# Quantum Integral Inequalities with Respect to Raina’s Function via Coordinated GeneralizedΨ-Convex Functions with Applications

Authors: Saima Rashid; Saad Ihsan Butt; Shazia Kanwal; Hijaz Ahmad; Miao-Kun Wang
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

In accordance with the quantum calculus, we introduced the two variable forms of Hermite-Hadamard- (HH-) type inequality over finite rectangles for generalized Ψ-convex functions. This novel framework is the convolution of quantum calculus, convexity, and special functions. Taking into account the q^1q^2-integral identity, we demonstrate the novel generalizations of the HH-type inequality for q^1q^2-differentiable function by acquainting Raina’s functions. Additionally, we present a different approach that can be used to characterize HH-type variants with respect to Raina’s function of coordinated generalized Ψ-convex functions within the quantum techniques. This new study has the ability to generate certain novel bounds and some well-known consequences in the relative literature. As application viewpoint, the proposed study for changing parametric values associated with Raina’s functions exhibits interesting results in order to show the applicability and supremacy of the obtained results. It is expected that this method which is very useful, accurate, and versatile will open a new venue for the real-world phenomena of special relativity and quantum theory.


---

## Body

## 1. Introduction

Recently, a nonrestricted analysis is recognized as quantum calculus (in short,q^-calculus) and has initiated numerous q^-mathematical formulation as q^↦1−. In 1707–1783, Euler proposed q^-calculus theory. Accordingly, Jackson [1] explored the investigation of q^-integrals efficiently. The previously mentioned outcomes prompted an escalated presentation on quantum theory in the 20th Century. As an application perspective, the concept of q^-calculus has been potentially utilized in quantum mechanics, special relativity theory, anomalous diffusion equations, orthogonal polynomials, fractional calculus, and henceforth. In [2, 3], authors contemplated the q^-derivatives on finite intervals of real line and amplified several new generalizations of classical convexity, q^-version of Grüss, q^-C˘ebyšev’s, and q^-Pólya-Szegö type inequalities. Over the most recent couple of years, the subject of q^-theory has become a fascinating theme for several researchers, and new developments have been investigated in the relative literature (see [4–6]).Within the framework ofq^-calculus, mechanothermodynamics, translimiting states, analysis, and generalization of experimental data, several special approaches are being developed to assess the quantum calculus in terms of a generalized energy states (see [7, 8]).Convex functions have potential applications in many intriguing and captivating fields of research and furthermore played a remarkable role in numerous areas, such as coding theory, optimization, physics, information theory, engineering, and inequality theory. Several new classes of classical convexity have been proposed in the literature (see [9–14]). Mathematical inequalities are viewed as the prominent framework for assembling the qualitative and quantitative characterization in the area of applied analysis. A persistent development of intrigue has emerged to address the prerequisites of issue for rich utilization of these variants. Numerous generalizations were investigated by several scientists who thus utilized different procedures for introducing and proposing these bounds [15–17]. Additionally, many authors demonstrated various forms of inequalities such as Ostrowski, Lyenger, Opial, Hardy, and Olsen, and the most distinguished one is the Hermite-Hadamard inequality. Here, we intend to find the novel version of HH-type inequality in the frame of q^1q^2-integral on coordinated generalized Ψ-convex functions that correlates with Raina’s function. Also, we shall represent the application of our findings in the Mittag-Leffler and hypergeometric functions which show the applicability of the suggested scheme.LetG:I⊆ℝ↦ℝ be a convex function such that φ1<φ2. Then,
(1)Gφ1+φ22≤1φ2−φ1∫φ1φ2Gzdz≤Gφ1+Gφ22.The inequality (1) is a well-known paramount in related literature and plays a pivotal role in optimization, coding, and fractional calculus theory [18, 19].In [20], Dragomir proposed the two-variable version of the HH-type inequality for convex functions as follows:Theorem 1. (see [20]).
LetG:Δ↦ℝ be the coordinated convex on Δ. Then, the following inequalities hold:
(2)Gφ1+φ22,ϕ1+ϕ22≤121φ2−φ1∫φ1φ2Gμ,ϕ1+ϕ22dμ+1ϕ2−ϕ1∫ϕ1ϕ2Gφ1+φ22,νdν≤1φ2−φ1ϕ2−ϕ1∫φ1φ2∫ϕ1ϕ2Gμ,νdμdv≤141φ2−φ1∫φ1φ2Gμ,ϕ1dμ+1φ2−φ1∫φ1φ2Gμ,ϕ2dμ+1ϕ2−ϕ1∫ϕ1ϕ2Gφ1,νdν+1ϕ2−ϕ1∫ϕ1ϕ2Gφ2,νdν≤Gφ1,ϕ1+Gφ1,ϕ2+Gφ2,ϕ1+Gφ2,ϕ24.In [21], Kunt et al. established the q^-HH-type inequality for functions of two variables utilizing convexity on rectangle from the plane ℝ2.Theorem 2.
LetG:Δ=φ1,φ2×ϕ1,ϕ2⊆ℝ2↦ℝ be convex on the coordinates on Δ with 0<q^1,q^2<1 and φ1<φ2,ϕ1<ϕ2. Then, one has the following inequalities:
(3)Gq^1φ1+φ2q^1+1,q^2ϕ1+ϕ2q^2+1≤121φ2−φ1∫φ1φ2Gμ,q^2ϕ1+ϕ2q^2+1φ1dq^1μ+1ϕ2−ϕ1∫ϕ1ϕ2Gq^1φ1+φ2q^1+1,νϕ1dq^2ν≤1φ2−φ1ϕ2−ϕ1∫φ1φ2∫ϕ1ϕ2Gμ,νφ1dq^1μϕ1dq^2ν≤12q^21+q2φ2−φ1∫φ1φ2Gμ,ϕ1φ1dq^1μ+q^21+q^2φ2−φ1∫φ1φ2Gμ,ϕ2φ1dq^1μ+q^11+q^1ϕ2−ϕ1∫ϕ1ϕ2Gφ1,νϕ1dq^2ν+q^11+q^1ϕ2−ϕ1∫ϕ1ϕ2Gφ2,νϕ1dq^2ν≤q^1q^2Gφ1,ϕ1+q^1Gφ1,ϕ2+q^2Gφ2,ϕ1+Gφ2,ϕ21+q^11+q^2.For many useful consequences on the coordinates on rectangle from the planeℝ2 with the various sorts of variants for mappings that hold numerous types of convex mappings, see [22–24] and the references cited therein.Owing to the above-mentioned work, this research is aimed at exploring the novel generalizations ofHH-type inequalities on the coordinates by the use of generalized Ψ-convex functions which are elaborated. An auxiliary identity is derived with respect to the q^1q^2-derivative by the correlation of Raina’s function. Considering this new approach, we derive certain novel quantum bounds of HH-type variants for coordinated generalized Ψ-convex mappings. Meanwhile, we recapture remarkable cases in the relative literature. For the change of parameter in Raina’s function, we generate numerous new outcomes depending on hypergeometric and Mittag-Leffler functions. This new study may stimulate further investigation in this dynamic field of inequality theory.


## 2. Prelude

This segment evokes certain earlier ideas and necessary details related to the notion of a coordinated generalizedΨ-convex set and coordinated generalized Ψ-convex function by considering Raina’s function.Assume that a finite interval of real numbersI, and we say that a mapping G:I↦ℝ is known to be convex if
(4)Gζx˘+1−ζy˘≤ζGx˘+1−ζGy˘,x˘,y˘∈I,ζ∈0,1.In [20], Dragomir introduced a new term in convexity theory, which is known as the coordinated convex function described as follows:Definition 3.
Let a mappingG:∇~→ℝ be said to be convex on the coordinates, for all ζ,θ∈0,1 with x˘,y˘,u˘,v˘∈∇~, if the partial functions
(5)Gζx˘+1−ζu˘,θy˘+1−θv˘≤ζθGx˘,y˘+ζ1−θGx˘,v˘+1−ζθGu˘,y˘+1−ζ1−θGu˘,v˘,holds for all ζ,θ∈0,1] and x˘,y˘,u˘,v˘∈∇~.
In [25], Raina contemplated the subsequent class of function
(6)Fγ,ρλt=Fγ,ρλ0,λ1,⋯t=∑p=0∞λpΓγp+ρtp,where γ,ρ>0,t<ℝ and
(7)λ=λ0,λ1,⋯λp,⋯,is a bounded sequence of ℝ+. Also, setting γ=1,ρ=0 in (6) and
(8)λp=ϑ1pϑ2pϑ3pforp=0,1,2,3⋯,
where the parametersϑi,i=1,2,3 are assumed to be real or complex (provided that ϑ3=0,−1,−2,⋯) and the symbol zp mentions the value
(9)zp=Γz+pΓz=zz+1⋯z+p−1,p=0,1,2,⋯,
and its domain is restricted as∣t∣≤1 (with t∈ℂ), then we attain the subsequent hypergeometric function,
(10)Fγ,ρλt=Fϑ1;ϑ2;ϑ3;t=∑p=0∞ϑ1pϑ2pp!ϑ3ptp.
Furthermore, ifλ=1,1,⋯ with γ=ϑ1,Rϑ1>0,λ=1 and its domain is restricted as t∈ℂ in (6), then we attain the subsequent Mittag-Leffler function
(11)Eϑ1t=∑p=0∞1Γ1+ϑ1ptp.Next, we mention a novel concept that reunites the coordinated convex function and Raina’s function as mentioned above.Definition 4.
Forγ,λ>0 and λ=λ0,λ1,⋯λp,⋯ is assumed to be a bounded sequence of ℝ+. A nonempty set Δ~ is known to be a coordinated generalized Ψ-convex set
(12)Gz˘+ζFγ,ρλx˘−z˘,w˘+θFσ,ρλy˘−w˘∈Δ~,
holds for allζθ∈0,1], x˘,y˘,z˘,w˘∈Δ~, and Fγ,ρλ. denotes Raina’s function.Definition 5.
Forγ,λ>0 and λ=λ0,λ1,⋯λp,⋯ is assumed to be a bounded sequence of ℝ+. A mapping G:Δ~→ℝ is said to be a coordinated generalized Ψ-convex, if
(13)Gz˘+ζFγ,ρλx˘−z˘,w˘+θFγ,ρλy˘−w˘≤ζθGx˘,y˘+ζ1−θGx˘,w˘+1−ζθGz˘,y˘+1−ζ1−θGz˘,w˘,
holds for allζ,θ∈0,1] and x˘,y˘,z˘,w˘∈Δ~.Remark 6.
SettingFγ,ρλx˘−φ1=x˘−φ1>0 and Fγ,ρλy˘−ϕ1=y˘−ϕ1>0 in Definition 5, we get Definition 3.
Furthermore, we demonstrate some essential ideas and preliminaries inq^-analog for a single and two-variable senses.
LetJ=ϱ1,ϱ2⊆ℝ, and let U=ϱ1,ϱ2×ϱ3,ϱ4⊆ℝ2 with constants q^,q^k∈0,1, k=1,2.Tariboon and Ntouyas [2, 3] studied the concept of q^-derivative, q^-integral, and characteristics for finite interval, which has been shown asDefinition 7.
Assume that a continuous mappingG:J→ℝ and t∈J. Then, one has q^-derivative of G on J at t which is stated as
(14)ϱ1Dq^Gt=Gt−Gqt+1−qϱ11−qt−ϱ1,t≠ϱ1.
Clearly, we see that(15)limt→ϱ1ϱ1Dq^Gt=ϱ1Dq^Gϱ1.We say that the mapping G is q^-differentiable over J, also ρ1Dq^Gt exists ∀t∈J.
Observe that ifϱ1=0 in (14), then 0Dq^G=Dq^G, where Dq^G is a well-defined q^-derivative of Gt, i.e, it is mentioned as
(16)Dq^Gt=Gt−Gqt1−qt.Definition 8.
Assume that a continuous mappingG:J→ℝ is symbolized as ϱ1Dq^2G, given that ϱ1Dq^2G is q^-differentiable from J→ℝ defined by
(17)ϱ1Dq^2G=ϱ1Dq^ϱ1Dq^G.
Therefore, the higher orderq^-differentiable is defined as ϱ1Dq^nG:J→ℝ.Definition 9.
Assume that a continuous mappingG:J→ℝ and the q^-integral on J is stated as
(18)∫ϱ1tGzϱ1dq^z=1−q^t−ϱ1∑n=0∞q∧nGq∧nt+1−q∧nϱ1,∀t∈J.
Next, ifϱ1=0 in (18), then we have a new formulation of q^-integral, which is pointed out as
(19)∫0tGz0dq^z=1−q^t∑n=0∞q∧nGq∧nt.Theorem 10.
Assuming that a continuous mappingG:J→ℝ, the following assumptions hold:
(20)ϱ1Dq^∫ϱ1tGzϱ1dq^z=Gt,∫ϱ1tϱ1Dq^Gzϱ1dq^z=Gt,∫ϱ2tϱ1Dq^Gzϱ1dq^z=Gt−Gϱ2,ϱ2∈ϱ1,t.Theorem 11.
Assuming that a continuous mappingG:J→ℝ and a∈ℝ, then the following assumptions hold:
(21)∫ϱ1tG1z+G2zϱ1dq^z=∫ϱ1tG1zϱ1dq^z+∫ϱ1tG2zϱ1dq^z,∫ϱ1taG1zϱ1dq^z=a∫ϱ1tG1zϱ1dq^z.In [26], Kalsoom et al. introduced the quantum integral identities in a two-variable sense as follows:Definition 12.
Consider a continuous mapping in two-variable senseG:U→ℝ, then the partial q^1-derivative, q^2-derivative, and q^1q^2-derivative at z,w∈ϱ1,ϱ2×ϱ3,ϱ4 are, respectively, stated as
(22)ϱ1∂q^1Gz,wϱ1∂q^1z=Gz,w-Gq^1z+1-q^1ϱ1,w1-q^1z-ϱ1,z≠ϱ1,ϱ3∂q^2Gz,wϱ3∂q^2w=Gz,w-Gz,q^2w+1-q^2ϱ31-q^2w-ϱ3,w≠ϱ3,ϱ1,ϱ3∂q^1,q^22Gz,wϱ1∂q^1zϱ3∂q^2w=11-q^11-q^2z-ϱ1w-ϱ3×Gq^1z+1-q^1ϱ1,q^2w+1-q^2ϱ3-Gq^1z+1-q^1ϱ1,w-Gz,q^2w+1-q^2ϱ3+Gz,w,z≠ϱ1,w≠ϱ3.Definition 13.
Consider a continuous mapping in two-variable senseG:U→ℝ, then the definite q^1q^2-integral on ϱ1,ϱ2×ϱ3,ϱ4 is stated as
(23)∫ϱ3t∫ϱ1tGz,wϱ1dq^1zϱ3dq^2w=1−q^11−q^2t−ϱ1t1−ϱ3∑m=0∞∑n=0∞q^1nq^2mGq^1nt+1−q^1nϱ1,q^2mt1+1−q^2mϱ3,for t,t1∈ϱ1,ϱ2×ϱ3,ϱ4.Theorem 14.
Consider a continuous mapping in two-variable senseG:B→ℝ, then the following assumptions hold:
(24)ϱ1,ϱ3∂q^1,q^22ϱ1∂q^1tϱ3∂q^2t1∫ϱ4t1∫ϱ1tGz,wϱ1dq^1zϱ3dq^2w=Gt,t1,∫ϱ3t1∫ϱ1tϱ1,ϱ3∂q^1,q^22Gz,wϱ1∂q^1zϱ3∂q^2wϱ1dq^1zϱ3dq^2w=Gt,t1,∫t2t1∫y1tϱ1,ϱ3∂q^1,q^22Gz,wϱ1∂q^1zϱ3∂q^2wϱ1dq^1zϱ3dq^2w;=Gt,t1−Gt,t2−Gy1,t1+Gy1,t2,y1,t2∈ϱ1,t×ϱ4,t1.Theorem 15.
Suppose thatG1,G2:U→ℝ are continuous mappings of two variables. Then, the following properties hold for t,t1∈ϱ1,ϱ2×ϱ3,ϱ4,(25)∫ϱ3t1∫ϱ1tG1z,w+G2z,wϱ1dq^1zϱ4dq^2w=∫ϱ3t1∫ϱ1tG1z,wϱ1dq^1zϱ3dq^2w+∫ϱ3t∫ϱ1tG2z,wϱ1dq^1zϱ3dq^2w,∫ϱ3t1∫ϱ1taGz,wϱ1dq^1zϱ3dq^2w=a∫ϱ3t1∫ϱ1tGz,wϱ1dq^1zϱ3dq^2w.


## 3. QuantumHH-Type Inequality for Generalized Ψ-Convex on the Coordinates

This section addresses theq^1-HH-type inequality on the coordinates via generalized Ψ-convex functions.Theorem 16.
Forγ,ρ>0 with λ=λ0,⋯,λp as the bounded sequence of positive real numbers and let G:Δ↦ℝ be the coordinated generalized Ψ-convex and partially differentiable function on Δ∘ with 0<q^1,q^2<1, then the following inequalities hold:
(26)Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,q^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^2≤12Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,q^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^2φ1dq^1μ+12Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,νϕ1dq^2ν≤1Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫φ1φ1+Fγ,ρλφ2−φ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gμ,νϕ1dq^2νφ1dq^1μ≤q^221+q^21Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,ϕ1φ1dq^1μ+q^221+q^21Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,ϕ2φ1dq^1μ+q^121+q^11Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gφ1,νϕ1dq^2μ+q^121+q^11Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gφ2,νϕ1dq^2ν≤q^1q^2Gφ1,ϕ1+q^1Gφ1,ϕ2+q^2Gφ2,ϕ1+Gφ2,ϕ21+q^11+q^2.Proof.
SinceG is the coordinated generalized Ψ-convex on Δ and partially differentiable mappings on Δ∘, clearly, we see that the mapping Gμ:ϕ1,ϕ1+Fγ,ρλϕ2−ϕ1↦ℝ,Gμν≔Gμ,ν is a generalized Ψ-convex on ϕ1,ϕ1+Fγ,ρλϕ2−ϕ1 and a differentiable function on ϕ1,ϕ1+Fγ,ρλϕ2−ϕ1 for all μ∈φ1,φ1+Fγ,ρλφ2−φ1. Then, by using the q^1-HH-type inequality, we obtain
(27)Gμq^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^2≤1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gμνμdq^2ν≤q^2Gμϕ1+Gϕ1ϕ21+q^2,μ∈φ1,φ1+Fγ,ρλφ2−φ1,which can be written as
(28)Gμ,q^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^2≤1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gμ,νϕ1dq^2ν≤q^2Gμ,ϕ1+Gμ,ϕ21+q^2,μ∈φ1,φ1+Fγ,ρλφ2−φ1.
Applyingq^1-inegration on the above inequalities over φ1,φ1+Fγ,ρλφ2−φ1, we have
(29)1Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,q^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^2φ1dq^1μ≤1Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫φ1φ1+Fγ,ρλφ2−φ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gμ,νϕ1dq^2νφ1dq^1μ≤11+q^2q^2Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,ϕ1φ1dq^1μ+1Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,ϕ2φ1dq^1μ.
Adopting the same procedure for the mappingGν:φ1,φ1+Fγ,ρλφ2−φ1↦ℝ,Gνμ≔Gμ,ν, we have
(30)1Fγ,ρλϕ2−ϕ1∫ϕ1φ1+Fγ,ρλϕ2−ϕ1Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,νϕ1dq^1μ≤1Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫φ1φ1+Fγ,ρλφ2−φ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gμ,νϕ1dq^2νφ1dq^1μ≤11+q^1q^1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gφ1,νϕ1dq^2ν+1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gφ2,νϕ1dq^2ν.
Adding (29) and (30), yields
(31)12Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,q^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^2φ1dq^1μ+12Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,νϕ1dq^2ν+≤1Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫φ1φ1+Fγ,ρλφ2−φ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gμ,νϕ1dq^2νφ1dq^1μ≤q^221+q^2Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,ϕ1φ1dq^1μ+121+q^2Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,ϕ2φ1dq^1μ+q^121+q^1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gφ1,νϕ1dq^2ν+121+q^1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gφ2,νϕ1dq^2ν.
Also, by considering theq^-HH-type inequality, we have
(32)Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,q^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^2≤1Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,q^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^2φ1dq^1μ,(33)Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,q^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^2≤1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,νϕ1dq^2ν.
Adding the inequalities (32) and (33), we have the following inequality:
(34)Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,q^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^2≤12Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,q^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^2φ1dq^1μ+1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,νϕ1dq^2ν.
Consequently, we have(35)q^221+q^21Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,ϕ1φ1dq^1μ≤q^221+q^2q^1Gφ1,ϕ1+Gφ2,ϕ11+q^1,q^221+q^21Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,ϕ2φ1dq^1μ≤q^221+q^2q^1Gφ1,ϕ2+Gφ2,ϕ21+q^1,1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gφ1,νϕ1dq^2μ≤q^121+q^1q^2Gφ1,ϕ1+Gφ1,ϕ21+q^2,q^121+q^11Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gφ2,νϕ1dq^2ν≤q^121+q^1q^2Gφ2,ϕ1+Gφ2,ϕ21+q^2.
Adding the above inequalities yields(36)q^221+q^21Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,ϕ1φ1dq^1μ+q^221+q^21Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,ϕ2φ1dq^1μ+q^121+q^11Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gφ1,νϕ1dq^2μ+q^121+q^11Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gφ2,νϕ1dq^2ν≤q^1q^2Gφ1,ϕ1+q^1Gφ1,ϕ2+q^2Gφ2,ϕ1+Gφ2,ϕ21+q^11+q^2.
A combination of (31), (34), and (36) gives (36). This completes the proof.Corollary 17.
In Theorem16, if we choose q^1,q^2↦1−, we have the following new double inequality:
(37)G2φ1+Fγ,ρλφ2−φ12,2ϕ1+Fγ,ρλϕ2−ϕ12≤12Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,2ϕ1+Fγ,ρλϕ2−ϕ12dμ+12Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,νdν≤1Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫φ1φ1+Fγ,ρλφ2−φ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gμ,νdνdμ≤141Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,ϕ1φ1dq^1μ+1Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,ϕ2φ1dq^1μ+1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gφ1,νϕ1dq^2μ+1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gφ2,νϕ1dq^2ν≤Gφ1,ϕ1+Gφ1,ϕ2+Gφ2,ϕ1+Gφ2,ϕ24.Remark 18.
In Theorem16,
(i)
lettingFγ,ρλφ2−φ1=φ2−φ1 and Fγ,ρλϕ2−ϕ1=ϕ2−ϕ1 along with q^1,q^2↦1−, then we attain Theorem 1 in [20](ii)
lettingFγ,ρλφ2−φ1=φ2−φ1 and Fγ,ρλϕ2−ϕ1=ϕ2−ϕ1, then we attain Theorem 4 in [21]


## 4. Quantum Integral Identity for Coordinated GeneralizedΨ-Convex Functions

The following identity plays a significant role in inaugurating the main consequences of this paper. The identification is expressed as follows.Lemma 19.
Forγ,ρ>0 with λ=λ0,⋯,λp as the bounded sequence of positive real numbers and let a twice partially q^1q^2-differentiable mapping G:Δ↦ℝ be defined on Δ∘ (the interior of Δ). If the second-order partial q^1q^2-derivatives are continuous and integrable over Δ with 0<q^1,q^2<1, then the following equality holds:
(38)Yq^1,q^2φ1,φ2,ϕ1,ϕ2G≔Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,q^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^2−1Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,q^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^2φ1dq^1μ−1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,νϕ1dq^2ν+1Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫φ1φ1+Fγ,ρλφ2−φ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gμ,νϕ1dq^2νφ1dq^1μ,where
(39)Yq^1,q^2φ1,φ2,ϕ1,ϕ2G≔q^1q^2Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01∫01Aζ,θφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ,Aζ,θ=ζθ,ζ,θ∈0,11+q^1×0,11+q^2,ζθ−1q^2,ζ,θ∈0,11+q^1×11+q^2,1,θζ−1q^1,ζ,θ∈11+q^1,1×0,11+q^2,ζ−1q^1θ−1q^2,ζ,θ∈11+q^1,1×11+q^2,1.Proof.
Consider(40)q^1q^2Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01∫01Aζ,θφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ=q^1q^2Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1×∫01/1+q^1∫01/1+q^2ζθφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ+∫11+q^11∫01/1+q^2θζ−1q^1φ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ+∫01/1+q^1∫1/1+q^21ζθ−1q^2φ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ+∫1/1+q^11∫1/1+q^21ζ−1q^1θ−1q^2φ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ=q^1q^2Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01/1+q^1∫01/1+q^2ζθφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ+∫1/1+q^11∫01/1+q^2ζθφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ+∫01/1+q^1∫1/1+q^21ζθφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ+∫1/1+q^11∫1/1+q^21ζθφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ−1q^1∫01/1+q^1∫01/1+q^2θφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ−1q^2∫1/1+q^11∫01/1+q^2ζφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ−1q^2∫01/1+q^1∫1/1+q^21ζφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θdq^10ζ1q^1q^2∫1/1+q^11∫1/1+q^21/1+q^21φ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρëϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ=q^1q^2Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01∫01ζθφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ−q^2Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01∫01θφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ−q^1Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01∫01ζφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ+q^2Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01/1+q^1∫01θφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ+q^1Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01∫01/1+q^1ζφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ+Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01∫01φ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ−Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01/1+q^1∫01φ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ−Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01∫01/1+q^1φ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ+Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01/1+q^1∫011+q^2φ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ.
In view of Definition12 and Definition 13, we conclude the following identities with the aid of the last nine integrals appearing in the aforementioned identities as follows:
(41)q^1q^2Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01∫01ζθφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1=−Gφ2,ϕ2−1Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,ϕ20dq^1μ−1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gφ2,ν0dq^2ν+1Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫φ1φ1+Fγ,ρλφ2−φ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gμ,νϕ1dq^2νφ1dq^1μ,(42)q^2Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01∫01θφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ=−Gφ2,ϕ2−1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gφ2,ν0dq^2ν,(43)q^1Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01∫01ζφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ=−Gφ2,ϕ2−1Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,ϕ20dq^1ν,(44)q^2Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01/1+q^1∫01θφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ=−Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,ϕ2−1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,ν0dq^1ν,(45)q^1Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01∫01/1+q^1ζφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ=−Gφ2,q^1+1ϕ1+Fγ,ρλϕ2−ϕ11+q^1−1Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,q^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^20dq^1μ,(46)Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01∫01/1+q^1φ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ=−Gφ2,ϕ2,(47)Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01/1+q^1∫01φ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ=−Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,ϕ2,(48)Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01∫01/1+q^1φ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ=−Gφ2,q^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^2,(49)Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01/1+q^1∫01/1+q^2φ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ=−Gq^1+1φ1+Fγ,ρλφ2−φ11+q^1,q^2+1ϕ1+Fγ,ρλϕ2−ϕ11+q^2.
Combining (42), (43), (44), (45), (46), (47), (48), and (49), we have the identity (38). This is the proof of Lemma 19.Corollary 20.
In Lemma19, if we choose q^1,q^2↦1−, we have the following new identity:
(50)Λφ1,φ2,ϕ1,ϕ2G≔G2φ1+Fγ,ρλφ2−φ12,2ϕ1+Fγ,ρλϕ2−ϕ12−1Fγ,ρλφ2−φ1∫φ1φ1+Fγ,ρλφ2−φ1Gμ,2ϕ1+Fγ,ρλϕ2−ϕ12dμ−1Fγ,ρλϕ2−ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1G2φ1+Fγ,ρλφ2−φ12,νdv+1Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫φ1φ1+Fγ,ρλφ2−φ1∫ϕ1ϕ1+Fγ,ρλϕ2−ϕ1Gμ,νdνdμ,where
(51)Λφ1,φ2,ϕ1,ϕ2G≔Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1∫01∫01Aζ,θ∂2Gφ1+ζFγ,ρλφ2−φ1,ϕ1+θFγ,ρλϕ2−ϕ1∂ζ∂θdθdζ,Aζ,θζθ,ζ−θ∈0,12×0,12,ζθ−1,ζ−θ∈0,12×12,1,θζ−1,ζ−θ∈12,1×0,12,ζ−1θ−1,ζ−θ∈12,1×12,1.


## 5. Certain Newq^1q^2-Integral Estimates for Generalized Ψ-Convex Functions

The following results exhibit some practice related to Lemma19 on quantum calculus for generalized Ψ-convex on coordinates.Theorem 21.
Forγ,ρ>0 with λ=λ0,⋯,λp as the bounded sequence of positive real numbers and let a mapping G:Δ↦ℝ be a twice partially q^1q^2-differentiable on Δ∘ such that continuous partial q^1q^2-derivatives φ1,ϕ1∂q^1,q^22G/φ1∂q^1ζϕ1∂q^2θ is integrable on Δ with 0<q^1,q^2<1. If φ1,ϕ1∂q∧1,q∧22G/φ1∂q∧1ζϕ1∂q∧2θσ is a generalized Ψ-convex on the coordinates on Δ for σ≥1, where σ−1+β−1=1. Then, the following inequality holds:
(52)Yq^1,q^2φ1,φ2,ϕ1,ϕ2G≤q^1q^2Fγ,ρλφ2−φ1Fγ,ρλϕ2−ϕ1B1q∧1,q∧21−1/σ×B2q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ+B3q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2φ1∂q∧1ζϕ1∂q∧2θσ+B4q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ+B5q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ1/σ,where
(53)B1q^1,q^2≔41+q∧131+q∧23,(54)B2q^1,q^2≔91+q∧131+q∧231+q^1+q^121+q^2+q^22,(55)B3q^1,q^2≔1+q^1+q^2+q^22-3q^1q^2-q^1q^23q^1q^21+q∧131+q∧231+q^1+q^121+q^2+q^22,(56)B4q^1,q^2≔1+q^1+q^2+q^12+q^22-3q^1q^2-q^2q^13-q^13q^22+6q^1q^22-q^12q^23-q^13q^23+5q^1q^23q^1q^21+q∧131+q∧231+q^1+q^121+q^2+q^22(57)B5q^1,q^2≔-2q^15-6q^14+2q^14q^23+2q^13q^24-4q^14q^22-4q^14q^2-2q^1q^25-2q^15q^2-2q^15q^23+16q^13q^23-4q^12+2q^13q^2-2q^12q^25-4q^12q^24+4q^12q^2+6q^13+8q^12q^22+10q^13q^22+10q^12q^23-6q^24-6q^23-4q^22-4q^1q^24+2q^1q^23+4q^1q^22+9q^1q^2-2q^25/q^1q^21+q∧131+q∧231+q^1+q^121+q^2+q^22Proof.
Taking into consideration theq^1q^2-integral power mean inequality, the generalized Ψ-convexity of φ1,ϕ1∂q∧1,q∧22G/φ1∂q∧1ζϕ1∂q∧2θσ on the coordinates on Δ with the aid of Lemma 19, we have
(58)Yq^1,q^2φ1,φ2,ϕ1,ϕ2G≤q^1q^2Fγ,ρλφ2-φ1Fγ,ρλϕ2-ϕ1×∫01∫01Aζ,θφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2-φ1,ϕ1+θFγ,ρλϕ2-ϕ1φ1∂q^1ζϕ1∂q^2θ0φ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2-φ1,ϕ1+θFγ,ρλϕ2-ϕ1φ1∂q^1ζϕ1∂q^2θ·dq^2θ0dq^1ζ≤q^1q^2Fγ,ρλφ2-φ1Fγ,ρλϕ2-ϕ1⋅∫01∫01Aζ,θ0dq∧2θ0dq∧1ζ1-1/σ×∫01∫01Aζ,θφ1,ϕ1∂q∧1,q∧22Gφ1+ζFγ,ρλφ2-φ1,ϕ1+θFγ,ρλϕ2-ϕ1φ1∂q∧1ζϕ1∂q∧2θσ0φ1,ϕ1∂q∧1,q∧22Gφ1+ζFγ,ρλφ2-φ1,ϕ1+θFγ,ρλϕ2-ϕ1φ1∂q∧1ζϕ1∂q∧2θσ0·dq∧2θ0dq∧1ζ1/σ=q^1q^2Fγ,ρλφ2-φ1Fγ,ρλϕ2-ϕ1·∫01/1+q^1∫01/1+q^1ζθ0dq^2θ0dq^1ζ+∫01/1+q^1∫1/1+q^21ζ1q^2-θ0dq^2θ0dq^1ζ+∫1/1+q^11∫01/1+q^2θ1q^1-ζ0dq^2θ0dq^1ζ+∫1/1+q^11∫1/1+q^211q^1-ζ1q^2-θ0dq^2θ0dq^1ζ×∫01/1+q∧1∫01/1+q∧1ζθζθφ1,ϕ1∂q∧1,q∧22Gφ2,ϕ2φ1∂q∧1ζϕ1∂q∧2θσθ1-ζφ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2φ1∂q∧1ζϕ1∂q∧2θσζ1-θφ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ1-ζ1-θφ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ0·dq∧2θ0dq∧1ζ+∫01/1+q∧1∫1/1+q∧21ζ1q∧2-θζθφ1,ϕ1∂q∧1,q∧22Gφ2,ϕ2φ1∂q∧1ζϕ1∂q∧2θσθ1-ζφ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2φ1∂q∧1ζϕ1∂q∧2θσζ1-θφ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ1-ζ1-θφ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ0·dq∧2θ0dq∧1ζ+∫1/1+q∧11∫01/1+q∧2θ1q∧1-ζζθφ1,ϕ1∂q∧1,q∧22Gφ2,ϕ2φ1∂q∧1ζϕ1∂q∧2θσθ1-ζφ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2φ1∂q∧1ζϕ1∂q∧2θσζ1-θφ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ1-ζ1-θφ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ0⋅dq∧2θ0dq∧1ζ+∫1/1+q∧11∫1/1+q∧211q∧1-ζ1q∧2-θ×ζθφ1,ϕ1∂q∧1,q∧22Gφ2,ϕ2φ1∂q∧1ζϕ1∂q∧2θσθ1-ζφ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2φ1∂q∧1ζϕ1∂q∧2θσζ1-θφ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ1-ζ1-θφ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ0dq∧2θ0dq∧1ζ1/σ=q^1q^2Fγ,ρλφ2-φ1Fγ,ρλϕ2-ϕ1B1q∧1,q∧21-1/σ×B2q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ+B3q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2φ1∂q∧1ζϕ1∂q∧2θσ+B4q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ+B5q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ1/σ.
This completes the proof of Theorem21.Corollary 22.
In Theorem21, if we choose q^1,q^2↦1−, we have the following new inequality:
(59)G2φ1+Fγ,ρλφ2-φ12,2ϕ1+Fγ,ρλϕ2-ϕ12-1Fγ,ρλφ2-φ1∫φ1φ1+Fy,ρλφ2-φ1Gμ,2ϕ1+Fγ,ρλϕ2-ϕ12dμ-1Fγ,ρλϕ2-ϕ1∫ϕ1ϕ1+Fy,ρλϕ2-ϕ1G2φ1+Fγ,ρλφ2-φ12,νdν-1Fγ,ρλφ2-φ1Fγ,ρλϕ2-ϕ1∫φ1φ1+Fy,ρλφ2-φ1∫ϕ1ϕ1+Fy,ρλϕ2-ϕ1Gμ,νdνdμ≤Fγ,ρλφ2-φ1Fγ,ρλϕ2-ϕ116·∂2Gφ2,ϕ2/∂ζ∂θσ+∂2Gφ1,ϕ2/∂ζ∂θσ+∂2Gφ2,ϕ1/∂ζ∂θσ+∂2Gφ1,ϕ1/∂ζ∂θσ4.Remark 23.
In Theorem21,
(i)
lettingFγ,ρλφ2−φ1=φ2−φ1 and Fγ,ρλϕ2−ϕ1=ϕ2−ϕ1, then we attain Theorem 5 in [21](ii)
lettingFγ,ρλφ2−φ1=φ2−φ1 and Fγ,ρλϕ2−ϕ1=ϕ2−ϕ1 along with q^1,q^2↦1−, then we attain Corollary 1 in [21] and Theorem 4 in [27], respectivelyTheorem 24.
Forγ,ρ>0 with λ=λ0,⋯,λ1 as the bounded sequence of positive real numbers and let a mapping G:Δ↦ℝ be a twice partially q^1q^2-differentiable on Δ∘(the interior of Δ) such that continuous partial q^1q^2-derivatives φ1,ϕ1∂q^1,q^22G/φ1∂q^1ζϕ1∂q^2θ is integrable on Δ with 0<q^1,q^2<1. If φ1,ϕ1∂q∧1,q∧22G/φ1∂q∧1ζϕ1∂q∧2θσ is a generalized Ψ-convex on the coordinates on Δ for σ>1 where σ−1+β−1=1. Then, the following inequality holds
(60)Yq^1,q^2φ1,φ2,ϕ1,ϕ2G≤q^1q^2Fγ,ρλφ2-φ1Fγ,ρλϕ2-ϕ1∫01∫01Aζ,θβ0dq∧2θ0dq∧1ζ1/β×φ1,φ1∂q∧1,q∧22Gφ2,ϕ2/ϕ1∂q∧1ζϕ1∂q∧2θσ+q∧1φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2/φ1∂q∧1ζϕ1∂q∧2θσ+q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1/φ1∂q∧1ζϕ1∂q∧2θσ+q∧1q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1/φ1∂q∧1ζϕ1∂q∧2θσ/1+q∧11+q∧21/σ,where Aζ,θ is defined as in (38).Proof.
Taking into consideration theq^1q^2-Hölder integral inequality, the generalized Ψ-convexity of φ1,ϕ1∂q∧1,q∧22G/φ1∂q∧1ζϕ1∂q∧2θσ on the coordinates on Δ with the aid of Lemma 19, we have
(61)Yq^1,q^2φ1,φ2,ϕ1,ϕ2G≤q^1q^2Fγ,ρλφ2-φ1Fγ,ρλϕ2-ϕ1×∫01∫01Aζ,θφ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2-φ1,ϕ1+θFγ,ρλϕ2-ϕ1σ1∂q^1ζϕ1∂q^2θ0φ1,ϕ1∂q^1,q^22Gφ1+ζFγ,ρλφ2-φ1,ϕ1+θFγ,ρλϕ2-ϕ1φ1∂q^1ζϕ1∂q^2θ·dq^2θ0dq^1ζ≤q^1q^2Fγ,ρλφ2-φ1Fγ,ρλϕ2-ϕ1×∫01∫01Aζ,θβ0dq∧2θ0dq∧1ζ1/β×∫01∫01Aζ,θφ1,ϕ1∂q∧1,q∧22Gφ1+ζFγ,ρλφ2-φ1,ϕ1+θFγ,ρλϕ2-ϕ1φ1∂q∧1ζϕ1∂q∧2θσ0φ1,ϕ1∂q∧1,q∧22Gφ1+ζFγ,ρλφ2-φ1,ϕ1+θFγ,ρλϕ2-ϕ1φ1∂q∧1ζϕ1∂q∧2θσ0·dq∧2θ0dq∧1ζ1/σ=q^1q^2Fγ,ρλφ2-φ1Fγ,ρλϕ2-ϕ1∫01∫01Aζ,θβ0dq∧2θ0dq∧1ζ1/β×φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ2φ1∂q∧1ζϕ1∂q∧2θσ∫01∫01ζθ0dq∧2θ0dq∧1ζ+φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2φ1∂q∧1ζϕ1∂q∧2θσ∫01∫01θ1-ζ0dq∧2θ0dq∧1ζ+φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ∫01∫01ζ1-θ0dq∧2θ0dq∧1ζ+φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ∫01∫011-ζ1-θ0dq∧2θ0dq∧1ζ1/σ=q^1q^2Fγ,ρλφ2-φ1Fγ,ρλϕ2-ϕ1∫01∫01Aζ,θβ0dq∧2θ0dq∧1ζ1/β×φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ2/φ1∂q∧1ζϕ1∂q∧2θσ+q∧1φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2/φ1∂q∧1ζϕ1∂q∧2θσ+q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1/φ1∂q∧1ζϕ1∂q∧2θσ+q∧1q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1/φ1∂q∧1ζϕ1∂q∧2θσ/1+q∧11+q∧21/σ.
This completes the proof of Theorem21.Corollary 25.
In Theorem21, if we choose q^1,q^2↦1−, we have the following new inequality:
(62)G2φ1+Fγ,ρλφ2-φ12,2ϕ1+Fγ,ρλϕ2-ϕ12-1Fγ,ρλφ2-φ1∫φ1φ1+Fγ,ρλφ2-φ1Gμ,2ϕ1+Fγ,ρλϕ2-ϕ12dμ-1Fγ,ρλϕ2-ϕ1∫ϕ1ϕ1+Fγ,ρλϕ2-ϕ1G2φ1+Fγ,ρλφ2-φ12,νdν-1Fγ,ρλφ2-φ1Fγ,ρλϕ2-ϕ1∫φ1φ1+Fγ,ρλφ2-φ1∫ϕ1ϕ1+Fγ,ρλϕ2-ϕ1Gμ,νdνdμ≤q^1q^2Fγ,ρλφ2-φ1Fγ,ρλϕ2-ϕ14β+12/β×φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ2/φ1∂q∧1ζϕ1∂q∧2θσ+φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2/φ1∂q∧1ζϕ1∂q∧2θσ+φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1/φ1∂q∧1ζϕ1∂q∧2θσ+φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1/φ1∂q∧1ζϕ1∂q∧2θσ/41/σ.Remark 26.
In Theorem21,
(i)
lettingFγ,ρλφ2−φ1=φ2−φ1 and Fγ,ρλϕ2−ϕ1=ϕ2−ϕ1, then we attain Theorem 6 in [21](ii)
lettingFγ,ρλφ2−φ1=φ2−φ1 and Fγ,ρλϕ2−ϕ1=ϕ2−ϕ1 along with q^1q^2↦1−, then we attain Theorem 3 in [27]


## 6. Applications

This section contains some useful utilities of our findings derived in the previous sections. For appropriate and suitable selections of parametersγ,ρ, and λ in the special functions stated in (6), (10), and (11). Taking into account Raina’s function (6), we shall derive outcomes for the hypergeometric function and Mittag-Leffler function as particular cases.


### 6.1. Hypergeometric Function

Lettingγ=1 and ρ=0, and
(63)λp=ϑ1pϑ2pϑ3p,forp=0,1,2,⋯,then for Theorem16, Lemma 19, and Theorems 21–24, the following results hold.Theorem 27.
Supposeλ=λ0,⋯,λp is the bounded sequence of positive real numbers and let G:O=φ1,φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1×ϕ1,ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1↦ℝ is the coordinated generalized Ψ-convex and partially differentiable function on O∘ with 0<q^1,q^2<1, then the following inequalities hold:
(64)Gq^1+1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ11+q^1,q^2+1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ11+q^2≤12Fϑ1;ϑ2;ϑ3,φ2−φ1∫φ1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1Gμ,q^2+1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ11+q^2φ1dq^1μ+12Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1∫ϕ1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1Gq^1+1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ11+q^1,νϕ1dq^2ν≤1Fϑ1;ϑ2;ϑ3,φ2−φ1Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1∫φ1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1∫ϕ1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1Gμ,νϕ1dq^2νφ1dq^1≤q^221+q^21Fϑ1;ϑ2;ϑ3,φ2−φ1∫φ1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1Gμ,ϕ1φ1dq^1μ+q^221+q^21Fϑ1;ϑ2;ϑ3,φ2−φ1∫φ1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1Gμ,ϕ2φ1dq^1μ+q^121+q^11Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1∫ϕ1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1Gφ1,νϕ1dq^2μ+q^121+q^11Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1∫ϕ1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1Gφ2,νϕ1dq^2ν≤q^1q^2Gφ1,ϕ1+q^1Gφ1,ϕ2+q^2Gφ2,ϕ1+Gφ2,ϕ21+q^11+q^2.Lemma 28.
Supposeλ=λ0,⋯,λp be the bounded sequence of positive real numbers and let a twice partially q^1q^2-differentiable mapping G:O=φ1,φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1×ϕ1,ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1↦ℝ defined on O∘(the interior of O). If the second-order partial q^1q^2-derivatives are continuous and integrable over O with 0<q^1,q^2<1, then the following equality holds:
(65)Y~q^1,q^2φ1,φ2,ϕ1,ϕ2G≔Gq^1+1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ11+q^1,q^2+1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−c11+q^2−1Fϑ1;ϑ2;ϑ3,φ2−φ1∫φ1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1Gμ,q^2+1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ11+q^2φ1dq^1μ−1Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1∫ϕ1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1Gq^1+1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ11+q^1,νϕ1dq^2ν+1Fϑ1;ϑ2;ϑ3,φ2−φ1Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1∫φ1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1∫ϕ1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1Gμ,νϕ1dq^2νφ1dq^1μ,where
(66)Y~q^1,q^2φ1,φ2,ϕ1,ϕ2G≔q^1q^2Fϑ1;ϑ2;ϑ3,φ2−φ1Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1×∫01∫01Aζ,θφ1,ϕ1∂q^1,q^22Gφ1+ζFϑ1;ϑ2;ϑ3,φ2−φ1,ϕ1+θFϑ1;ϑ2;ϑ3,ϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζfv,
andAζ,θ given in (38).Theorem 29.
Supposeλ=λ0,⋯,λp is the bounded sequence of positive real numbers and let a mapping G:O=φ1,φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1×ϕ1,ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1↦ℝ be a twice partially q^1q^2-differentiable on O∘ such that continuous partial q^1q^2-derivatives φ1,ϕ1∂q^1,q^22G/φ1∂q^1ζϕ1∂q^2θ is integrable on O with 0<q^1,q^2<1. If φ1,ϕ1∂q∧1,q∧22G/φ1∂q∧1ζϕ1∂q∧2θσ is a generalized Ψ-convex on the coordinates on O for σ≥1 where σ−1+β−1=1. Then, the following inequality holds:
(67)Y~q^1,q^2φ1,φ2,ϕ1,ϕ2G≤q^1q^2Fϑ1;ϑ2;ϑ3,φ2−φ1Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1B1q∧1,q∧21−1/σ×B2q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ+B3q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2φ1∂q∧1ζϕ1∂q∧2θσ+B4q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ+B5q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ1/σ,where B1q^1,q^2,B2q^1,q^2,B3q^1,q^2,B4q^1,q^2, and B5q^1,q^2 are given in (53), (54), (55), (56), and (57), respectively.Theorem 30.
Supposeλ=λ0,⋯,λp is the bounded sequence of positive real numbers and let a mapping G:O=φ1,φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1×ϕ1,ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1↦ℝ be a twice partially q^1q^2-differentiable on O∘ such that continuous partial q^1q^2-derivatives φ1,ϕ1∂q^1,q^22G/φ1∂q^1ζϕ1∂q^2θ is integrable on O with 0<q^1,q^2<1. If φ1,ϕ1∂q∧1,q∧22G/φ1∂q∧1ζϕ1∂q∧2θσ is a generalized Ψ-convex on the coordinates on O for σ>1 where σ−1+β−1=1. Then, the following inequality holds:
(68)Y~q^1,q^2φ1,φ2,ϕ1,ϕ2G≤q^1q^2Fϑ1;ϑ2;ϑ3,φ2-φ1Fϑ1;ϑ2;ϑ3,ϕ2-ϕ1∫01∫01Aζ,θβ0dq∧2θ0dq∧1ζ1/β×φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ2/φ1∂q∧1ζϕ1∂q∧2θσ+q∧1φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2/φ1∂q∧1ζϕ1∂q∧2θσ+q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1/φ1∂q∧1ζϕ1∂q∧2θσ+q∧1q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1/φ1∂q∧1ζϕ1∂q∧2θσ/1+q∧11+q∧21/σ,
whereAζ,θ is defined as in (38).


### 6.2. Mittag-Leffler Function

Settingν=1,1,⋯ having γ=ϑ1,Rϑ1>0 and ρ=1, then from Theorem 16, Lemma 19, and Theorems 21–24, the following results hold.Theorem 31.
LetG:S=φ1,φ1+Eϑ1φ2−φ1×ϕ1,ϕ1+Eϑ1ϕ2−ϕ1↦ℝ be the coordinated generalized Ψ-convex and partially differentiable function on S∘ with 0<q^1,q^2<1, then the following inequalities hold:
(69)Gq^1+1φ1+Eϑ1φ2−φ11+q^1,q^2+1ϕ1+Eϑ1ϕ2−ϕ11+q^2≤12Eϑ1φ2−φ1∫φ1φ1+Eϑ1φ2−φ1Gμ,q^2+1ϕ1+Eϑ1ϕ2−ϕ11+q^2φ1dq^1μ+12Eϑ1ϕ2−ϕ1∫ϕ1ϕ1+Eϑ1ϕ2−ϕ1Gq^1+1φ1+Eϑ1ϕ2−ϕ11+q^1,νϕ1dq^2ν≤1Eϑ1φ2−φ1Eϑ1ϕ2−ϕ1∫φ1φ1+Eϑ1φ2−φ1∫ϕ1ϕ1+Eϑ1ϕ2−ϕ1Gμ,νϕ1dq^2νφ1dq^1μ≤q^221+q^21Eϑ1ϕ2−ϕ1∫φ1φ1+Eϑ1φ2−φ1Gμ,ϕ1φ1dq^1μ+q^221+q^21Eϑ1φ2−φ1∫φ1φ1+Eϑ1φ2−φ1Gμ,ϕ2φ1dq^1μ+q^121+q^11Eϑ1ϕ2−ϕ1∫ϕ1ϕ1+Eϑ1ϕ2−ϕ1Gφ1,νϕ1dq^2μ+q^121+q^11Eϑ1ϕ2−ϕ1∫ϕ1ϕ1+Eϑ1ϕ2−ϕ1Gφ2,νϕ1dq^2ν≤q^1q^2Gφ1,ϕ1+q^1Gφ1,ϕ2+q^2Gφ2,ϕ1+Gφ2,ϕ21+q^11+q^2.Lemma 32.
Let a twice partiallyq^1q^2-differentiable mapping G:S=φ1,φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1×ϕ1,ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1↦ℝ defined on S∘(the interior of S). If the second-order partial q^1q^2-derivatives are continuous and integrable over S with 0<q^1,q^2<1, then the following equality holds:
(70)Y~~q^1,q^2φ1,φ2,ϕ1,ϕ2G≔Gq^1+1φ1+Eϑ1φ2−φ11+q^1,q^2+1ϕ1+Eϑ1ϕ2−c11+q^2−1Eϑ1φ2−φ1∫φ1φ1+Eϑ1φ2−φ1Gμ,q^2+1ϕ1+Eϑ1ϕ2−ϕ11+q^2φ1dq^1μ−1Eϑ1ϕ2−ϕ1∫ϕ1ϕ1+Eϑ1ϕ2−ϕ1Gq^1+1φ1+Eϑ1φ2−φ11+q^1,νϕ1dq^2ν+1Eϑ1φ2−φ1Eϑ1ϕ2−ϕ1∫φ1φ1+Eϑ1φ2−φ1∫ϕ1ϕ1+Eϑ1ϕ2−ϕ1Gμ,νϕ1dq^2νφ1dq^1μ,where
(71)Y~~q^1,q^2φ1,φ2,ϕ1,ϕ2G≔q^1q^2Eϑ1φ2−φ1Eϑ1ϕ2−ϕ1×∫01∫01Aζ,θφ1,ϕ1∂q^1,q^22Gφ1+ζEϑ1φ2−φ1,ϕ1+θEϑ1ϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ,
andAζ,θ given in (38).Theorem 33.
Let a mappingG:S=φ1,φ1+Eϑ1φ2−φ1×ϕ1,ϕ1+Eϑ1ϕ2−ϕ1↦ℝ be a twice partially q^1q^2-differentiable on S∘ such that continuous partial q^1q^2-derivatives φ1,ϕ1∂q^1,q^22G/φ1∂q^1ζϕ1∂q^2θ is integrable on S with 0<q^1,q^2<1. If φ1,ϕ1∂q∧1,q∧22G/φ1∂q∧1ζϕ1∂q∧2θσ is a generalized Ψ-convex on the coordinates on S for σ≥1 where σ−1+β−1=1. Then, the following inequality holds:
(72)Y~~q^1,q^2φ1,φ2,ϕ1,ϕ2G≤q^1q^2Eϑ1φ2−φ1Eϑ1ϕ2−ϕ1B1q∧1,q∧21−1/σ×B2q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ+B3q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2φ1∂q∧1ζϕ1∂q∧2θσ+B4q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ+B5q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ1/σ,
whereB1q^1,q^2,B2q^1,q^2,B3q^1,q^2,B4q^1,q^2, and B5q^1,q^2 are given in (53), (54), (55), (56), and (57), respectively.Theorem 34.
Let a mappingG:S=φ1,φ1+Eϑ1φ2−φ1×ϕ1,ϕ1+Eϑ1ϕ2−ϕ1↦ℝ be a twice partially q^1q^2-differentiable on S∘ such that continuous partial q^1q^2-derivatives φ1,ϕ1∂q^1,q^22G/φ1∂q^1ζϕ1∂q^2θ is integrable on S with 0<q^1,q^2<1. If φ1,ϕ1∂q∧1,q∧22G/φ1∂q∧1ζϕ1∂q∧2θσ is a generalized Ψ-convex on the coordinates on S for σ>1 where σ−1+β−1=1. Then, the following inequality holds:
(73)Y~~q^1,q^2φ1,φ2,ϕ1,ϕ2G≤q^1q^2Eϑ1φ2−φ1Eϑ1ϕ2−ϕ1∫01∫01Aζ,θβ0dq∧2θ0dq∧1ζ1/β×φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ2/φ1∂q∧1ζϕ1∂q∧2θσ+q∧1φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2/φ1∂q∧1ζϕ1∂q∧2θσ+q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1/φ1∂q∧1ζϕ1∂q∧2θσ+q∧1q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1/φ1∂q∧1ζϕ1∂q∧2θσ1+q∧11+q∧21/σ,
whereAζ,θ is defined as in (38).


## 6.1. Hypergeometric Function

Lettingγ=1 and ρ=0, and
(63)λp=ϑ1pϑ2pϑ3p,forp=0,1,2,⋯,then for Theorem16, Lemma 19, and Theorems 21–24, the following results hold.Theorem 27.
Supposeλ=λ0,⋯,λp is the bounded sequence of positive real numbers and let G:O=φ1,φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1×ϕ1,ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1↦ℝ is the coordinated generalized Ψ-convex and partially differentiable function on O∘ with 0<q^1,q^2<1, then the following inequalities hold:
(64)Gq^1+1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ11+q^1,q^2+1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ11+q^2≤12Fϑ1;ϑ2;ϑ3,φ2−φ1∫φ1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1Gμ,q^2+1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ11+q^2φ1dq^1μ+12Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1∫ϕ1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1Gq^1+1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ11+q^1,νϕ1dq^2ν≤1Fϑ1;ϑ2;ϑ3,φ2−φ1Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1∫φ1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1∫ϕ1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1Gμ,νϕ1dq^2νφ1dq^1≤q^221+q^21Fϑ1;ϑ2;ϑ3,φ2−φ1∫φ1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1Gμ,ϕ1φ1dq^1μ+q^221+q^21Fϑ1;ϑ2;ϑ3,φ2−φ1∫φ1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1Gμ,ϕ2φ1dq^1μ+q^121+q^11Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1∫ϕ1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1Gφ1,νϕ1dq^2μ+q^121+q^11Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1∫ϕ1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1Gφ2,νϕ1dq^2ν≤q^1q^2Gφ1,ϕ1+q^1Gφ1,ϕ2+q^2Gφ2,ϕ1+Gφ2,ϕ21+q^11+q^2.Lemma 28.
Supposeλ=λ0,⋯,λp be the bounded sequence of positive real numbers and let a twice partially q^1q^2-differentiable mapping G:O=φ1,φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1×ϕ1,ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1↦ℝ defined on O∘(the interior of O). If the second-order partial q^1q^2-derivatives are continuous and integrable over O with 0<q^1,q^2<1, then the following equality holds:
(65)Y~q^1,q^2φ1,φ2,ϕ1,ϕ2G≔Gq^1+1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ11+q^1,q^2+1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−c11+q^2−1Fϑ1;ϑ2;ϑ3,φ2−φ1∫φ1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1Gμ,q^2+1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ11+q^2φ1dq^1μ−1Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1∫ϕ1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1Gq^1+1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ11+q^1,νϕ1dq^2ν+1Fϑ1;ϑ2;ϑ3,φ2−φ1Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1∫φ1φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1∫ϕ1ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1Gμ,νϕ1dq^2νφ1dq^1μ,where
(66)Y~q^1,q^2φ1,φ2,ϕ1,ϕ2G≔q^1q^2Fϑ1;ϑ2;ϑ3,φ2−φ1Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1×∫01∫01Aζ,θφ1,ϕ1∂q^1,q^22Gφ1+ζFϑ1;ϑ2;ϑ3,φ2−φ1,ϕ1+θFϑ1;ϑ2;ϑ3,ϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζfv,
andAζ,θ given in (38).Theorem 29.
Supposeλ=λ0,⋯,λp is the bounded sequence of positive real numbers and let a mapping G:O=φ1,φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1×ϕ1,ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1↦ℝ be a twice partially q^1q^2-differentiable on O∘ such that continuous partial q^1q^2-derivatives φ1,ϕ1∂q^1,q^22G/φ1∂q^1ζϕ1∂q^2θ is integrable on O with 0<q^1,q^2<1. If φ1,ϕ1∂q∧1,q∧22G/φ1∂q∧1ζϕ1∂q∧2θσ is a generalized Ψ-convex on the coordinates on O for σ≥1 where σ−1+β−1=1. Then, the following inequality holds:
(67)Y~q^1,q^2φ1,φ2,ϕ1,ϕ2G≤q^1q^2Fϑ1;ϑ2;ϑ3,φ2−φ1Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1B1q∧1,q∧21−1/σ×B2q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ+B3q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2φ1∂q∧1ζϕ1∂q∧2θσ+B4q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ+B5q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ1/σ,where B1q^1,q^2,B2q^1,q^2,B3q^1,q^2,B4q^1,q^2, and B5q^1,q^2 are given in (53), (54), (55), (56), and (57), respectively.Theorem 30.
Supposeλ=λ0,⋯,λp is the bounded sequence of positive real numbers and let a mapping G:O=φ1,φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1×ϕ1,ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1↦ℝ be a twice partially q^1q^2-differentiable on O∘ such that continuous partial q^1q^2-derivatives φ1,ϕ1∂q^1,q^22G/φ1∂q^1ζϕ1∂q^2θ is integrable on O with 0<q^1,q^2<1. If φ1,ϕ1∂q∧1,q∧22G/φ1∂q∧1ζϕ1∂q∧2θσ is a generalized Ψ-convex on the coordinates on O for σ>1 where σ−1+β−1=1. Then, the following inequality holds:
(68)Y~q^1,q^2φ1,φ2,ϕ1,ϕ2G≤q^1q^2Fϑ1;ϑ2;ϑ3,φ2-φ1Fϑ1;ϑ2;ϑ3,ϕ2-ϕ1∫01∫01Aζ,θβ0dq∧2θ0dq∧1ζ1/β×φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ2/φ1∂q∧1ζϕ1∂q∧2θσ+q∧1φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2/φ1∂q∧1ζϕ1∂q∧2θσ+q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1/φ1∂q∧1ζϕ1∂q∧2θσ+q∧1q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1/φ1∂q∧1ζϕ1∂q∧2θσ/1+q∧11+q∧21/σ,
whereAζ,θ is defined as in (38).


## 6.2. Mittag-Leffler Function

Settingν=1,1,⋯ having γ=ϑ1,Rϑ1>0 and ρ=1, then from Theorem 16, Lemma 19, and Theorems 21–24, the following results hold.Theorem 31.
LetG:S=φ1,φ1+Eϑ1φ2−φ1×ϕ1,ϕ1+Eϑ1ϕ2−ϕ1↦ℝ be the coordinated generalized Ψ-convex and partially differentiable function on S∘ with 0<q^1,q^2<1, then the following inequalities hold:
(69)Gq^1+1φ1+Eϑ1φ2−φ11+q^1,q^2+1ϕ1+Eϑ1ϕ2−ϕ11+q^2≤12Eϑ1φ2−φ1∫φ1φ1+Eϑ1φ2−φ1Gμ,q^2+1ϕ1+Eϑ1ϕ2−ϕ11+q^2φ1dq^1μ+12Eϑ1ϕ2−ϕ1∫ϕ1ϕ1+Eϑ1ϕ2−ϕ1Gq^1+1φ1+Eϑ1ϕ2−ϕ11+q^1,νϕ1dq^2ν≤1Eϑ1φ2−φ1Eϑ1ϕ2−ϕ1∫φ1φ1+Eϑ1φ2−φ1∫ϕ1ϕ1+Eϑ1ϕ2−ϕ1Gμ,νϕ1dq^2νφ1dq^1μ≤q^221+q^21Eϑ1ϕ2−ϕ1∫φ1φ1+Eϑ1φ2−φ1Gμ,ϕ1φ1dq^1μ+q^221+q^21Eϑ1φ2−φ1∫φ1φ1+Eϑ1φ2−φ1Gμ,ϕ2φ1dq^1μ+q^121+q^11Eϑ1ϕ2−ϕ1∫ϕ1ϕ1+Eϑ1ϕ2−ϕ1Gφ1,νϕ1dq^2μ+q^121+q^11Eϑ1ϕ2−ϕ1∫ϕ1ϕ1+Eϑ1ϕ2−ϕ1Gφ2,νϕ1dq^2ν≤q^1q^2Gφ1,ϕ1+q^1Gφ1,ϕ2+q^2Gφ2,ϕ1+Gφ2,ϕ21+q^11+q^2.Lemma 32.
Let a twice partiallyq^1q^2-differentiable mapping G:S=φ1,φ1+Fϑ1;ϑ2;ϑ3,φ2−φ1×ϕ1,ϕ1+Fϑ1;ϑ2;ϑ3,ϕ2−ϕ1↦ℝ defined on S∘(the interior of S). If the second-order partial q^1q^2-derivatives are continuous and integrable over S with 0<q^1,q^2<1, then the following equality holds:
(70)Y~~q^1,q^2φ1,φ2,ϕ1,ϕ2G≔Gq^1+1φ1+Eϑ1φ2−φ11+q^1,q^2+1ϕ1+Eϑ1ϕ2−c11+q^2−1Eϑ1φ2−φ1∫φ1φ1+Eϑ1φ2−φ1Gμ,q^2+1ϕ1+Eϑ1ϕ2−ϕ11+q^2φ1dq^1μ−1Eϑ1ϕ2−ϕ1∫ϕ1ϕ1+Eϑ1ϕ2−ϕ1Gq^1+1φ1+Eϑ1φ2−φ11+q^1,νϕ1dq^2ν+1Eϑ1φ2−φ1Eϑ1ϕ2−ϕ1∫φ1φ1+Eϑ1φ2−φ1∫ϕ1ϕ1+Eϑ1ϕ2−ϕ1Gμ,νϕ1dq^2νφ1dq^1μ,where
(71)Y~~q^1,q^2φ1,φ2,ϕ1,ϕ2G≔q^1q^2Eϑ1φ2−φ1Eϑ1ϕ2−ϕ1×∫01∫01Aζ,θφ1,ϕ1∂q^1,q^22Gφ1+ζEϑ1φ2−φ1,ϕ1+θEϑ1ϕ2−ϕ1φ1∂q^1ζϕ1∂q^2θ0dq^2θ0dq^1ζ,
andAζ,θ given in (38).Theorem 33.
Let a mappingG:S=φ1,φ1+Eϑ1φ2−φ1×ϕ1,ϕ1+Eϑ1ϕ2−ϕ1↦ℝ be a twice partially q^1q^2-differentiable on S∘ such that continuous partial q^1q^2-derivatives φ1,ϕ1∂q^1,q^22G/φ1∂q^1ζϕ1∂q^2θ is integrable on S with 0<q^1,q^2<1. If φ1,ϕ1∂q∧1,q∧22G/φ1∂q∧1ζϕ1∂q∧2θσ is a generalized Ψ-convex on the coordinates on S for σ≥1 where σ−1+β−1=1. Then, the following inequality holds:
(72)Y~~q^1,q^2φ1,φ2,ϕ1,ϕ2G≤q^1q^2Eϑ1φ2−φ1Eϑ1ϕ2−ϕ1B1q∧1,q∧21−1/σ×B2q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ+B3q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2φ1∂q∧1ζϕ1∂q∧2θσ+B4q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ+B5q∧1,q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1φ1∂q∧1ζϕ1∂q∧2θσ1/σ,
whereB1q^1,q^2,B2q^1,q^2,B3q^1,q^2,B4q^1,q^2, and B5q^1,q^2 are given in (53), (54), (55), (56), and (57), respectively.Theorem 34.
Let a mappingG:S=φ1,φ1+Eϑ1φ2−φ1×ϕ1,ϕ1+Eϑ1ϕ2−ϕ1↦ℝ be a twice partially q^1q^2-differentiable on S∘ such that continuous partial q^1q^2-derivatives φ1,ϕ1∂q^1,q^22G/φ1∂q^1ζϕ1∂q^2θ is integrable on S with 0<q^1,q^2<1. If φ1,ϕ1∂q∧1,q∧22G/φ1∂q∧1ζϕ1∂q∧2θσ is a generalized Ψ-convex on the coordinates on S for σ>1 where σ−1+β−1=1. Then, the following inequality holds:
(73)Y~~q^1,q^2φ1,φ2,ϕ1,ϕ2G≤q^1q^2Eϑ1φ2−φ1Eϑ1ϕ2−ϕ1∫01∫01Aζ,θβ0dq∧2θ0dq∧1ζ1/β×φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ2/φ1∂q∧1ζϕ1∂q∧2θσ+q∧1φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ2/φ1∂q∧1ζϕ1∂q∧2θσ+q∧2φ1,ϕ1∂q∧1,q∧22Gφ2,ϕ1/φ1∂q∧1ζϕ1∂q∧2θσ+q∧1q∧2φ1,ϕ1∂q∧1,q∧22Gφ1,ϕ1/φ1∂q∧1ζϕ1∂q∧2θσ1+q∧11+q∧21/σ,
whereAζ,θ is defined as in (38).


## 7. Conclusion

The main objective of this paper will be a motivation source for future studies. An auxiliary result inq^1q^2-integrals has been derived. We established some new generalizations for the HH-type inequality pertaining to q^1q^2-differentiable mappings for generalized Ψ-convex functions on coordinates in the special Raina’s function sense that correlates with the q^1q^2-identity. Some useful applications of our findings have been illustrated with the association of the well-known special functions (hypergeometric and Mittag-Leffler function). Moreover, our findings are essentially applicable for obtaining the solution of integral equations that interact withnbodies subject to mixed boundary conditions (see [7, 8]). For further potential investigation, we left the details for futuristic research. Every aspect of the suggested scheme is versatile and simple to execute. We apprehended noteworthy special cases for varying the parametric values in the involvement of special functions. This new study is explicit and viable and can be effectively utilized in inequality theory, special relativity theory, and quantum mechanics.



---

*Source: 6631474-2021-01-27.xml*