱ,ρ0dq2v+1η1ξ2ξ1∫ξ1ξ1+η1ϱ,ξ1Φu,ρ0dq1u+∫ξ2ξ2+η1ϱ,ξ2Φu,ρ0dq1u−Q≤q1q2η1ϱ,ξ12η2ρ,ξ32η1ξ2ξ1η2ξ4ξ3∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ3d0q1z0dq2w+q1q2η1ϱ,ξ12η2ξ4,ρ2η1ξ2ξ1η2ξ4ξ3∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ4+wη2ρ,ξ4d0q1z0dq2w+q1q2η1ξ2,ϱ2η2ρ,ξ32η1ξ2ξ1η2ξ4ξ3∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ3+wη2ρ,ξ3d0q1z0dq2w+q1q2η1ξ2,ϱ2η2ξ4,ρ2η1ξ2ξ1η2ξ4ξ3∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ4+wη2ρ,ξ4d0q1z0dq2w.Considering the first integral
(33)∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ3d0q1z0dq2w≤∫01w∫01z1n∑p=1n1−1−zpξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦϱ,ξ3+wη2ρ,ξ3+1n∑p=1n1−zpξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1,ξ3+wη2ρ,ξ3d0q1z.d0q2w,in view of the Definition 6 for k=1,2, we get
(34)Aqk=1n∑p=1n∫01z1−1−zpd0qkz=11+qk−1−qkn∑p=1n∑e=0∞qk2e1−qkep,Bqk=1n∑p=1n∫01z1−zpd0qkz=11+qk−1n∑p=1n1−qk1−qkp+2.
From (33) and computing the q1-integral, we get
(35)∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ3d0q1z0dq2w≤∫01wAq1ξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦϱ,ξ3+wη2ρ,ξ3+Bq1ξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1,ξ3+wη2ρ,ξ3d0q2w.
Computing theq2-integral and utilizing the fact ξ1,ξ3∂q1,q22/ξ1∂q1z∂q2wΦϱ,ρ≤Mforϱ,ρ∈N, inequality (35) leads to the conclusion that
(36)∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1z∂q2wΦξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ3d0q1z0dq2w≤MAq1+Bq1Aq2+Bq2.
Analogously, we also have(37)∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ4+wη2ρ,ξ4d0q1z0dq2w≤MAq1+Bq1Aq2+Bq2,(38)∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ3+wη2ρ,ξ3d0q1z0dq2w≤MAq1+Bq1Aq2+Bq2,(39)∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ4+wη2ρ,ξ4d0q1z0dq2w≤MAq1+Bq1Aq2+Bq2.
Making use of the inequalities (36), (37), (38), and (39) and the fact that
(40)η1ϱ,ξ12η2ρ,ξ32+η1ϱ,ξ12η2ξ4,ρ2+η1ξ2,ϱ2η2ρ,ξ32+η1ξ2,ϱ2η2ξ4,ρ2=η1ϱ,ξ12+η1ξ2,ϱ2η2ρ,ξ32+η2ξ4,ρ2,we get the desired inequality (31).Theorem 15.
Letγ1,γ2>1 with 1/γ2+1/γ1=1,q1,q2∈0,1, and Φ:N⊆ℝ2⟶ℝ be mixed partial q1q2-differentiable on N° such that its mixed partial q1q2-derivative is continuous and integrable on ξ1,ξ1+η1ξ2,ξ1×ξ3,ξ3+η2ξ4,ξ3⊂N∘ for η1ξ2,ξ1,η2ξ4,ξ3>0. If ξ1,ξ3∂q1,q22/ξ1∂q1zξ3∂q2wΦγ2 is an n-polynomial preinvex function on the coordinates on ξ1,ξ1+η1ξ2,ξ1×ξ3,ξ3+η2ξ4,ξ3 and ξ1,ξ3∂q1,q22/ξ1∂q1zξ3∂q2wΦϱ,ρ≤Mforϱ,ρ,∈N, then we have
(41)Φϱ,ρ+1η2ξ4,ξ3∫ξ3ξ3+η2ρ,ξ3Φϱ,ρd0q2v+∫ξ4ξ3+η2ρ,ξ3Φϱ,ρd0q2v1η1ξ2,ξ1∫ξ1ξ1+η1ϱ,ξ1Φu,ρd0q1u+∫ξ2ξ2+η1ϱ,ξ2Φu,ρd0q1u−Q≤q1q2MCq1+Dq1Cq2+Dq2γ2γ1+1q1γ1+1q2γ1η1ϱ,ξ12+η1ξ2,ϱ2η1ξ2,ξ1×η2ρ,ξ32+η2ξ4,ρ2η2ξ4,ξ3,where Q is defined in Lemma 13.Proof.
It follows from (19), the Hölder inequality and the property of n-polynomial preinvexity of the function ∂q1,q22Φ/ξ1∂q1zξ3∂q2wγ2 on coordinates that
(42)Φϱ,ρ+1η2ξ4,ξ3∫ξ3ξ3+η2ρ,ξ3Φϱ,vd0q2v+∫ξ4ξ4+η2ρ,ξ4Φϱ,vd0q2v+1η1ξ2,ξ1∫ξ1ξ1+η1ϱ,ξ1Φu,ρd0q1u+∫ξ2ξ2+η1ϱ,ξ2Φu,ρd0q1u−Q≤∫01∫01zγ1wγ1d0q1z0dq2w1/γ1×q1q2η1ϱ,ξ12η2ρ,ξ32η1ξ2,ξ1η2ξ4,ξ3∫01∫01ξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ3γ2d0q1z0dq2w1/γ2+q1q2η1ϱ,ξ12η2ξ3,ρ2η1ξ2,ξ1η2ξ4,ξ3∫01∫01ξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ4+wη2ρ,ξ4γ2d0q1z0dq2w1/γ2+q1q2η1ξ2,ϱ2η2ρ,ξ32η1ξ2,ξ1η2ξ4,ξ3∫01∫01ξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ3+wη2ρ,ξ3γ2d0q1z0dq2w1/γ2+q1q2η1ξ2,ϱ2η2ξ4,ρ2η1ξ2,ξ1η2ξ4,ξ3∫01∫01ξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ4+wη2ρ,ξ4γ2d0q1z0dq2w1γ2,for all ϱ,ρ∈N.
From then-polynomial preinvexity of the function ξ1,ξ3∂q1,q22/ξ1∂q1zξ3∂q2wΦγ2, we get
(43)∫01∫01ξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ3γ2d0q1z0dq2w≤∫01∫011n∑p=1n1−1−zpξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦϱ,ξ3+wη2ρ,ξ3γ2+1n∑p=1n1−zpξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1,ξ3+wη2ρ,ξ3γ2d0q1zd0q2w.
Computing theq1-integral on the right-hand side of (43), we have
(44)∫01∫01ξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ3γ2d0q1z0dq2w≤∫011n∑p=1n1−1−zpξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦϱ,ξ3+wη2ρ,ξ3γ2+1n∑p=1n1−zpξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1,ξ3+wη2ρ,ξ3γ2d0q1z.
In view of the Definition6 for k=1,2, we obtain
(45)Cqk=1n∑p=1n∫011−1−zpd0qkz=1−1−qkn∑p=1n∑e=0∞qke1−qkep,Dqk=1n∑p=1n∫011−zpd0qkz=1−1n∑p=1n1−qk1−qkp+2.
Therefore, we get(46)∫01∫01ξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ3γ2d0q1z0dq2w≤∫01Cq1ξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦϱ,ξ3+wη2ρ,ξ3γ2+Dq1ξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1,ξ3+wη2ρ,ξ3γ2d0q2w.
Similarly, computing theq2-integral and utilizing the fact ξ1,ξ3∂q1,q22/ξ1∂q1z∂q2wΦϱ,ρ≤Mforϱ,ρ∈N on the right-hand side of (46), one has
(47)∫01∫01ξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ3γ2d0q1z0dq2w≤Mγ2Cq1+Dq1Cq2+Dq2.
Analogously, we also can get(48)∫01∫01ξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ4+wη2ρ,ξ4γ2d0q1z0dq2w,≤Mγ2Cq1+Dq1Cq2+Dq2,(49)∫01∫01ξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ3+wη2ρ,ξ3γ2d0q1z0dq2w≤Mγ2Cq1+Dq1Cq2+Dq2,(50)∫01∫01ξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ4+wη2ρ,ξ4γ2d0q1z0dq2w≤Mγ2Cq1+Dq1Cq2+Dq2.
Therefore, the desired inequality (41) follows from (47), (48), (49), and (50) and the fact that
(51)∫01∫01zγ1wγ1d0q1z0dq2w=1γ1+1q1γ1+1q2,where γ1+1q1 and γ1+1q2 are the q1- and q2-analogues of γ1+1 and γ2+1, respectively.Theorem 16.
Letq1,q2∈0,1,γ>1andΦ:N⊆ℝ2⟶ℝ be mixed partial q1q2-differentiable over N∘ such that its partial q1q2-derivative is continuous and integrable on ξ1,ξ1+η1ξ2,ξ1×ξ3,ξ3+η2ξ4,ξ3⊂N∘forη1ξ2,ξ1,η2ξ4,ξ3>0. If ξ1,ξ3∂q1,q22/ξ1∂q1zξ3∂q2wΦ is an n-polynomial preinvex function on the coordinates on ξ1,ξ1+η1ξ2,ξ1×ξ3,ξ3+η2ξ4,ξ3andξ1,ξ3∂q1,q22/ξ1∂q1zξ3∂q2wΦϱ,ρ≤Mforϱ,ρ∈N, then we have the inequality
(52)Φϱ,ρ+1η2ξ4,ξ3∫ξ3ξ3+η2ρ,ξ3Φϱ,vd0q2v+∫ξ4ξ3+η2ρ,ξ3Φϱ,vd0q2v+1η1ξ2,ξ1∫ξ1ξ1+η1ϱ,ξ1Φu,ρd0q1u+∫ξ2ξ2+η1ϱ,ξ2Φu,ρd0q1u−Q≤q1q2MAq1+Bq1Aq2+Bq2γ1+q11+q21−1/γ×η1ϱ,ξ12+η1ξ2,ϱ2η1ξ2,ξ1η2ρ,ξ32+η2ξ4,ρ2η2ξ4,ξ3,where Q is defined in Lemma 13.Proof.
It follows from (19), the power mean inequality and the property of n-polynomial preinvexity of the function ξ1,ξ3∂q1,q22Φ/ξ1∂q1zξ3∂q2w on coordinates that
(53)Φϱ,ρ+1η2ξ4,ξ3∫ξ3ξ3+η2ρ,ξ3Φϱ,ρd0q2v+∫ξ4ξ4+η2ρ,ξ4Φϱ,vd0q2v+1η1ξ2,ξ1∫ξ1ξ1+η1ϱ,ξ1Φu,ρd0q1u+∫ξ2ξ2+η1ϱ,ξ2Φu,ρd0q1u−Q≤∫01∫01zw0dq1z0dq2w1−1/γ×q1q2η1ϱ,ξ12η2ρ,ξ32η1ξ2,ξ1η2ξ4,ξ3∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ3γd0q1z0dq2w1/γ+q1q2η1ϱ,ξ12η2ξ4,ρ2η1ξ2,ξ1η2ξ4,ξ3∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ4+wη2ρ,ξ4γd0q1z0dq2w1/γ+q1q2η1ξ2,ϱ2η2ρ,ξ32η1ξ2,ξ1η2ξ4,ξ3∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ3+wη2ρ,ξ3γd0q1z0dq2w1/γ+q1q2η1ξ2,ϱ2η2ξ4,ρ2η1ξ2,ξ1η2ξ4,ξ3∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ4+wη2ρ,ξ4γd0q1z0dq2w1/γ,for all ϱ,ρ∈N.
By similar argument as in Theorem14, we can prove that
(54)∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ3+wη2ρ,ξ3γd0q1z0dq2w≤MγAq1+Bq1Aq2+Bq2,(55)∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ1+zη1ϱ,ξ1,ξ4+wη2ρ,ξ4γd0q1z0dq2w≤MγAq1+Bq1Aq2+Bq2,(56)∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ3+wη2ρ,ξ3γd0q1z0dq2w≤MγAq1+Bq1Aq2+Bq2,(57)∫01∫01zwξ1,ξ3∂q1,q22ξ1∂q1zξ3∂q2wΦξ2+zη1ϱ,ξ2,ξ4+wη2ρ,ξ4γd0q1z0dq2w≤MγAq1+Bq1Aq2+Bq2.
Now by making use of the inequalities (54), (55), (56), and (57) and the fact that
(58)∫01∫01zw0dq1z0dq2w=11+q11+q2,we get the desired inequality (52).


## 3. Conclusion

In the article, we have introduced a new class of preinvex functions which is namedn-polynomial preinvex functions and discovered a new quantum integral identity involving second-order mixed partial differentiable function. By using the obtained quantum integral identity as an auxiliary result, we have established several q1q2-Ostrowski-type inequalities for the class of n-polynomial preinvex functions on coordinates, which have improved and unified many previously known results. Our given ideas and approaches may lead to a lot of follow-up research for the interested readers.



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