On Fractional Newton Inequalities via Coordinated Convex Functions

Abstract

In this paper, firstly, we present an integral identity for functions of two variables via Riemann–Liouville fractional integrals. Then, a Newton-type inequality via partially differentiable coordinated convex mappings is derived by taking the absolute value of the obtained identity. Moreover, several inequalities are obtained with the aid of the Hölder and power mean inequality. In addition, we investigate some Newton-type inequalities utilizing mappings of two variables with bounded variation. Finally, we gave some mathematical examples and their graphical behavior to validate the obtained inequalities.

1. Introduction

Inequalities are widely recognized as one of the main drivers behind the development of mathematics and various branches of applied mathematics. Fundamental inequalities that have taken their place in the literature over the last decade have greatly contributed to applications in many fields of mathematics. Since inequalities and convex functions play an important role in all areas of mathematics and are an active research area, they have become the focus of attention of researchers, especially in recent years. Among these inequalities, the Simpson and Newton-type inequality has directed much research. The inequality obtained from Simpson’s $1/3$ rule, known as Simpson’s type inequality in the literature, is as follows.

Suppose that $ϝ:\left[\sigma ,\rho \right]\to \mathbb{R}$ is a four times continuously differentiable mapping on $\left(\sigma ,\rho \right)$ and let ${∥{ϝ}^{\left(4\right)}∥}_{\infty }={\displaystyle \underset{\kappa \in \left(\sigma ,\rho \right)}{\sup }}\left|{ϝ}^{\left(4\right)}\left(\kappa \right)\right|<\infty .$ Then, one has the inequality

$$\left|\frac{1}{3}\left[\frac{ϝ\left(\sigma \right)+ϝ\left(\rho \right)}{2}+2ϝ\left(\frac{\sigma +\rho }{2}\right)\right]-\frac{1}{\rho -\sigma }{\int }_{\sigma }^{\rho }ϝ\left(\kappa \right)d\kappa \right|\le \frac{1}{2880}{∥{ϝ}^{\left(4\right)}∥}_{\infty }{\left(\rho -\sigma \right)}^4.$$

The Simpson’s $1/3$ type inequality intrigued researchers. For instance, Dragomir et al. [1] proved some new Simpson’s type results and their applications to quadrature formulas in numerical integration. Alomari et al. investigated some of Simpson’s type inequalities based on the s-convex functions in [2]. Sarikaya et al. gave the variants of Simpson’s type inequalities via convexity in [3]. In [4], a Simpson-type inequality via an n-times continuously differentiable function is given. New Simpson-type inequalities are presented based on $(s,m)$-convexity with the help of the differentiable mappings in [5]. Du et al. introduced the concepts of an m-invex set, generalized $(s,m)$-preinvex mapping, and explicitly $(s,m)$-preinvex mapping, provided some properties for the newly introduced mappings, and obtained new Hadamard–Simpson-type integral inequalities via a mapping of which the power of the absolute of the first derivative is generalized $(s,m)$-preinvex mapping in [6]. Hezenci et al. obtained several fractional Simpson-type inequalities via functions whose second derivatives in modulus are convex in [7]. Sarıkaya et al. obtained Simpson’s type inequality via the mapping whose second derivatives modulus is F-convex in [8].

The inequality obtained from Simpson’s $3/8$ rule and known as Newtonian inequality in the literature is as follows.

If $ϝ:\left[\sigma ,\rho \right]\to \mathbb{R}$ is a four times continuously differentiable mapping on $\left(\sigma ,\rho \right)$ and let ${∥{ϝ}^{\left(4\right)}∥}_{\infty }={\displaystyle \underset{\kappa \in \left(\sigma ,\rho \right)}{\sup }}\left|{ϝ}^{\left(4\right)}\left(\kappa \right)\right|<\infty .$ Then, one has the inequality

$$\left|\frac{1}{8}\left[ϝ\left(\sigma \right)+3ϝ\left(\frac{2\sigma +\rho }{3}\right)+3ϝ\left(\frac{\sigma +2\rho }{3}\right)+ϝ\left(\rho \right)\right]-\frac{1}{\rho -\sigma }{\int }_{\sigma }^{\rho }ϝ\left(\kappa \right)d\kappa \right|\le \frac{1}{6480}{∥{ϝ}^{\left(4\right)}∥}_{\infty }{\left(\rho -\sigma \right)}^4.$$

There are many studies in the literature on Newton-type inequalities. Gao and Shi derived inequalities of Newton’s type with the help of the functions, whose second derivatives moduli are convex in [9]. Erden et al. presented some error estimates of the Newton-type cubature formula with the aid of the bounded variation and Lipschitzian mappings in [10]. Noor et al. gave Newton’s type inequalities based on harmonic convex and p-harmonic convex mappings in [11,12], respectively. Iftikhar et al. investigated some new Newton-type integral inequalities on coordinates in [13].

Liouville first introduced the concepts of fractional derivative and fractional integral. The idea of fractional derivative and fractional integral emerged from the question of whether derivatives and integrals exist only for integers. Since the 17th century, it has developed with the pioneering studies of Leibniz, Euler, Lagrange, Abel, Liouville, and many other mathematicians based on the generalization of differential and integration for fractional order. Many researchers have focused on this issue. Sarikaya et al. obtained new inequalities of Hermite–Hadamard type and trapezoid type based on Riemann–Liouville fractional integrals in [14] for the first time. Set proved inequalities of Ostrowski-type inequalities utilizing the Riemann–Liouville fractional integrals via differentiable mappings in [15]. İşcan and Wu obtained inequalities of Hermite–Hadamard type with the aid of the harmonic convexity in [16]. Sarikaya and Yıldrım gave new inequalities of Hermite–Hadamard type and midpoint type inequalities based on Riemann–Liouville fractional integrals in [17]. Chen and Huang established Simpson $1/3$ rule type inequality via s-convex mappings with the aid of the Riemann–Liouville fractional integrals in [18].

Moreover, after Camille Jordan introduced the mappings of bounded variation of a single variable, various studies on mapping this bounded variation were put forward. In p, functions of bounded variations have been the subject of new research in inequality theory. For instance, Dragomir investigated midpoint-type inequalities with the help of the functions of bounded variation in [19]. Then, Dragomir was also obtained for trapezoid-type inequalities in [20]. What is more, Dragomir proved new Simpson’s type inequalities based on functions of bounded variations in [21]. Jawarneh and Noorani established results for some inequalities based on mappings of bounded mapping on coordinates in [22]. However, there are minor errors in Lemma 1, which he established here, and Moricz corrected this error in [23]. Then, Budak and Sarıkaya, with the help of the Lemma established by Moricz, obtained the corrections of these results in [22].

Here are the articles that inspire us: Sitthiwirattham et al. [24] gave new Newton’s type inequalities with the help of the convex mappings via Riemann–Liouville fractional integrals. The authors gave new fractional Simpson’s second formula inequalities via mappings of bounded variation in [24]. Hezenci et al. proved some of Newton’s type inequalities with the help of differentiable convex mappings based on the well-known Riemann–Liouville fractional integrals in [25]. For the other paper devoted to Simpson-type inequalities, please refer to [26,27,28]. With the motivation from these studies, we will establish new Simpson’s second rule inequalities via convex mappings on coordinates by using Riemann–Liouville fractional integrals. We will also investigate new fractional Newton formula-type inequalities based on mappings of two variables with bounded variations.

2. Preliminaries

In this section, fundamental definitions of Riemann–Liouville integrals via one and two variables in the literature are given. In addition, Riemann–Liouville fractional Newton-type inequalities are mentioned via a variable. What is more, the two lemmas we will use in the Newton-type inequalities based on bounded variations section will be addressed.

Definition 1

([29,30]). If $ϝ\in L_1\left[\sigma ,\rho \right]$ is a mapping with $\alpha >0$ and $\sigma \ge 0$, then the integrals ${\mathcal{J}}_{\sigma +}^{\alpha }ϝ\left(\kappa \right)$ and ${\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\kappa \right)$ are described by

$${\mathcal{J}}_{\sigma +}^{\alpha }ϝ\left(\kappa \right)=\frac{1}{\mathsf{\Gamma }\left(\sigma \right)}{\int }_{\sigma }^{\kappa }{\left(\kappa -\gamma \right)}^{\alpha -1}ϝ\left(\gamma \right)d\gamma ,\kappa >\sigma$$

and

$${\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\kappa \right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{\kappa }^{\rho }{\left(\gamma -\kappa \right)}^{\alpha -1}ϝ\left(\gamma \right)d\gamma ,\kappa <\rho ,$$

where $\mathsf{\Gamma }(·)$ is the well-known Gamma function. These integrals are called Riemann–Liouville fractional integrals in the literature.

Definition 2

([31]). If $ϝ\in L_1\left(\Delta =\left[\sigma ,\rho \right]\times \left[\varsigma ,d\right]\right)$ is a mapping and $\alpha ,\beta >0$ and $\sigma ,\varsigma \ge 0$, then the impressions ${\mathcal{J}}_{\sigma +,\varsigma +}^{\alpha ,\beta },{\mathcal{J}}_{\sigma +,d-}^{\alpha ,\beta },+{\mathcal{J}}_{\rho -,\varsigma +}^{\alpha ,\beta }$, and ${\mathcal{J}}_{\rho -,d-}^{\alpha ,\beta }$ are defined by

$$\begin{array}{cc}\hfill {\mathcal{J}}_{\sigma +,\varsigma +}^{\alpha ,\beta }ϝ(\kappa ,y)& =\frac{1}{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}\underset{\sigma }{\overset{\kappa }{\int }}\underset{\varsigma }{\overset{y}{\int }}{\left(\kappa -\gamma \right)}^{\alpha -1}{\left(y-\delta \right)}^{\beta -1}ϝ\left(\gamma ,\delta \right)d\delta d\gamma ,\kappa >\sigma ,y>\varsigma ,\hfill \\ \hfill {\mathcal{J}}_{\sigma +,d-}^{\alpha ,\beta }ϝ(\kappa ,y)& =\frac{1}{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}\underset{\sigma }{\overset{\kappa }{\int }}\underset{y}{\overset{d}{\int }}{\left(\kappa -\gamma \right)}^{\alpha -1}{\left(\delta -y\right)}^{\beta -1}ϝ\left(\gamma ,\delta \right)d\delta d\gamma ,\kappa >\sigma ,y>d,\hfill \\ \hfill {\mathcal{J}}_{\rho -,\varsigma +}^{\alpha ,\beta }ϝ(\kappa ,y)& =\frac{1}{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}\underset{\kappa }{\overset{\rho }{\int }}\underset{\varsigma }{\overset{y}{\int }}{\left(\gamma -\kappa \right)}^{\alpha -1}{\left(y-\delta \right)}^{\beta -1}ϝ\left(\gamma ,\delta \right)d\delta d\gamma ,\kappa <\rho ,y>\varsigma ,\hfill \\ \hfill {\mathcal{J}}_{\rho -,d-}^{\alpha ,\beta }ϝ(\kappa ,y)& =\frac{1}{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}\underset{\kappa }{\overset{\rho }{\int }}\underset{y}{\overset{d}{\int }}{\left(\gamma -\kappa \right)}^{\alpha -1}{\left(\delta -y\right)}^{\beta -1}ϝ\left(\gamma ,\delta \right)d\delta d\gamma ,quad\kappa <\rho ,y<d,\hfill \end{array}$$

where Γ is the well-known Gamma function. These impressions are called double Riemann–Liouville fractional integrals in the literature.

For more information and recent results for fractional calculus, one can refer to [32,33,34].

With the aid of ${\mathcal{J}}_{\sigma +}^{\alpha }ϝ\left(\kappa \right)$ and ${\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\kappa \right)$, Sitthiwirattham et al. [24] obtained the new Newton-type inequalities as follows:

Theorem 1.

Let $ϝ:\left[\sigma ,\rho \right]\to \mathbb{R}$ be a differentiable mapping on $\left(\sigma ,\rho \right)$ with $ϝ\in L_1\left[\sigma ,\rho \right]$. If $\left|{ϝ}^{\prime }\right|$ is convex mapping, then we have the following Newton’s type inequality:

$$\begin{array}{cc}\hfill & \left|\frac{3^{\alpha -1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha }}\left[{\mathcal{J}}_{\sigma +}^{\alpha }ϝ\left(\frac{2\sigma +\rho }{3}\right)+{\mathcal{J}}_{\frac{2\sigma +\rho }{3}+}^{\alpha }ϝ\left(\frac{\sigma +2\rho }{3}\right)+{\mathcal{J}}_{\frac{\sigma +2\rho }{3}+}^{\alpha }ϝ\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-\frac{1}{8}\left[ϝ\left(\sigma \right)+3ϝ\left(\frac{2\sigma +\rho }{3}\right)+3ϝ\left(\frac{\sigma +2\rho }{3}\right)+ϝ\left(\rho \right)\right]\right|\hfill \\ \hfill & \le \frac{\rho -\sigma }{27}\left[\left|{ϝ}^{\prime }\left(\rho \right)\right|\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)+2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\right.\hfill \\ \hfill & \left.+\left|{ϝ}^{\prime }\left(\sigma \right)\right|\left({\mathsf{\Omega }}_1\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)+{\mathsf{\Omega }}_3\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)+{\mathsf{\Omega }}_5\left(\alpha \right)\right)\right],\hfill \end{array}$$

(1)

where

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_1\left(\upsilon \right)& ={\int }_0^1\tau \left|{\tau }^{\upsilon }-\frac{3}{8}\right|d\tau =\frac{\upsilon }{\upsilon +2}{\left(\frac{3}{8}\right)}^{\frac{\upsilon +2}{\upsilon }}+\frac{1}{\upsilon +2}-\frac{3}{16},\hfill \\ \hfill {\mathsf{\Omega }}_2\left(\upsilon \right)& ={\int }_0^1\left|{\tau }^{\upsilon }-\frac{3}{8}\right|d\tau =\frac{2\upsilon }{\upsilon +1}{\left(\frac{3}{8}\right)}^{\frac{\upsilon +1}{\upsilon }}+\frac{1}{\upsilon +1}-\frac{3}{8},\hfill \\ \hfill {\mathsf{\Omega }}_3\left(\upsilon \right)& ={\int }_0^1\tau \left|{\tau }^{\upsilon }-\frac{1}{2}\right|d\tau =\frac{\upsilon }{\upsilon +2}{\left(\frac{1}{2}\right)}^{\frac{\upsilon +2}{\upsilon }}+\frac{1}{\upsilon +2}-\frac{1}{4},\hfill \\ \hfill {\mathsf{\Omega }}_4\left(\upsilon \right)& ={\int }_0^1\left|{\tau }^{\upsilon }-\frac{1}{2}\right|d\tau =\frac{2\upsilon }{\upsilon +1}{\left(\frac{1}{2}\right)}^{\frac{\upsilon +1}{\upsilon }}+\frac{1}{\upsilon +1}-\frac{1}{2},\hfill \\ \hfill {\mathsf{\Omega }}_5\left(\upsilon \right)& ={\int }_0^1\tau \left|{\tau }^{\upsilon }-\frac{5}{8}\right|d\tau =\frac{\upsilon }{\upsilon +2}{\left(\frac{5}{8}\right)}^{\frac{\upsilon +2}{\upsilon }}+\frac{1}{\upsilon +2}-\frac{5}{16},\hfill \\ \hfill {\mathsf{\Omega }}_6\left(\upsilon \right)& ={\int }_0^1\left|{\tau }^{\upsilon }-\frac{5}{8}\right|d\tau =\frac{2\upsilon }{\upsilon +1}{\left(\frac{5}{8}\right)}^{\frac{\upsilon +1}{\upsilon }}+\frac{1}{\upsilon +1}-\frac{5}{8}.\hfill \end{array}$$

(2)

The next definition will be used several times in the proofs of the main results:

Definition 3

([35]). A function $ϝ:\Delta \to \mathbb{R}$ is called coordinated convex on $\Delta ,$ for all $(\kappa ,u),(y,v)\in \Delta$ and $\gamma ,\delta \in [0,1]$, if it satisfies the following inequality:

$$\begin{array}{cc}\hfill & f(\gamma \kappa +(1-\gamma )y,\gamma u+(1-\gamma \left)v\right)\hfill \\ \hfill \le & \gamma \delta ϝ(\kappa ,u)+\gamma (1-\delta )ϝ(\kappa ,v)+\delta (1-\gamma )ϝ(y,u)+(1-\gamma )(1-\gamma )ϝ(y,v).\hfill \end{array}$$

(3)

The mapping f is a coordinated concave on $\Delta$ if the inequality (3) holds in the reversed direction for all $\gamma ,\delta \in [0,1]$ and $(\kappa ,u),(y,v)\in \Delta$.

For example, the function $ϝ:\left[0,1\right]\times \left[0,1\right]\to \mathbb{R}$ defined by $ϝ\left(\kappa ,y\right)={\kappa }^2{e}^y$ is coordinated convex on $\left[0,1\right]\times \left[0,1\right].$

The following lemmas will be used in the inequalities we will establish based on the mappings of bounded variation.

Lemma 1

([23]). If $ϝ(\gamma ,\delta )$ is continuous on renctangle Δ and $\alpha (\gamma ,\delta )$ is a bounded variation on Δ, then $\alpha (\gamma ,\delta )$ is integrable with respect to $ϝ(\gamma ,\delta )$ over Δ in the Riemann-Stieltjes sense, and

$$\begin{array}{cc}\hfill \underset{\sigma }{\overset{\rho }{\int }}\underset{\varsigma }{\overset{d}{\int }}ϝ(\gamma ,\delta )d_{\gamma }{d}_{\delta }\alpha (\gamma ,\delta )& =\underset{\sigma }{\overset{\rho }{\int }}\underset{\varsigma }{\overset{d}{\int }}\alpha (\gamma ,\delta )d_{\gamma }{d}_{\delta }ϝ(\gamma ,\delta )\hfill \\ \hfill & -\underset{\sigma }{\overset{\rho }{\int }}\alpha (\gamma ,d)d_{\gamma }ϝ(\gamma ,d)+\underset{\sigma }{\overset{\rho }{\int }}\alpha (\gamma ,\varsigma )d_{\gamma }ϝ(\gamma ,\varsigma )\hfill \\ \hfill & -\underset{\varsigma }{\overset{d}{\int }}\alpha (\rho ,\delta )d_{\delta }ϝ(\rho ,\delta )+\underset{\varsigma }{\overset{d}{\int }}\alpha (\sigma ,\delta )d_{\delta }ϝ(\sigma ,\delta )\hfill \\ \hfill & +ϝ(\rho ,d)\alpha (\rho ,d)-ϝ(\rho ,\varsigma )\alpha (\rho ,\varsigma )-ϝ(\sigma ,d)\alpha (\sigma ,d)+ϝ(\sigma ,\varsigma )\alpha (\sigma ,\varsigma ).\hfill \end{array}$$

Lemma 2

([22]). Assume that ϝ is integrable with respect to $g(\gamma ,\delta )$ over Δ in the Riemann–Stieltjes sense on Δ and g is of bounded variation on Δ, then

$$\left|\underset{\sigma }{\overset{\rho }{\int }}\underset{\varsigma }{\overset{d}{\int }}ϝ(\kappa ,y)d_{\kappa }{d}_{y}g(\kappa ,y)\right|\le {\displaystyle \underset{(\kappa ,y)\in \Delta }{\sup }}\left|ϝ(\kappa ,y)\right|\underset{\sigma }{\overset{\rho }{\bigvee }}\underset{\varsigma }{\overset{d}{\bigvee }}\left(g\right).$$

(4)

3. An Identity

In this section, by using ${\mathcal{J}}_{\sigma +}^{\alpha }ϝ\left(\kappa \right),{\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\kappa \right),{\mathcal{J}}_{\sigma +,\varsigma +}^{\alpha ,\beta },{\mathcal{J}}_{\sigma +,d-}^{\alpha ,\beta },{\mathcal{J}}_{\rho -,\varsigma +}^{\alpha ,\beta }$, and ${\mathcal{J}}_{\rho -,d-}^{\alpha ,\beta },$ we will consider a convex function in differentiable coordinates and obtain a lemma that we will use throughout the article.

Lemma 3.

Let $ϝ:\Delta \to \mathbb{R}$ be a partially differentiable on ${\Delta }^{\circ }$; then, the following Riemann–Liouville fractional integrals identity yield:

$${\mathsf{\Theta }}^{\alpha ,\beta }\left(\sigma ,\rho ;\varsigma ,d\right)=\frac{\left(\rho -\sigma \right)\left(d-\varsigma \right)}{81}\sum _{k=1}^9{\mathsf{\Phi }}_k,$$

(5)

where

$$\begin{array}{cc}\hfill & {\mathsf{\Theta }}^{\alpha ,\beta }\left(\sigma ,\rho ;\varsigma ,d\right)\hfill \\ \hfill & :=\frac{\left[ϝ\left(\sigma ,\varsigma \right)+ϝ\left(\sigma ,d\right)+ϝ\left(\rho ,\varsigma \right)+ϝ\left(\rho ,d\right)\right]}{64}\hfill \\ \hfill & +\frac{3}{64}\left[ϝ\left(\sigma ,\frac{2\varsigma +d}{3}\right)+ϝ\left(\sigma ,\frac{\varsigma +2d}{3}\right)+ϝ\left(\rho ,\frac{2\varsigma +d}{3}\right)+ϝ\left(\rho ,\frac{\varsigma +2d}{3}\right)\right.\hfill \\ \hfill & +\left.ϝ\left(\frac{2\sigma +\rho }{3},\varsigma \right)+ϝ\left(\frac{\sigma +2\rho }{3},\varsigma \right)+\left(\frac{2\sigma +\rho }{3},d\right)+ϝ\left(\frac{\sigma +2\rho }{3},d\right)\right]\hfill \\ \hfill & +\frac{9}{64}\left[ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)+ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)+ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)+ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)\right]\hfill \\ \hfill & -\frac{3^{\alpha -1}\mathsf{\Gamma }\left(\alpha +1\right)}{8{\left(\rho -\sigma \right)}^{\alpha }}\left[{\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\frac{\sigma +2\rho }{3},d\right)+{\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\frac{\sigma +2\rho }{3},\varsigma \right)\right.\hfill \\ \hfill & +\left.{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-}^{\alpha }ϝ\left(\frac{2\sigma +\rho }{3},d\right)+{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-}^{\alpha }ϝ\left(\frac{2\sigma +\rho }{3},\varsigma \right)+{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-}^{\alpha }ϝ\left(\sigma ,d\right)+{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-}^{\alpha }ϝ\left(\sigma ,\varsigma \right)\right]\hfill \\ \hfill & -\frac{3^{\alpha }\mathsf{\Gamma }\left(\alpha +1\right)}{8{\left(\rho -\sigma \right)}^{\alpha }}\left[{\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-}^{\alpha }ϝ\left(\sigma ,\frac{\varsigma +2d}{3}\right)\right.\hfill \\ \hfill & +\left.{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-}^{\alpha }ϝ\left(\sigma ,\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-}^{\alpha }ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-}^{\alpha }ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)\right]\hfill \\ \hfill & -\frac{3^{\beta -1}\mathsf{\Gamma }\left(\beta +1\right)}{8{\left(d-\varsigma \right)}^{\beta }}\left[{\mathcal{J}}_{d-}^{\beta }ϝ\left(\rho ,\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\frac{\varsigma +2d}{3}-}^{\beta }ϝ\left(\rho ,\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{2\varsigma +d}{3}-}^{\beta }ϝ\left(\rho ,\varsigma \right)\right.\hfill \\ \hfill & +\left.{\mathcal{J}}_{d-}^{\beta }ϝ\left(\sigma ,\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\frac{\varsigma +2d}{3}-}^{\beta }ϝ\left(\sigma ,\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{2\varsigma +d}{3}-}^{\beta }ϝ\left(\sigma ,\varsigma \right)\right]\hfill \\ \hfill & -\frac{3^{\beta }\mathsf{\Gamma }\left(\beta +1\right)}{8{\left(d-\varsigma \right)}^{\beta }}\left[{\mathcal{J}}_{d-}^{\beta }ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\frac{\varsigma +2d}{3}-}^{\beta }ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{2\varsigma +d}{3}-}^{\beta }ϝ\left(\frac{\sigma +2\rho }{3},\varsigma \right)\right.\hfill \\ \hfill & +\left.{\mathcal{J}}_{d-}^{\beta }ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\frac{\varsigma +2d}{3}-}^{\beta }ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{2\varsigma +d}{3}-}^{\beta }ϝ\left(\frac{2\sigma +\rho }{3},\varsigma \right)\right]\hfill \\ \hfill & +\frac{3^{\alpha -1}{3}^{\beta -1}\mathsf{\Gamma }\left(\alpha +1\right)\mathsf{\Gamma }\left(\beta +1\right)}{{\left(\rho -\sigma \right)}^{\alpha }{\left(d-\varsigma \right)}^{\beta }}\hfill \\ \hfill & \times \left[{\mathcal{J}}_{\rho -,d-}^{\alpha ,\beta }ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\rho -,\frac{\varsigma +2d}{3}-}^{\alpha ,\beta }ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\rho -,\frac{2\varsigma +d}{3}-}^{\alpha ,\beta }ϝ\left(\frac{\sigma +2\rho }{3},\varsigma \right)\right.\hfill \\ \hfill & +{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-,d-}^{\alpha ,\beta }ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-,\frac{\varsigma +2d}{3}-}^{\alpha ,\beta }ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-,\frac{2\varsigma +d}{3}-}^{\alpha ,\beta }ϝ\left(\frac{2\sigma +\rho }{3},\varsigma \right)\hfill \\ \hfill & +\left.{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-,d-}^{\alpha ,\beta }ϝ\left(\sigma ,\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-,\frac{\varsigma +2d}{3}-}^{\alpha ,\beta }ϝ\left(\sigma ,\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-,\frac{2\varsigma +d}{3}-}^{\alpha ,\beta }ϝ\left(\sigma ,\varsigma \right)\right]\hfill \end{array}$$

(6)

and here,

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_1& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{5}{8}\right)\left({\delta }^{\beta }-\frac{5}{8}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{1-\gamma }{3}\right)\sigma +\left(\frac{\gamma +2}{3}\right)\rho ,\left(\frac{1-\delta }{3}\right)\varsigma +\left(\frac{\delta +2}{3}\right)d\right)d\delta d\gamma ,\hfill \\ \hfill {\mathsf{\Phi }}_2& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{5}{8}\right)\left({\delta }^{\beta }-\frac{1}{2}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{1-\gamma }{3}\right)\sigma +\left(\frac{\gamma +2}{3}\right)\rho ,\left(\frac{2-\delta }{3}\right)\varsigma +\left(\frac{\delta +1}{3}\right)d\right)d\delta d\gamma ,\hfill \\ \hfill {\mathsf{\Phi }}_3& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{5}{8}\right)\left({\delta }^{\beta }-\frac{3}{8}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{1-\gamma }{3}\right)\sigma +\left(\frac{\gamma +2}{3}\right)\rho ,\left(\frac{3-\delta }{3}\right)\varsigma +\frac{\delta }{3}d\right)d\delta d\gamma ,\hfill \\ \hfill {\mathsf{\Phi }}_4& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{1}{2}\right)\left({\delta }^{\beta }-\frac{5}{8}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{2-\gamma }{3}\right)\sigma +\left(\frac{\gamma +1}{3}\right)\rho ,\left(\frac{1-\delta }{3}\right)\varsigma +\left(\frac{\delta +2}{3}\right)d\right)d\delta d\gamma ,\hfill \\ \hfill {\mathsf{\Phi }}_5& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{1}{2}\right)\left({\delta }^{\beta }-\frac{1}{2}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{2-\gamma }{3}\right)\sigma +\left(\frac{\gamma +1}{3}\right)\rho ,\left(\frac{2-\delta }{3}\right)\varsigma +\left(\frac{\delta +1}{3}\right)d\right)d\delta d\gamma ,\hfill \\ \hfill {\mathsf{\Phi }}_6& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{1}{2}\right)\left({\delta }^{\beta }-\frac{3}{8}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{2-\gamma }{3}\right)\sigma +\left(\frac{\gamma +1}{3}\right)\rho ,\left(\frac{3-\delta }{3}\right)\varsigma +\frac{\delta }{3}d\right)d\delta d\gamma ,\hfill \\ \hfill {\mathsf{\Phi }}_7& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{3}{8}\right)\left({\delta }^{\beta }-\frac{5}{8}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{3-\gamma }{3}\right)\sigma +\frac{\gamma }{3}\rho ,\left(\frac{1-\delta }{3}\right)\varsigma +\left(\frac{\delta +2}{3}\right)d\right)d\delta d\gamma ,\hfill \\ \hfill {\mathsf{\Phi }}_8& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{3}{8}\right)\left({\delta }^{\beta }-\frac{1}{2}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{3-\gamma }{3}\right)\sigma +\frac{\gamma }{3}\rho ,\left(\frac{2-\delta }{3}\right)\varsigma +\left(\frac{\delta +1}{3}\right)d\right)d\delta d\gamma ,\hfill \\ \hfill {\mathsf{\Phi }}_9& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{3}{8}\right)\left({\delta }^{\beta }-\frac{3}{8}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{3-\gamma }{3}\right)\sigma +\frac{\gamma }{3}\rho ,\left(\frac{3-\delta }{3}\right)\varsigma +\frac{\delta }{3}d\right)d\delta d\gamma .\hfill \end{array}$$

Proof. 

By utilizing integration by parts and change of variables, we derive

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_1& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{5}{8}\right)\left({\delta }^{\beta }-\frac{5}{8}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{1-\gamma }{3}\right)\sigma +\left(\frac{\gamma +2}{3}\right)\rho ,\left(\frac{1-\delta }{3}\right)\varsigma +\left(\frac{\delta +2}{3}\right)d\right)d\delta d\gamma \hfill \\ \hfill & =\frac{81}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\rho ,d\right)+\frac{135}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{\sigma +2\rho }{3},d\right)\hfill \\ \hfill & +\frac{135}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\rho ,\frac{\varsigma +2d}{3}\right)+\frac{225}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)\hfill \\ \hfill & -\frac{9}{8\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\frac{\sigma +2\rho }{3},d\right)-\frac{15}{8\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)\hfill \\ \hfill & -\frac{9}{8\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{d-}^{\beta }ϝ\left(\rho ,\frac{\varsigma +2d}{3}\right)-\frac{15}{8\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{d-}^{\beta }ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)\hfill \\ \hfill & +\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\rho -,d-}^{\alpha ,\beta }ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_2& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{5}{8}\right)\left({\delta }^{\beta }-\frac{1}{2}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{1-\gamma }{3}\right)\sigma +\left(\frac{\gamma +2}{3}\right)\rho ,\left(\frac{2-\delta }{3}\right)\varsigma +\left(\frac{\delta +1}{3}\right)d\right)d\delta d\gamma \hfill \\ \hfill & =\frac{27}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\rho ,\frac{\varsigma +2d}{3}\right)+\frac{45}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)\hfill \\ \hfill & +\frac{27}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\rho ,\frac{2\varsigma +d}{3}\right)+\frac{45}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)\hfill \\ \hfill & -\frac{3}{2\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)-\frac{3}{2\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)\hfill \\ \hfill & -\frac{9}{8\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{\varsigma +2d}{3}-}^{\beta }ϝ\left(\rho ,\frac{2\varsigma +d}{3}\right)-\frac{15}{8\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{\varsigma +2d}{3}-}^{\beta }ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)\hfill \\ \hfill & +\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\rho -,\frac{\varsigma +2d}{3}-}^{\alpha ,\beta }ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_3& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{5}{8}\right)\left({\delta }^{\beta }-\frac{3}{8}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{1-\gamma }{3}\right)\sigma +\left(\frac{\gamma +2}{3}\right)\rho ,\left(\frac{3-\delta }{3}\right)\varsigma +\frac{\delta }{3}d\right)d\delta d\gamma \hfill \\ \hfill & =\frac{135}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\rho ,\frac{2\varsigma +d}{3}\right)+\frac{225}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)\hfill \\ \hfill & +\frac{81}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\rho ,\varsigma \right)+\frac{135}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{\sigma +2\rho }{3},\varsigma \right)\hfill \\ \hfill & -\frac{15}{8\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)-\frac{9}{8\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\frac{\sigma +2\rho }{3},\varsigma \right)\hfill \\ \hfill & -\frac{9}{8\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{2\varsigma +d}{3}-}^{\beta }ϝ\left(\rho ,\varsigma \right)-\frac{15}{8\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{2\varsigma +d}{3}-}^{\beta }ϝ\left(\frac{\sigma +2\rho }{3},\varsigma \right)\hfill \\ \hfill & +\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\rho -,\frac{2\varsigma +d}{3}-}^{\alpha ,\beta }ϝ\left(\frac{\sigma +2\rho }{3},\varsigma \right),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_4& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{1}{2}\right)\left({\delta }^{\beta }-\frac{5}{8}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{2-\gamma }{3}\right)\sigma +\left(\frac{\gamma +1}{3}\right)\rho ,\left(\frac{1-\delta }{3}\right)\varsigma +\left(\frac{\delta +2}{3}\right)d\right)d\delta d\gamma \hfill \\ \hfill & =\frac{27}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{\sigma +2\rho }{3},d\right)+\frac{27}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{2\sigma +\rho }{3},d\right)\hfill \\ \hfill & +\frac{45}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)+\frac{45}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)\hfill \\ \hfill & -\frac{9}{8\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-}^{\alpha }ϝ\left(\frac{2\sigma +\rho }{3},d\right)-\frac{15}{8\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-}^{\alpha }ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)\hfill \\ \hfill & -\frac{3}{2\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{d-}^{\beta }ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)-\frac{3}{2\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{d-}^{\beta }ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)\hfill \\ \hfill & +\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-,d-}^{\alpha ,\beta }ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_5& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{1}{2}\right)\left({\delta }^{\beta }-\frac{1}{2}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{2-\gamma }{3}\right)\sigma +\left(\frac{\gamma +1}{3}\right)\rho ,\left(\frac{2-\delta }{3}\right)\varsigma +\left(\frac{\delta +1}{3}\right)d\right)d\delta d\gamma \hfill \\ \hfill & =\frac{9}{4\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)+\frac{9}{4\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)\hfill \\ \hfill & +\frac{9}{4\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)+\frac{9}{4\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)\hfill \\ \hfill & -\frac{3}{2\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-}^{\alpha }ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)-\frac{3}{2\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-}^{\alpha }ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)\hfill \\ \hfill & -\frac{3}{2\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{\varsigma +2d}{3}-}^{\beta }ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)-\frac{3}{2\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{\varsigma +2d}{3}-}^{\beta }ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)\hfill \\ \hfill & +\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-,\frac{\varsigma +2d}{3}-}^{\alpha ,\beta }ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_6& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{1}{2}\right)\left({\delta }^{\beta }-\frac{3}{8}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{2-\gamma }{3}\right)\sigma +\left(\frac{\gamma +1}{3}\right)\rho ,\left(\frac{3-\delta }{3}\right)\varsigma +\frac{\delta }{3}d\right)d\delta d\gamma \hfill \\ \hfill & =\frac{45}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)+\frac{15}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)\hfill \\ \hfill & +\frac{27}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{\sigma +2\rho }{3},\varsigma \right)+\frac{27}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{2\sigma +\rho }{3},\varsigma \right)\hfill \\ \hfill & -\frac{15}{8\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-}^{\alpha }ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)-\frac{9}{8\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-}^{\alpha }ϝ\left(\frac{2\sigma +\rho }{3},\varsigma \right)\hfill \\ \hfill & -\frac{3}{2\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{2\varsigma +d}{3}-}^{\beta }ϝ\left(\frac{\sigma +2\rho }{3},\varsigma \right)-\frac{3}{2\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{\varsigma +2d}{3}+}^{\beta }ϝ\left(\frac{2\sigma +\rho }{3},\varsigma \right)\hfill \\ \hfill & +\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-,\frac{2\varsigma +d}{3}-}^{\alpha ,\beta }ϝ\left(\frac{2\sigma +\rho }{3},\varsigma \right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_7& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{3}{8}\right)\left({\delta }^{\beta }-\frac{5}{8}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{3-\gamma }{3}\right)\sigma +\frac{\gamma }{3}\rho ,\left(\frac{1-\delta }{3}\right)\varsigma +\left(\frac{\delta +2}{3}\right)d\right)d\delta d\gamma \hfill \\ \hfill & =\frac{135}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{2\sigma +\rho }{3},d\right)+\frac{81}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)\hfill \\ \hfill & +\frac{225}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)+\frac{135}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\sigma ,\frac{\varsigma +2d}{3}\right)\hfill \\ \hfill & -\frac{9}{8\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-}^{\alpha }ϝ\left(\sigma ,d\right)-\frac{15}{8\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-}^{\alpha }ϝ\left(\sigma ,\frac{\varsigma +2d}{3}\right)\hfill \\ \hfill & -\frac{15}{8\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{d-}^{\beta }ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)-\frac{9}{8\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{d-}^{\beta }ϝ\left(\sigma ,\frac{\varsigma +2d}{3}\right)\hfill \\ \hfill & +\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-,d-}^{\alpha ,\beta }ϝ\left(\sigma ,\frac{\varsigma +2d}{3}\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_8& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{3}{8}\right)\left({\delta }^{\beta }-\frac{1}{2}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{3-\gamma }{3}\right)\sigma +\frac{\gamma }{3}\rho ,\left(\frac{2-\delta }{3}\right)\varsigma +\left(\frac{\delta +1}{3}\right)d\right)d\delta d\gamma \hfill \\ \hfill & =\frac{45}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)+\frac{27}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\sigma ,\frac{\varsigma +2d}{3}\right)\hfill \\ \hfill & -\frac{45}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)+\frac{27}{16\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\sigma ,\frac{2\varsigma +d}{3}\right)\hfill \\ \hfill & -\frac{3}{2\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-}^{\alpha }ϝ\left(\sigma ,\frac{\varsigma +2d}{3}\right)-\frac{3}{2\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-}^{\alpha }ϝ\left(\sigma ,\frac{2\varsigma +d}{3}\right)\hfill \\ \hfill & -\frac{15}{8\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{\varsigma +2d}{3}-}^{\beta }ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)-\frac{9}{8\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{\varsigma +2d}{3}-}^{\beta }ϝ\left(\sigma ,\frac{2\varsigma +d}{3}\right)\hfill \\ \hfill & +\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-,\frac{\varsigma +2d}{3}-}^{\alpha ,\beta }ϝ\left(\sigma ,\frac{2\varsigma +d}{3}\right)\hfill \end{array}$$

(14)

and

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_9& ={\int }_0^1{\int }_0^1\left({\gamma }^{\alpha }-\frac{3}{8}\right)\left({\delta }^{\beta }-\frac{3}{8}\right)\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{3-\gamma }{3}\right)\sigma +\frac{\gamma }{3}\rho ,\left(\frac{3-\delta }{3}\right)\varsigma +\frac{\delta }{3}d\right)d\delta d\gamma \hfill \\ \hfill & =\frac{225}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)+\frac{135}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\sigma ,\frac{2\varsigma +d}{3}\right)\hfill \\ \hfill & -\frac{135}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\frac{2\sigma +\rho }{3},\varsigma \right)+\frac{81}{64\left(\rho -\sigma \right)\left(d-\varsigma \right)}ϝ\left(\sigma ,\varsigma \right)\hfill \\ \hfill & -\frac{15}{8\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-}^{\alpha }ϝ\left(\sigma ,\frac{2\varsigma +d}{3}\right)-\frac{9}{8\left(d-\varsigma \right)}\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-}^{\alpha }ϝ\left(\sigma ,\varsigma \right)\hfill \\ \hfill & -\frac{15}{8\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{2\varsigma +d}{3}-}^{\beta }ϝ\left(\frac{2\sigma +\rho }{3},\varsigma \right)-\frac{9}{8\left(\rho -\sigma \right)}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{2\varsigma +d}{3}-}^{\beta }ϝ\left(\sigma ,\varsigma \right)\hfill \\ \hfill & +\frac{3^{\alpha +1}\mathsf{\Gamma }\left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha +1}}\frac{3^{\beta +1}\mathsf{\Gamma }\left(\beta +1\right)}{{\left(d-\varsigma \right)}^{\beta +1}}{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-,\frac{2\varsigma +d}{3}-}^{\alpha ,\beta }ϝ\left(\sigma ,\varsigma \right).\hfill \end{array}$$

(15)

Thus, we obtain the required identity by adding (7)–(15) and multiplying the resultant one by $\frac{\left(\rho -\sigma \right)\left(d-\varsigma \right)}{81}$. □

4. Fractional Newton-Type Inequalities for Coordinated Convex Functions

In this section, we will present fractional Newton-type inequalities via differentiable coordinated convex mapping.

Theorem 2.

Let the conditions of Lemma 3 be satisfied. If $\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\right|$ is a coordinated convex function, then we obtain the following inequality:

$$\begin{array}{cc}\hfill \left|{\mathsf{\Theta }}^{\alpha ,\beta }\left(\sigma ,\rho ;\varsigma ,d\right)\right|& \le \frac{\left(\rho -\sigma \right)\left(d-\varsigma \right)}{729}\left[\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|\left[{\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)+3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right]\right.\hfill \\ \hfill & \times \left[{\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)+2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)+3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right]\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|\left[{\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)+3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right]\hfill \\ \hfill & \times \left[2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)+{\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)+{\mathsf{\Omega }}_1\left(\beta \right)\right]\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|\left[2{\mathsf{\Omega }}_6\left(\alpha \right)+{\mathsf{\Omega }}_5\left(\alpha \right)+{\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)+{\mathsf{\Omega }}_1\left(\alpha \right)\right]\hfill \\ \hfill & \times \left[{\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)+2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)+3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right]\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|\left[2{\mathsf{\Omega }}_6\left(\alpha \right)+{\mathsf{\Omega }}_5\left(\alpha \right)+{\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)+{\mathsf{\Omega }}_1\left(\alpha \right)\right]\hfill \\ \hfill & \times \left.\left[2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)+{\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)+{\mathsf{\Omega }}_1\left(\beta \right)\right]\right],\hfill \end{array}$$

(16)

where ${\mathsf{\Omega }}_{\mathsf{\Phi }},\mathsf{\Phi }=1,2,\dots ,6$ are described in (2) and ${\mathsf{\Theta }}^{\alpha ,\beta }\left(\sigma ,\rho ;\varsigma ,d\right)$ is defined in (6).

Proof. 

Taking the modulus of (5), we obtain

$$\left|{\mathsf{\Theta }}^{\alpha ,\beta }\left(\sigma ,\rho ;\varsigma ,d\right)\right|\le \frac{\left(\rho -\sigma \right)\left(d-\varsigma \right)}{81}\sum _{k=1}^9\left|{\mathsf{\Phi }}_k\right|.$$

With the help of the coordinated convexity of $\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\right|$, we possess

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_1\right|& \le {\int }_0^1{\int }_0^1\left|{\gamma }^{\alpha }-\frac{5}{8}\right|\left|{\delta }^{\beta }-\frac{5}{8}\right|\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{1-\gamma }{3}\right)\sigma +\left(\frac{\gamma +2}{3}\right)\rho ,\left(\frac{1-\delta }{3}\right)\varsigma +\left(\frac{\delta +2}{3}\right)d\right)\right|d\delta d\gamma \hfill \\ \hfill & \le {\int }_0^1{\int }_0^1\left|{\gamma }^{\alpha }-\frac{5}{8}\right|\left|{\delta }^{\beta }-\frac{5}{8}\right|\left[\left(\frac{1-\gamma }{3}\right)\left(\frac{1-\delta }{3}\right)\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|+\left(\frac{1-\gamma }{3}\right)\left(\frac{\delta +2}{3}\right)\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|\right.\hfill \\ \hfill & +\left.\left(\frac{\gamma +2}{3}\right)\left(\frac{1-\delta }{3}\right)\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|+\left(\frac{\gamma +2}{3}\right)\left(\frac{\delta +2}{3}\right)\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|\right]d\delta d\gamma \hfill \\ \hfill & =\frac{1}{9}\left[\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)\right.\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)\hfill \\ \hfill & +\left.\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)\right].\hfill \end{array}$$

(17)

Similarly, we derive

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_2\right|& \le \frac{1}{9}\left[\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)\right)\right.\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)\right)\hfill \\ \hfill & +\left.\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right)\left({\mathsf{\Omega }}_3\left(\beta \right)-{\mathsf{\Omega }}_4\left(\beta \right)\right)\right],\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_3\right|& \le \frac{1}{9}\left[\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)\right.\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right){\mathsf{\Omega }}_1\left(\beta \right)\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)\hfill \\ \hfill & +\left.\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right){\mathsf{\Omega }}_1\left(\beta \right)\right],\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_4\right|& \le \frac{1}{9}\left[\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)\right.\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|\left({\mathsf{\Omega }}_4\left(\alpha \right)+{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)\hfill \\ \hfill & +\left.\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|\left({\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)\right],\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_5\right|& \le \frac{1}{9}\left[\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)\right)\right.\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|\left({\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)\right)\hfill \\ \hfill & +\left.\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|\left({\mathsf{\Omega }}_4\left(\alpha \right)+{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)\right],\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_6\right|& \le \frac{1}{9}\left[\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)\right.\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right){\mathsf{\Omega }}_1\left(\beta \right)\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|\left({\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)\right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)\hfill \\ \hfill & +\left.\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|\left({\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)\right){\mathsf{\Omega }}_1\left(\beta \right)\right],\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_7\right|& \le \frac{1}{9}\left[\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)\right.\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|{\mathsf{\Omega }}_1\left(\alpha \right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)\hfill \\ \hfill & +\left.\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|{\mathsf{\Omega }}_1\left(\alpha \right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)\right],\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_8\right|& \le \frac{1}{9}\left[\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)\right)\right.\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|{\mathsf{\Omega }}_1\left(\alpha \right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)\hfill \\ \hfill & +\left.\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|{\mathsf{\Omega }}_1\left(\alpha \right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)\right]\hfill \end{array}$$

(24)

and

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_9\right|& \le \frac{1}{9}\left[\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)\right.\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right){\mathsf{\Omega }}_1\left(\beta \right)\hfill \\ \hfill & +\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|{\mathsf{\Omega }}_1\left(\alpha \right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)\hfill \\ \hfill & +\left.\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|{\mathsf{\Omega }}_1\left(\alpha \right){\mathsf{\Omega }}_1\left(\beta \right)\right].\hfill \end{array}$$

(25)

Thus, we establish the desired inequality by adding (17) to (25) and then multiplying by $\frac{\left(\rho -\sigma \right)\left(d-\varsigma \right)}{81}$. □

Example 1.

Define a mapping $ϝ:\left[0,1\right]\times \left[0,1\right]\to \mathbb{R}$ by $ϝ\left(\gamma ,\delta \right)={\gamma }^2{\delta }^2$. The right-hand side of the inequality (16) reduces the following equality

$$\begin{array}{cc}\hfill \frac{\left(\rho -\sigma \right)\left(d-\varsigma \right)}{729}& \left[\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|\left[{\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)+3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right]\right.\hfill \\ \hfill & \times \left[{\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)+2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)+3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right]\hfill \\ \hfill & \times \left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|\left[{\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)+3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right]\hfill \\ \hfill & \times \left[2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)+{\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)+{\mathsf{\Omega }}_1\left(\beta \right)\right]\hfill \\ \hfill & \times \left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|\left[2{\mathsf{\Omega }}_6\left(\alpha \right)+{\mathsf{\Omega }}_5\left(\alpha \right)+{\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)+{\mathsf{\Omega }}_1\left(\alpha \right)\right]\hfill \\ \hfill & \times \left[{\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)+2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)+3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right]\hfill \\ \hfill & \times \left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|\left[2{\mathsf{\Omega }}_6\left(\alpha \right)+{\mathsf{\Omega }}_5\left(\alpha \right)+{\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)+{\mathsf{\Omega }}_1\left(\alpha \right)\right]\hfill \\ \hfill & \times \left.\left[2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)+{\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)+{\mathsf{\Omega }}_1\left(\beta \right)\right]\right]\hfill \\ \hfill & =\frac{4}{729}\left[2{\mathsf{\Omega }}_6\left(\alpha \right)+{\mathsf{\Omega }}_5\left(\alpha \right)+{\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)+{\mathsf{\Omega }}_1\left(\alpha \right)\right]\hfill \\ \hfill & \times \left.\left[2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)+{\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)+{\mathsf{\Omega }}_1\left(\beta \right)\right]\right]:={\mathsf{\Psi }}_1.\hfill \end{array}$$

(26)

Then, we obtain the following expressions

$$\frac{\left[ϝ\left(0,0\right)+ϝ\left(0,1\right)+ϝ\left(1,0\right)+ϝ\left(1,1\right)\right]}{64}=\frac{1}{64},$$

(27)

$$\begin{array}{cc}\hfill \frac{3}{64}& \left[ϝ\left(\sigma ,\frac{2\varsigma +d}{3}\right)+ϝ\left(\sigma ,\frac{\varsigma +2d}{3}\right)+ϝ\left(\rho ,\frac{2\varsigma +d}{3}\right)+ϝ\left(\rho ,\frac{\varsigma +2d}{3}\right)\right.\hfill \\ \hfill & +\left.ϝ\left(\frac{2\sigma +\rho }{3},\varsigma \right)+ϝ\left(\frac{\sigma +2\rho }{3},\varsigma \right)+\left(\frac{2\sigma +\rho }{3},d\right)+ϝ\left(\frac{\sigma +2\rho }{3},d\right)\right]\hfill \\ \hfill & =\frac{3}{64}·\frac{10}{9}=\frac{5}{96}\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}\hfill \frac{9}{64}& \left[ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)+ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)+ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)+ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)\right]\hfill \\ \hfill & =\frac{25}{576}.\hfill \end{array}$$

(29)

By utilizing the definition of Riemann–Liouville fractional integrals, we derive

$$\begin{array}{cc}\hfill -\frac{3^{\alpha -1}\mathsf{\Gamma }\left(\alpha +1\right)}{8{\left(\rho -\sigma \right)}^{\alpha }}& \left[{\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\frac{\sigma +2\rho }{3},d\right)+{\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\frac{\sigma +2\rho }{3},\varsigma \right)+{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-}^{\alpha }ϝ\left(\frac{2\sigma +\rho }{3},d\right)\right.\hfill \\ \hfill & +\left.{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-}^{\alpha }ϝ\left(\frac{2\sigma +\rho }{3},\varsigma \right)+{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-}^{\alpha }ϝ\left(\sigma ,d\right)+{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-}^{\alpha }ϝ\left(\sigma ,\varsigma \right)\right]\hfill \\ \hfill & =-\frac{3^{\alpha -1}\mathsf{\Gamma }\left(\alpha +1\right)}{8}\left[{\mathcal{J}}_{1-}^{\alpha }ϝ\left(\frac{2}{3},1\right)+{\mathcal{J}}_{\frac{2}{3}-}^{\alpha }ϝ\left(\frac{1}{3},1\right)+{\mathcal{J}}_{\frac{1}{3}-}^{\alpha }ϝ\left(0,1\right)\right]\hfill \\ \hfill & =-\frac{3^{\alpha -1}\alpha }{8}\left[{\int }_{\frac{2}{3}}^1{\left(\gamma -\frac{2}{3}\right)}^{\alpha -1}{\gamma }^{2}d\gamma +{\int }_{\frac{1}{3}}^{\frac{2}{3}}{\left(\gamma -\frac{1}{3}\right)}^{\alpha -1}{\gamma }^{2}d\gamma +{\int }_0^{\frac{1}{3}}{\gamma }^{\alpha +1}d\gamma \right]\hfill \\ \hfill & =-\frac{3^{\alpha -1}}{8}\left[\frac{3^{-\alpha -2}\left(9{\alpha }^2+21\alpha +8\right)}{\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{3^{-\alpha -2}\left(4{\alpha }^2+8\alpha +2\right)}{\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{3^{-\alpha -2}\left({\alpha }^2+\alpha \right)}{\left(\alpha +1\right)\left(\alpha +2\right)}\right]\hfill \\ \hfill & =-\frac{1}{108}\left(\frac{7{\alpha }^2+15\alpha +5}{\left(\alpha +1\right)\left(\alpha +2\right)}\right),\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill -\frac{3^{\alpha }\mathsf{\Gamma }\left(\alpha +1\right)}{8{\left(\rho -\sigma \right)}^{\alpha }}& \left[{\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\rho -}^{\alpha }ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-}^{\alpha }ϝ\left(\sigma ,\frac{\varsigma +2d}{3}\right)\right.\hfill \\ \hfill & +\left.{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-}^{\alpha }ϝ\left(\sigma ,\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-}^{\alpha }ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-}^{\alpha }ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)\right]\hfill \\ \hfill & =-\frac{3^{\alpha }\mathsf{\Gamma }\left(\alpha +1\right)}{8{\left(\rho -\sigma \right)}^{\alpha }}\left[{\mathcal{J}}_{1-}^{\alpha }ϝ\left(\frac{2}{3},\frac{2}{3}\right)+{\mathcal{J}}_{1-}^{\alpha }ϝ\left(\frac{2}{3},\frac{1}{3}\right)+{\mathcal{J}}_{\frac{1}{3}-}^{\alpha }ϝ\left(0,\frac{2}{3}\right)\right.\hfill \\ \hfill & +\left.{\mathcal{J}}_{\frac{1}{3}-}^{\alpha }ϝ\left(0,\frac{1}{3}\right)+{\mathcal{J}}_{\frac{2}{3}-}^{\alpha }ϝ\left(\frac{1}{3},\frac{2}{3}\right)+{\mathcal{J}}_{\frac{2}{3}-}^{\alpha }ϝ\left(\frac{1}{3},\frac{1}{3}\right)\right]\hfill \\ \hfill & =-\frac{3^{\alpha }\alpha }{72}\left[4\underset{\frac{2}{3}}{\overset{1}{\int }}{\left(\gamma -\frac{2}{3}\right)}^{\alpha -1}{\gamma }^{2}d\gamma +\underset{\frac{2}{3}}{\overset{1}{\int }}{\left(\gamma -\frac{2}{3}\right)}^{\alpha -1}{\gamma }^{2}d\gamma +4\underset{0}{\overset{\frac{1}{3}}{\int }}{\gamma }^{\alpha +1}d\gamma \right.\hfill \\ \hfill & +\left.\underset{0}{\overset{\frac{1}{3}}{\int }}{\gamma }^{\alpha +1}d\gamma +4\underset{\frac{1}{3}}{\overset{\frac{2}{3}}{\int }}{\left(\gamma -\frac{1}{3}\right)}^{\alpha -1}{\gamma }^{2}d\gamma +\underset{\frac{1}{3}}{\overset{\frac{2}{3}}{\int }}{\left(\gamma -\frac{1}{3}\right)}^{\alpha -1}{\gamma }^{2}d\gamma \right]\hfill \\ \hfill & =-\frac{5}{648}\left[\frac{9{\alpha }^2+21\alpha +8}{\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{{\alpha }^2+\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{4{\alpha }^2+8\alpha +2}{\left(\alpha +1\right)\left(\alpha +2\right)}\right]\hfill \\ \hfill & =-\frac{5}{324}\left[\frac{7{\alpha }^2+15\alpha +5}{\left(\alpha +1\right)\left(\alpha +2\right)}\right],\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill -\frac{3^{\beta -1}\mathsf{\Gamma }\left(\beta +1\right)}{8{\left(d-\varsigma \right)}^{\beta }}& \left[{\mathcal{J}}_{d-}^{\beta }ϝ\left(\rho ,\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\frac{\varsigma +2d}{3}-}^{\beta }ϝ\left(\rho ,\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{2\varsigma +d}{3}-}^{\beta }ϝ\left(\rho ,\varsigma \right)\right.\hfill \\ \hfill & +\left.{\mathcal{J}}_{d-}^{\beta }ϝ\left(\sigma ,\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\frac{\varsigma +2d}{3}-}^{\beta }ϝ\left(\sigma ,\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{2\varsigma +d}{3}-}^{\beta }ϝ\left(\sigma ,\varsigma \right)\right]\hfill \\ \hfill & -\frac{3^{\beta -1}\mathsf{\Gamma }\left(\beta +1\right)}{8{\left(d-\varsigma \right)}^{\beta }}\left[{\mathcal{J}}_{1-}^{\beta }ϝ\left(1,\frac{2}{3}\right)+{\mathcal{J}}_{\frac{2}{3}-}^{\beta }ϝ\left(1,\frac{1}{3}\right)+{\mathcal{J}}_{\frac{1}{3}-}^{\beta }ϝ\left(1,0\right)\right]\hfill \\ \hfill & =-\frac{1}{108}\left(\frac{{\beta }^2+15\beta +5}{\left(\beta +1\right)\left(\beta +2\right)}\right)\hfill \end{array}$$

(32)

and

$$\begin{array}{cc}\hfill -\frac{3^{\beta }\mathsf{\Gamma }\left(\beta +1\right)}{8{\left(d-\varsigma \right)}^{\beta }}& \left[{\mathcal{J}}_{d-}^{\beta }ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\frac{\varsigma +2d}{3}-}^{\beta }ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{2\varsigma +d}{3}-}^{\beta }ϝ\left(\frac{\sigma +2\rho }{3},\varsigma \right)\right.\hfill \\ \hfill & +\left.{\mathcal{J}}_{d-}^{\beta }ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\frac{\varsigma +2d}{3}-}^{\beta }ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{2\varsigma +d}{3}-}^{\beta }ϝ\left(\frac{2\sigma +\rho }{3},\varsigma \right)\right]\hfill \\ \hfill & =-\frac{5}{324}\left[\frac{7{\beta }^2+15\beta +5}{\left(\beta +1\right)\left(\beta +2\right)}\right].\hfill \end{array}$$

(33)

With the help of the definition of double Riemann–Liouville fractional integrals, we possess

$$\begin{array}{cc}\hfill & \frac{3^{\alpha -1}{3}^{\beta -1}\mathsf{\Gamma }\left(\alpha +1\right)\mathsf{\Gamma }\left(\beta +1\right)}{{\left(\rho -\sigma \right)}^{\alpha }{\left(d-\varsigma \right)}^{\beta }}\hfill \\ \hfill & \times \left[{\mathcal{J}}_{\rho -,d-}^{\alpha ,\beta }ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\rho -,\frac{\varsigma +2d}{3}-}^{\alpha ,\beta }ϝ\left(\frac{\sigma +2\rho }{3},\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\rho -,\frac{2\varsigma +d}{3}-}^{\alpha ,\beta }ϝ\left(\frac{\sigma +2\rho }{3},\varsigma \right)\right.\hfill \\ \hfill & +{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-,d-}^{\alpha ,\beta }ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-,\frac{\varsigma +2d}{3}-}^{\alpha ,\beta }ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{\sigma +2\rho }{3}-,\frac{2\varsigma +d}{3}-}^{\alpha ,\beta }ϝ\left(\frac{2\sigma +\rho }{3},\varsigma \right)\hfill \\ \hfill & +\left.{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-,d-}^{\alpha ,\beta }ϝ\left(\sigma ,\frac{\varsigma +2d}{3}\right)+{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-,\frac{\varsigma +2d}{3}-}^{\alpha ,\beta }ϝ\left(\sigma ,\frac{2\varsigma +d}{3}\right)+{\mathcal{J}}_{\frac{2\sigma +\rho }{3}-,\frac{2\varsigma +d}{3}-}^{\alpha ,\beta }ϝ\left(\sigma ,\varsigma \right)\right]\hfill \\ \hfill & =3^{\alpha -1}{3}^{\beta -1}\mathsf{\Gamma }\left(\alpha +1\right)\mathsf{\Gamma }\left(\beta +1\right)\hfill \\ \hfill & \times \left[{\mathcal{J}}_{1-,1-}^{\alpha ,\beta }ϝ\left(\frac{2}{3},\frac{2}{3}\right)+{\mathcal{J}}_{1-,\frac{2}{3}-}^{\alpha ,\beta }ϝ\left(\frac{2}{3},\frac{1}{3}\right)+{\mathcal{J}}_{1-,\frac{1}{3}-}^{\alpha ,\beta }ϝ\left(\frac{2}{3},0\right)+{\mathcal{J}}_{\frac{2}{3}-,1-}^{\alpha ,\beta }ϝ\left(\frac{1}{3},\frac{2}{3}\right)\right.\hfill \\ \hfill & +\left.{\mathcal{J}}_{\frac{2}{3}-,\frac{2}{3}-}^{\alpha ,\beta }ϝ\left(\frac{1}{3},\frac{1}{3}\right)+{\mathcal{J}}_{\frac{2}{3}-,\frac{1}{3}-}^{\alpha ,\beta }ϝ\left(\frac{1}{3},0\right)+{\mathcal{J}}_{\frac{1}{3}-,1-}^{\alpha ,\beta }ϝ\left(0,\frac{2}{3}\right)+{\mathcal{J}}_{\frac{1}{3}-,\frac{2}{3}-}^{\alpha ,\beta }ϝ\left(0,\frac{1}{3}\right)+{\mathcal{J}}_{\frac{1}{3}-,\frac{1}{3}-}^{\alpha ,\beta }ϝ\left(0,0\right)\right]\hfill \\ \hfill & =3^{\alpha -1}{3}^{\beta -1}\alpha \beta \left[\underset{\frac{2}{3}}{\overset{1}{\int }}\underset{\frac{2}{3}}{\overset{1}{\int }}{\left(\gamma -\frac{2}{3}\right)}^{\alpha -1}{\left(\delta -\frac{2}{3}\right)}^{\beta -1}{\gamma }^2{\delta }^{2}d\delta d\gamma +\underset{\frac{2}{3}}{\overset{1}{\int }}\underset{\frac{1}{3}}{\overset{\frac{2}{3}}{\int }}{\left(\gamma -\frac{2}{3}\right)}^{\alpha -1}{\left(\delta -\frac{1}{3}\right)}^{\beta -1}{\gamma }^2{\delta }^{2}d\delta d\gamma \right.\hfill \\ \hfill & +\underset{\frac{2}{3}}{\overset{1}{\int }}\underset{0}{\overset{\frac{1}{3}}{\int }}{\left(\gamma -\frac{2}{3}\right)}^{\alpha -1}{\left(\delta \right)}^{\beta -1}{\gamma }^2{\delta }^{2}d\delta d\gamma +\underset{\frac{1}{3}}{\overset{\frac{2}{3}}{\int }}\underset{\frac{2}{3}}{\overset{1}{\int }}{\left(\gamma -\frac{1}{3}\right)}^{\alpha -1}{\left(\delta -\frac{2}{3}\right)}^{\beta -1}{\gamma }^2{\delta }^{2}d\delta d\gamma \hfill \\ \hfill & +\underset{\frac{1}{3}}{\overset{\frac{2}{3}}{\int }}\underset{\frac{1}{3}}{\overset{\frac{2}{3}}{\int }}{\left(\gamma -\frac{1}{3}\right)}^{\alpha -1}{\left(\delta -\frac{1}{3}\right)}^{\beta -1}{\gamma }^2{\delta }^{2}d\delta d\gamma +\underset{\frac{1}{3}}{\overset{\frac{2}{3}}{\int }}\underset{0}{\overset{\frac{1}{3}}{\int }}{\left(\gamma -\frac{1}{3}\right)}^{\alpha -1}{\left(\delta \right)}^{\beta -1}{\gamma }^2{\delta }^{2}d\delta d\gamma \hfill \\ \hfill & +\underset{0}{\overset{\frac{1}{3}}{\int }}\underset{\frac{2}{3}}{\overset{1}{\int }}{\left(\gamma \right)}^{\alpha -1}{\left(\delta -\frac{2}{3}\right)}^{\beta -1}{\gamma }^2{\delta }^{2}d\delta d\gamma +\underset{0}{\overset{\frac{1}{3}}{\int }}\underset{\frac{1}{3}}{\overset{\frac{2}{3}}{\int }}{\left(\gamma \right)}^{\alpha -1}{\left(\delta -\frac{1}{3}\right)}^{\beta -1}{\gamma }^2{\delta }^{2}d\delta d\gamma +\underset{0}{\overset{\frac{1}{3}}{\int }}\underset{0}{\overset{\frac{1}{3}}{\int }}{\left(\gamma \right)}^{\alpha -1}{\left(\delta \right)}^{\beta -1}{\gamma }^2{\delta }^{2}d\delta d\gamma \hfill \\ \hfill & =\frac{4\left(7{\alpha }^2+15\alpha +5\right)\left(7{\beta }^2+15\beta +5\right)}{729\left(\alpha +1\right)\left(\alpha +2\right)\left(\beta +1\right)\left(\beta +2\right)}.\hfill \end{array}$$

(34)

If we add the expressions (27)–(34) and we have the left-hand side of (16),

$$\begin{array}{cc}\hfill \left|{\mathsf{\Theta }}^{\alpha ,\beta }\left(0,1;0,1\right)\right|& =\left|\frac{1}{9}-\frac{2}{81}\left[\left(\frac{7{\alpha }^2+15\alpha +5}{\left(\alpha +1\right)\left(\alpha +2\right)}\right)+\left(\frac{7{\beta }^2+15\beta +5}{\left(\beta +1\right)\left(\beta +2\right)}\right)\right]\right.\hfill \\ \hfill & \left.+\frac{4\left(7{\alpha }^2+15\alpha +5\right)\left(7{\beta }^2+15\beta +5\right)}{729\left(\alpha +1\right)\left(\alpha +2\right)\left(\beta +1\right)\left(\beta +2\right)}\right|:={\mathsf{\Psi }}_2.\hfill \end{array}$$

(35)

If we substitute (35) and (26) in (16), we derive

$$\begin{array}{cc}\hfill \left|{\mathsf{\Theta }}^{\alpha ,\beta }\left(0,1;0,1\right)\right|& \le \frac{1}{729}\left[2{\mathsf{\Omega }}_6\left(\alpha \right)+{\mathsf{\Omega }}_5\left(\alpha \right)+{\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)+{\mathsf{\Omega }}_1\left(\alpha \right)\right]\hfill \\ \hfill & \times \left.\left[2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)+{\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)+{\mathsf{\Omega }}_1\left(\beta \right)\right]\right].\hfill \end{array}$$

If we demonstrate Example 1 on the graph as Figure 1:

Remark 1.

If we take $\alpha =\beta =1$ in Theorem 2, then we obtain

$$\begin{array}{cc}\hfill & \left|{\mathsf{\Theta }}^{\alpha ,\beta }\left(\sigma ,\rho ;\varsigma ,d\right)\right|\hfill \\ \hfill & \le \frac{\left(\rho -\sigma \right)\left(d-\varsigma \right)}{729}\left[\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|\left[{\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)+3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right]\right.\hfill \\ \hfill & \times \left[{\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)+2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)+3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right]\hfill \\ \hfill & \times \left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|\left[{\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)+3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right]\hfill \\ \hfill & \times \left[2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)+{\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)+{\mathsf{\Omega }}_1\left(\beta \right)\right]\hfill \\ \hfill & \times \left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|\left[2{\mathsf{\Omega }}_6\left(\alpha \right)+{\mathsf{\Omega }}_5\left(\alpha \right)+{\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)+{\mathsf{\Omega }}_1\left(\alpha \right)\right]\hfill \\ \hfill & \times \left[{\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)+2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)+3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right]\hfill \\ \hfill & \times \left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|\left[2{\mathsf{\Omega }}_6\left(\alpha \right)+{\mathsf{\Omega }}_5\left(\alpha \right)+{\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)+{\mathsf{\Omega }}_1\left(\alpha \right)\right]\hfill \\ \hfill & \times \left.\left[2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)+{\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)+{\mathsf{\Omega }}_1\left(\beta \right)\right]\right],\hfill \end{array}$$

(36)

and

$$\begin{array}{cc}\hfill & \left|{\rm Y}\left(\sigma ,\rho ;\varsigma ,d\right)\right|\hfill \\ \hfill & \le \left(\rho -\sigma \right)\left(d-\varsigma \right)\frac{625}{331776}\left[\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|+\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|+\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|+\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|\right],\hfill \end{array}$$

which is given by Iftikhar et al. in [13] and ${\rm Y}\left(\sigma ,\rho ;\varsigma ,d\right)$ is described by

$$\begin{array}{cc}\hfill & {\rm Y}\left(\sigma ,\rho ;\varsigma ,d\right)\hfill \\ \hfill & :=\frac{\left[ϝ\left(\sigma ,\varsigma \right)+ϝ\left(\sigma ,d\right)+ϝ\left(\rho ,\varsigma \right)+ϝ\left(\rho ,d\right)\right]}{64}\hfill \\ \hfill & +\frac{3}{64}\left[ϝ\left(\sigma ,\frac{2\varsigma +d}{3}\right)+ϝ\left(\sigma ,\frac{\varsigma +2d}{3}\right)+ϝ\left(\rho ,\frac{2\varsigma +d}{3}\right)+ϝ\left(\rho ,\frac{\varsigma +2d}{3}\right)\right.\hfill \\ \hfill & +\left.ϝ\left(\frac{2\sigma +\rho }{3},\varsigma \right)+ϝ\left(\frac{\sigma +2\rho }{3},\varsigma \right)+\left(\frac{2\sigma +\rho }{3},d\right)+ϝ\left(\frac{\sigma +2\rho }{3},d\right)\right]\hfill \\ \hfill & +\frac{9}{64}\left[ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)+ϝ\left(\frac{2\sigma +\rho }{3},\frac{\varsigma +2d}{3}\right)\right.\hfill \\ \hfill & \left.+ϝ\left(\frac{2\sigma +\rho }{3},\frac{2\varsigma +d}{3}\right)+ϝ\left(\frac{\sigma +2\rho }{3},\frac{\varsigma +2d}{3}\right)\right]\hfill \\ \hfill & -\frac{1}{8\left(\rho -\sigma \right)}\left[\underset{\sigma }{\overset{\rho }{\int }}ϝ\left(\kappa ,\varsigma \right)d\kappa +\underset{\sigma }{\overset{\rho }{\int }}ϝ\left(\kappa ,d\right)d\kappa +3\underset{\sigma }{\overset{\rho }{\int }}ϝ\left(\kappa ,\frac{2\varsigma +d}{3}\right)d\kappa +3\underset{\sigma }{\overset{\rho }{\int }}ϝ\left(\kappa ,\frac{\varsigma +2d}{3}\right)d\kappa \right]\hfill \\ \hfill & -\frac{1}{8\left(d-\varsigma \right)}\left[\underset{\varsigma }{\overset{d}{\int }}ϝ\left(\sigma ,y\right)dy+\underset{\varsigma }{\overset{d}{\int }}ϝ\left(\rho ,y\right)dy+3\underset{\varsigma }{\overset{d}{\int }}ϝ\left(\frac{2\sigma +\rho }{3},y\right)dy+3\underset{\varsigma }{\overset{d}{\int }}ϝ\left(\frac{\sigma +2\rho }{3},y\right)dy\right]\hfill \\ \hfill & +\frac{1}{\left(\rho -\sigma \right)\left(d-\varsigma \right)}\underset{\sigma }{\overset{\rho }{\int }}\underset{\varsigma }{\overset{d}{\int }}ϝ\left(\kappa ,y\right)dyd\kappa .\hfill \end{array}$$

(37)

Theorem 3.

Let the conditions of Lemma 3 be satisfied. If ${\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\right|}^q$,$q>1$ and $p^{-1}+q^{-1}=1$ is a convex function on coordinates, then we establish the following Newton’s type inequality:

$$\begin{array}{cc}\hfill & \left|{\mathsf{\Theta }}^{\alpha ,\beta }\left(\sigma ,\rho ;\varsigma ,d\right)\right|\hfill \\ \hfill & \le \frac{\left(\rho -\sigma \right)\left(d-\varsigma \right)}{81}\hfill \\ \hfill & \times \left[{\mathsf{\Omega }}_9^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_9^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+25{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{36}\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +{\mathsf{\Omega }}_9^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_8^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{12}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\mathsf{\Omega }}_9^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_7^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+25{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{36}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\mathsf{\Omega }}_8^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_9^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{12}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\mathsf{\Omega }}_8^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_8^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{4}\right)}^{\frac{1}{q}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\mathsf{\Omega }}_8^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_7^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{12}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\mathsf{\Omega }}_7^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_9^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+25{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{36}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\mathsf{\Omega }}_7^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_8^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{12}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\mathsf{\Omega }}_7^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_7^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{25{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{36}\right)}^{\frac{1}{q}},\hfill \end{array}$$

where $q^{-1}+p^{-1}=1$, ${\mathsf{\Omega }}_{\mathsf{\Phi }},\mathsf{\Phi }=1,2,\dots ,6$ are as in (2), ${\mathsf{\Theta }}^{\alpha ,\beta }\left(\sigma ,\rho ;\varsigma ,d\right)$ is defined by (6) and

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_7\left(\upsilon ,p\right)& ={\int }_0^1{\left|{\tau }^{\upsilon }-\frac{3}{8}\right|}^{p}d\tau ,\hfill \\ \hfill {\mathsf{\Omega }}_8\left(\upsilon ,p\right)& ={\int }_0^1{\left|{\tau }^{\upsilon }-\frac{1}{2}\right|}^{p}d\tau ,\hfill \\ \hfill {\mathsf{\Omega }}_9\left(\upsilon ,p\right)& ={\int }_0^1{\left|{\tau }^{\upsilon }-\frac{5}{8}\right|}^{p}d\tau .\hfill \end{array}$$

(38)

Proof. 

With the help of the Hölder inequality and coordinated convexity of ${\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\gamma ,\delta \right)\right|}^q$, the following inequalities hold:

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_1\right|& \le {\int }_0^1{\int }_0^1\left|{\gamma }^{\alpha }-\frac{5}{8}\right|\left|{\delta }^{\beta }-\frac{5}{8}\right|\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{1-\gamma }{3}\right)\sigma +\left(\frac{\gamma +2}{3}\right)\rho ,\left(\frac{1-\delta }{3}\right)\varsigma +\left(\frac{\delta +2}{3}\right)d\right)\right|d\delta d\gamma \hfill \\ \hfill & \le {\left({\int }_0^1{\int }_0^1{\left|{\gamma }^{\alpha }-\frac{5}{8}\right|}^p{\left|{\delta }^{\beta }-\frac{5}{8}\right|}^p\right)}^{\frac{1}{p}}\hfill \\ \hfill & \times {\left({\int }_0^1{\int }_0^1{\left|\frac{{\partial }^2}{\partial \delta \partial \gamma }ϝ\left(\left(\frac{1-\gamma }{3}\right)\sigma +\left(\frac{\gamma +2}{3}\right)\rho ,\left(\frac{1-\delta }{3}\right)\varsigma +\left(\frac{\delta +2}{3}\right)d\right)\right|}^{q}d\delta d\gamma \right)}^{\frac{1}{q}}\hfill \\ \hfill & \le {\left({\int }_0^1{\int }_0^1{\left|{\gamma }^{\alpha }-\frac{5}{8}\right|}^p{\left|{\delta }^{\beta }-\frac{5}{8}\right|}^p\right)}^{\frac{1}{p}}\hfill \\ \hfill & \times \left({\int }_0^1{\int }_0^1\left[\left(\frac{1-\gamma }{3}\right)\left(\frac{1-\delta }{3}\right){\left|\frac{{\partial }^2}{\partial \delta \partial \gamma }ϝ\left(\sigma ,\varsigma \right)\right|}^q+\left(\frac{1-\gamma }{3}\right)\left(\frac{\delta +2}{3}\right){\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\right.\right.\hfill \\ \hfill & +{\left.{\left.\left(\frac{\gamma +2}{3}\right)\left(\frac{1-\delta }{3}\right){\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+\left(\frac{\gamma +2}{3}\right)\left(\frac{\delta +2}{3}\right){\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\right]}^{q}d\delta d\gamma \right)}^{\frac{1}{q}}\hfill \\ \hfill & ={\mathsf{\Omega }}_9^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_9^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+25{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{36}\right)}^{\frac{1}{q}}.\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_2\right|& \le {\mathsf{\Omega }}_9^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_8^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{12}\right)}^{\frac{1}{q}},\hfill \\ \hfill \left|{\mathsf{\Phi }}_3\right|& \le {\mathsf{\Omega }}_9^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_7^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+25{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{36}\right)}^{\frac{1}{q}},\hfill \\ \hfill \left|{\mathsf{\Phi }}_4\right|& \le {\int }_0^1{\mathsf{\Omega }}_8^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_9^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{12}\right)}^{\frac{1}{q}},\hfill \\ \hfill \left|{\mathsf{\Phi }}_5\right|& \le {\mathsf{\Omega }}_8^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_8^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{4}\right)}^{\frac{1}{q}},\hfill \\ \hfill \left|{\mathsf{\Phi }}_6\right|& \le {\mathsf{\Omega }}_8^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_7^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{12}\right)}^{\frac{1}{q}},\hfill \\ \hfill \left|{\mathsf{\Phi }}_7\right|& \le {\mathsf{\Omega }}_7^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_9^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+25{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{36}\right)}^{\frac{1}{q}},\hfill \\ \hfill \left|{\mathsf{\Phi }}_8\right|& \le {\mathsf{\Omega }}_7^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_8^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{12}\right)}^{\frac{1}{q}}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_9\right|& \le {\mathsf{\Omega }}_7^{\frac{1}{p}}\left(\alpha ,p\right){\mathsf{\Omega }}_7^{\frac{1}{p}}\left(\beta ,p\right){\left(\frac{25{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{36}\right)}^{\frac{1}{q}}.\hfill \end{array}$$

This proof is completed. □

Corollary 1.

If we choose $\alpha =\beta =1$ in Theorem 3, then we derive:

$$\begin{array}{cc}\hfill & \left|{\rm Y}\left(\sigma ,\rho ;\varsigma ,d\right)\right|\hfill \\ \hfill & \le \frac{\left(\rho -\sigma \right)\left(d-\varsigma \right)}{81}\hfill \\ \hfill & \times \left[{\left(\frac{5^{p+1}+3^{p+1}}{8^{p+1}\left(p+1\right)}\right)}^{\frac{2}{p}}{\left(\frac{{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+25{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{36}\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +{\left(\frac{5^{p+1}+3^{p+1}}{8^{p+1}\left(p+1\right)}\right)}^{\frac{1}{p}}{\left(\frac{1}{2^p\left(p+1\right)}\right)}^{\frac{1}{p}}{\left(\frac{{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{12}\right)}^{\frac{1}{q}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\left(\frac{5^{p+1}+3^{p+1}}{8^{p+1}\left(p+1\right)}\right)}^{\frac{2}{p}}{\left(\frac{5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+25{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{36}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{2^p\left(p+1\right)}\right)}^{\frac{2}{p}}{\left(\frac{{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{12}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{2^p\left(p+1\right)}\right)}^{\frac{1}{p}}{\left(\frac{5^{p+1}+3^{p+1}}{8^{p+1}\left(p+1\right)}\right)}^{\frac{1}{p}}{\left(\frac{{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{4}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{5^{p+1}+3^{p+1}}{8^{p+1}\left(p+1\right)}\right)}^{\frac{2}{p}}{\left(\frac{{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{12}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{5^{p+1}+3^{p+1}}{8^{p+1}\left(p+1\right)}\right)}^{\frac{1}{p}}{\left(\frac{1}{2^p\left(p+1\right)}\right)}^{\frac{1}{p}}{\left(\frac{5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{12}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{5^{p+1}+3^{p+1}}{8^{p+1}\left(p+1\right)}\right)}^{\frac{2}{p}}{\left(\frac{25{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q+5{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q}{36}\right)}^{\frac{1}{q}},\hfill \end{array}$$

where ${\rm Y}\left(\sigma ,\rho ;\varsigma ,d\right)$ is given in (37).

Theorem 4.

Let the conditions of Lemma 3 be satisfied. If ${\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\right|}^q$, $q\ge 1$, is a coordinated convex mapping, then we obtain the following Newton-type inequality:

$$\begin{array}{cc}\hfill & \left|{\mathsf{\Theta }}^{\alpha ,\beta }\left(\sigma ,\rho ;\varsigma ,d\right)\right|\hfill \\ \hfill & \le \frac{\left(\rho -\sigma \right)\left(d-\varsigma \right)}{81}\left[{\mathsf{\Omega }}_6^{1-\frac{1}{q}}\left(\alpha \right){\mathsf{\Omega }}_6^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\right.\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\mathsf{\Omega }}_6^{1-\frac{1}{q}}\left(\alpha \right){\mathsf{\Omega }}_4^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)}{9}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)}{9}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\mathsf{\Omega }}_6^{1-\frac{1}{q}}\left(\alpha \right){\mathsf{\Omega }}_2^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right){\mathsf{\Omega }}_1\left(\beta \right)}{9}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)}{9}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right){\mathsf{\Omega }}_1\left(\beta \right)}{9}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\mathsf{\Omega }}_4^{1-\frac{1}{q}}\left(\alpha \right){\mathsf{\Omega }}_6^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_4\left(\alpha \right)+{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\mathsf{\Omega }}_4^{1-\frac{1}{q}}\left(\alpha \right){\mathsf{\Omega }}_4^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)}{9}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_4\left(\alpha \right)+{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)}{9}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\mathsf{\Omega }}_4^{1-\frac{1}{q}}\left(\alpha \right).{\mathsf{\Omega }}_2^{1-\frac{1}{q}}\left(\beta \right)\left(\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|\frac{\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right){\mathsf{\Omega }}_1\left(\beta \right)}{9}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)\right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)\right){\mathsf{\Omega }}_1\left(\beta \right)}{9}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\mathsf{\Omega }}_2^{1-\frac{1}{q}}\left(\alpha \right){\mathsf{\Omega }}_6^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{{\mathsf{\Omega }}_1\left(\alpha \right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{{\mathsf{\Omega }}_1\left(\alpha \right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\mathsf{\Omega }}_2^{1-\frac{1}{q}}\left(\alpha \right){\mathsf{\Omega }}_4^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)}{9}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{{\mathsf{\Omega }}_1\left(\alpha \right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{{\mathsf{\Omega }}_1\left(\alpha \right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)}{9}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\mathsf{\Omega }}_2^{1-\frac{1}{q}}\left(\alpha \right){\mathsf{\Omega }}_4^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right){\mathsf{\Omega }}_1\left(\beta \right)}{9}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{{\mathsf{\Omega }}_1\left(\alpha \right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{{\mathsf{\Omega }}_1\left(\alpha \right){\mathsf{\Omega }}_1\left(\beta \right)}{9}\right)}^{\frac{1}{q}},\hfill \end{array}$$

where ${\mathsf{\Omega }}_{\mathsf{\Phi }}$ for $\mathsf{\Phi }=1,2\dots 6$ are shown in (2) and ${\mathsf{\Theta }}^{\alpha ,\beta }\left(\sigma ,\rho ;\varsigma ,d\right)$ is as noted in (6).

Proof. 

By taking the modulus of Lemma 3 and by using of the power mean inequality, we possess:

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_1\right|& \le {\int }_0^1{\int }_0^1\left|{\gamma }^{\alpha }-\frac{5}{8}\right|\left|{\delta }^{\beta }-\frac{5}{8}\right|\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\left(\frac{1-\gamma }{3}\right)\sigma +\left(\frac{\gamma +2}{3}\right)\rho ,\left(\frac{1-\delta }{3}\right)\varsigma +\left(\frac{\delta +2}{3}\right)d\right)\right|d\delta d\gamma \hfill \\ \hfill & \le {\left({\int }_0^1{\int }_0^1\left|{\gamma }^{\alpha }-\frac{5}{8}\right|\left|{\delta }^{\beta }-\frac{5}{8}\right|\right)}^{1-\frac{1}{q}}\hfill \\ \hfill & \times {\left({\int }_0^1{\int }_0^1\left|{\gamma }^{\alpha }-\frac{5}{8}\right|\left|{\delta }^{\beta }-\frac{5}{8}\right|{\left|\frac{{\partial }^2}{\partial \delta \partial \gamma }ϝ\left(\left(\frac{1-\gamma }{3}\right)\sigma +\left(\frac{\gamma +2}{3}\right)\rho ,\left(\frac{1-\delta }{3}\right)\varsigma +\left(\frac{\delta +2}{3}\right)d\right)\right|}^{q}d\delta d\gamma \right)}^{\frac{1}{q}}\hfill \\ \hfill & \le {\left({\int }_0^1{\int }_0^1\left|{\gamma }^{\alpha }-\frac{5}{8}\right|\left|{\delta }^{\beta }-\frac{5}{8}\right|\right)}^{1-\frac{1}{q}}\hfill \\ \hfill & \times \left({\int }_0^1{\int }_0^1\left|{\gamma }^{\alpha }-\frac{5}{8}\right|\left|{\delta }^{\beta }-\frac{5}{8}\right|\left[\left(\frac{1-\gamma }{3}\right)\left(\frac{1-\delta }{3}\right){\left|\frac{{\partial }^2}{\partial \delta \partial \gamma }ϝ\left(\sigma ,\varsigma \right)\right|}^q+\left(\frac{1-\gamma }{3}\right)\left(\frac{\delta +2}{3}\right){\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\right.\right.\hfill \\ \hfill & +{\left.\left.\left(\frac{\gamma +2}{3}\right)\left(\frac{1-\delta }{3}\right){\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q+\left(\frac{\gamma +2}{3}\right)\left(\frac{\delta +2}{3}\right){\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\right]d\delta d\gamma \right)}^{\frac{1}{q}}\hfill \\ \hfill & ={\mathsf{\Omega }}_6^{1-\frac{1}{q}}\left(\alpha \right){\mathsf{\Omega }}_6^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\hfill \\ \hfill & {\left.+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\right)}^{\frac{1}{q}}.\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_2\right|& \le {\mathsf{\Omega }}_6^{1-\frac{1}{q}}\left(\alpha \right){\mathsf{\Omega }}_4^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)}{9}\right)}^{\frac{1}{q}},\hfill \\ \hfill \left|{\mathsf{\Phi }}_3\right|& \le {\mathsf{\Omega }}_6^{1-\frac{1}{q}}\left(\alpha \right){\mathsf{\Omega }}_2^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_6\left(\alpha \right)-{\mathsf{\Omega }}_5\left(\alpha \right)\right){\mathsf{\Omega }}_1\left(\beta \right)}{9}\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_5\left(\alpha \right)+2{\mathsf{\Omega }}_6\left(\alpha \right)\right){\mathsf{\Omega }}_1\left(\beta \right)}{9}\right)}^{\frac{1}{q}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_4\right|& \le {\mathsf{\Omega }}_4^{1-\frac{1}{q}}\left(\alpha \right).{\mathsf{\Omega }}_6^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_4\left(\alpha \right)+{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\right)}^{\frac{1}{q}},\hfill \\ \hfill \left|{\mathsf{\Phi }}_5\right|& \le {\mathsf{\Omega }}_4^{1-\frac{1}{q}}\left(\alpha \right){\mathsf{\Omega }}_4^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_4\left(\alpha \right)+{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)}{9}\right)}^{\frac{1}{q}},\hfill \\ \hfill \left|{\mathsf{\Phi }}_6\right|& \le {\mathsf{\Omega }}_4^{1-\frac{1}{q}}\left(\alpha \right).{\mathsf{\Omega }}_2^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left(2{\mathsf{\Omega }}_4\left(\alpha \right)-{\mathsf{\Omega }}_3\left(\alpha \right)\right){\mathsf{\Omega }}_1\left(\beta \right)}{9}\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{\left({\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)\right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{\left({\mathsf{\Omega }}_3\left(\alpha \right)+{\mathsf{\Omega }}_4\left(\alpha \right)\right){\mathsf{\Omega }}_1\left(\beta \right)}{9}\right)}^{\frac{1}{q}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_7\right|& \le {\mathsf{\Omega }}_2^{1-\frac{1}{q}}\left(\alpha \right).{\mathsf{\Omega }}_6^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{{\mathsf{\Omega }}_1\left(\alpha \right)\left({\mathsf{\Omega }}_6\left(\beta \right)-{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{{\mathsf{\Omega }}_1\left(\alpha \right)\left(2{\mathsf{\Omega }}_6\left(\beta \right)+{\mathsf{\Omega }}_5\left(\beta \right)\right)}{9}\right)}^{\frac{1}{q}},\hfill \\ \hfill \left|{\mathsf{\Phi }}_8\right|& \le {\mathsf{\Omega }}_2^{1-\frac{1}{q}}\left(\alpha \right).{\mathsf{\Omega }}_4^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right)\left(2{\mathsf{\Omega }}_4\left(\beta \right)-{\mathsf{\Omega }}_3\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)}{9}\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{{\mathsf{\Omega }}_1\left(\alpha \right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)}{9}\hfill \\ \hfill & {\left.+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{{\mathsf{\Omega }}_1\left(\alpha \right)\left({\mathsf{\Omega }}_3\left(\beta \right)+{\mathsf{\Omega }}_4\left(\beta \right)\right)}{9}\right)}^{\frac{1}{q}},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left|{\mathsf{\Phi }}_9\right|& \le {\mathsf{\Omega }}_2^{1-\frac{1}{q}}\left(\alpha \right).{\mathsf{\Omega }}_4^{1-\frac{1}{q}}\left(\beta \right)\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)}{9}\right.\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{\left(3{\mathsf{\Omega }}_2\left(\alpha \right)-{\mathsf{\Omega }}_1\left(\alpha \right)\right){\mathsf{\Omega }}_1\left(\beta \right)}{9}\hfill \\ \hfill & +{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{{\mathsf{\Omega }}_1\left(\alpha \right)\left(3{\mathsf{\Omega }}_2\left(\beta \right)-{\mathsf{\Omega }}_1\left(\beta \right)\right)}{9}\hfill \\ \hfill & {\left.+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{{\mathsf{\Omega }}_1\left(\alpha \right){\mathsf{\Omega }}_1\left(\beta \right)}{9}\right)}^{\frac{1}{q}}.\hfill \end{array}$$

Therefore, the proof is completed. □

Corollary 2.

If we consider $\alpha =\beta =1$ in Theorem 4, then we obtain

$$\begin{array}{cc}\hfill & \left|{\rm Y}\left(\sigma ,\rho ;\varsigma ,d\right)\right|\hfill \\ \hfill & \le \frac{\left(\rho -\sigma \right)\left(d-\varsigma \right)}{6^4}\left[{\left(\frac{{17}^2}{2^8}\right)}^{1-\frac{1}{q}}\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{{251}^2}{2^{14}·3^4}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{7·139.251}{2^{14}·3^4}\right.\right.\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{7·139.251}{2^{14}·3^4}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{7^2·{139}^2}{2^{14}·3^4}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{17}{2^4}\right)}^{1-\frac{1}{q}}\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{251}{2^8·3^2}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{251}{2^8·3^2}\right.\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{7·139}{2^8·3^2}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{7·139}{2^8·3^2}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{{17}^2}{2^8}\right)}^{1-\frac{1}{q}}\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{7·139.251}{2^{14}·3^4}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{{251}^2}{2^{14}·3^4}\right.\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{7^2·{139}^2}{2^{14}·3^4}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{7·139·251}{2^{14}·3^4}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{17}{2^4}\right)}^{1-\frac{1}{q}}\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{157}{2^8·3^2}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{7·139}{2^8·3^2}\right.\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{157}{2^8·3^2}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{7·139}{2^8·3^2}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{1}{4}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{1}{4}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{1}{4}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{1}{4}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{17}{2^4}\right)}^{1-\frac{1}{q}}\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{973}{2304}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{157}{2304}\right.\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{7·139}{2^8·3^2}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{157}{2^8·3^2}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{{17}^2}{2^8}\right)}^{1-\frac{1}{q}}\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{973.251}{2^{14}·3^4}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{157.973}{2^{14}·3^4}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{{251}^2}{2^{14}·3^4}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{973.251}{2^{14}·3^4}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{17}{2^4}\right)}^{1-\frac{1}{q}}\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{973}{2304}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{973}{2304}\right.\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{251}{2304}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{251}{2304}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{17}{2^4}\right)}^{1-\frac{1}{q}}\left({\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,\varsigma \right)\right|}^q\frac{{973}^2}{2^{14}·3^4}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\sigma ,d\right)\right|}^q\frac{251.973}{2^{14}·3^4}\right.\hfill \\ \hfill & +{\left.{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,\varsigma \right)\right|}^q\frac{973.251}{2^{14}·3^4}+{\left|\frac{{\partial }^2ϝ}{\partial \delta \partial \gamma }\left(\rho ,d\right)\right|}^q\frac{{251}^2}{2^{14}·3^4}\right)}^{\frac{1}{q}},\hfill \end{array}$$

where ${\rm Y}\left(\sigma ,\rho ;\varsigma ,d\right)$ is stated as in (37).

5. Fractional Newton Inequality Based on Functions of Two Variables with Bounded Variation

In this section, with the aid of Riemann–Liouville fractional integrals, we will give Newton-type inequality via the mapping of two variables with bounded variation.

Theorem 5.

If $ϝ:\Delta \to \mathbb{R}$ is a mapping of bounded variation on $\Delta ,$ then we obtain the following inequality

$$\left|{\mathsf{\Theta }}^{\alpha ,\beta }\left(\sigma ,\rho ;\varsigma ,d\right)\right|\le \frac{25}{576}\underset{\sigma }{\overset{\rho }{\bigvee }}\underset{\varsigma }{\overset{d}{\bigvee }}\left(ϝ\right),$$

where $\underset{\sigma }{\overset{\rho }{\bigvee }}\underset{\varsigma }{\overset{d}{\bigvee }}\left(ϝ\right)$ denotes the total variation of ϝ on interval $\left[\sigma ,\rho \right]\times \left[\varsigma ,d\right].$

Proof. 

Describe the impressions $K_{\alpha }\left(\kappa \right)$ and $L_{\beta }\left(y\right)$ by

$$K_{\alpha }\left(\kappa \right)=\left\{\begin{array}{cc}{\displaystyle {\left(\kappa -\sigma \right)}^{\alpha }-\frac{{\left(\rho -\sigma \right)}^{\alpha }}{8·3^{\alpha -1}},}\hfill & \mathrm{for}\sigma \le \kappa \le \frac{2\sigma +\rho }{3};\hfill \\ {\displaystyle {\left(\kappa -\frac{2\sigma +\rho }{3}\right)}^{\alpha }-\frac{{\left(\rho -\sigma \right)}^{\alpha }}{2·3^{\alpha }},}\hfill & \mathrm{for}\frac{2\sigma +\rho }{3}<\kappa \le \frac{\sigma +2\rho }{3};\hfill \\ {\displaystyle {\left(\kappa -\frac{\sigma +2\rho }{3}\right)}^{\alpha }-\frac{5{\left(\rho -\sigma \right)}^{\alpha }}{8·3^{\alpha }}}\hfill & \mathrm{for}\frac{\sigma +2\rho }{3}<\kappa \le \rho \hfill \end{array}\right.$$

and

$$L_{\beta }\left(y\right)=\left\{\begin{array}{cc}{\left(y-\varsigma \right)}^{\beta }-\frac{{\left(d-\varsigma \right)}^{\alpha }}{8·3^{\alpha -1}},\hfill & \mathrm{for}\varsigma \le \kappa \le \frac{2\varsigma +d}{3};\hfill \\ {\left(y-\frac{2\varsigma +d}{3}\right)}^{\alpha }-\frac{{\left(d-\varsigma \right)}^{\alpha }}{2·3^{\alpha }}\hfill & \mathrm{for}\frac{2\varsigma +d}{3}<y\le \frac{\varsigma +2d}{3};\hfill \\ {\left(y-\frac{\varsigma +2d}{3}\right)}^{\alpha }-\frac{5{\left(d-\varsigma \right)}^{\alpha }}{8·3^{\alpha }}\hfill & \mathrm{for}\frac{\varsigma +2d}{3}<y\le d,\hfill \end{array}\right.$$

respectively. By Lemma 1, one can easily see that

$${\mathsf{\Theta }}^{\alpha ,\beta }\left(\sigma ,\rho ;\varsigma ,d\right)=\frac{3^{\alpha +\beta -2}}{{\left(\rho -\sigma \right)}^{\alpha }{\left(d-\varsigma \right)}^{\beta }}\underset{\sigma }{\overset{\rho }{\int }}\underset{\varsigma }{\overset{d}{\int }}{K}_{\alpha }\left(\kappa \right)L_{\beta }\left(y\right)d_{\kappa }{d}_yϝ(\kappa ,y),$$

(39)

where ${\mathsf{\Theta }}^{\alpha ,\beta }\left(\sigma ,\rho ;\varsigma ,d\right)$ is stated as in (6). Taking the modulus of (39) and with the aid of Lemma 2, we derive:

$$\begin{array}{cc}\hfill & \left|{\mathsf{\Theta }}^{\alpha ,\beta }\left(\sigma ,\rho ;\varsigma ,d\right)\right|\hfill \\ \hfill & =\frac{3^{\alpha +\beta -2}}{{\left(\rho -\sigma \right)}^{\alpha }{\left(d-\varsigma \right)}^{\beta }}\left|\underset{\sigma }{\overset{\rho }{\int }}\underset{\varsigma }{\overset{d}{\int }}{K}_{\alpha }\left(\kappa \right)L_{\beta }\left(y\right)d_{\kappa }{d}_yϝ(\kappa ,y)\right|\hfill \\ \hfill & \le \frac{3^{\alpha +\beta -2}}{{\left(\rho -\sigma \right)}^{\alpha }{\left(d-\varsigma \right)}^{\beta }}{\displaystyle \underset{(\kappa ,y)\in \Delta }{\sup }}\left|K_{\alpha }\left(\kappa \right)L_{\beta }\left(y\right)\right|\underset{\sigma }{\overset{\rho }{\bigvee }}\underset{\varsigma }{\overset{d}{\bigvee }}\left(ϝ\right)\hfill \\ \hfill & =\frac{3^{\alpha +\beta -2}}{{\left(\rho -\sigma \right)}^{\alpha }{\left(d-\varsigma \right)}^{\beta }}{\displaystyle \underset{\kappa \in \left[\sigma ,\rho \right]}{sup}}\left|K_{\alpha }\left(\kappa \right)\right|{\displaystyle \underset{y\in \left[\varsigma ,d\right]}{sup}}\left|L_{\beta }\left(y\right)\right|\underset{\sigma }{\overset{\rho }{\bigvee }}\underset{\varsigma }{\overset{d}{\bigvee }}\left(ϝ\right)\hfill \\ \hfill & =\frac{3^{\alpha +\beta -2}}{{\left(\rho -\sigma \right)}^{\alpha }{\left(d-\varsigma \right)}^{\beta }}max\left\{{\displaystyle \underset{\kappa \in \left[\sigma ,\frac{2\sigma +\rho }{3}\right]}{\sup }}\left|{\left(\kappa -\sigma \right)}^{\alpha }-\frac{{\left(\rho -\sigma \right)}^{\alpha }}{8·3^{\alpha -1}}\right|,\right.\hfill \\ \hfill & \left.{\displaystyle \underset{\kappa \in \left[\frac{2\sigma +\rho }{3},\frac{\sigma +2\rho }{3}\right]}{\sup }}\left|{\left(\kappa -\frac{2\sigma +\rho }{3}\right)}^{\alpha }-\frac{{\left(\rho -\sigma \right)}^{\alpha }}{2·3^{\alpha }}\right|,{\displaystyle \underset{\kappa \in \left[\frac{\sigma +2\rho }{3},\rho \right]}{\sup }}\left|{\left(\kappa -\frac{\sigma +2\rho }{3}\right)}^{\alpha }-\frac{5{\left(\rho -\sigma \right)}^{\alpha }}{8·3^{\alpha }}\right|\right\}\hfill \\ \hfill & \times max\left\{{\displaystyle \underset{y\in \left[\varsigma ,\frac{2\varsigma +d}{3}\right]}{\sup }}\left|{\left(y-\varsigma \right)}^{\beta }-\frac{{\left(d-\varsigma \right)}^{\alpha }}{8·3^{\alpha -1}}\right|,{\displaystyle \underset{y\in \left[\frac{2\varsigma +d}{3},\frac{\varsigma +2d}{3}\right]}{\sup }}\left|{\left(y-\frac{2\varsigma +d}{3}\right)}^{\alpha }-\frac{{\left(d-\varsigma \right)}^{\alpha }}{2·3^{\alpha }}\right|,\right\}\hfill \\ \hfill & \left.{\displaystyle \underset{y\in \left[\frac{\varsigma +2d}{3},d\right]}{\sup }}\left|{\left(y-\frac{\varsigma +2d}{3}\right)}^{\alpha }-\frac{5{\left(d-\varsigma \right)}^{\alpha }}{8·3^{\alpha }}\right|\right\}\underset{\sigma }{\overset{\rho }{\bigvee }}\underset{\varsigma }{\overset{d}{\bigvee }}\left(ϝ\right)\hfill \\ \hfill & =\frac{3^{\alpha +\beta -2}}{{\left(\rho -\sigma \right)}^{\alpha }{\left(d-\varsigma \right)}^{\beta }}max\left\{\frac{5{\left(\rho -\sigma \right)}^{\alpha }}{8·3^{\alpha }},\frac{{\left(\rho -\sigma \right)}^{\alpha }}{2·3^{\alpha }}\right\}\hfill \\ \hfill & \times max\left\{\frac{5{\left(d-\varsigma \right)}^{\alpha }}{8·3^{\alpha }},\frac{{\left(d-\varsigma \right)}^{\alpha }}{2·3^{\alpha }}\right\}\underset{\sigma }{\overset{\rho }{\bigvee }}\underset{\varsigma }{\overset{d}{\bigvee }}\left(ϝ\right)=\frac{25}{576}\underset{\sigma }{\overset{\rho }{\bigvee }}\underset{\varsigma }{\overset{d}{\bigvee }}\left(ϝ\right).\hfill \end{array}$$

This completes the proof. □

Corollary 3.

If we take $\alpha =1$ and $\rho =1$ in Theorem 5, and we possess

$$\left|{\rm Y}\left(\sigma ,\rho ;\varsigma ,d\right)\right|\le \frac{25}{576}\underset{\sigma }{\overset{\rho }{\bigvee }}\underset{\varsigma }{\overset{d}{\bigvee }}\left(ϝ\right),$$

where ${\rm Y}\left(\sigma ,\rho ;\varsigma ,d\right)$ is expressed as in (37).

6. Conclusions

In this presented paper, we proved Simpson’s second rule formula type inequalities via Riemann–Liouville fractional integrals for differentiable coordinated convex mappings. Moreover, fractional Simpson’s $3/8$ rule inequalities were obtained via bounded variation functions. The results for symmetric functions can be reached by employing the notions of symmetric convex functions, which will be explored further in future work. Curious readers can investigate new inequalities via inequalities of Newton type utilizing other kinds via fractional integrals. Different types of convexity of these resulting inequalities can be researched in the future.