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On fractional Simpson type integral inequalities for co-ordinated convex functions

Abstract

In this study, we prove equality for twice partially differentiable mappings involving the double generalized fractional integral. Using the established identity, we offer some Simpson’s type inequalities for partially differentiable co-ordinated convex functions in a rectangle from the plane \(\mathbb{R} ^{2}\).

Introduction

The following inequality is one of the well-known results in the literature called Simpson’s inequality.

Theorem 1

Let \(\digamma:[\kappa _{1},\kappa _{2}]\rightarrow{\mathbb{R}}\) be a four times continuously differentiable mapping on \((\kappa _{1},\kappa _{2})\) and \(\| \digamma ^{4}\|_{\infty}=\sup |\digamma ^{4}(x)|<\infty \). Then, the following inequality holds:

$$\begin{aligned} &\biggl\vert \frac{1}{6} \biggl[\digamma (\kappa _{1})+ \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +\digamma (\kappa _{2}) \biggr]- \frac{1}{\kappa _{2}-\kappa _{1}} \int _{\kappa _{1}}^{\kappa _{2}}\digamma (x)\,dx \biggr\vert \\ &\quad\leq \frac{1}{2880} \bigl\Vert \digamma ^{(4)} \bigr\Vert _{\infty}(\kappa _{2}- \kappa _{1})^{4}. \end{aligned}$$

For recent refinements, counterparts, generalizations, and new Simpson’s type inequalities, see [1, 48, 1117, 1928].

A formal definition for co-ordinated convex function may be stated as follows:

Definition 1

A function \(\digamma:\bigtriangleup:{[\kappa _{1},\kappa _{2}]\times {}[ \kappa _{3},\kappa _{4}]}\rightarrow {\mathbb{R}}\) is called co-ordinated convex on for all \((x,u)\), \((y,v)\in {\bigtriangleup }\) and \(t,s\in {[0,1]}\) if it satisfies the following inequality:

$$\begin{aligned} &\digamma \bigl(tx+(1-t)y,su+(1-s)v\bigr) \\ &\quad \leq ts\digamma (x,u)+t(1-s)\digamma (x,v)+s(1-t) \digamma (y,u)+(1-s) (1-t)\digamma (y,v). \end{aligned}$$
(1.1)

The mapping Ϝ is co-ordinated concave on , and the inequality (1.1) holds in reversed direction for all \(t,s\in {[0,1]}\) and \((x,u),(y,v)\in {\bigtriangleup }\).

In [5], Dragomir et al. proved the following some recent developments on Simpson’s inequality for which the remainder is expressed in terms of lower derivatives than the fourth.

Theorem 2

Suppose \(\digamma:[\kappa _{1},\kappa _{2}]\rightarrow {\mathbb{R}}\) is an absolutely continuous mapping on \([\kappa _{1},\kappa _{2}]\) whose derivative belongs to \(L_{p}[\kappa _{1},\kappa _{2}]\). Then, the following inequality holds:

$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[\digamma (\kappa _{1})+4 \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr)+\digamma (\kappa _{2}) \biggr]- \frac{1}{\kappa _{2}-\kappa _{1}} \int _{\kappa _{1}}^{\kappa _{2}} \digamma (x)\,dx \biggr\vert \\ & \quad\leq \frac{1}{6} \biggl[\frac{2^{q+1}+1}{3(q+1)} \biggr]^{\frac{1}{q}}( \kappa _{2}-\kappa _{1})^{\frac{1}{q}} \bigl\Vert \digamma ^{\prime } \bigr\Vert _{p}, \end{aligned}$$

where \(\frac{1}{p}+\frac{1}{q}=1\).

In [19], Sarikaya et al. obtained inequalities for differentiable convex mappings that are connected with Simpson’s inequality, and they used the following lemma to prove it.

Lemma 1

Let \(\digamma:I\subset {\mathbb{R}}\rightarrow {\mathbb{R}}\) be an absolutely continuous mapping on \(I^{\circ }\) (\(I^{\circ }\) is the interior of I) such that \(\digamma ^{\prime }\in {L_{1}[\kappa _{1},\kappa _{2}]}\), where \(\kappa _{1},\kappa _{2}\in {I^{\circ }}\) with \(\kappa _{1}<\kappa _{2}\), then the following equality holds:

$$\begin{aligned} & \frac{1}{6} \biggl[\digamma (\kappa _{1})+4\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr)+\digamma (\kappa _{2}) \biggr]- \frac{1}{(\kappa _{2}-\kappa _{1})} \int _{\kappa _{1}}^{\kappa _{2}} \digamma (x)\,dx \\ &\quad =\frac{\kappa _{2}-\kappa _{1}}{2} \int _{0}^{1} \biggl[ \biggl( \frac{t}{2}-\frac{1}{3} \biggr)\digamma ^{\prime } \biggl(\frac{(1+t)\kappa _{1}}{2}+ \frac{(1-t)\kappa _{2}}{2} \biggr)\\ &\qquad{}+ \biggl(\frac{1}{3}-\frac{t}{2} \biggr) \digamma ^{\prime } \biggl(\frac{(1+t)\kappa _{1}}{2}+ \frac{(1-t)\kappa _{2}}{2} \biggr) \biggr]\,dt. \end{aligned}$$

The main aim of this article is to set up new Simpson’s type inequalities for mappings whose twice partially derivatives in absolute value are co-ordinated convex via generalized fractional integral operators.

Generalized fractional integrals operators

In this section, we summarize the generalized fractional integrals defined by Sarikaya and Ertuğral in [18].

Let us define a function \(\phi:[0,\infty )\rightarrow {}[ 0,\infty )\) satisfying the following condition:

$$\begin{aligned} \int _{0}^{1}\frac{\phi ( t ) }{t}\,dt< \infty. \end{aligned}$$

We consider the following left-sided and right-sided generalized fractional integral operators

$$\begin{aligned} {}_{a+}I_{\phi }f(x)= \int _{a}^{x}\frac{\phi ( x-t ) }{x-t}f(t)\,dt, \quad x>a \end{aligned}$$
(2.1)

and

$$\begin{aligned} {}_{b-}I_{\phi }f(x)= \int _{x}^{b}\frac{\phi ( t-x ) }{t-x}f(t)\,dt,\quad x< b, \end{aligned}$$
(2.2)

respectively.

Some forms of fractional integrals, namely, Riemann–Liouville fractional integrals, k-Riemann–Liouville fractional integrals, Katugampola fractional integrals, conformable fractional integrals, Hadamard fractional integrals, etc. are generalized as the most significant feature of generalized fractional integrals. These important special cases of the integral operators (2.1) and (2.2) are mentioned below:

  1. (1)

    If we choose \(\phi ( t ) =t\), the operators (2.1) and (2.2) reduce to the Riemann integral.

  2. (2)

    Considering \(\phi ( t ) = \frac{t^{\alpha }}{\Gamma ( \alpha ) }\) and \(\alpha >0\), the operators (2.1) and (2.2) reduce to the Riemann–Liouville fractional integrals \(J_{{a+}}^{\alpha }f(x) \) and \(J_{b-}^{\alpha }f(x)\), respectively. Here, Γ is the Gamma function.

  3. (3)

    For \(\phi ( t ) = \frac{1}{k\Gamma _{k} ( \alpha ) }t^{\frac{\alpha }{k}}\) and \(\alpha,k>0\), the operators (2.1) and (2.2) reduce to the k-Riemann–Liouville fractional integrals \(J_{{a+,k}}^{\alpha }f(x)\) and \(J_{b-,k}^{\alpha }f(x)\), respectively. Here, \(\Gamma _{k}\) is the k-Gamma function.

There are several papers on inequalities for generalized fractional integrals in the literature. In [18], Sarikaya and Ertuğral also proved Hermite–Hadamard inequalities for generalized fractional integrals. In addition, Budak et al. proved Midpoint type inequalities and extensions of Hermite–Hadamard inequalities in the papers [2] and [3], respectively. In [9], Ertuğral and Sarikaya presented some Simpson-type inequalities for these fractional integral operators.

Generalized double fractional integrals are given by Turkay et al. in [26], as follows:

Definition 2

The Generalized double fractional integrals \({}_{a+,c+}I_{\phi,\psi }\), \({}_{a+,d-}I_{\phi,\psi }\), \({}_{b-,c+}I_{\phi,\psi }\), \({}_{b-,d-}I_{\phi,\psi }\) are defined by

$$\begin{aligned} &{}_{a+,c+}I_{\phi,\psi }f ( x,y ) = \int _{a}^{x} \int _{c}^{y} \frac {\phi ( x-t ) }{x-t}\frac {\psi ( y-s ) }{y-s}f ( t,s ) \,ds\,dt,\quad x>a, y>c, \end{aligned}$$
(2.3)
$$\begin{aligned} &{}_{a+,d-}I_{\phi,\psi }f ( x,y ) = \int _{a}^{x} \int _{y}^{d} \frac {\phi ( x-t ) }{x-t}\frac {\psi ( s-y ) }{s-y}f ( t,s ) \,ds\,dt,\quad x>a, y< d, \end{aligned}$$
(2.4)
$$\begin{aligned} &{}_{b-,c+}I_{\phi,\psi }f ( x,y ) = \int _{x}^{b} \int _{c}^{y} \frac {\phi ( t-x ) }{t-x}\frac {\psi ( y-s ) }{y-s}f ( t,s ) \,ds\,dt,\quad x< b, y>c, \end{aligned}$$
(2.5)

and

$$\begin{aligned} {}_{b-,d-}I_{\phi,\psi }f ( x,y ) = \int _{x}^{b} \int _{y}^{d} \frac {\phi ( t-x ) }{t-x}\frac {\psi ( s-y ) }{s-y}f ( t,s ) \,ds\,dt,\quad x< b, y< d. \end{aligned}$$
(2.6)

Here, \(f\in L_{1}([a,b]\times {}[ c,d])\), and the functions \(\phi,\psi: [ 0,\infty ) \rightarrow [ 0,\infty ) \) satisfy \(\int _{0}^{1}\frac {\phi ( t ) }{t}\,dt<\infty \) and \(\int _{0}^{1}\frac {\psi ( s ) }{s}\,ds<\infty \), respectively.

Using Definition 2, well-known fractional integrals can be obtained by some special choices. For example,

  1. (1)

    If we choose \(\phi ( t ) =t\) and \(\psi ( s ) =s\), the operators (2.3), (2.4), (2.5) and (2.6) reduce to the double Riemann integral.

  2. (2)

    Considering \(\phi ( t ) =\frac{t^{\alpha }}{\Gamma (\alpha )} \), \(\psi ( s ) =\frac{s^{\beta }}{\Gamma (\beta )}\), then for \(\alpha,\beta >0\), the operators (2.3), (2.4), (2.5) and (2.6) reduce to the Riemann–Liouville fractional integrals \(J_{a+,c+}^{\alpha,\beta }f ( x,y ) \), \(J_{a+,d-}^{\alpha,\beta }f ( x,y ) \) \(J_{b-,c+}^{\alpha,\beta }f ( x,y ) \) and \(J_{b-,d-}^{\alpha,\beta }f ( x,y ) \), respectively.

  3. (3)

    For \(\phi ( t ) = \frac{t^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) and \(\psi ( s ) = \frac{s^{\frac{\beta }{k}}}{k\Gamma _{k}(\beta )}\), for \(\alpha,\beta,k>0\), the operators (2.3), (2.4), (2.5) and (2.6) reduce to the k-Riemann–Liouville fractional integrals \(J_{a+,c+}^{\alpha,\beta,k}f ( x,y ) \), \(J_{a+,d-}^{\alpha,\beta,k}f ( x,y ) \), \(J_{b-,c+}^{\alpha,\beta,k}f ( x,y ) \) and \(J_{b-,d-}^{\alpha,\beta,k}f ( x,y ) \), respectively ([10]).

In this paper, utilizing generalized double fractional integrals, we extend the results proved in [9] to co-ordinated convex functions.

An identity for generalized double fractional integrals

Throughout this paper, for conciseness, we define

$$\begin{aligned} \Lambda ({t})= \int _{0}^{t} \frac{\phi ( \frac{b-a}{2}u ) }{u}\,du,\qquad \bigtriangledown ({s})= \int _{0}^{s} \frac{\psi ( \frac{d-c}{2}u ) }{u}\,du. \end{aligned}$$

In this section, we start the usage of generalized fractional essential operators by subsequent lemma:

Lemma 2

Let \(\digamma:\Delta:= [ \kappa _{1},\kappa _{2} ] \times [ \kappa _{3},\kappa _{4} ] \rightarrow \mathbb{R} \) be a twice partially differentiable mapping on \(\Delta ^{\circ }\) (\(\Delta ^{\circ }\) is the interior of Δ). If \(\frac {\partial ^{2}\digamma }{\partial t\,\partial s}\in L(\Delta )\), then we have the following equality for generalized fractional integrals:

$$\begin{aligned} & \Re (\kappa _{1},\kappa _{2};\kappa _{3}, \kappa _{4}) \\ &\quad = \frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4\Lambda (1)\bigtriangledown (1)} \\ &\qquad{} \times \biggl\{ \int _{0}^{1} \int _{0}^{1} \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggl(\frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \frac{\partial ^{2}}{\partial t\,\partial {s}} \\ &\qquad{}\times \digamma \biggl(\frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2}\kappa _{2}, \frac{1-s}{2}\kappa _{3}+ \frac{1+s}{2}\kappa _{4} \biggr)\,ds\,dt \\ & \qquad{}+ \int _{0}^{1} \int _{0}^{1} \biggl(\frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggl(\frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr) \frac{\partial ^{2}}{\partial t\,\partial {s}} \\ &\qquad{}\times\digamma \biggl(\frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2}\kappa _{2},\frac{1+s}{2}\kappa _{3}+ \frac{1-s}{2}\kappa _{4} \biggr)\,ds\,dt \\ &\qquad{} + \int _{0}^{1} \int _{0}^{1} \biggl(\frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggl(\frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \frac{\partial ^{2}}{\partial t\,\partial {s}} \\ &\qquad{}\times\digamma \biggl(\frac{1+t}{2}\kappa _{1}+ \frac{1-t}{2}\kappa _{2},\frac{1-s}{2}\kappa _{3}+ \frac{1+s}{2}\kappa _{4} \biggr)\,ds\,dt \\ & \qquad{}+ \int _{0}^{1} \int _{0}^{1} \biggl(\frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggl(\frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr) \frac{\partial ^{2}}{\partial t\,\partial {s}} \\ &\qquad{}\times\digamma \biggl(\frac{1+t}{2}\kappa _{1}+ \frac{1-t}{2}\kappa _{2},\frac{1+s}{2}\kappa _{3}+ \frac{1-s}{2}\kappa _{4} \biggr)\,ds\,dt \biggr\} , \end{aligned}$$

where

$$\begin{aligned} &\Re (\kappa _{1},\kappa _{2};\kappa _{3}, \kappa _{4}) \\ &\quad= \frac{\digamma (\kappa _{1},\kappa _{3})+\digamma (\kappa _{2},\kappa _{4})+\digamma (\kappa _{2},\kappa _{3})+\digamma (\kappa _{1},\kappa _{4})}{36} \\ &\qquad{}+\frac{1}{9} \biggl[\digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2}, \kappa _{4} \biggr)+\digamma \biggl(\kappa _{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+\digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{3} \biggr)+ \digamma \biggl(\kappa _{1},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &\qquad{}+4\digamma \biggl(\frac{\kappa _{2}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \biggr]+ \frac{1}{4\Lambda (1)\bigtriangledown (1)} \biggl[_{\kappa _{2}^{-},\kappa _{4}^{-}}I_{\phi,\psi }\digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &\qquad{}+{}_{\kappa _{2}^{-},\kappa _{3}^{+}}I_{\phi,\psi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+{}_{ \kappa _{1}^{+},\kappa _{4}^{-}}I_{\phi,\psi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &\qquad{}+{}_{\kappa _{1}^{+},\kappa _{3}^{+}}I_{\phi,\psi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \biggr] \\ &\qquad{}-\frac{1}{12\Lambda (1)} \biggl[_{\kappa _{2}^{-}}I_{\phi }\digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{4} \biggr)+{}_{\kappa _{2}^{-}}I_{ \phi } \digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{3} \biggr)+{}_{\kappa _{1}^{+}}I_{\phi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{4} \biggr) \\ &\qquad{}+{}_{\kappa _{1}^{+}}I_{\phi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{3} \biggr)+4_{\kappa _{2}^{-}}I_{\phi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &\qquad{}+4_{\kappa _{1}^{+}}I_{\phi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \biggr] \\ &\qquad{}-\frac{1}{12\bigtriangledown (1)} \biggl[_{\kappa _{4}^{-}}I_{\psi } \digamma \biggl(\kappa _{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+{}_{\kappa _{4}^{-}}I_{ \psi } \digamma \biggl(\kappa _{1},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+{}_{\kappa _{3}^{+}}I_{\psi } \digamma \biggl(\kappa _{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &\qquad{}+{}_{\kappa _{3}^{+}}I_{\psi }\digamma \biggl(\kappa _{1}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+4_{\kappa _{4}^{-}}I_{\psi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &\qquad{}+4_{\kappa _{3}^{+}}I_{\psi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \biggr]. \end{aligned}$$

Proof

It suffices to be aware that

$$\begin{aligned} I ={}& \int _{0}^{1} \int _{0}^{1} \biggl(\frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggl(\frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \frac{\partial ^{2}}{\partial t\,\partial {s}} \\ &{}\times\digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2},\frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr)\,ds \,dt \\ &{}+ \int _{0}^{1} \int _{0}^{1} \biggl(\frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggl(\frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr) \frac{\partial ^{2}}{\partial t\,\partial {s}} \\ &{}\times\digamma \biggl(\frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2}\kappa _{2},\frac{1+s}{2}\kappa _{3}+ \frac{1-s}{2}\kappa _{4} \biggr)\,ds\,dt \\ &{}+ \int _{0}^{1} \int _{0}^{1} \biggl(\frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggl(\frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \frac{\partial ^{2}}{\partial t\,\partial {s}} \\ &{}\times\digamma \biggl(\frac{1+t}{2}\kappa _{1}+ \frac{1-t}{2}\kappa _{2},\frac{1-s}{2}\kappa _{3}+ \frac{1+s}{2}\kappa _{4} \biggr)\,ds\,dt \\ &{}+ \int _{0}^{1} \int _{0}^{1} \biggl(\frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggl(\frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr) \frac{\partial ^{2}}{\partial t\,\partial {s}} \\ &{}\times\digamma \biggl(\frac{1+t}{2}\kappa _{1}+ \frac{1-t}{2}\kappa _{2},\frac{1+s}{2}\kappa _{3}+ \frac{1-s}{2}\kappa _{4} \biggr)\,ds\,dt \\ ={}&I_{1}+I_{2}+I_{3}+I_{4}. \end{aligned}$$

Integrating with the aid of using by parts, we obtain

$$\begin{aligned} I_{1} ={}& \int _{0}^{1} \int _{0}^{1} \biggl(\frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggl(\frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \frac{\partial ^{2}}{\partial t\,\partial {s}} \\ &{}\times\digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2},\frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr)\,ds \,dt \\ ={}& \int _{0}^{1} \biggl(\frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggl[\frac{2}{\kappa _{4}-\kappa _{3}} \biggl( \frac{\bigtriangledown ({1})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr) \\ &{}\times\frac{\partial \digamma }{\partial t} \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2},\frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr)\vert _{0}^{1} \\ &{}-\frac{1}{\kappa _{4}-\kappa _{3}} \int _{0}^{1} \frac{\Psi (\frac{\kappa _{4}-\kappa _{3}}{2}s )}{s} \frac{\partial \digamma }{\partial t} \biggl(\frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2}\kappa _{2},\frac{1-s}{2} \kappa _{3}+ \frac{1+s}{2}\kappa _{4} \biggr)\,ds \biggr]\,dt \\ ={}&\frac{2}{\kappa _{4}-\kappa _{3}} \int _{0}^{1} \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggl(\frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \frac{\partial \digamma }{\partial t} \biggl(\frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2}\kappa _{2},\kappa _{4} \biggr)\,dt \\ &{}+\frac{2}{\kappa _{4}-\kappa _{3}} \int _{0}^{1} \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggl(\frac{\bigtriangledown ({1})}{3} \biggr)\frac{\partial \digamma }{\partial t} \biggl(\frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2}\kappa _{2},\frac{\kappa _{4}+\kappa _{3}}{2} \biggr)\,dt \\ &{}-\frac{1}{\kappa _{4}-\kappa _{3}} \int _{0}^{1} \int _{0}^{1} \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \frac{\psi (\frac{\kappa _{4}-\kappa _{3}}{2}s )}{s} \\ &{}\times\frac{\partial \digamma }{\partial t} \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2} \kappa _{2},\frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr)\,ds\,dt. \end{aligned}$$

By changing the variables and applying integration on integrals, we obtain

$$\begin{aligned} I_{1} ={}& \frac{\Lambda ({1})\triangledown (1)}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})} \digamma (\kappa _{2},\kappa _{4})+ \frac{2\Lambda ({1})\bigtriangledown ({1})}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})} \digamma \biggl(\frac{\kappa _{2}+\kappa _{1}}{2},\kappa _{4} \biggr) \\ &{}- \frac{\bigtriangledown ({1})}{3(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}_{ \kappa _{2}^{-}}I_{\phi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{4} \biggr)+ \frac{2\Lambda ({1})\bigtriangledown ({1})}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}\digamma \biggl(\kappa _{2},\frac{\kappa _{4}+\kappa _{3}}{2} \biggr) \\ &{}+\frac{4\Lambda ({1})\bigtriangledown ({1})}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})} \digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{4}+\kappa _{3}}{2} \biggr)\\ &{}- \frac{2\bigtriangledown ({1})}{3(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}_{\kappa _{2}^{-}}I_{ \phi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &{}- \frac{\Lambda ({1})}{3(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}_{\kappa _{4}^{-}}I_{\psi }\digamma \biggl(\kappa _{2}, \frac{\kappa _{4}+\kappa _{3}}{2} \biggr)\\ &{}- \frac{2\Lambda ({1})}{3(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}_{ \kappa _{4}^{-}}I_{\psi } \digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &{}+\frac{1}{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}_{ \kappa _{2}^{-}\kappa _{4}^{-}}I_{\phi,\psi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr). \end{aligned}$$

Similarly, we obtain

$$\begin{aligned} I_{2} ={}& \int _{0}^{1} \int _{0}^{1} \biggl(\frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggl(\frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr) \frac{\partial ^{2}}{\partial t\,\partial {s}} \\ &{}\times\digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2},\frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr)\,ds \,dt \\ ={}& \frac{\Lambda ({1})\triangledown (1)}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})} \digamma (\kappa _{2},\kappa _{3})+ \frac{2\Lambda ({1})\bigtriangledown ({1})}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}\digamma \biggl(\frac{\kappa _{2}+\kappa _{1}}{2},\kappa _{3} \biggr) \\ &{}- \frac{\bigtriangledown ({1})}{3(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})} { }_{\kappa _{2}^{-}}I_{\phi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{3} \biggr)+ \frac{2\Lambda ({1})\bigtriangledown ({1})}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}\digamma \biggl(\kappa _{2},\frac{\kappa _{4}+\kappa _{3}}{2} \biggr) \\ &{}+ \frac{4\Lambda ({1})\bigtriangledown ({1})}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})} \digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{4}+\kappa _{3}}{2} \biggr)\\ &{}- \frac{2\bigtriangledown ({1})}{3(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{ }_{\kappa _{2}^{-}}I_{ \phi }\digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &{}+ \frac{\Lambda ({1})}{3(\kappa _{3}-\kappa _{4})(\kappa _{2}-\kappa _{1})}_{\kappa _{3}^{+}}I_{\psi }\digamma \biggl(\kappa _{2}, \frac{\kappa _{4}+\kappa _{3}}{2} \biggr)\\ &{}+ \frac{2\Lambda ({1})}{3(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})} { }_{\kappa _{3}^{+}}I_{\psi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &{}+\frac{1}{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}_{ \kappa _{2}^{-},\kappa _{3}^{+}}I_{\phi,\psi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr), \\ I_{3} ={}& \int _{0}^{1} \int _{0}^{1} \biggl(\frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggl(\frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \frac{\partial ^{2}}{\partial t\,\partial {s}} \\ &{}\times\digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2},\frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr)\,ds \,dt \\ ={}& \frac{\Lambda ({1})\triangledown (1)}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})} \digamma (\kappa _{1},\kappa _{4})+ \frac{2\Lambda ({1})\bigtriangledown ({1})}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}\digamma \biggl(\frac{\kappa _{2}+\kappa _{1}}{2},\kappa _{4} \biggr) \\ &{}- \frac{\bigtriangledown ({1})}{3(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}_{ \kappa _{1}^{+}}I_{\phi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{4} \biggr)+ \frac{2\Lambda ({1})\bigtriangledown ({1})}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}\digamma \biggl(\kappa _{1},\frac{\kappa _{4}+\kappa _{3}}{2} \biggr) \\ &{}+ \frac{4\Lambda ({1})\bigtriangledown ({1})}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})} \digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{4}+\kappa _{3}}{2} \biggr)\\ &{}- \frac{2\bigtriangledown ({1})}{3(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}_{\kappa _{1}^{+}}I_{ \phi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &{}- \frac{\Lambda ({1})}{3(\kappa _{3}-\kappa _{4})(\kappa _{2}-\kappa _{1})}_{\kappa _{4}^{-}}I_{\psi }\digamma \biggl(\kappa _{1}, \frac{\kappa _{4}+\kappa _{3}}{2} \biggr)\\ &{}- \frac{2\Lambda ({1})}{3(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}_{ \kappa _{4}^{-}}I_{\psi } \digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &{}+\frac{1}{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}_{ \kappa _{1}^{+},\kappa _{4}^{-}}I_{\phi,\psi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr), \end{aligned}$$

and

$$\begin{aligned} I_{4} ={}& \int _{0}^{1} \int _{0}^{1} \biggl(\frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggl(\frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr) \frac{\partial ^{2}}{\partial t\,\partial {s}} \\ &{}\times\digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2},\frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr)\,ds \,dt \\ ={}& \frac{\Lambda ({1})\triangledown (1)}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})} \digamma (\kappa _{1},\kappa _{3})+ \frac{2\Lambda ({1})\bigtriangledown ({1})}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}\digamma \biggl( \frac{\kappa _{2}+\kappa _{1}}{2},\kappa _{3} \biggr) \\ &{}- \frac{\bigtriangledown ({1})}{3(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}_{ \kappa _{1}^{+}}I_{\phi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{3} \biggr)+ \frac{2\Lambda ({1})\bigtriangledown ({1})}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}\digamma \biggl(\kappa _{1},\frac{\kappa _{4}+\kappa _{3}}{2} \biggr) \\ &{}+\frac{4\Lambda ({1})\bigtriangledown ({1})}{9(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})} \digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{4}+\kappa _{3}}{2} \biggr)\\ &{}- \frac{2\bigtriangledown ({1})}{3(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}_{\kappa _{1}^{+}}I_{ \phi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &{}- \frac{\Lambda ({1})}{3(\kappa _{3}-\kappa _{4})(\kappa _{2}-\kappa _{1})}_{\kappa _{3}^{+}}I_{\psi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{4}+\kappa _{3}}{2} \biggr)\\ &{}- \frac{2\Lambda ({1})}{3(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}_{ \kappa _{3}^{+}}I_{\psi }\digamma \biggl(\kappa _{1}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &{}+\frac{1}{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}_{ \kappa _{1}^{+},\kappa _{3}^{+}}I_{\phi,\psi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr). \end{aligned}$$

By adding the above integrals and multiply by \(\frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{\Lambda (1)\bigtriangledown (1)}\), we can write

$$\begin{aligned} \frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4\Lambda (1)\bigtriangledown (1)} [ I_{1}+I_{2}+I_{3}+I_{4} ] =\Re (\kappa _{1},\kappa _{2}; \kappa _{3}, \kappa _{4}), \end{aligned}$$

which completes the proof. □

Corollary 1

Under the assumption of Lemma 2, with \(\phi (t)=t\) and \(\psi (s)=s\), then we obtain the following equality for Riemann integrals:

$$\begin{aligned} & \Upsilon (\kappa _{1},\kappa _{2};\kappa _{3}, \kappa _{4}) \\ &\quad =\frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4}\int _{0}^{1} \int _{0}^{1} \biggl( \frac{t}{2}- \frac{1}{3} \biggr) \biggl( \frac{s}{2}-\frac{1}{3} \biggr) \\ &\qquad{}\times \frac{\partial ^{2}\digamma }{\partial t\,\partial s} \biggl(\frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2}\kappa _{2},\frac{1-s}{2} \kappa _{3}+ \frac{1+s}{2}\kappa _{4} \biggr)\,ds\,dt \\ &\qquad{} +\frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4}\int _{0}^{1} \int _{0}^{1} \biggl( \frac{t}{2}- \frac{1}{3} \biggr) \biggl( \frac{1}{3}-\frac{s}{2} \biggr) \\ &\qquad{}\times\frac{\partial ^{2}\digamma }{\partial t\,\partial s} \biggl(\frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2}\kappa _{2},\frac{1+s}{2} \kappa _{3}+ \frac{1-s}{2}\kappa _{4} \biggr)\,ds\,dt \\ &\qquad{} +\frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4}\int _{0}^{1} \int _{0}^{1} \biggl( \frac{1}{3}- \frac{t}{2} \biggr) \biggl( \frac{s}{2}-\frac{1}{3} \biggr) \\ &\qquad{}\times\frac{\partial ^{2}\digamma }{\partial t\,\partial s} \biggl(\frac{1+t}{2}\kappa _{1}+ \frac{1-t}{2}\kappa _{2},\frac{1-s}{2} \kappa _{3}+ \frac{1+s}{2}\kappa _{4} \biggr)\,ds\,dt \\ & \qquad{}+\frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4}\int _{0}^{1} \int _{0}^{1} \biggl( \frac{1}{3}- \frac{t}{2} \biggr) \biggl( \frac{1}{3}-\frac{s}{2} \biggr) \\ &\qquad{}\times\frac{\partial ^{2}\digamma }{\partial t\,\partial s} \biggl(\frac{1+t}{2}\kappa _{1}+ \frac{1-t}{2}\kappa _{2},\frac{1+s}{2} \kappa _{3}+ \frac{1-s}{2}\kappa _{4} \biggr)\,ds\,dt, \end{aligned}$$

where \(\Upsilon (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})\) is defined by

$$\begin{aligned} & \Upsilon (\kappa _{1},\kappa _{2};\kappa _{3}, \kappa _{4}) \\ &\quad = \frac{\digamma (\kappa _{1},\frac{\kappa _{3}+\kappa _{4}}{2})+\digamma (\kappa _{2},\frac{\kappa _{3}+\kappa _{4}}{2})+4\digamma (\frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2})+\digamma (\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{3})+\digamma (\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{4})}{9} \\ &\qquad{}+ \frac{\digamma (\kappa _{1},\kappa _{3})+\digamma (\kappa _{2},\kappa _{3})+\digamma (\kappa _{2},\kappa _{4})+\digamma (\kappa _{2},\kappa _{4})}{36} \\ & \qquad{}-\frac{1}{6(\kappa _{2}-\kappa _{1})} \int _{\kappa _{1}}^{\kappa _{2}} \biggl[\digamma (x,\kappa _{3})+4\digamma \biggl(x, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+ \digamma (x,\kappa _{4}) \biggr]\,dx \\ &\qquad{} -\frac{1}{6(\kappa _{4}-\kappa _{3})} \int _{\kappa _{3}}^{\kappa _{4}} \biggl[\digamma (\kappa _{1},y)+4\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},y \biggr)+ \digamma (x,\kappa _{4}) \biggr]\,dy \\ & \qquad{}+\frac{1}{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})} \int _{ \kappa _{1}}^{\kappa _{2}} \int _{\kappa _{3}}^{\kappa _{4}}\digamma (x,y)\,dy\,dx. \end{aligned}$$

Corollary 2

Under the assumption of Lemma 2, with \(\phi (t)=\frac{t^{\alpha }}{\Gamma {(\alpha )}}\) and \(\psi (s)=\frac{s^{\beta }}{\Gamma {(\beta )}}\), then we obtain the following equality for Riemann–Liouville fractional integrals:

$$\begin{aligned} &\Omega (\kappa _{1},\kappa _{2};\kappa _{3}, \kappa _{4}) \\ &\quad=\frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4} \\ &\qquad{}\times \biggl\{ \int _{0}^{1} \int _{0}^{1} \biggl( \frac{t^{\alpha }}{2}- \frac{1}{3} \biggr) \biggl(\frac{s^{\beta }}{2}-\frac{1}{3} \biggr) \\ &\qquad{}\times\frac{\partial ^{2}}{\partial t\,\partial {s}}\digamma \biggl(\frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2}\kappa _{2},\frac{1-s}{2}\kappa _{3}+ \frac{1+s}{2}\kappa _{4} \biggr)\,ds\,dt \\ &\qquad{}+ \int _{0}^{1} \int _{0}^{1} \biggl(\frac{t^{\alpha }}{2}- \frac{1}{3} \biggr) \biggl(\frac{1}{3}- \frac{s^{\beta }}{2} \biggr) \\ &\qquad{}\times \frac{\partial ^{2}}{\partial {t}\partial {s}}\digamma \biggl(\frac{1-t}{2} \kappa _{1}+\frac{1+t}{2} \kappa _{2},\frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr)\,ds\,dt \\ &\qquad{}+ \int _{0}^{1} \int _{0}^{1} \biggl(\frac{1}{3}- \frac{t^{\alpha }}{2} \biggr) \biggl(\frac{s^{\beta }}{2}- \frac{1}{3} \biggr) \\ &\qquad{}\times\frac{\partial ^{2}}{\partial {t}\partial {s}}\digamma \biggl(\frac{1+t}{2} \kappa _{1}+\frac{1-t}{2} \kappa _{2},\frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr)\,ds\,dt \\ &\qquad{}+ \int _{0}^{1} \int _{0}^{1} \biggl(\frac{1}{3}- \frac{t^{\alpha }}{2} \biggr) \biggl(\frac{1}{3}- \frac{s^{\beta }}{2} \biggr) \\ &\qquad{}\times\frac{\partial ^{2}}{\partial {t}\partial {s}}\digamma \biggl(\frac{1+t}{2} \kappa _{1}+\frac{1-t}{2} \kappa _{2},\frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr)\,ds\,dt \biggr\} , \end{aligned}$$

where

$$\begin{aligned} &\Omega (\kappa _{1},\kappa _{2};\kappa _{3}, \kappa _{4}) \\ &\quad= \frac{\digamma (\kappa _{1},\kappa _{3})+\digamma (\kappa _{2},\kappa _{4})+\digamma (\kappa _{2},\kappa _{3})+\digamma (\kappa _{1},\kappa _{4})}{36} \\ &\qquad{}+\frac{1}{9} \biggl[\digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2}, \kappa _{4} \biggr)+\digamma \biggl(\kappa _{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+\digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{3} \biggr)+ \digamma \biggl(\kappa _{1},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &\qquad{}+4\digamma \biggl(\frac{\kappa _{2}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \biggr] \\ &\qquad{}+ \frac{2^{\alpha +\beta -2}\Gamma (\alpha +1)\Gamma (\beta +1)}{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha } ( \kappa _{4}-\kappa _{3} ) ^{\beta }} \biggl[J_{\kappa _{2}^{-},\kappa _{4}^{-}}^{\alpha,\beta } \digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &\qquad{}+J_{\kappa _{2}^{-},\kappa _{3}^{+}}^{\alpha,\beta }\digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &\qquad{}+J_{\kappa _{1}^{+},\kappa _{4}^{-}}^{\alpha,\beta }\digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+J_{\kappa _{1}^{+},\kappa _{3}^{+}\phi,\psi }^{\alpha,\beta } \digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \biggr] \\ &\qquad{}- \frac{2^{\alpha -1}\Gamma (\alpha +1)}{6 ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \biggl[J_{\kappa _{2}^{-}}^{\alpha }\digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{4} \biggr)+J_{\kappa _{2}^{-}}^{ \alpha } \digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{3} \biggr)+J_{\kappa _{1}^{+}}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{4} \biggr) \\ &\qquad{}+J_{\kappa _{1}^{+}}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{3} \biggr)+4J_{\kappa _{2}^{-}}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+4J_{\kappa _{1}^{+}}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \biggr] \\ &\qquad{}- \frac{2^{\beta -1}\Gamma (\beta +1)}{6 ( \kappa _{4}-\kappa _{3} ) ^{\beta }} \biggl[J_{\kappa _{4}^{-}}^{\beta }\digamma \biggl(\kappa _{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+J_{\kappa _{4}^{-}}^{ \beta } \digamma \biggl(\kappa _{1},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+J_{\kappa _{3}^{+}}^{\beta } \digamma \biggl(\kappa _{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &\qquad{}+J_{\kappa _{3}^{+}}^{\beta }\digamma \biggl(\kappa _{1}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+4J_{\kappa _{4}^{-}}^{\beta } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+4J_{\kappa _{3}^{+}}^{\beta } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \biggr]. \end{aligned}$$

Corollary 3

Under the assumption of Lemma 2, with \(\phi (t)=\frac{t^{\frac{\alpha }{k}}}{k\Gamma _{k}{(\alpha )}}\) and \(\psi (s)=\frac{s^{\frac{\beta }{k}}}{k\Gamma _{k}({\beta )}}\), then we obtain the following equality for k-Riemann–Liouville fractional integrals:

$$\begin{aligned} &\$(\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \\ &\quad=\frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4} \\ &\qquad{}\times \biggl\{ \int _{0}^{1} \int _{0}^{1} \biggl( \frac{t^{\frac{\alpha }{k}}}{2}- \frac{1}{3} \biggr) \biggl(\frac{s^{\frac{\beta }{k}}}{2}-\frac{1}{3} \biggr) \\ &\qquad{}\times\frac{\partial ^{2}}{\partial t\,\partial {s}}\digamma \biggl( \frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2},\frac{1-s}{2} \kappa _{3}+ \frac{1+s}{2}\kappa _{4} \biggr) \,ds\,dt \\ &\qquad{}+ \int _{0}^{1} \int _{0}^{1} \biggl(\frac{t^{\frac{\alpha }{k}}}{2}- \frac{1}{3} \biggr) \biggl(\frac{1}{3}- \frac{s^{\frac{\beta }{k}}}{2} \biggr) \\ &\qquad{}\times\frac{\partial ^{2}}{\partial t\,\partial {s}}\digamma \biggl( \frac{1-t}{2} \kappa _{1}+\frac{1+t}{2}\kappa _{2},\frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr)\,ds\,dt \\ &\qquad{}+ \int _{0}^{1} \int _{0}^{1} \biggl(\frac{1}{3}- \frac{t^{\frac{\alpha }{k}}}{2} \biggr) \biggl(\frac{s^{\frac{\beta }{k}}}{2}- \frac{1}{3} \biggr) \\ &\qquad{}\times \frac{\partial ^{2}}{\partial t\,\partial {s}}\digamma \biggl( \frac{1+t}{2} \kappa _{1}+\frac{1-t}{2}\kappa _{2},\frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr)\,ds\,dt \\ &\qquad{}+ \int _{0}^{1} \int _{0}^{1} \biggl(\frac{1}{3}- \frac{t^{\frac{\alpha }{k}}}{2} \biggr) \biggl(\frac{1}{3}- \frac{s^{\frac{\beta }{k}}}{2} \biggr) \\ &\qquad{}\times \frac{\partial ^{2}}{\partial t\,\partial {s}}\digamma \biggl( \frac{1+t}{2} \kappa _{1}+\frac{1-t}{2}\kappa _{2},\frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr)\,ds\,dt \biggr\} , \end{aligned}$$

where

$$\begin{aligned} &\$(\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \\ &\quad= \frac{\digamma (\kappa _{1},\kappa _{3})+\digamma (\kappa _{2},\kappa _{4})+\digamma (\kappa _{2},\kappa _{3})+\digamma (\kappa _{1},\kappa _{4})}{36} \\ &\qquad{}+\frac{1}{9} \biggl[\digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2}, \kappa _{4} \biggr)+\digamma \biggl(\kappa _{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+\digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{3} \biggr)+ \digamma \biggl(\kappa _{1},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &\qquad{}+4\digamma \biggl(\frac{\kappa _{2}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \biggr] \\ &\qquad{}+ \frac{2^{\frac{\alpha +\beta }{k}-2}\Gamma _{k}(\alpha +k)\Gamma _{k}(\beta +k)}{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha } ( \kappa _{4}-\kappa _{3} ) ^{\beta }} \biggl[J_{\kappa _{2}^{-}, \kappa _{4}^{-}}^{\alpha,\beta,k}\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &\qquad{}+J_{\kappa _{2}^{-},\kappa _{3}^{+}}^{\alpha,\beta,k}\digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &\qquad{}+J_{\kappa _{1}^{+},\kappa _{4}^{-}}^{\alpha,\beta,k}\digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+J_{\kappa _{1}^{+},\kappa _{3}^{+}\phi,\psi }^{\alpha,\beta,k} \digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \biggr] \\ &\qquad{}- \frac{2^{\frac{\alpha }{k}-1}\Gamma _{k}(\alpha +k)}{6 ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[J_{\kappa _{2}^{-}}^{\alpha,k}\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{4} \biggr)+J_{\kappa _{2}^{-}}^{ \alpha,k} \digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{3} \biggr)+J_{\kappa _{1}^{+}}^{\alpha,k} \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{4} \biggr) \\ &\qquad{}+J_{\kappa _{1}^{+}}^{\alpha,k}\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{3} \biggr)+4J_{\kappa _{2}^{-}}^{ \alpha,k}\digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+4J_{\kappa _{1}^{+}}^{\alpha,k} \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \biggr] \\ &\qquad{}- \frac{2^{\frac{\beta }{k}-1}\Gamma _{k}(\beta +k)}{6 ( \kappa _{4}-\kappa _{3} ) ^{\frac{\beta }{k}}} \biggl[J_{\kappa _{4}^{-}}^{\beta,k}\digamma \biggl( \kappa _{2}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+J_{\kappa _{4}^{-}}^{\beta,k} \digamma \biggl(\kappa _{1}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+J_{\kappa _{3}^{+}}^{\beta,k} \digamma \biggl(\kappa _{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \\ &\qquad{}+J_{\kappa _{3}^{+}}^{\beta,k}\digamma \biggl(\kappa _{1}, \frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+4J_{\kappa _{4}^{-}}^{\beta,k} \digamma \biggl(\frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr)+4J_{\kappa _{3}^{+}}^{\beta,k} \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2},\frac{\kappa _{3}+\kappa _{4}}{2} \biggr) \biggr]. \end{aligned}$$

Simpson type inequalities for generalized double fractional integrals

In this section, we present some Simpson-type inequalities for generalized fractional integrals.

Theorem 3

We assume that the conditions of Lemma 2hold. If the mapping \(\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s} \vert \) is convex on , then we have the following inequality for generalized fractional integrals:

$$\begin{aligned} & \bigl\vert \Re (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad\leq \frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4\Lambda (1)\triangledown (1)} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda (t)}{2}- \frac{\Lambda (1)}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown (s)}{2}- \frac{\bigtriangledown (1)}{3} \biggr\vert \,ds\,dt \biggr) \\ &\qquad{}\times \biggl[ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert \biggr], \end{aligned}$$

where \(\Re (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})\) is defined as in Lemma 2.

Proof

By taking modulus in Lemma 2, we obtain

$$\begin{aligned} & \bigl\vert \Re (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad\leq \frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4\Lambda (1)\bigtriangledown (1)} \\ &\qquad{}\times \biggl\{ \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}-\frac{\Lambda ({1})}{3} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\qquad{}+ \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({1})}{3}-\frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\qquad{}+ \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\qquad{}+ \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({1})}{3}-\frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \biggr\} . \end{aligned}$$
(4.1)

Since the mapping \(|\frac{\partial \digamma }{\partial t\,\partial s} |\) is co-ordinated convex on , we obtain

$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert \biggl\vert \biggl(\frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad\leq \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert \biggl( \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert \\ &\qquad{}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert + \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert \\ &\qquad{}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert \biggr)\,ds\,dt. \end{aligned}$$
(4.2)

Similarly, we obtain

$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert \biggl\vert \biggl(\frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert \end{aligned}$$
(4.3)
$$\begin{aligned} &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad\leq \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({1})}{3}-\frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert \biggl( \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert \\ &\qquad{}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert + \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert \\ &\qquad{}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert \biggr)\,ds\,dt, \\ & \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert \biggl\vert \biggl(\frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad\leq \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert \biggl( \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert \\ &\qquad{}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert + \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert \\ &\qquad{}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert \biggr)\,ds\,dt \end{aligned}$$
(4.4)

and

$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert \biggl\vert \biggl(\frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad\leq \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({1})}{3}-\frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert \biggl( \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert \\ &\qquad{}\times \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert + \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert \\ &\qquad{}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert \biggr)\,ds\,dt. \end{aligned}$$
(4.5)

Using the inequalities (4.2)–(4.5) in (4.1), the proof is completed. □

Corollary 4

If we take \(\phi (t)=t\) and \(\psi (s)=s\) in Theorem 3, then Theorem 3reduces to [15, Theorem 3].

Corollary 5

In Theorem 3, if we use \(\phi (t)=\frac{t^{\alpha }}{\Gamma {(\alpha )}}\) and \(\psi (s)=\frac{s^{\beta }}{\Gamma (\beta )}\), then we obtain the following inequality for Riemann–Loiuville fractional integrals:

$$\begin{aligned} & \bigl\vert \Omega (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad \leq \frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4} \\ &\qquad{}\times \biggl( \frac{\alpha }{\alpha +1} \biggl( \frac{2}{3} \biggr) ^{ \frac{1}{\alpha }+1}+\frac{1}{2(\alpha +1)}- \frac{1}{3} \biggr) \biggl( \frac{\beta }{\beta +1} \biggl( \frac{2}{3} \biggr) ^{\frac{1}{\beta }+1}+ \frac{1}{2(\beta +1)}- \frac{1}{3} \biggr) \\ & \qquad{}\times \biggl[ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert \biggr], \end{aligned}$$

where \(|\Omega (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})|\) is defined as in Corollary 2.

Corollary 6

If we take \(\phi (t)=\frac{t^{\frac{\alpha }{k}}}{k\Gamma _{k}{(\alpha )}}\) and \(\psi (s)=\frac{s^{\frac{\beta }{k}}}{k\Gamma _{k}{(\beta )}}\) in Theorem 3, we obtain the following inequality for k-Riemann–Louville fractional integrals:

$$\begin{aligned} & \bigl\vert \$(\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad\leq \frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4} \biggl( \frac{\alpha }{\alpha +k} \biggl( \frac{2}{3} \biggr) ^{ \frac{k}{\alpha }+1}+\frac{k}{2(\alpha +k)}- \frac{1}{3} \biggr) \\ &\qquad{}\times \biggl( \frac{\beta }{\beta +k} \biggl( \frac{2}{3} \biggr) ^{\frac{k}{\beta }+1}+ \frac{k}{2(\beta +k)}- \frac{1}{3} \biggr) \\ &\qquad{}\times \biggl[ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert \biggr], \end{aligned}$$

where \(\$(\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})\) is defined as in Corollary 3.

Theorem 4

Suppose that the assumptions of Lemma 2hold. If the mapping \(\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s} \vert ^{q}\) is co-ordinated convex on Δ, then we have the following inequality for generalized fractional integrals:

$$\begin{aligned} & \bigl\vert \Re (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad\leq \frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4\Lambda (1)\bigtriangledown (1)} \\ &\qquad{}\times \biggl[ \biggl( \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \biggl(\frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \frac{ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}} \\ &\qquad{}+ \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \frac{\bigtriangledown ({1})}{3}-\frac{\bigtriangledown ({s})}{2} \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \frac{3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}} \\ &\qquad{}+ \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \frac{3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}} \\ &\qquad{}+ \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{1} \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \frac{9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}} \biggr], \end{aligned}$$

where \(\Re (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})\) is defined as in Lemma 2and \(\frac{1}{p}+\frac{1}{q}=1\).

Proof

From Hölder’s inequality and co-ordinated convexity of \(|\frac{\partial ^{2}\digamma }{\partial t\,\partial s} |^{q}\), we have

$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert \biggl\vert \biggl(\frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert \\ & \qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ & \quad\leq \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ & \qquad{}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}}{\partial {t}\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2} \kappa _{2},\frac{1-s}{2}\kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert ^{q}\,dtds \biggr)^{\frac{1}{q}} \\ & \quad\leq \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ & \qquad{}\times \int _{0}^{1} \int _{0}^{1} \biggl( \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ & \qquad{}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}}\,ds\,dt \\ & \quad= \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{} \times \biggl( \frac{ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}}. \end{aligned}$$
(4.6)

Similarly, we obtain

$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert \end{aligned}$$
(4.7)
$$\begin{aligned} &\qquad{} \times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad \leq \biggl( \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \biggl( \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ & \qquad{}\times \biggl( \frac{3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}}, \\ & \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ & \qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad \leq \biggl( \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \biggl( \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{} \times \biggl( \frac{3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}}, \end{aligned}$$
(4.8)

and

$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert \\ & \qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ & \quad\leq \biggl( \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{1} \biggl\vert \biggl(\frac{\bigtriangledown ({1})}{3}-\frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert ^{p} \,ds \biggr)^{\frac{1}{p}} \\ &\qquad{} \times \biggl( \frac{9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}}. \end{aligned}$$
(4.9)

Using the inequalities (4.6)–(4.9) in (4.1), we obtain the required result. □

Corollary 7

If we take \(\phi (t)=t\) and \(\psi (s)=s\) in Theorem 3, we have the following Simpson inequality for Riemann integrals:

$$\begin{aligned} & \bigl\vert \Upsilon (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad\leq \frac{(\kappa _{2}-\kappa _{1})(\kappa _{4}-\kappa _{3})}{144} \biggl(\frac{1+2^{p+1}}{3(p+1)} \biggr)^{\frac{2}{p}} \biggl(\frac{1}{16} \biggr)^{ \frac{1}{q}} \\ &\qquad{}\times \biggl[ \biggl( \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{2},\kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr], \end{aligned}$$

where \(\Upsilon (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})\) is defined as in Corollary 1.

Corollary 8

If we take \(\phi (t)=\frac{t^{\alpha }}{\Gamma {(\alpha )}}\) and \(\psi (s)=\frac{t^{\beta }}{\Gamma {(\beta )}}\) in Theorem 3, we obtain the following Simpson inequality for Riemann–Liouville fractional integrals:

$$\begin{aligned} & \bigl\vert \Omega (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad\leq \biggl( \int _{0}^{1} \biggl\vert \frac{t^{\alpha }}{2}- \frac{1}{3} \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{1} \biggl\vert \frac{s^{\alpha }}{2}- \frac{1}{3} \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \biggl(\frac{1}{16} \biggr)^{\frac{1}{q}} \\ &\qquad{}\times \biggl[ \biggl( \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{2},\kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr], \end{aligned}$$

where \(\Omega (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})\) is defined as in Corollary 2.

Corollary 9

If we take \(\phi (t)=\frac{t^{\frac{\alpha }{k}}}{k\Gamma _{k}{(\alpha )}}\) and \(\psi (s)=\frac{t^{\frac{\beta }{k}}}{k\Gamma _{k}{(\beta )}}\) in Theorem 3, we obtain the following Simpson inequality for k-Riemann–Liouville fractional integrals:

$$\begin{aligned} & \bigl\vert \$(\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad\leq \biggl( \biggl\vert \frac{t^{\frac{\alpha }{k}}}{2}-\frac{1}{3} \biggr\vert \,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \frac{s^{\frac{\beta }{k}}}{2}-\frac{1}{3} \biggr\vert \,ds \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl[ \biggl( \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{2},\kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr], \end{aligned}$$

where \(\$(\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})\) is defined as in Corollary 3.

Theorem 5

Suppose that the assumptions of Lemma 2hold. If the mapping \(\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s} \vert ^{q}\), \(q>1\), is co-ordinated convex on Δ, then we have the following inequality for generalized fractional integrals:

$$\begin{aligned} & \bigl\vert \Re (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad \leq \frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{\Lambda (1)\bigtriangledown (1)} \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}-\frac{\Lambda ({1})}{3} \biggr\vert \,dt \biggr)^{1-\frac{1}{q}} \biggl( \biggl\vert \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr\vert \,ds \biggr)^{1-\frac{1}{q}} \\ & \qquad{}\times \biggl\{ ( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}-\frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ & \qquad{}\times \biggl[ \biggl( \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ &\qquad{} + \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}} \\ &\qquad{} + ( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ &\qquad{} \times \biggl[ \biggl( \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ &\qquad{} + \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}} \\ &\qquad{} + ( \int _{0}^{1} \int _{0}^{1} ( \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ & \qquad{}\times \biggl[ \biggl( \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ & \qquad{}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}} \\ & \qquad{}+ ( \int _{0}^{1} \int _{0}^{1} ( \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert \\ & \qquad{}\times \biggl[ \biggl( \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ & \qquad{} + \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}} \biggr\} , \end{aligned}$$

where \(\Re (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})\) is defined as in Lemma 2.

Proof

Using the power mean inequality and coordinate-convexity of \(|\frac{\partial ^{2}\digamma }{\partial t\,\partial s} |^{q}\), we have

$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2}\kappa _{2},\frac{1-s}{2} \kappa _{3}+ \frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad \leq \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \,dt \biggr)^{1-\frac{1}{q}} \biggl( \int _{0}^{1} \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \,ds \biggr)^{1-\frac{1}{q}} \\ & \qquad{}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ & \qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert ^{q}\,dtds \biggr)^{\frac{1}{q}} \\ & \quad\leq \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \,dt \biggr)^{1-\frac{1}{q}} \biggl( \int _{0}^{1} \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \,ds \biggr)^{1-\frac{1}{q}} \\ & \qquad{}\times ( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ &\qquad{} \times \biggl[ \biggl( \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ & \qquad{}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}}. \end{aligned}$$
(4.10)

Similarly, we obtain

$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert \end{aligned}$$
(4.11)
$$\begin{aligned} & \qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ & \quad\leq \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \,dt \biggr)^{1-\frac{1}{q}} \biggl( \int _{0}^{1} \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert \,ds \biggr)^{1-\frac{1}{q}} \\ &\qquad{} \times ( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ &\qquad{} \times \biggl[ \biggl( \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ & \quad\qquad{}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}}, \\ & \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ &\qquad{} \times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad \leq \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \,dt \biggr)^{1-\frac{1}{q}} \biggl( \int _{0}^{1} \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \,ds \biggr)^{1-\frac{1}{q}} \\ & \qquad{}\times ( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ &\qquad{} \times \biggl[ \biggl( \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ & \qquad{}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}}, \end{aligned}$$
(4.12)

and

$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert \\ & \qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ & \quad\leq \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \,dt \biggr)^{1-\frac{1}{q}} \biggl( \int _{0}^{1} \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert \,ds \biggr)^{1-\frac{1}{q}} \\ & \qquad{}\times ( \int _{0}^{1} \int _{0}^{1} ( \biggl\vert \frac{\Lambda ({t})}{2}-\frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ & \qquad{}\times \biggl[ \biggl( \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ & \qquad{}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}}. \end{aligned}$$
(4.13)

By substituting the inequalities (4.10)–(4.12) in (4.1), we obtain the desired result. □

Remark 1

By special choices of the functions ϕ and ψ in Theorem 5, one can obtain several new Simpson-type inequalities. These are left to the reader.

Concluding remarks

In this paper, we present several generalized fractional Simpson-type inequalities for functions whose partial derivatives in absolute value are co-ordinated convex functions. We also show that the results given here are a strong generalization of some already published ones. In the forthcoming papers, researchers can use the techniques of this work to obtain similar inequalities for other kinds of co-ordinated convexity.

Availability of data and materials

Data sharing not applicable to this paper as no data sets were generated or analysed during the current study.

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Acknowledgements

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

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SK: computation, writing original draft. HB: supervision, writing-review, and editing. All authors read and approved the final manuscript.

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Correspondence to Hüseyin Budak.

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Khan, S., Budak, H. On fractional Simpson type integral inequalities for co-ordinated convex functions. J Inequal Appl 2022, 94 (2022). https://doi.org/10.1186/s13660-022-02830-z

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  • DOI: https://doi.org/10.1186/s13660-022-02830-z

MSC

  • 26D07
  • 26D10
  • 26D15
  • 26B15
  • 26B25

Keywords

  • Simpson’s inequality
  • Integral inequalities
  • Fractional calculus
  • Co-ordinated convex functions