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New trapezium type inequalities of coordinated distance-disturbed convex type functions of higher orders via extended Katugampola fractional integrals

Abstract

In this paper we establish some new results on trapezium type inequalities of coordinated distance-disturbed \((\ell _{1},h_{1})\)\((\ell _{2},h_{2})\)-convex functions of higher orders \((\sigma _{1},\sigma _{2})\) by using the Katugampola \((k_{1},k_{2})\)-fractional integrals. As special cases of our general results, we recapture some earlier proved results.

Introduction

During the most recent couple of decades, the theory of convex functions has been widely considered because of its applications in the theory of optimization and biological systems [15, 31]. In modern days many generalizations of different convexities and combinations of such concepts appears in the literature. These notions adapted and generalized those inequalities which belong to the classical convexity. Hermite–Hadamard type inequalities are significant and very important on the basis of geometric interpretation. Dragomir in [7], presented the definition of convex functions on \(\mathbb{R}^{2}\), with coordinates in a rectangle. He considered a bi-dimensional interval \(\Lambda =[\alpha ,\beta ]\times [\phi ,\varphi ]\) with \(0\leq \alpha <\beta <\infty \), \(0\leq \phi <\varphi <\infty \). Indeed, a function \(\rho :\Lambda \rightarrow \mathbb{R}\), will be called convex on the coordinates on Λ, if the partial mappings \(\rho _{y}:[\phi ,\varphi ]\rightarrow \mathbb{R}\), \(\rho _{y}(u)=\rho (u,y)\) and \(\rho _{x}:[\alpha ,\beta ]\rightarrow \mathbb{R}\), \(\rho _{x}(v)=\rho (x,v)\) are convex for all \(y,v\in [\alpha ,\beta ]\) and for all \(x,u\in [\phi ,\varphi ]\), respectively. A function on Λ is said to be convex if it satisfies the following inequality:

$$\begin{aligned} \rho \bigl(\hat{a} x+(1-\hat{a})u,\hat{a} y+(1-\hat{a})v \bigr) \leq \hat{a} \rho (x,y)+(1-\hat{a})\rho (u,v) \end{aligned}$$
(1.1)

for all \((x,y),(u,v)\in \Lambda \) and \(\hat{a}\in [0,1]\). Every convex function is coordinated convex but the converse is not true [7]. Dragomir in [7], presented Hadamard type inequalities related to one dimensional case. Further the researchers present many generalizations of the coordinated convex functions and introduced new inequalities, we refer the reader to [231, 3335, 37]. The contributions of Noor [27], Yang [38], Sarikaya [32], Chen [6] and Set et al. [36] in this regards are remarkable.

The aim of this paper is to develop new trapezium type inequalities of coordinated distance-disturbed \((\ell _{1},h_{1})\)\((\ell _{2},h_{2})\)-convex functions of higher orders \((\sigma _{1},\sigma _{2})\) by using the Katugampola \((k_{1},k_{2})\)-fractional integrals. We establish our results for many special cases like coordinated distance-disturbed \((\ell _{1} ,s_{1})\)\((\ell _{2} ,s_{2})\)-convex functions, coordinated distance-disturbed \(\ell _{1}\ell _{2}\)-convex functions, coordinated distance-disturbed \((h_{1} ,h_{2})\)-convex functions, coordinated distance-disturbed \((s_{1} ,s_{2})\)-convex functions. Here results are proved for coordinated distance-disturbed h-convex functions, coordinated distance-disturbed s-convex functions and coordinated distance-disturbed convex functions. At the end, a brief conclusion is given.

Preliminaries

Definition 2.1

([26])

Let \(\psi :[\alpha ,\beta ]\to \mathbb{R}\) be termed convex if the inequality holds on an interval \([\alpha ,\beta ]\subseteq \mathbb{R}\) as

$$ \psi \bigl(t l+(1-t)r\bigr)\leq t\psi (l)+(1-t)\psi (r), $$

where \(l,r \in [\alpha ,\beta ]\) and \(t\in [0,1]\).

This inequality provides bounds of the mean value of a continuous convex function is given by the following theorem.

Theorem 2.2

If \(\Psi : U\rightarrow \mathbb{R}\) is a convex function on the interval U of real numbers, such that \({\alpha ,\beta }\in U\) with \(\alpha <\beta \), then

$$ \Psi \biggl(\frac{\alpha +\beta }{2} \biggr)\leq \frac{1}{\beta -\alpha } \int _{\alpha }^{\beta }\Psi (\xi )\,d\xi \leq \frac{\Psi (\alpha )+\Psi (\beta )}{2}. $$

Theorem 2.3

([7])

Suppose that \(\rho :\Lambda \rightarrow \mathbb{R}\) is convex on the coordinates on Λ. Then the following inequalities hold:

$$\begin{aligned} &\rho \biggl( \frac{\alpha +\beta }{2},\frac{\phi +\varphi }{2} \biggr) \\ &\quad \leq \frac{1}{2} \biggl[ \frac{1}{\beta -\alpha } \int _{\alpha }^{\beta }\rho \biggl( x, \frac{\phi +\varphi }{2} \biggr) \,dx+\frac{1}{\varphi -\phi }\int _{\phi }^{\varphi }\rho \biggl( \frac{\alpha +\beta }{2},y \biggr) \,dy \biggr] \\ &\quad \leq \frac{1}{ ( \beta -\alpha ) ( \varphi -\phi ) } \int _{\alpha }^{\beta }\int _{\phi }^{\varphi }\rho ( x,y ) \,dy \,dx \\ &\quad \leq \frac{1}{4} \biggl[ \frac{1}{\beta -\alpha } \int _{\alpha }^{\beta } \bigl[ \rho ( x,\phi ) +\rho ( x, \varphi ) \bigr] \,dx+\frac{1}{\varphi -\phi } \int _{\phi }^{\varphi } \bigl[ \rho ( \alpha ,y ) +\rho ( \beta ,y ) \bigr] \,dy \biggr] \\ &\quad \leq \frac{\rho ( \alpha ,\phi ) +\rho ( \alpha ,\varphi ) +\rho ( \beta ,\phi ) +\rho ( \beta ,\varphi ) }{4}. \end{aligned}$$

A formal definition of coordinated convex function may be stated as follows.

Definition 2.4

Let \(\rho :\Lambda \rightarrow \mathbb{R}\) be coordinated convex function on Λ, then the inequality

$$\begin{aligned} \rho \bigl( tx+ ( 1-t ) u,sy+ ( 1-s ) v \bigr) \leq & ts\rho ( x,y ) +t ( 1-s ) \rho ( x,v ) \\ &{}+ ( 1-t )s\rho ( u,y ) + ( 1-t ) ( 1-s ) \rho ( u,v ) \end{aligned}$$

holds for all \(( x,y ) , ( x,v ) , ( u,y ) , ( u,v ) \in \Lambda \) and \(s,t\in {}[ 0,1]\).

Let us recall some fundamental definitions and results which are helpful in developing main results. Further details can be found in [2124, 2635].

Definition 2.5

The left- and right-sided Riemann Liouville fractional integrals \(I_{\alpha ^{+}}^{\mu }\psi \) and \(I_{\beta ^{-}}^{\mu }\psi \) of order with \(\mu >0\), on a finite interval \([\alpha ,\beta ]\), are defined as

$$ I_{\alpha ^{+}}^{\mu }\psi (x)=\frac{1}{\Gamma (\mu )} \int _{\alpha }^{x} (x-t)^{\mu -1}\psi (t) \,dt, \quad x>\alpha , $$

and

$$ I_{\beta ^{-}}^{\mu }\psi (x)=\frac{1}{\Gamma (\mu )} \int _{x}^{\beta }(t-x)^{\mu -1}\psi (t) \,dt, \quad x< \beta , $$

respectively. Here Γ represents the usual Gamma function defined by

$$ \Gamma (t)= \int _{0}^{\infty }x^{t-1}e^{-x}\,dx, \quad \mathbb{R}(t)>0. $$

Definition 2.6

Let \(\rho \in L_{1} [\alpha ,\beta ]\) and \(k>0\). The left and right k-Riemann–Liouville integrals of order \(\mu >0\) with \(\alpha \geq 0\) are denoted by

$$ I_{\alpha +}^{\mu , k }\rho ( x ) = \frac{1}{k\Gamma _{k} ( \mu ) }\int _{\alpha }^{x} ( x-\tau ) ^{\frac{\mu }{k} -1}\rho ( \tau ) \,d\tau ,\quad x>\alpha , $$

and

$$ I_{\beta -}^{\mu , k }\rho ( x ) = \frac{1}{k\Gamma _{k} ( \mu ) }\int _{x}^{\beta } ( \tau -x ) ^{\frac{\mu }{k} -1}\rho ( \tau ) \,d\tau ,\quad x< \beta , $$

respectively. Note that when \(k\to 1\), then it reduces to the classical Riemann–Liouville fractional integral.

Definition 2.7

Let \(\rho \in L_{1} ( \Lambda )\). The Riemann–Liouville integrals \(J_{\hat{a}+,\hat{c}+}^{\mu ,\nu }\), \(J_{\hat{a}+,\hat{d}-}^{\mu ,\nu }\), \(J_{\hat{b}-,\hat{c}+}^{\mu ,\nu }\) and \(J_{\hat{b}-,\hat{d}-}^{\mu ,\nu }\) of order \(\mu ,\nu >0\) with \(\hat{a},\hat{c}\geq 0\) are defined by

$$\begin{aligned}& J_{\hat{a}+,\hat{c}+}^{\mu ,\nu }\rho ( x,y ) = \frac{1}{\Gamma ( \mu ) \Gamma ( \nu ) } \int _{\hat{a}}^{x} \int _{\hat{c}}^{y} ( x-\jmath _{1} )^{\mu -1} ( y-\jmath _{2} ) ^{\nu -1}\rho ( \jmath _{1}, \jmath _{2} ) \,d\jmath _{2}\,d\jmath _{1},\quad x>\hat{a}, y>\hat{c}, \\& J_{\hat{a}+,\hat{d}-}^{\mu ,\nu }\rho ( x,y ) = \frac{1}{\Gamma ( \mu ) \Gamma ( \nu ) } \int _{\hat{a}}^{x} \int _{y}^{\hat{d}} ( x-\jmath _{1} )^{\mu -1} ( \jmath _{2}-y ) ^{\nu -1}\rho ( \jmath _{1}, \jmath _{2} ) \,d\jmath _{2}\,d\jmath _{1},\quad x>\hat{a}, y< \hat{d}, \\& J_{\hat{b}-,\hat{c}+}^{\mu ,\nu }\rho ( x,y ) = \frac{1}{\Gamma ( \mu ) \Gamma ( \nu ) } \int _{x}^{\hat{b}} \int _{\hat{c}}^{y} ( \jmath _{1}-x )^{\mu -1} ( y-\jmath _{2} ) ^{\nu -1}\rho ( \jmath _{1}, \jmath _{2} ) \,d\jmath _{2}\,d\jmath _{1},\quad x< \hat{b}, y>\hat{c}, \end{aligned}$$

and

$$ J_{\hat{b}-,\hat{d}-}^{\mu ,\nu }\rho ( x,y ) = \frac{1}{\Gamma ( \mu ) \Gamma ( \nu ) } \int _{x}^{\hat{b}} \int _{y}^{\hat{d}} ( \jmath _{1}-x )^{\mu -1} ( \jmath _{2}-y ) ^{\nu -1}\rho ( \jmath _{1}, \jmath _{2} ) \,d\jmath _{2}\,d\jmath _{1},\quad x< \hat{b}, y< \hat{d}, $$

respectively. Furthermore,

$$ J_{\hat{a}+,\hat{c}+}^{0,0}\rho ( x,y ) =J_{\hat{a}+, \hat{d}-}^{0,0} \rho ( x,y ) =J_{\hat{b}-,\hat{c}+}^{0,0} \rho ( x,y ) =J_{\hat{b}-,\hat{d}-}^{0,0} \rho ( x,y ) =\rho ( x,y ) $$

and

$$ J_{\hat{a}+,\hat{d}-}^{1,1}\rho ( x,y ) = \frac{1}{\Gamma ( \mu ) \Gamma ( \nu ) } \int _{\hat{a}}^{x} \int _{y}^{\hat{d}} \rho ( \jmath _{1},\jmath _{2} ) \,d\jmath _{2}\,d\jmath _{1}. $$

Similar to Definition 2.5, Sarikaya [32] introduced the following fractional integrals:

$$\begin{aligned}& J_{\hat{a}+}^{\mu }\rho \biggl( x,\frac{\hat{c}+\hat{d}}{2} \biggr) = \frac{1}{\Gamma ( \mu ) } \int _{\hat{a}}^{x} ( x-\jmath _{1} )^{\mu -1} \rho \biggl( \jmath _{1},\frac{\hat{c}+\hat{d}}{2} \biggr) \,d \jmath _{1}, \quad x>\hat{a}, \\& J_{\hat{b}-}^{\mu }\rho \biggl( x,\frac{\hat{c}+\hat{d}}{2} \biggr) = \frac{1}{\Gamma ( \mu ) } \int _{x}^{\hat{b}} ( \jmath _{1}-x )^{\mu -1} \rho \biggl( \jmath _{1},\frac{\hat{c}+\hat{d}}{2} \biggr) \,d \jmath _{1}, \quad x< \hat{b}, \\& J_{\hat{c}+}^{\nu }\rho \biggl( \frac{\hat{a}+\hat{b}}{2},y \biggr) = \frac{1}{\Gamma (\nu ) } \int _{\hat{c}}^{y} ( y- \jmath _{2} )^{\nu -1}\rho \biggl( \frac{\hat{a}+\hat{b}}{2},\jmath _{2} \biggr) \,d\jmath _{2},\quad y>\hat{c}, \\& J_{\hat{d}-}^{\nu }\rho \biggl( \frac{\hat{a}+\hat{b}}{2},y \biggr) = \frac{1}{\Gamma ( \nu ) } \int _{y}^{\hat{d}} ( \jmath _{2}-y )^{\nu -1}\rho \biggl( \frac{\hat{a}+\hat{b}}{2},\jmath _{2} \biggr)\,d\jmath _{2},\quad y< \hat{d}. \end{aligned}$$

Sarikaya gave the following remarkable results in [32].

Theorem 2.8

Let \(\rho :\Omega \subset \mathbb{R}^{2}\rightarrow \mathbb{R}\) be a coordinated convex function on \(\Omega := [ \hat{a},\hat{b} ] \times [\hat{c},\hat{d} ] \in \mathbb{R}^{2}\) with \(0\leq \hat{a}<\hat{b}\), \(0\leq \hat{c}<\hat{d}\) and \(\rho \in L_{1} ( \Omega ) \). Then one has the inequalities:

$$\begin{aligned} \rho \biggl( \frac{\hat{a}+\hat{b}}{2},\frac{\hat{c}+\hat{d}}{2} \biggr) \leq & \frac{\Gamma ( \mu +1 ) \Gamma ( \nu +1 ) }{4 ( \hat{b}-\hat{a} ) ^{\mu } ( \hat{d}-\hat{c} ) ^{\nu }} \\ &{}\times \bigl[ J_{\hat{a}+,\hat{c}+}^{\mu ,\nu }\rho ( \hat{b},\hat{d} ) +J_{\hat{a}+,\hat{d}-}^{\mu ,\nu }\rho ( \hat{b},\hat{c} )+J_{\hat{b}-,\hat{c}+}^{\mu ,\nu } \rho ( \hat{a},\hat{d} ) +J_{\hat{b}-,\hat{d}-}^{\mu ,\nu }\rho ( \hat{a},\hat{c} ) \bigr] \\ \leq &\frac{\rho ( \hat{a},\hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) }{4}. \end{aligned}$$

Now, we are in a position to introduce the following extended Riemann–Liouville integrals.

Definition 2.9

Let \(\rho \in L_{1} ( \Omega )\) and \(k_{1}, k_{2}>0\). The \((k_{1},k_{2})\)-Riemann–Liouville integrals \(I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1},k_{2} }\), \(I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1},k_{2} }\), \(I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1},k_{2} }\) and \(I_{\hat{b}-,\hat{d}}^{\mu ,\nu , k_{1},k_{2} }\) of order \(\mu ,\nu >0\) with \(\hat{a},\hat{c}\geq 0\) are defined by

$$\begin{aligned} &I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1},k_{2} }\rho ( x,y ) \\ &\quad = \frac{1}{k_{1}k_{2}\Gamma _{k_{1}} ( \mu ) \Gamma _{k_{2}} ( \nu ) } \int _{\hat{a}}^{x} \int _{\hat{c}}^{y} ( x-\jmath _{1} )^{\frac{\mu }{k_{1}} -1} ( y-\jmath _{2} ) ^{\frac{\nu }{k_{2}} -1}\rho ( \jmath _{1},\jmath _{2} ) \,d \jmath _{2}\,d\jmath _{1},\quad x>\hat{a}, y>\hat{c}, \\ &I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1},k_{2} }\rho ( x,y ) \\ &\quad = \frac{1}{k_{1}k_{2}\Gamma _{k_{1}} ( \mu ) \Gamma _{k_{2}} ( \nu ) } \int _{\hat{a}}^{x} \int _{y}^{\hat{d}} ( x-\jmath _{1} )^{\frac{\mu }{k_{1}} -1} ( \jmath _{2}-y ) ^{\frac{\nu }{k_{2}} -1}\rho ( \jmath _{1},\jmath _{2} ) \,d \jmath _{2}\,d\jmath _{1},\quad x>\hat{a}, y< \hat{d}, \\ &I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1},k_{2} }\rho ( x,y ) \\ &\quad = \frac{1}{k_{1}k_{2}\Gamma _{k_{1}} ( \mu ) \Gamma _{k_{2}} ( \nu ) } \int _{x}^{\hat{b}} \int _{\hat{c}}^{y} ( \jmath _{1}-x )^{\frac{\mu }{k_{1}} -1} ( y-\jmath _{2} ) ^{\frac{\nu }{k_{2}} -1}\rho ( \jmath _{1},\jmath _{2} ) \,d \jmath _{2}\,d\jmath _{1},\quad x< \hat{b}, y>\hat{c} \end{aligned}$$

and

$$\begin{aligned} &I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1},k_{2} }\rho ( x,y ) \\ &\quad = \frac{1}{k_{1}k_{2}\Gamma _{k_{1}} ( \mu ) \Gamma _{k_{2}} ( \nu ) } \int _{x}^{\hat{b}} \int _{y}^{\hat{d}} ( \jmath _{1}-x )^{\frac{\mu }{k_{1}} -1} ( \jmath _{2}-y ) ^{\frac{\nu }{k_{2}} -1}\rho ( \jmath _{1},\jmath _{2} ) \,d \jmath _{2}\,d\jmath _{1},\quad x< \hat{b}, y< \hat{d}, \end{aligned}$$

respectively. Note that when \(k_{1}, k_{2}\to 1\), then it reduces to Definition 2.7.

Noor et al. in [27], introduced the notion of coordinated \(\ell _{1}\ell _{2}\)-convex functions to generalize the \(\ell _{1}\)-convex functions as follows.

Definition 2.10

Let \(\Lambda \subset \mathbb{R}^{2}\) be a rectangle. A function \(\rho :\Lambda \rightarrow \mathbb{R}\) is said to be two dimensional (coordinated) \(\ell _{1}\ell _{2}\)-convex function, if

$$\begin{aligned} &\rho \bigl(\bigl[tx^{\ell _{1}}+(1-t)u^{\ell _{1}}\bigr]^{\frac{1}{\ell _{1}}}, \bigl[ry^{\ell _{2}}+(1-r)v^{\ell _{2}}\bigr]^{\frac{1}{\ell _{2}}} \bigr) \\ &\quad \leq tr\rho (x,y)+t(1-r)\rho (x,v)+(1-t)r\rho (u,y)+(1-t) (1-r)\rho (u,v) \end{aligned}$$

for all \((x,y),(x,v),(u,y),(u,v)\in \Lambda \) and \(r,t\in [0,1]\).

Theorem 2.11

([27])

Let \(\rho :\Omega \subset \mathbb{R}^{2}\rightarrow \mathbb{R}\) be a \(\ell _{1}\ell _{2}\)-convex function on the coordinates on Ω, then the following inequalities hold:

$$\begin{aligned} &\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\ell _{1}\ell _{2}}{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) }\int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}}x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) \,dy \,dx \\ &\quad \leq \frac{\rho ( \hat{a},\hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) }{4}. \end{aligned}$$

Yang in [38], generalized this concept by defining a larger class of coordinated convex functions termed coordinated \(( \ell _{1} ,h_{1})\)\(( \ell _{2},h_{2})\)-convex function as follows:

Definition 2.12

Let \(\hat{h}_{1},\hat{h}_{2}:J\rightarrow \mathbb{R}\) be two non-negative and non-zero mappings. A mapping \(\rho :\Lambda \rightarrow \mathbb{R}\) is said to be \(( \ell _{1} ,\hat{h}_{1})\)\(( \ell _{2},\hat{h}_{2})\)-convex function on the coordinates on Λ, if the mappings \(\rho _{y}:[\hat{a},\hat{b}]\rightarrow \mathbb{R}\), \(\rho _{y}(u)=\rho (u,y)\) and \(\rho _{x}:[\hat{c},\hat{d}]\rightarrow \mathbb{R}\), \(\rho _{x}(v)=\rho (x,v)\) are \((\ell _{1},\hat{h}_{1})\)-convex with respect to u on \([\hat{a},\hat{b}]\) and \((\ell _{2},\hat{h}_{2})\)-convex with respect to v on \([\hat{c},\hat{d}]\), respectively, for all \(y\in [\hat{c},\hat{d}]\) and \(x\in [\hat{a},\hat{b}]\).

From the above definition, we can say that, if ρ is a coordinated \((\ell _{1} ,\hat{h}_{1})\)\(( \ell _{2},\hat{h}_{2})\)-convex function, then the following inequality holds:

$$\begin{aligned} &\rho \bigl(\bigl[tx^{\ell _{1}}+(1-t)u^{\ell _{1}} \bigr]^{\frac{1}{\ell _{1}}},\bigl[ry^{\ell _{2}}+(1-r)v^{\ell _{2}} \bigr]^{\frac{1}{\ell _{2}}} \bigr) \\ &\quad \leq \hat{h}_{1}(t)\hat{h}_{2}(r)\rho (x,y)+ \hat{h}_{1}(t)\hat{h}_{2}(1-r) \rho (x,v) \\ &\qquad {}+\hat{h}_{1}(1-t)\hat{h}_{2}(r)\rho (u,y) \\ &\qquad {}+\hat{h}_{1}(1-t)\hat{h}_{2}(1-r)\rho (u,v). \end{aligned}$$

Remark 2.13

If \(\ell _{1}=\ell _{2}=1\), then the function ρ will be reduced to coordinated \((\hat{h}_{1},\hat{h}_{2})\)-convex function.

Remark 2.14

If \(\hat{h}_{1}(t)=t^{s_{1}}\) and \(\hat{h}_{2}(t)=t^{s_{2}}\), then the function ρ will be called a coordinated \((\ell _{1},s_{1})\)\(( \ell _{2},s_{2})\)-convex function.

Remark 2.15

If \(\hat{h}_{1}(t)=t^{s_{1}}\), \(\hat{h}_{2}(t)=t^{s_{2}}\) and \(\ell _{1}=\ell _{2}=1\), then the function ρ will be called a coordinated \((s_{1},s_{2})\)-convex function.

Yang in [38], gave the following two interesting results.

Theorem 2.16

Let \(\rho :\Omega \rightarrow \mathbb{R}\) be a \(( \ell _{1},\hat{h}_{1} ) \)\(( \ell _{2},\hat{h}_{2} ) \)-convex function on the coordinates on Ω. Then one has the inequalities

$$\begin{aligned} & \frac{1}{4\hat{h}_{1} ( \frac{1}{2} ) \hat{h}_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\ell _{1}\ell _{2}}{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) } \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}}x^{\ell _{1}-1}y^{\ell _{2}-1}\rho (x,y)\,dy \,dx \\ &\quad \leq \bigl[\rho ( \hat{a},\hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho (\hat{b},\hat{d} ) \bigr] \int _{0}^{1}\hat{h}_{1}(t)\,dt \int _{0}^{1}\hat{h}_{2}(t)\,dt. \end{aligned}$$

Theorem 2.17

Let \(\rho :\Omega \rightarrow \mathbb{R}\) be a \(( \ell _{1},\hat{h}_{1} ) \)\(( \ell _{2},\hat{h}_{2} ) \)-convex function on the coordinates on Ω. Then one has the inequalities

$$\begin{aligned} &\frac{1}{4\hat{h}_{1} ( \frac{1}{2} ) \hat{h}_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\ell _{1}}{4\hat{h}_{1} ( \frac{1}{2} ) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) } \int _{\hat{a}}^{\hat{b}}x^{\ell _{1}-1}\rho \biggl( x, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \,dx \\ &\qquad {}+ \frac{\ell _{2}}{4\hat{h}_{2} ( \frac{1}{2} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) } \int _{\hat{c}}^{\hat{d}}y^{\ell _{2}-1}\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},y \biggr) \,dy \\ &\quad \leq \frac{\ell _{1}\ell _{2}}{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) } \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}}x^{\ell _{1}-1}y^{\ell _{2}-1}\rho (x,y)\,dy \,dx \\ &\quad \leq \frac{\ell _{1}}{2 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) } \biggl[ \int _{\hat{a}}^{\hat{b}}x^{\ell _{1}-1}\rho ( x, \hat{c} ) \,dx+ \int _{\hat{a}}^{\hat{b}}x^{\ell _{1}-1} \rho ( x,\hat{d} ) \,dx \biggr] \int _{0}^{1} \hat{h}_{2}(t)\,dt \\ &\qquad {}+ \frac{\ell _{2}}{2 ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) } \biggl[ \int _{\hat{c}}^{\hat{d}}y^{\ell _{2}-1}\rho ( \hat{a},y ) \,dy+ \int _{\hat{c}}^{\hat{d}}y^{\ell _{2}-1} \rho ( \hat{b},y ) \,dy \biggr] \int _{0}^{1} \hat{h}_{1}(t)\,dt \\ &\quad \leq \bigl[ \rho ( \hat{a},\hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \int _{0}^{1}\hat{h}_{1}(t)\,dt \int _{0}^{1}\hat{h}_{2}(t)\,dt. \end{aligned}$$

Definition 2.18

([21])

\(X_{\hat{c}}^{p}(\hat{a},\hat{b})\) (\(\hat{c}\in \mathbb{R}\), \(1\leq p\leq \infty \)) is the set of those complex valued Lebesgue measurable functions ρ of \([\hat{a},\hat{b}]\) for which \(\Vert \rho \Vert _{X_{\hat{c}}^{p}}<\infty \), where the norm is defined by

$$ \Vert \rho \Vert _{X_{\hat{c}}^{p}}= \biggl( \int _{\hat{a}}^{\hat{b}} \bigl\vert t^{\hat{c}}\rho (t) \bigr\vert ^{p}\frac{dt}{t} \biggr) ^{\frac{1}{p}}< \infty \quad \text{for } 1\leq p< \infty , \hat{c} \in \mathbb{R}, $$

and for the case \(p=\infty \),

$$ \Vert \rho \Vert _{X_{\hat{c}}^{p}}=ess \sup_{\hat{a}\leq t\leq \hat{b}} \bigl[t^{\hat{c}} \bigl\vert \rho (t) \bigr\vert \bigr], \quad \hat{c}\in \mathbb{R}. $$

Katugampola introduced a new fractional integral which generalizes the Riemann–Liouville and Hadamard fractional integrals in a single form as follows; see [1824, 2630].

Definition 2.19

Let \([\hat{a},\hat{b}]\subseteq \mathbb{R} \) be a finite interval. Then the left- and right-sided Katugampola fractional integrals of order \(\mu >0\) of \(\in X_{\hat{c}}^{p}(\hat{a},\hat{b})\) with \(\hat{a}\geq 0\) are defined by

$$ {}^{r}I_{\hat{a}+}^{\mu }\rho ( x ) = \frac{r ^{1-\mu }}{\Gamma ( \mu ) } \int _{\hat{a}}^{x}\frac{t^{r -1}}{ ( x^{r }-t^{r } ) ^{1-\mu }}\rho ( t ) \,dt $$

and

$$ {}^{r}I_{\hat{b}-}^{\mu }\rho ( x ) = \frac{r ^{1-\mu }}{\Gamma ( \mu ) } \int _{x}^{\hat{b}}\frac{t^{r -1}}{ ( t^{r }-x^{r } ) ^{1-\mu }}\rho ( t ) \,dt $$

with \(\hat{a}< x<\hat{b}\) and \(r>0\), provided the integrals exist.

Definition 2.20

Let \([\hat{a},\hat{b}]\subseteq \mathbb{R} \) be a finite interval and \(k>0\). Then the left- and right-sided Katugampola k-fractional integrals of order \(\mu >0\) of \(\rho \in X_{\hat{c}}^{p}(\hat{a},\hat{b})\) with \(\hat{a}\geq 0\) are defined by

$$ {}^{r}I_{\hat{a}+}^{\mu , k }\rho ( x ) = \frac{r ^{1-\frac{\mu }{k} }}{k\Gamma _{k} ( \mu ) } \int _{\hat{a}}^{x} \frac{t^{r -1}}{ ( x^{r }-t^{r } ) ^{1-\frac{\mu }{k} }}\rho ( t ) \,dt $$

and

$$ {}^{r}I_{\hat{b}-}^{\mu , k }\rho ( x ) = \frac{r ^{1-\frac{\mu }{k} }}{k\Gamma _{k} ( \mu ) } \int _{x}^{\hat{b}} \frac{t^{r -1}}{ ( t^{r }-x^{r } ) ^{1-\frac{\mu }{k} }}\rho ( t ) \,dt $$

with \(\hat{a}< x<\hat{b}\) and \(\rho >0\), provided the integrals exist. Note that when \(k\to 1\), then it reduces to Definition 2.19.

Katugampola fractional integrals into two dimensional case may be given as follows.

Definition 2.21

Let \(\rho \in X^{p} ( \Omega ) \). The Katugampola fractional integrals \({}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu ,\nu }\), \({}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu }\), \({}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu }\) and \({}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu }\) of order \(\mu ,\nu >0\) with \(\hat{a},\hat{c}\geq 0\) are defined by

$$\begin{aligned}& {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu ,\nu }\rho ( x,y ) \\& \quad = \frac{\ell _{1}^{1-\mu } \ell _{2}^{1-\nu }}{\Gamma ( \mu ) \Gamma ( \nu ) } \int _{\hat{a}}^{x} \int _{\hat{c}}^{y} \frac{t^{\ell _{1}-1}s^{\ell _{2}-1}}{ ( x^{\ell _{1}}-t^{\ell _{1}} ) ^{1-\mu } ( y^{\ell _{2}}-s^{\ell _{2}} ) ^{1-\nu }}\rho ( t,s ) \,ds \,dt,\quad x>\hat{a}, y>\hat{c}, \\& {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu }\rho ( x,y ) \\& \quad= \frac{\ell _{1}^{1-\mu } \ell _{2}^{1-\nu }}{\Gamma ( \mu ) \Gamma ( \nu ) } \int _{\hat{a}}^{x} \int _{y}^{\hat{d}} \frac{t^{\ell _{1}-1}s^{\ell _{2}-1}}{ ( x^{\ell _{1}}-t^{\ell _{1}} ) ^{1-\mu } ( s^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\nu }}\rho ( t,s ) \,ds \,dt,\quad x>\hat{a}, y< \hat{d}, \\& {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu }\rho ( x,y ) \\& \quad= \frac{\ell _{1}^{1-\mu } \ell _{2}^{1-\nu }}{\Gamma ( \mu ) \Gamma ( \nu ) } \int _{x}^{\hat{b}} \int _{\hat{c}}^{y} \frac{t^{\ell _{1}-1}s^{\ell _{2}-1}}{ ( t^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\mu } ( y^{\ell _{2}}-s^{\ell _{2}} ) ^{1-\nu }}\rho ( t,s ) \,ds \,dt,\quad x< \hat{b}, y>\hat{c}, \end{aligned}$$

and

$$ \begin{gathered} {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu }\rho ( x,y ) \\ \quad= \frac{\ell _{1}^{1-\mu } \ell _{2}^{1-\nu }}{\Gamma ( \mu ) \Gamma ( \nu ) } \int _{x}^{\hat{b}} \int _{y}^{\hat{d}} \frac{t^{\ell _{1}-1}s^{\ell _{2}-1}}{ ( t^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\mu } ( s^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\nu }}\rho ( t,s ) \,ds \,dt,\quad x< \hat{b}, y< \hat{d}, \end{gathered} $$

respectively, and \(\ell _{1},\ell _{2}>0\). Moreover,

$$ {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{0,0}\rho ( x,y ) = {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{0,0}\rho ( x,y ) = {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{0,0} \rho ( x,y ) = {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{0,0} \rho ( x,y ) =\rho ( x,y ) $$

and

$$ {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{1,1}\rho ( x,y ) = \int _{\hat{a}}^{x} \int _{y}^{\hat{d}}t^{\ell _{1}-1}s^{\ell _{2}-1}\rho ( t,s ) \,ds \,dt. $$

Similar to Definition 2.19, we introduce the following fractional integrals:

$$\begin{aligned}& {}^{\ell _{1}}I_{\hat{a}+}^{\mu }\rho \biggl( x, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) = \frac{\ell _{1}^{1-\mu }}{\Gamma ( \mu ) } \int _{\hat{a}}^{x} \frac{t^{\ell _{1}-1}}{ ( x^{\ell _{1}}-t^{\ell _{1}} ) ^{1-\mu }}\rho \biggl( t, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \,dt,\quad x>\hat{a}, \\& {}^{\ell _{1}}I_{\hat{b}-}^{\mu }\rho \biggl( x, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) = \frac{\ell _{1}^{1-\mu }}{\Gamma ( \mu ) } \int _{x}^{\hat{b}} \frac{t^{\ell _{1}-1}}{ ( t^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\mu }}\rho \biggl( t, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \,dt,\quad x< \hat{b}, \\& {}^{\ell _{2}}I_{\hat{c}+}^{\nu }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},y \biggr) = \frac{\ell _{2}^{1-\nu }}{\Gamma ( \nu ) } \int _{\hat{c}}^{y} \frac{s^{\ell _{2}-1}}{ ( y^{\ell _{2}}-s^{\ell _{2}} ) ^{1-\nu }}\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},s \biggr) \,ds,\quad y>\hat{c}, \\& {}^{\ell _{2}}I_{\hat{d}-}^{\nu }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},y \biggr) = \frac{\ell _{2}^{1-\nu }}{\Gamma ( \nu ) } \int _{y}^{\hat{d}} \frac{s^{\ell _{2}-1}}{ ( s^{\ell _{2}}-y^{\ell _{2}} )^{1-\nu }}\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},s \biggr) \,ds,\quad y< \hat{d}. \end{aligned}$$

It is important to notice that, if \(\ell _{1} =\ell _{2} =1\), then the Katugampola fractional integrals reduce to Riemann–Liouville fractional integrals given in Definition 2.7.

Similarly, we can define the extended Katugampola fractional integrals in the two dimensional case as follows.

Definition 2.22

Let \(\rho \in X^{p} ( \Omega )\) and \(k_{1}, k_{2}>0\). The Katugampola \((k_{1}, k_{2})\)-fractional integrals \({}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\), \({}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\), \({}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\) and \({}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\) of order \(\mu ,\nu >0\) with \(\hat{a},\hat{c}\geq 0\) are defined by

$$\begin{aligned} &{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( x,y ) \\ &\quad = \frac{\ell _{1}^{1-\frac{\mu }{k_{1}} } \ell _{2}^{1-\frac{\nu }{k_{2}} }}{k_{1}k_{2}\Gamma _{k_{1}} ( \mu ) \Gamma _{k_{2}} ( \nu ) } \int _{\hat{a}}^{x} \int _{\hat{c}}^{y} \frac{t^{\ell _{1}-1}s^{\ell _{2}-1}}{ ( x^{\ell _{1}}-t^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( y^{\ell _{2}}-s^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\rho ( t,s ) \,ds \,dt, \quad x>\hat{a}, y>\hat{c}, \\ &{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( x,y ) \\ &\quad = \frac{ \ell _{1}^{1-\frac{\mu }{k_{1}} } \ell _{2}^{1-\frac{\nu }{k_{2}} }}{k_{1}k_{2}\Gamma _{k_{1}} ( \mu ) \Gamma _{k_{2}} ( \nu ) } \int _{\hat{a}}^{x} \int _{y}^{\hat{d}} \frac{t^{\ell _{1}-1}s^{\ell _{2}-1}}{ ( x^{\ell _{1}}-t^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( s^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\rho ( t,s ) \,ds \,dt, \quad x>\hat{a}, y< \hat{d}, \\ &{}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( x,y ) \\ &\quad = \frac{\ell _{1}^{1-\frac{\mu }{k_{1}} } \ell _{2}^{1-\frac{\nu }{k_{2}} }}{k_{1}k_{2}\Gamma _{k_{1}} ( \mu ) \Gamma _{k_{2}} ( \nu ) } \int _{x}^{\hat{b}} \int _{\hat{c}}^{y} \frac{t^{\ell _{1}-1}s^{\ell _{2}-1}}{ ( t^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( y^{\ell _{2}}-s^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\rho ( t,s ) \,ds \,dt, \quad x< \hat{b},y>\hat{c}, \end{aligned}$$

and

$$\begin{aligned} &{}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( x,y ) \\ &\quad = \frac{\ell _{1}^{1-\frac{\mu }{k_{1}} } \ell _{2}^{1-\frac{\nu }{k_{2}} }}{k_{1}k_{2}\Gamma _{k_{1}} ( \mu ) \Gamma _{k_{2}} ( \nu ) } \int _{x}^{\hat{b}} \int _{y}^{\hat{d}} \frac{t^{\ell _{1}-1}s^{\ell _{2}-1}}{ ( t^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( s^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\rho ( t,s ) \,ds \,dt, \quad x< \hat{b}, y< \hat{d}, \end{aligned}$$

respectively, and \(\ell _{1},\ell _{2}>0\). Note that when \(k_{1}, k_{2}\to 1\), then it reduces to Definition 2.21.

We also introduce the following useful fractional integrals:

$$\begin{aligned}& {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} }\rho \biggl( x, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\& \quad = \frac{\ell _{1}^{1-\frac{\mu }{k_{1}} }}{k_{1}\Gamma _{k_{1}} ( \mu ) } \int _{\hat{a}}^{x} \frac{t^{\ell _{1}-1}}{ ( x^{\ell _{1}}-t^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }}\rho \biggl( t, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \,dt,\quad x>\hat{a}, \\& {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} }\rho \biggl( x, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\& \quad= \frac{\ell _{1}^{1-\frac{\mu }{k_{1}} }}{k_{1}\Gamma _{k_{1}} ( \mu ) } \int _{x}^{\hat{b}} \frac{t^{\ell _{1}-1}}{ ( t^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }}\rho \biggl( t, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \,dt,\quad x< \hat{b}, \\& {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},y \biggr)\\& \quad = \frac{\ell _{2}^{1-\frac{\nu }{k_{2}} }}{k_{2}\Gamma _{k_{2}} ( \nu ) } \int _{\hat{c}}^{y} \frac{s^{\ell _{2}-1}}{ ( y^{\ell _{2}}-s^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},s \biggr) \,ds,\quad y>\hat{c}, \\& {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},y \biggr) \\& \quad= \frac{\ell _{2}^{1-\frac{\nu }{k_{2}} }}{k_{2}\Gamma _{k_{2}} ( \nu ) } \int _{y}^{\hat{d}} \frac{s^{\ell _{2}-1}}{ ( s^{\ell _{2}}-y^{\ell _{2}} )^{1-\frac{\nu }{k_{2}} }}\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},s \biggr) \,ds,\quad y< \hat{d}. \end{aligned}$$

Definition 2.23

Let \(h_{1},h_{2}:J\rightarrow \mathbb{R}\) be two non-negative and non-zero functions. Assume that \(\sigma _{1}, \sigma _{2}>0\). A mapping \(\rho :\Omega \rightarrow \mathbb{R}\) is said to be a distance-disturbed \(( \ell _{1} ,h_{1})\)\(( \ell _{2},h_{2})\)-convex function on the coordinates on Ω with modulus \(\mu _{1}, \mu _{2} >0\) of higher orders \((\sigma _{1},\sigma _{2})\), if the partial mappings \(\rho _{y}:[\hat{a},\hat{b}]\rightarrow \mathbb{R}\), \(\rho _{y}(u)=\rho (u,y)\) and \(\rho _{x}:[\hat{c},\hat{d}]\rightarrow \mathbb{R}\), \(\rho _{x}(v)=\rho (x,v)\) are, respectively, distance-disturbed \((\ell _{1},h_{1})\)-convex with modulus \(\mu _{1}>0\) of order \(\sigma _{1}>0\) with respect to u on \([\hat{a},\hat{b}]\) and distance-disturbed \((\ell _{2},h_{2})\)-convex with modulus \(\mu _{2}>0\) of order \(\sigma _{2}>0\) with respect to v on \([\hat{c},\hat{d}]\), for all \(y\in [\hat{c},\hat{d}]\) and \(x\in [\hat{a},\hat{b}]\).

From the above definition, we can say that, if f is a coordinated distance-disturbed \(( \ell _{1} ,h_{1})\)\(( \ell _{2},h_{2})\)-convex function with modulus \(\mu _{1}, \mu _{2} >0\) of higher orders \((\sigma _{1},\sigma _{2})\), then the following inequality holds:

$$\begin{aligned} &\rho \bigl(\bigl[tx^{\ell _{1}}+(1-t)u^{\ell _{1}} \bigr]^{\frac{1}{\ell _{1}}},\bigl[ry^{\ell _{2}}+(1-r)v^{\ell _{2}} \bigr]^{\frac{1}{\ell _{2}}} \bigr) \\ &\qquad {}+\mu _{1}t(1-t) \bigl(u^{\ell _{1}}-x^{\ell _{1}} \bigr)^{\sigma _{1}}+ \mu _{2}r(1-r) \bigl(v^{\ell _{2}}-y^{\ell _{2}} \bigr)^{\sigma _{2}} \\ &\quad \leq h_{1}(t)h_{2}(r)\rho (x,y)+h_{1}(t)h_{2}(1-r) \rho (x,v) +h_{1}(1-t)h_{2}(r) \rho (u,y) \\ &\qquad {}+h_{1}(1-t)h_{2}(1-r)\rho (u,v). \end{aligned}$$

Remark 2.24

If \(\mu _{1},\mu _{2}\to 0^{+}\), then Definition 2.23 will be reduced to Definition 2.12.

Remark 2.25

If \(\ell _{1}=\ell _{2}=\ell \), then the function f will be reduced to coordinated distance-disturbed \((\ell ,h_{1})\)\((\ell ,h_{2})\)-convex function of higher orders. If \(\ell _{1}=\ell _{2}=1\), then the function f will be reduced to coordinated distance-disturbed \((h_{1},h_{2})\)-convex function of higher orders.

Remark 2.26

If \(h_{1}(t)=t^{s_{1}}\) and \(h_{2}(t)=t^{s_{2}}\), then the function f will be called a coordinated distance-disturbed \((\ell _{1},s_{1})\)\((\ell _{2},s_{2})\)-convex function of higher orders. If \(h_{1}(t)=t^{s_{1}}\), \(h_{2}(t)=t^{s_{2}}\) and \(\ell _{1}=\ell _{2}=\ell \), then the function f will be called a coordinated distance-disturbed \((\ell ,s_{1})\)\((\ell ,s_{2})\)-convex function of higher orders. If \(h_{1}(t)=t^{s_{1}}\), \(h_{2}(t)=t^{s_{2}}\) and \(\ell _{1}=\ell _{2}=1\), then the function f will be called a coordinated distance-disturbed \((s_{1},s_{2})\)-convex function of higher orders.

Main results

In this section we give the trapezium type inequalities by using distance-disturbed \(( \ell _{1} ,h_{1})\)-\(( \ell _{2},h_{2})\)-convex functions with modulus \(\mu _{1}, \mu _{2} >0\) of higher orders \((\sigma _{1},\sigma _{2})\), where \(\sigma _{1},\sigma _{2}>0\) on the coordinates on Ω.

Theorem 3.1

Suppose that \(\rho :\Omega \rightarrow \mathbb{R}\) is a distance-disturbed \(( \ell _{1} ,h_{1})\)-\(( \ell _{2},h_{2})\)-convex function of higher orders on the coordinates on Ω and \(\rho \in L_{1}(\Omega )\). Then one has the inequalities

$$\begin{aligned} &\frac{1}{4h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) + A \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{4 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu , k_{1}, k_{2} }\rho ( \hat{b},\hat{d} ) \\ &\qquad {}+{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2}}\rho ( \hat{b},\hat{c} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a}, \hat{d} ) + {}^{{\ell _{1},\ell _{2}}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a}, \hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{1}\,d \imath _{2} \\ &\qquad {}- \frac{\mu _{1} [ (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}+ (\hat{a}^{\ell _{1}} -\hat{b}^{\ell _{1}} )^{\sigma _{1}} ]+\mu _{2} [ (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}+ (\hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} )^{\sigma _{2}} ]}{2}, \end{aligned}$$
(3.1)

where

$$ A=-\frac{\mu \nu }{16k_{1}k_{2}} \biggl[\frac{\mu _{1}k_{2}}{\nu } \biggl( \frac{1}{2\hat{b}^{\ell _{1}}} \biggr)^{\frac{\mu }{k_{1}}}C_{1}+ \frac{\mu _{2}k_{1}}{\mu } \biggl( \frac{1}{2\hat{d}^{\ell _{2}}} \biggr)^{\frac{\nu }{k_{2}}}C_{2} \biggr] $$

and

$$ C_{1}= \int _{\hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}}}^{\hat{a}^{\ell _{1}}+ \hat{b}^{\ell _{1}}}\imath _{1}^{\sigma _{1}} \bigl[\imath _{1}-\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}} \bigr]^{\frac{\mu }{k_{1}}-1} \,d\imath _{1}, \qquad C_{2}= \int _{\hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}}}^{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}\imath _{2}^{\sigma _{2}} \bigl[\imath _{2}- \hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}} \bigr]^{\frac{\nu }{k_{2}}-1} \,d \imath _{2}. $$

Proof

Let \(x^{\ell _{1}}=\imath _{1}\hat{a}^{\ell _{1}}+ ( 1-\imath _{1} ) \hat{b}^{\ell _{1}}\), \(y^{\ell _{1}}= ( 1-\imath _{1} ) \hat{a}^{\ell _{1}}+ \imath _{1}\hat{b}^{\ell _{1}}\) and \(u^{\ell _{2}}=\imath _{2}\hat{c}^{\ell _{2}}+ ( 1-\imath _{2} ) \hat{d}^{\ell _{2}}\), \(v^{\ell _{2}}= ( 1-\imath _{2} ) \hat{c}^{\ell _{2}}+ \imath _{2}\hat{d}^{\ell _{2}}\), then, by coordinated distance-disturbed \(( \ell _{1} ,h_{1})\)-\(( \ell _{2},h_{2})\)-convexity of higher orders of ρ, we have

$$\begin{aligned} &\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad = \rho \biggl( \biggl[ \frac{x^{\ell _{1}}+y^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}}, \biggl[ \frac{u^{\ell _{2}}+v^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq h_{1} \biggl( \frac{1}{2} \biggr) h_{2} \biggl( \frac{1}{2} \biggr) \bigl[ \rho ( x,u ) +\rho ( x,v ) + \rho ( y,u ) +\rho ( y,v ) \bigr] \\ &\qquad {}-\frac{\mu _{1}}{4} \bigl(y^{\ell _{1}}-x^{\ell _{1}} \bigr)^{\sigma _{1}}-\frac{\mu _{2}}{4} \bigl(v^{\ell _{2}}-u^{\ell _{2}} \bigr)^{\sigma _{2}}. \end{aligned}$$

Multiplying by \(\frac{\mu \nu }{4k_{1}k_{2}}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1}\) and integrating over \(( [ 0,1 ] \times [ 0,1 ] ) \), one has

$$\begin{aligned} &\frac{1}{4h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr]^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\mu \nu }{4k_{1}k_{2}} \int _{0}^{1} \int _{0}^{1} \imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ \rho ( x,u ) +\rho ( x,v )+\rho ( y,u ) +\rho ( y,v ) \bigr] \,d\imath _{1}\,d \imath _{2} \\ &\qquad {}- \frac{\mu \nu }{16k_{1}k_{2}h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) } \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[\mu _{1} \bigl(y^{\ell _{1}}-x^{\ell _{1}} \bigr)^{\sigma _{1}}+\mu _{2} \bigl(v^{\ell _{2}}-u^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr]\,d \imath _{1}\,d\imath _{2}. \end{aligned}$$
(3.2)

Note that by the change of variable, we have for the first integral on the right-hand side of the inequality (3.2)

$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ \rho ( x,u ) +\rho ( x,v ) +\rho ( y,u ) +\rho ( y,v ) \bigr] \,d\imath _{1}\,d\imath _{2} \\ &\quad =\frac{\ell _{1}\ell _{2}}{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \biggl[ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}}{ (\hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} )^{1-\frac{\nu }{k_{2}} }}\rho ( x,y ) \,dy \,dx \\ &\qquad {}+ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}}{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{1-\frac{\nu }{k_{2}} }}\rho ( x,y ) \,dy \,dx \\ &\qquad {}+ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}}{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} )^{1-\frac{\nu }{k_{2}} }}\rho ( x,y ) \,dy \,dx \\ &\qquad {}+ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}}{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{1-\frac{\nu }{k_{2}} }}\rho ( x,y ) \,dy \,dx \biggr]. \end{aligned}$$

Now applying Definition 2.22 of the Katugampola \((k_{1}, k_{2})\)-fractional integral, the first inequality of (3.1) is obtained. For the second inequality on the right-hand side of (3.1), we use the coordinated distance-disturbed \(( \ell _{1} ,h_{1})\)-\(( \ell _{2},h_{2})\)-convexity of higher orders of ρ as follows:

$$\begin{aligned}& \begin{aligned}[b] \rho ( x,u ) &=\rho \bigl( \bigl[ \imath _{1} \hat{a}^{\ell _{1}}+ ( 1-\imath _{1} ) \hat{b}^{\ell _{1}} \bigr]^{\frac{1}{\ell _{1}}}, \bigl[ \imath _{2}\hat{c}^{\ell _{2}}+ ( 1- \imath _{2} ) \hat{d}^{\ell _{2}} \bigr] ^{\frac{1}{\ell _{2}}} \bigr) \\ &\leq h_{1} ( \imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{a},\hat{c} ) +h_{1} ( \imath _{1} ) h_{2} ( 1-\imath _{2} ) \rho ( \hat{a}, \hat{d} ) +h_{1} ( 1-\imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{b},\hat{c} ) \\ &\quad {}+h_{1} ( 1-\imath _{1} ) h_{2} ( 1-\imath _{2} ) \rho ( \hat{b},\hat{d} )-\mu _{1} \bigl(\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}} -\mu _{2} \bigl( \hat{d}^{\ell _{2}}- \hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}} , \end{aligned} \end{aligned}$$
(3.3)
$$\begin{aligned}& \begin{aligned}[b] \rho ( x,v ) &=\rho \bigl( \bigl[ \imath _{1}\hat{a}^{\ell _{1}}+ ( 1-\imath _{1} ) \hat{b}^{\ell _{1}} \bigr] ^{\frac{1}{\ell _{1}}}, \bigl[ ( 1-\imath _{2} ) \hat{c}^{\ell _{2}}+\imath _{2}\hat{d}^{\ell _{2}} \bigr]^{\frac{1}{\ell _{2}}} \bigr) \\ &\leq h_{1} ( \imath _{1} ) h_{2} ( 1-\imath _{2} ) \rho ( \hat{a},\hat{c} ) +h_{1} ( \imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{a}, \hat{d} ) +h_{1} ( 1-\imath _{1} ) h_{2} ( 1- \imath _{2} ) \rho ( \hat{b},\hat{c} ) \\ &\quad {}+h_{1} ( 1-\imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{b},\hat{d} )-\mu _{1} \bigl( \hat{b}^{\ell _{1}}- \hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}-\mu _{2} \bigl(\hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} , \end{aligned} \end{aligned}$$
(3.4)
$$\begin{aligned}& \begin{aligned}[b] \rho ( y,u ) &=\rho \bigl( \bigl[ ( 1-\imath _{1} ) \hat{a}^{\ell _{1}}+\imath _{1}\hat{b}^{\ell _{1}} \bigr] ^{\frac{1}{\ell _{1}}}, \bigl[ \imath _{2}\hat{c}^{\ell _{2}}+ ( 1-\imath _{2} ) \hat{d}^{\ell _{2}} \bigr] ^{\frac{1}{\ell _{2}}} \bigr) \\ &\leq h_{1} ( 1-\imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{a},\hat{c} ) +h_{1} ( 1-\imath _{1} ) h_{2} ( 1-\imath _{2} ) \rho ( \hat{a}, \hat{d} ) +h_{1} ( \imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{b},\hat{c} ) \\ &\quad {}+h_{1} ( \imath _{1} ) h_{2} ( 1-\imath _{2} ) \rho ( \hat{b},\hat{d} )-\mu _{1} \bigl( \hat{a}^{\ell _{1}}- \hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}}-\mu _{2} \bigl(\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}} ,\end{aligned} \end{aligned}$$
(3.5)

and

$$\begin{aligned} \rho ( y,v ) =&\rho \bigl( \bigl[ ( 1-\imath _{1} ) \hat{a}^{\ell _{1}}+\imath _{1} \hat{b}^{\ell _{1}} \bigr] ^{\frac{1}{\ell _{1}}}, \bigl[ ( 1-\imath _{2} ) \hat{c}^{\ell _{2}}+\imath _{2}\hat{d}^{\ell _{2}} \bigr]^{\frac{1}{\ell _{2}}} \bigr) \\ \leq &h_{1} ( 1-\imath _{1} ) h_{2} ( 1-\imath _{2} ) \rho ( \hat{a},\hat{c} ) +h_{1} ( 1-\imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{a}, \hat{d} ) +h_{1} ( \imath _{1} ) h_{2} ( 1- \imath _{2} ) \rho ( \hat{b},\hat{c} ) \\ &{} +h_{1} ( \imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{b},\hat{d} )-\mu _{1} \bigl( \hat{a}^{\ell _{1}}- \hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}}-\mu _{2} \bigl(\hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} . \end{aligned}$$
(3.6)

Adding inequalities (3.3), (3.4), (3.5) and (3.6), we arrive at the result

$$\begin{aligned} &\rho ( x,u ) +\rho ( x,v ) +\rho ( y,u ) +\rho ( y,v ) \\ &\quad \leq \bigl[ \rho ( \hat{a},\hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \bigl[ h_{1} ( \imath _{1} ) h_{2} ( \imath _{2} ) +h_{1} ( \imath _{1} ) h_{2} ( 1- \imath _{2} ) \\ &\qquad {}+ h_{1} ( 1-\imath _{1} ) h_{2} ( \imath _{2} ) +h_{1} ( 1-\imath _{1} ) h_{2} ( 1-\imath _{2} ) \bigr] \\ &\qquad {}-2\mu _{1} \bigl[ \bigl(\hat{b}^{\ell _{1}}- \hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl(\hat{a}^{\ell _{1}}- \hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] \\ &\qquad {}-2\mu _{2} \bigl[ \bigl(\hat{d}^{\ell _{2}}- \hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl(\hat{c}^{\ell _{2}}- \hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr]. \end{aligned}$$
(3.7)

Multiplying (3.7) by \(\frac{\mu \nu }{4k_{1}k_{2}}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1}\) and integrating over \(( [0,1]\times {}[ 0,1] ) \), one has the second inequality of (3.1) by applying Definition 2.22, which then completes the proof. □

Corollary 3.2

Taking \(k_{1}, k_{2}\to 1\) in Theorem 3.1, we have

$$\begin{aligned} &\frac{1}{4h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} )} \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) + A^{\star } \\ &\quad \leq \frac{\ell _{1}^{\mu }\ell _{2}^{\nu }\Gamma ( \mu +1 ) \Gamma ( \nu +1 ) }{4 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\mu } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu }\rho ( \hat{b},\hat{d} ) \\ &\qquad {}+{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu } \rho ( \hat{b},\hat{c} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu }\rho ( \hat{a},\hat{d} ) + {}^{{\ell _{1},\ell _{2}}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu }\rho ( \hat{a}, \hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4} \bigl[ \rho ( \hat{a},\hat{c} ) + \rho ( \hat{a}, \hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\mu -1} \imath _{2}^{\nu -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{1}\,d \imath _{2} \\ &\qquad {}- \frac{\mu _{1} [ (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}+ (\hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} )^{\sigma _{1}} ]+\mu _{2} [ (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}+ (\hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} )^{\sigma _{2}} ]}{2}, \end{aligned}$$
(3.8)

where

$$ A^{\star }=-\frac{\mu \nu }{16} \biggl[\frac{\mu _{1}}{\nu } \biggl( \frac{1}{2\hat{b}^{\ell _{1}}} \biggr)^{\mu }C_{1}^{\star }+ \frac{\mu _{2}}{\mu } \biggl(\frac{1}{2\hat{d}^{\ell _{2}}} \biggr)^{\nu }C_{2}^{\star } \biggr] $$

and

$$ C_{1}^{\star }= \int _{\hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}}}^{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}\imath _{1}^{\sigma _{1}} \bigl[\imath _{1}-\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}} \bigr]^{\mu -1} \,d\imath _{1},\qquad C_{2}^{\star }= \int _{\hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}}}^{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}\imath _{2}^{\sigma _{2}} \bigl[\imath _{2}-\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}} \bigr]^{\nu -1} \,d\imath _{2}. $$

Corollary 3.3

Taking \(\mu _{1}, \mu _{2}\to 0^{+}\) in Theorem 3.1, we get

$$\begin{aligned} &\frac{1}{4h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} )} \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{4 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu , k_{1}, k_{2} }\rho ( \hat{b},\hat{d} ) \\ &\qquad {}+{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{b},\hat{c} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a}, \hat{d} ) + {}^{{\ell _{1},\ell _{2}}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1- \imath _{2} ) \bigr] \,d\imath _{1}\,d\imath _{2}. \end{aligned}$$
(3.9)

Corollary 3.4

Taking \(k_{1}, k_{2}\to 1\) in Corollary 3.3, we obtain

$$\begin{aligned} &\frac{1}{4h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr]^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\ell _{1}^{\mu }\ell _{2}^{\nu }\Gamma ( \mu +1 ) \Gamma ( \nu +1 ) }{4 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu }\rho ( \hat{b},\hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu }\rho ( \hat{b},\hat{c} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu }\rho ( \hat{a}, \hat{d} ) + {}^{{\ell _{1},\ell _{2}}}I_{\hat{b}-, \hat{d}-}^{\mu ,\nu }\rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4} \bigl[ \rho ( \hat{a},\hat{c} ) + \rho ( \hat{a}, \hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\mu -1} \imath _{2}^{\nu -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{1}\,d \imath _{2}. \end{aligned}$$
(3.10)

Remark 3.5

If \(\mu =1=\nu \), then Corollary 3.4 becomes Theorem 2.16 which was proved in [38].

Remark 3.6

If \(h_{1}(t)=t=h_{2}(t)\), then Remark 3.5 coincides with Theorem 2.11 which was proved in [27].

Corollary 3.7

Taking \(\ell _{1}=\ell _{2}=1\) in Corollary 3.4, we have

$$\begin{aligned} &\frac{1}{4h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \frac{\hat{a}+b}{2}, \frac{\hat{c}+\hat{d}}{2} \biggr) \\ &\quad \leq \frac{\Gamma ( \mu +1 ) \Gamma ( \nu +1 ) }{4 ( \hat{b}-\hat{a} ) ^{\mu } ( \hat{d}-\hat{c} ) ^{\nu }} \bigl[ I_{\hat{a}+,\hat{c}+}^{\mu ,\nu } \rho ( \hat{b},\hat{d} ) + I_{\hat{a}+,\hat{d}-}^{\mu ,\nu }\rho ( \hat{b},\hat{c} ) + I_{\hat{b}-,\hat{c}+}^{\mu ,\nu }\rho ( \hat{a},\hat{d} ) + I_{\hat{b}-,\hat{d}-}^{\mu ,\nu } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4} \bigl[ \rho ( \hat{a},\hat{c} ) + \rho ( \hat{a}, \hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\mu -1} \imath _{2}^{\nu -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1- \imath _{2} ) \bigr] \,d\imath _{1}\,d\imath _{2}. \end{aligned}$$

This result coincides with Theorem 2.1 of [36], if \(h_{1}(t)=h_{2}(t)=h(t)\). Furthermore, if \(\mu =\nu =1\), it reduces to Theorem 7 of [23].

Remark 3.8

If \(h_{1}(t)=h_{2}(t)=t\) then Corollary 3.7 coincides with Theorem 2.8.

Corollary 3.9

Suppose that \(\rho :\Omega \rightarrow \mathbb{R}\) is distance-disturbed \(( \ell _{1} ,s_{1})\)-\(( \ell _{2},s_{2})\)-convex function of higher orders on the coordinates on Ω and \(\rho \in L_{1}(\Omega )\). Then one has the inequalities

$$\begin{aligned} &2^{s_{1}+s_{2}-2}\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) + A \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{4 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{\ell _{1}, \ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{b},\hat{c} ) \\ &\qquad {}+ {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{d} ) + {}^{{\ell _{1},\ell _{2}}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{1}{(\mu +s_{1})(\nu +s_{2})}+ \frac{B(\nu ,s_{2}+1)}{(\mu +s_{1})} \\ &\qquad {}+\frac{B(\mu ,s_{1}+1)}{(\nu +s_{2})} + B(\nu ,s_{2}+1)B( \mu ,s_{1}+1) \biggr\} \\ &\qquad {}- \frac{\mu _{1} [ (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}+ (\hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} )^{\sigma _{1}} ]+\mu _{2} [ (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}+ (\hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} )^{\sigma _{2}} ]}{2}, \end{aligned}$$

where \(B(x,y)=\int _{0}^{1}\jmath ^{x-1}(1-\jmath )^{y-1}\,d\jmath \), for all \(x,y>0\), is the Beta function.

Corollary 3.10

Suppose that \(\rho :\Omega \rightarrow \mathbb{R} \) is distance-disturbed \(( \ell ,s_{1})\)-\(( \ell ,s_{2})\)-convex function of higher orders on the coordinates on Ω and \(\rho \in L_{1}(\Omega )\). Then one has the inequalities

$$\begin{aligned} &2^{s_{1}+s_{2}-2}\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr]^{\frac{1}{\ell }}, \biggl[ \frac{\hat{c}^{\ell }+\hat{d}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr) + B \\ &\quad \leq \frac{\ell ^{\frac{\mu }{k_{1}} + \frac{\nu }{k_{2}}}\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{4 ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[^{\ell ,\ell }I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{\ell ,\ell }I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{c} ) \\ &\qquad {}+{}^{\ell ,\ell }I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{d} ) + {}^{{\ell ,\ell }}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4k_{1}k_{2}} \bigl[ \rho ( \hat{a},\hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{1}{(\mu +s_{1})(\nu +s_{2})}+ \frac{B(\nu ,s_{2}+1)}{(\mu +s_{1})}+ \frac{B(\mu ,s_{1}+1)}{(\nu +s_{2})} + B(\nu ,s_{2}+1)B(\mu ,s_{1}+1) \biggr\} \\ &\qquad {}- \frac{\mu _{1} [ (\hat{b}^{\ell }-\hat{a}^{\ell } )^{\sigma _{1}}+ (\hat{a}^{\ell }-\hat{b}^{\ell } )^{\sigma _{1}} ]+\mu _{2} [ (\hat{d}^{\ell }-\hat{c}^{\ell } )^{\sigma _{2}}+ (\hat{c}^{\ell }-\hat{d}^{\ell } )^{\sigma _{2}} ]}{2}, \end{aligned}$$

where

$$ B=-\frac{\mu \nu }{16k_{1}k_{2}} \biggl[\frac{\mu _{1}k_{2}}{\nu } \biggl( \frac{1}{2\hat{b}^{\ell }} \biggr)^{\frac{\mu }{k_{1}}}B_{1}+ \frac{\mu _{2}k_{1}}{\mu } \biggl( \frac{1}{2\hat{d}^{\ell }} \biggr)^{\frac{\nu }{k_{2}}}B_{2} \biggr] $$

and

$$ B_{1}= \int _{\hat{a}^{\ell }-\hat{b}^{\ell }}^{\hat{a}^{\ell }+\hat{b}^{\ell }}\imath _{1}^{\sigma _{1}} \bigl[\imath _{1}-\hat{a}^{\ell }+ \hat{b}^{\ell } \bigr]^{\frac{\mu }{k_{1}}-1} \,d\imath _{1},\qquad B_{2}= \int _{\hat{c}^{\ell }-\hat{d}^{\ell }}^{\hat{c}^{\ell }+\hat{d}^{\ell }} \imath _{2}^{\sigma _{2}} \bigl[\imath _{2}-\hat{c}^{\ell }+\hat{d}^{\ell } \bigr]^{\frac{\nu }{k_{2}}-1} \,d\imath _{2}. $$

Corollary 3.11

Suppose that \(\rho :\Omega \rightarrow \mathbb{R}\) is distance-disturbed \(( s_{1},s_{2} ) \)-convex function of order 2 on the coordinates on Ω and \(\rho \in L_{1}(\Omega )\). Then one has the inequalities:

$$\begin{aligned} &2^{s_{1}+s_{2}-2}\rho \biggl( \frac{\hat{a}+\hat{b}}{2}, \frac{\hat{c}+\hat{d}}{2} \biggr) + D \\ &\quad \leq \frac{\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{4 ( \hat{b}-\hat{a} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}-\hat{c} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[ {}^{1,1}I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{1,1}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{c} )\\ &\qquad {} + {}^{1,1}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{d} ) + {}^{{1,1}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{1}{(\mu +s_{1})(\nu +s_{2})}+ \frac{B(\nu ,s_{2}+1)}{(\mu +s_{1})}+ \frac{B(\mu ,s_{1}+1)}{(\nu +s_{2})} + B(\nu ,s_{2}+1)B(\mu ,s_{1}+1) \biggr\} \\ &\qquad {} - \bigl[\mu _{1} (\hat{b}-\hat{a} )^{2}+\mu _{2} ( \hat{d}-\hat{c} )^{2} \bigr], \end{aligned}$$

where

$$\begin{aligned}& D=-\frac{\mu \nu }{16k_{1}k_{2}} \biggl[\frac{\mu _{1}k_{2}}{\nu } \biggl( \frac{1}{2\hat{b}} \biggr)^{\frac{\mu }{k_{1}}}D_{1}+ \frac{\mu _{2}k_{1}}{\mu } \biggl( \frac{1}{2\hat{d}} \biggr)^{\frac{\nu }{k_{2}}}D_{2} \biggr], \\& D_{1}= \int _{\hat{a}-\hat{b}}^{\hat{a}+\hat{b}}\imath _{1}^{2} [ \imath _{1}-\hat{a}+\hat{b} ]^{\frac{\mu }{k_{1}}-1} \,d\imath _{1}, \qquad D_{2}= \int _{\hat{c}-\hat{d}}^{\hat{c}+\hat{d}}\imath _{2}^{2} [\imath _{2}-\hat{c}+\hat{d} ]^{\frac{\nu }{k_{2}}-1} \,d\imath _{2}. \end{aligned}$$

To prove our next result, we need Proposition 3.12.

Proposition 3.12

Let \(\rho :I=[\hat{a},\hat{b}]\subseteq ( 0,\infty ) \rightarrow \mathbb{R} \) be a distance-disturbed \(( \ell ,h ) \)-convex function of higher order \(\sigma >0\) and \(\rho \in L_{1}[\hat{a},\hat{b}]\). Then, for \(\alpha , \mu , k>0\), the following double inequality holds:

$$\begin{aligned} &\frac{1}{h ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr) +\frac{\alpha \mu }{4k} \bigl( \hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma }W \\ &\quad \leq \frac{\ell ^{\frac{\mu }{k} }\Gamma _{k} ( \alpha +k ) }{ ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k} }} \bigl[ ^{\ell }I_{\hat{a}+}^{\alpha , k } \rho ( \hat{b}) + {}^{\ell }I_{\hat{b}-}^{\alpha , k }\rho ( \hat{a} ) \bigr] \\ &\quad \leq \alpha \biggl[ \frac{\rho ( \hat{a} ) +\rho ( \hat{b} )}{k} \biggr] \int _{0}^{1}t^{\frac{\alpha }{k} -1} \bigl[ h ( t ) +h ( 1-t ) \bigr] \,dt \\ &\qquad {}-\mu \frac{\alpha k}{(\alpha +k)(\alpha +2k)} \bigl[ \bigl(\hat{b}^{\ell }- \hat{a}^{\ell } \bigr)^{\sigma }+ \bigl(\hat{a}^{\ell }- \hat{b}^{\ell } \bigr)^{\sigma } \bigr], \end{aligned}$$
(3.11)

where

$$ W= \int _{0}^{1}t^{\frac{\alpha }{k} -1}(2t-1)^{\sigma } \,dt. $$

Proof

Since ρ is a distance-disturbed \(( \ell ,h ) \)-convex function of higher order \(\sigma >0\) on \([\hat{a},\hat{b}]\), taking \(x^{\ell }=t\hat{a}^{\ell }+ ( 1-t ) \hat{b}^{\ell }\) and \(y^{\ell }= ( 1-t ) \hat{a}^{\ell }+t\hat{b}^{\ell }\), for all \(t\in {}[ 0,1]\), we have

$$\begin{aligned} \frac{1}{h ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr)&\leq \rho \bigl( \bigl[t\hat{a}^{\ell }+(1-t) \hat{b}^{\ell }\bigr]^{\frac{1}{\ell }} \bigr) +\rho \bigl( \bigl[(1-t) \hat{a}^{\ell }+t\hat{b}^{\ell }\bigr]^{\frac{1}{\ell }} \bigr) \\ &\quad {}-\frac{\mu }{4}(2t-1)^{\sigma } \bigl(\hat{b}^{\ell }- \hat{a}^{\ell } \bigr)^{\sigma }. \end{aligned}$$
(3.12)

Multiplying both sides of (3.12) by \(t^{\frac{\alpha }{k} -1}\) and integrating w.r.t. t over \([0,1]\), we obtain

$$\begin{aligned} &\frac{k}{\alpha h ( \frac{1}{2} ) }f \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr]^{\frac{1}{\ell }} \biggr) \\ &\quad \leq \int _{0}^{1}t^{\frac{\alpha }{k} -1}\rho \bigl( \bigl[ t \hat{a}^{\ell }+ ( 1-t ) \hat{b}^{\ell } \bigr] ^{\frac{1}{\ell }} \bigr)\,dt \\ &\qquad {}+ \int _{0}^{1}t^{\frac{\alpha }{k} -1}\rho \bigl( \bigl[ ( 1-t ) \hat{a}^{\ell }+t\hat{b}^{\ell } \bigr] ^{\frac{1}{\ell }} \bigr) \,dt-\frac{\mu }{4} \bigl(\hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma } \int _{0}^{1}t^{\frac{\alpha }{k} -1}(2t-1)^{\sigma } \,dt. \end{aligned}$$
(3.13)

By a change of variable in (3.13), we get

$$\begin{aligned} \frac{k}{\alpha h ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr]^{\frac{1}{\ell }} \biggr) &\leq \frac{\ell }{ ( \hat{b}^{\ell }-\hat{a}^{\ell } )^{\frac{\alpha }{k} }} \biggl[ \int _{\hat{a}}^{\hat{b}} \frac{x^{\frac{\alpha }{k}-1}}{ ( \hat{b}^{\ell }-x^{\ell } ) ^{\frac{\alpha }{k} }}\rho ( x ) \,dx+ \int _{\hat{a}}^{\hat{b}} \frac{x^{\frac{\alpha }{k}-1}}{ ( x^{\ell }-\hat{a}^{\ell } ) ^{\frac{\alpha }{k}}} \rho ( x ) \,dx \biggr] \\ &\quad {}-\frac{\mu }{4} \bigl(\hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma }W. \end{aligned}$$

Applying Definition 2.20 of Katugampola k-fractional integrals, one has the first inequality of (3.11). For the second inequality on the right-hand side of (3.11), by using the distance-disturbed \(( \ell ,h ) \)-convexity of higher order \(\sigma >0\) of ρ, we have

$$\begin{aligned} \rho \bigl( \bigl[ t\hat{a}^{\ell }+ ( 1-t ) \hat{b}^{\ell } \bigr] ^{\frac{1}{\ell }} \bigr) +\rho \bigl( \bigl[ ( 1-t ) \hat{a}^{\ell }+t \hat{b}^{\ell } \bigr] ^{\frac{1}{\ell }} \bigr) &\leq \bigl[ \rho ( \hat{a}) +\rho ( \hat{b}) \bigr] \bigl( h ( t ) +h ( 1-t ) \bigr) \\ &\quad {}-\mu t(1-t) \bigl[ \bigl(\hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma }+ \bigl(\hat{a}^{\ell }-\hat{b}^{\ell } \bigr)^{\sigma } \bigr]. \end{aligned}$$

Multiplying by \(t^{\frac{\alpha }{k} -1}\) on both sides and integrating over \([ 0,1 ]\), we obtained the second inequality of (3.11). □

Corollary 3.13

Taking \(\mu \to 0^{+}\) in Proposition 3.12, we have

$$\begin{aligned} \frac{1}{h ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr) &\leq \frac{\ell ^{\frac{\alpha }{k} }\Gamma _{k} ( \alpha +k ) }{ ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\alpha }{k} }} \bigl[^{\ell }I_{\hat{a}+}^{\alpha , k }\rho ( \hat{b} ) +{}^{\ell }I_{\hat{b}-}^{\mu , k }\rho ( \hat{a} ) \bigr] \\ &\leq \alpha \biggl[ \frac{\rho ( \hat{a} ) +\rho ( \hat{b})}{k} \biggr] \int _{0}^{1}t^{\frac{\alpha }{k} -1} \bigl[ h ( t ) +h ( 1-t ) \bigr] \,dt. \end{aligned}$$
(3.14)

Remark 3.14

If \(\alpha =k=1\), then Corollary 3.13 coincides with Theorem 5 of [8].

Remark 3.15

If \(\ell =k=1\) and \(h(t)=t\), then Corollary 3.13 coincides with Theorem 2 of [35].

Now, using Proposition 3.12 we can give the following result.

Theorem 3.16

Suppose that \(\rho :\Omega \rightarrow \mathbb{R}\) is a distance-disturbed \(( \ell _{1} ,h_{1})\)-\(( \ell _{2},h_{2})\)-convex function of higher orders on the coordinates on Ω and \(\rho \in L_{1}(\Omega )\). Then one has the inequalities

$$\begin{aligned} &\frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \biggl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr]^{\frac{1}{\ell _{2}}} \biggr) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \biggr] \\ &\qquad {}+ \frac{\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \biggl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},d \biggr) +{}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},c \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \ell _{1}\mu _{2} (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}}{8k_{1}k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }}W_{2} (F_{1}+F_{2} )+ \frac{\mu \nu \ell _{2}\mu _{1} (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}}{8k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }}W_{1} (F_{3}+F_{4} ) \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu , k_{1}, k_{2} }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b}, \hat{c} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a}, \hat{d} ) \\ &\qquad {} +{}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a}, \hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a}, \hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}}-1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (F_{1}+F_{2} ) \\ &\qquad {}+ \frac{\mu \ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{a},\hat{d} ) + {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{a}, \hat{c} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (F_{3}+F_{4} ) \\ &\quad \leq \frac{\mu \nu }{k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1- \imath _{1} ) \bigr] \,d\imath _{2}\,d\imath _{1} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (F_{1}+F_{2} ) \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (F_{3}+F_{4} ), \end{aligned}$$
(3.15)

where

$$\begin{aligned}& F_{1}= \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \,dx,\qquad F_{2}= \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \,dx, \\& F_{3}= \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \,dy,\qquad F_{4}= \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \,dy, \end{aligned}$$

and

$$ W_{1}= \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1}(2 \imath _{1}-1)^{\sigma _{1}}\,d\imath _{1},\qquad W_{2}= \int _{0}^{1} \imath _{2}^{\frac{\nu }{k_{2}} -1}(2 \imath _{2}-1)^{\sigma _{2}}\,d \imath _{2}. $$

Proof

Since \(\rho :\Omega \rightarrow \mathbb{R}\) is a distance-disturbed \(( \ell _{1},h_{1} ) \)-\(( \ell _{2},h_{2} ) \)-convex function of higher orders \((\sigma _{1}, \sigma _{2})\), then partial mapping \(\rho _{x}: [ \hat{c},\hat{d} ] \rightarrow \mathbb{R}\) defined by \(\rho _{x} ( v ) =\rho ( x,v ) \) for all \(x\in [ \hat{a},\hat{b} ] \) is distance-disturbed \(( \ell _{2},h_{2} ) \)-convex of order \(\sigma _{1}\) on \([ \hat{c},\hat{d} ] \). Similarly, \(\rho _{y}: [ \hat{a},\hat{b} ] \rightarrow \mathbb{R}\) defined by \(\rho _{y} ( u ) =\rho ( u,y ) \) for all \(y\in [ \hat{c},\hat{d} ] \) is distance-disturbed \(( \ell _{1},h_{1} ) \)-convex of order \(\sigma _{2}\) on \([ \hat{a},\hat{b} ] \). Then, by Proposition 3.12 and applying the distance-disturbed \(( \ell _{2},h_{2} ) \)-convexity of \(\rho _{x}\), we have

$$\begin{aligned} & \frac{1}{h_{2} ( \frac{1}{2} ) }\rho _{x} \biggl( \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) +\frac{\nu \mu _{2}}{4k_{2}} \bigl(\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}W_{2} \\ &\quad \leq \frac{\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho _{x}( \hat{d}) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho _{x}( \hat{c}) \bigr] \\ &\quad \leq \nu \biggl[ \frac{\rho _{x}( \hat{c}) +\rho _{x}( \hat{d})}{k_{2}} \biggr] \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d \imath _{2} \\ &\qquad {} -\mu _{2}\frac{\nu k_{2}}{(\nu +k_{2})(\nu +2k_{2})} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr], \\ & \frac{1}{h_{2} ( \frac{1}{2} ) }\rho \biggl(x, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) +\frac{\nu \mu _{2}}{4k_{2}} \bigl(\hat{d}^{\ell _{2}}- \hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}W_{2} \\ &\quad \leq \frac{\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( x,\hat{d} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho ( x, \hat{c} ) \bigr] . \end{aligned}$$

Or

$$\begin{aligned} &\leq \nu \biggl[ \frac{\rho ( x,\hat{c} ) +\rho ( x,\hat{d} )}{k_{2}} \biggr] \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\quad {}-\mu _{2}\frac{\nu k_{2}}{(\nu +k_{2})(\nu +2k_{2})} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr]. \end{aligned}$$
(3.16)

Integrating inequality (3.16) w.r.t. x over \([\hat{a},\hat{b}]\) after multiplying by

$$\frac{\mu \ell _{1}x^{\ell _{1}-1}}{2k_{1} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \quad \mbox{and}\quad \frac{\mu \ell _{1}x^{\ell _{1}-1}}{2k_{1} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }}, $$

respectively, we obtain

$$\begin{aligned} &\frac{\mu \ell _{1}}{2k_{1}h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \rho \biggl( x, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \,dx \\ &\qquad {}+ \frac{\mu \nu \ell _{1}\mu _{2} (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}}{8k_{1}k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }}W_{2} \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \,dx \\ &\quad \leq \frac{\mu \nu \ell _{1}\ell _{2}}{2k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\frac{\nu }{k_{2}} } ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \\ &\qquad {}\times \biggl[ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \\ &\qquad {}+ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \biggr] \\ &\quad \leq \frac{\mu \nu \ell _{1}}{2k_{1}k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \biggl[ \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}\rho ( x,\hat{c}) }{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }}\,dx+ \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}\rho ( x,\hat{d} ) }{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }}\,dx \biggr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] \\ &\qquad {}\times \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \,dx \end{aligned}$$
(3.17)

and

$$\begin{aligned} &\frac{\mu \ell _{1}}{2k_{1}h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \rho \biggl( x, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \,dx \\ &\qquad {}+ \frac{\mu \nu \ell _{1}\mu _{2} (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}}{8k_{1}k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }}W_{2} \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \,dx \\ &\quad \leq \frac{\mu \nu \ell _{1}\ell _{2}}{2k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\frac{\nu }{k_{2}} } ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \\ &\qquad {}\times \biggl[ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \\ &\qquad {} + \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \biggr] \\ &\quad \leq \frac{\mu \nu \ell _{1}}{2k_{1}k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \biggl[ \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}\rho ( x,\hat{c} ) }{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }}\,dx+ \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}\rho ( x,\hat{d} ) }{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }}\,dx \biggr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] \\ &\qquad {}\times \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \,dx . \end{aligned}$$
(3.18)

Now again by Proposition 3.12 and applying the distance-disturbed \(( \ell _{1},h_{1} ) \)-convexity of \(\rho _{y}\), we have

$$\begin{aligned} & \frac{1}{h_{1} ( \frac{1}{2} ) }\rho _{y} \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}} \biggr)+\frac{\mu \mu _{1}}{4k_{1}} \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}W_{1} \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho _{y}( \hat{b}) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho _{y}( \hat{a}) \bigr] \\ &\quad \leq \mu \biggl[ \frac{\rho _{y} ( \hat{a} ) +\rho _{y}( \hat{b})}{k_{1}} \biggr] \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1}\frac{\mu k_{1}}{(\mu +k_{1})(\mu +2k_{1})} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}} + \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr]. \end{aligned}$$

Or

$$\begin{aligned} & \frac{1}{h_{1} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2},y \biggr] ^{\frac{1}{\ell _{1}}} \biggr)+\frac{\mu \mu _{1}}{4k_{1}} \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}W_{1} \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},y ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} }\rho ( \hat{a},y ) \bigr] \\ &\quad \leq \mu \biggl[ \frac{\rho ( \hat{a},y ) +\rho ( \hat{b},y )}{k_{1}} \biggr] \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1}\frac{\mu k_{1}}{(\mu +k_{1})(\mu +2k_{1})} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}} + \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr]. \end{aligned}$$
(3.19)

Integrating (3.19) w.r.t. y over \([\hat{c},\hat{d}]\) after multiplying by

$$\frac{\nu \ell _{2}y^{\ell _{2}-1}}{2k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} } ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \quad \mbox{and}\quad \frac{\nu \ell _{2}y^{\ell _{2}-1}}{2k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} } ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}, $$

respectively, we have

$$\begin{aligned} &\frac{\nu \ell _{2}}{2k_{2}h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \int _{c}^{d} \frac{y^{\ell _{2}-1}}{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},y \biggr) \,dy \\ &\qquad {}+ \frac{\mu \nu \ell _{2}\mu _{1} (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}}{8k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }}W_{1} \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \,dy \\ &\quad \leq \frac{\mu \nu \ell _{1}\ell _{2}}{2k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\frac{\nu }{k_{2}} } ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \\ &\qquad {}\times \biggl[ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \\ &\qquad {} + \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \biggr] \\ &\quad \leq \frac{\mu \nu \ell _{2}}{2k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \biggl[ \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}\rho ( \hat{a},y ) }{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy+ \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}\rho ( \hat{b},y ) }{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \biggr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}} -\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] \\ &\qquad {}\times \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \,dy \end{aligned}$$
(3.20)

and

$$\begin{aligned} &\frac{\nu \ell _{2}}{2k_{2}h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},y \biggr) \,dy \\ &\qquad {}+ \frac{\mu \nu \ell _{2}\mu _{1} (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}}{8k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }}W_{1} \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \,dy \\ &\quad \leq \frac{\mu \nu \ell _{1}\ell _{2}}{2k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\frac{\nu }{k_{2}} } ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \\ &\qquad {}\times \biggl[ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \\ &\qquad {} + \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \biggr] \\ &\quad \leq \frac{\mu \nu \ell _{2}}{2k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \biggl[ \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}\rho ( a,y ) }{ ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy+ \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}\rho ( \hat{b},y ) }{ ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \biggr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl(\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl(\hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] \\ &\qquad {}\times \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \,dy. \end{aligned}$$
(3.21)

Adding inequalities (3.17), (3.18), (3.20), (3.21) and applying Definition 2.22, one obtains

$$\begin{aligned} &\frac{\ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \biggl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) +{}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \biggr] \\ &\qquad {}+ \frac{\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \biggl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{d} \biggr) +{}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{c} \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \ell _{1}\mu _{2} (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}}{8k_{1}k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }}W_{2} (F_{1}+F_{2} )+ \frac{\mu \nu \ell _{2}\mu _{1} (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}}{8k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }}W_{1} (F_{3}+F_{4} ) \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[ {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b}, \hat{c} ) \\ &\qquad {} + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a}, \hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a}, \hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho (\hat{a}, \hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (F_{1}+F_{2} ) \\ &\qquad {}+ \frac{\mu \ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{a}, \hat{d} ) + {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{a}, \hat{c} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{b}, \hat{c} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (F_{3}+F_{4} ), \end{aligned}$$

which are the second and third inequalities of (3.15). For the last inequality of (3.15), applying Proposition 3.12 to the last part of the above inequality, we have

$$\begin{aligned} &\frac{\nu \ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} }\rho ( \hat{b}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a}, \hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (F_{1}+F_{2} ) \\ &\qquad {}+ \frac{\mu \ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{a}, \hat{d} ) + {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{a}, \hat{c} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{b}, \hat{c} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (F_{3}+F_{4} ) \\ &\quad \leq \frac{\mu \nu }{k_{1}k_{2}} \bigl[ \rho ( \hat{b}, \hat{c} ) +\rho ( \hat{a}, \hat{c} ) +\rho ( \hat{b}, \hat{d} ) +\rho ( \hat{a}, \hat{d} ) \bigr] \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (F_{1}+F_{2} ) \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (F_{3}+F_{4} ). \end{aligned}$$

For the first inequality of (3.15), we again use Proposition 3.12, which then completes the proof. □

Corollary 3.17

Taking \(k_{1}, k_{2}\to 1\) in Theorem 3.16, we have

$$\begin{aligned} &\frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\ell _{1}^{\mu }\Gamma ( \mu +1 ) }{2h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu }} \biggl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr]^{\frac{1}{\ell _{2}}} \biggr) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \biggr] \\ &\qquad {}+ \frac{\ell _{2}^{\nu }\Gamma ( \nu +1 ) }{2h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \biggl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{d} \biggr) +{}^{\ell _{2}}I_{\hat{d}-}^{\nu }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{c} \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \ell _{1}\mu _{2} (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}}{8 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu }}U_{2} (G_{1}+G_{2} )+ \frac{\mu \nu \ell _{2}\mu _{1} (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}}{8 ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }}U_{1} (G_{3}+G_{4} ) \\ &\quad \leq \frac{\ell _{1}^{\mu }\ell _{2}^{\nu }\Gamma ( \mu +1 ) \Gamma ( \nu +1 ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu } \rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu } \rho ( \hat{b}, \hat{c} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-, \hat{c}+}^{\mu ,\nu }\rho ( \hat{a}, \hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu } \rho ( \hat{a}, \hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell _{1}^{\mu }\Gamma ( \mu +1 ) }{2 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu } \rho ( \hat{b}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{a}+}^{\mu }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu }\rho ( \hat{a}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu } \rho ( \hat{a}, \hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\nu -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d \imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}}{2(\nu +1)(\nu +2) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (G_{1}+G_{2} ) \\ &\qquad {}+ \frac{\mu \ell _{2}^{\nu }\Gamma ( \nu +1 ) }{2 ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu } \rho ( \hat{a}, \hat{d} ) + {}^{\ell _{2}}I_{\hat{c}+}^{\nu }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu } \rho ( \hat{a}, \hat{c} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu } \rho ( \hat{b}, \hat{c} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\mu -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d \imath _{1} \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}}{2(\mu +1)(\mu +2) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (G_{3}+G_{4} ) \\ &\quad \leq \mu \nu \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a}, \hat{d} ) +\rho ( \hat{b}, \hat{c} ) +\rho ( \hat{b}, \hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\mu -1} \imath _{2}^{\nu -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{2}\,d \imath _{1} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}}{2(\nu +1)(\nu +2) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (G_{1}+G_{2} ) \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}}{2(\mu +1)(\mu +2) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (G_{3}+G_{4} ), \end{aligned}$$

where

$$\begin{aligned}& G_{1}= \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\mu }} \,dx,\qquad G_{2}= \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\mu }} \,dx, \\& G_{3}= \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\nu }} \,dy,\qquad G_{4}= \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\nu }} \,dy, \end{aligned}$$

and

$$ U_{1}= \int _{0}^{1}\imath _{1}^{\mu -1}(2 \imath _{1}-1)^{\sigma _{1}}\,d\imath _{1},\qquad U_{2}= \int _{0}^{1}\imath _{2}^{\nu -1}(2 \imath _{2}-1)^{\sigma _{2}}\,d\imath _{2}. $$

Corollary 3.18

Taking \(\mu _{1}, \mu _{2}\to 0^{+}\) in Theorem 3.16, we get

$$\begin{aligned} &\frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \biggl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr]^{\frac{1}{\ell _{2}}} \biggr) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \biggr] \\ &\qquad {}+ \frac{\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \biggl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{d} \biggr) +{}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{c} \biggr) \biggr] \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu , k_{1}, k_{2} }\rho ( \hat{b},\hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{c} ) \\ &\qquad {}+ {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a}, \hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},\hat{c} ) + {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},\hat{d} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} }\rho ( \hat{a}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad \times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}}-1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}+ \frac{\mu \ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[{}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} }\rho ( \hat{a},\hat{d} ) +{}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{b},\hat{d} ) \\ &\qquad {}+{}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{a}, \hat{c} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\quad \leq \frac{\mu \nu }{k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1- \imath _{1} ) \bigr] \,d\imath _{2}\,d\imath _{1}. \end{aligned}$$
(3.22)

Corollary 3.19

Taking \(k_{1}, k_{2}\to 1\) in Corollary 3.18, we obtain

$$\begin{aligned} &\frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\ell _{1}^{\mu }\Gamma ( \mu +1 ) }{2h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu }} \biggl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr]^{\frac{1}{\ell _{2}}} \biggr) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \biggr] \\ &\qquad {}+ \frac{\ell _{2}^{\nu }\Gamma ( \nu +1 ) }{2h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \biggl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{d} \biggr) +{}^{\ell _{2}}I_{\hat{d}-}^{\nu }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{c} \biggr) \biggr] \\ &\quad \leq \frac{\ell _{1}^{\mu }\ell _{2}^{\nu }\Gamma ( \mu +1 ) \Gamma ( \nu +1 ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu }\rho ( \hat{b},\hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu } \rho ( \hat{b},\hat{c} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-, \hat{c}+}^{\mu ,\nu }\rho ( \hat{a},\hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell _{1}^{\mu }\Gamma ( \mu +1 ) }{2 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu } \rho ( \hat{b},\hat{c} ) + {}^{\ell _{1}}I_{\hat{a}+}^{\mu }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu }\rho ( \hat{a},\hat{c} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu } \rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\nu -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d \imath _{2} \\ &\qquad {}+ \frac{\mu \ell _{2}^{\nu }\Gamma ( \nu +1 ) }{2 ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu } \rho ( \hat{a},\hat{d} ) + {}^{\ell _{2}}I_{\hat{c}+}^{\nu }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu } \rho ( \hat{a},\hat{c} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\mu -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d \imath _{1} \\ &\quad \leq \mu \nu \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) + \rho ( \hat{b}, \hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\mu -1} \imath _{2}^{\nu -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{2}\,d \imath _{1}. \end{aligned}$$
(3.23)

Remark 3.20

If \(\mu =1=\nu \), then Corollary 3.19 gives Theorem 2.17, which was proved in [38].

Remark 3.21

If \(\ell _{1}=\ell _{2}=1\) and \(h_{1}(t)=h_{2}(t)=t\), then Remark 3.20 reduced to Theorem 2.8, which was proved in [32].

Corollary 3.22

Let ρ be a distance-disturbed \((\ell ,h_{1})\)\((\ell ,h_{2})\)-convex function on the coordinates on Ω and \(\rho \in L_{1}(\Omega )\), then one has following inequalities:

$$\begin{aligned} &\frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr]^{\frac{1}{\ell }}, \biggl[ \frac{\hat{c}^{\ell }+\hat{d}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr) \\ &\quad \leq \frac{\ell ^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \biggl[ ^{\ell }I_{\hat{a}+}^{\mu , k_{1} } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell }+\hat{d}^{\ell }}{2} \biggr]^{\frac{1}{\ell }} \biggr) + {}^{\ell }I_{\hat{b}-}^{\mu , k_{1} }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell }+\hat{d}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr) \biggr] \\ &\qquad {}+ \frac{\ell ^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \biggl[ ^{\ell }I_{\hat{c}+}^{\nu , k_{2} } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }},\hat{d} \biggr) + {}^{\ell }I_{\hat{d}-}^{\nu , k_{2} }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }},\hat{c} \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \ell \mu _{2} (\hat{d}^{\ell }-\hat{c}^{\ell } )^{\sigma _{2}}}{8k_{1}k_{2} ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }}W_{2} (H_{1}+H_{2} )+ \frac{\mu \nu \ell \mu _{1} (\hat{b}^{\ell }-\hat{a}^{\ell } )^{\sigma _{1}}}{8k_{1}k_{2} ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }}W_{1} (H_{3}+H_{4} ) \\ &\quad \leq \frac{\ell ^{\frac{\mu }{k_{1}} +\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[^{\ell ,\ell }I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{b},\hat{d} ) + {}^{\ell ,\ell }I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{c} ) + {}^{\ell ,\ell }I_{\hat{b}-, \hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{d} ) + {}^{\ell ,\ell }I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell ^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \bigl[ ^{\ell }I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},\hat{c} ) + {}^{\ell }I_{\hat{a}+}^{\mu , k_{1} }\rho ( \hat{b},\hat{d} ) + {}^{\ell }I_{\hat{b}-}^{\mu , k_{1} }\rho ( \hat{a},\hat{c} ) + {}^{\ell }I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}}-1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl(\hat{d}^{\ell }-\hat{c}^{\ell } \bigr)^{\sigma _{2}}+ \bigl(\hat{c}^{\ell }-\hat{d}^{\ell } \bigr)^{\sigma _{2}} \bigr] (H_{1}+H_{2} ) \\ &\qquad {}+ \frac{\mu \ell ^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \bigl[ ^{\ell }I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{a},\hat{d} ) + {}^{\ell }I_{\hat{c}+}^{\nu , k_{2} }\rho ( \hat{b},\hat{d} ) + {}^{\ell }I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{a},\hat{c} ) + {}^{\ell }I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell }-\hat{b}^{\ell } \bigr)^{\sigma _{1}} \bigr] (H_{3}+H_{4} ) \\ &\quad \leq \frac{\mu \nu }{k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1- \imath _{1} ) \bigr] \,d\imath _{2}\,d\imath _{1} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell }-\hat{c}^{\ell } \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell }-\hat{d}^{\ell } \bigr)^{\sigma _{2}} \bigr] (H_{1}+H_{2} ) \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell }-\hat{b}^{\ell } \bigr)^{\sigma _{1}} \bigr] (H_{3}+H_{4} ), \end{aligned}$$
(3.24)

where

$$ H_{1}= \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell -1}}{ ( \hat{b}^{\ell }-x^{\ell } ) ^{1-\frac{\mu }{k_{1}} }} \,dx, \qquad H_{2}= \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell -1}}{ ( x^{\ell }-\hat{a}^{\ell } ) ^{1-\frac{\mu }{k_{1}} }} \,dx, $$

and

$$ H_{3}= \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell -1}}{ ( \hat{d}^{\ell }-y^{\ell } ) ^{1-\frac{\nu }{k_{2}} }} \,dy, \qquad H_{4}= \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell -1}}{ ( y^{\ell }-\hat{c}^{\ell } ) ^{1-\frac{\nu }{k_{2}} }} \,dy. $$

Corollary 3.23

Let \(\rho : \Omega \rightarrow \mathbb{R}\) be a distance-disturbed \((\ell _{1},s_{1})\)-\((\ell _{2},s_{2})\)-convex function on the coordinates on Ω and \(\rho \in L_{1}(\Omega )\). Then one has the inequalities

$$\begin{aligned} &2^{s_{1}+s_{2}}\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{2^{s_{2}-1}\ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \biggl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr]^{\frac{1}{\ell _{2}}} \biggr) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \biggr] \\ &\qquad {}+ \frac{2^{s_{1}-1}\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times\biggl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{d} \biggr) +{}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{c} \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \ell _{1}\mu _{2} (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}}{8k_{1}k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }}W_{2} (F_{1}+F_{2} )+ \frac{\mu \nu \ell _{2}\mu _{1} (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}}{8k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }}W_{1} (F_{3}+F_{4} ) \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{c} ) \\ &\qquad {}+ {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a}, \hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},\hat{c} ) + {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},\hat{d} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} }\rho ( \hat{a}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{k_{2}}{\nu +k_{2}s_{2}}+B \biggl( \frac{\nu }{k_{2}} ,s_{2}+1 \biggr) \biggr] \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (F_{1}+F_{2} ) \\ &\qquad {}+ \frac{\mu \ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{a},\hat{d} ) + {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{a}, \hat{c} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{k_{1}}{\mu +k_{1}s_{1}}+B \biggl( \frac{\mu }{k_{1}} ,s_{1}+1 \biggr) \biggr] \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (F_{3}+F_{4} ) \\ &\quad \leq \frac{\mu \nu }{k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho (\hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{k_{1}k_{2}}{(\mu +k_{1}s_{1})(\nu +k_{2}s_{2})}+ \frac{k_{2}B (\frac{\mu }{k_{1}} ,s_{1}+1 )}{\nu +k_{2}s_{2}}+ \frac{k_{1}B (\frac{\nu }{k_{2}} ,s_{2}+1 )}{\mu +k_{1}s_{1}} \\ &\qquad {} +B \biggl(\frac{\mu }{k_{1}} ,s_{1}+1 \biggr)B \biggl(\frac{\nu }{k_{2}} ,s_{2}+1 \biggr) \biggr\} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (F_{1}+F_{2} ) \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (F_{3}+F_{4} ). \end{aligned}$$

Corollary 3.24

Taking \(\ell _{1}=\ell _{2}=\ell \) in Corollary 3.23, we have

$$\begin{aligned} &2^{s_{1}+s_{2}}\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr]^{\frac{1}{\ell }}, \biggl[ \frac{\hat{c}^{\ell }+\hat{d}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr) \\ &\quad \leq \frac{2^{s_{2}-1}\ell ^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{ ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \biggl[ ^{\ell }I_{\hat{a}+}^{\mu , k_{1} } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell }+\hat{d}^{\ell }}{2} \biggr]^{\frac{1}{\ell }} \biggr) + {}^{\ell }I_{\hat{b}-}^{\mu , k_{1} }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell }+\hat{d}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr) \biggr] \\ &\qquad {}+ \frac{2^{s_{1}-1}\ell ^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \biggl[ ^{\ell }I_{\hat{c}+}^{\nu , k_{2} } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }},\hat{d} \biggr) + {}^{\ell }I_{\hat{d}-}^{\nu , k_{2} }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }},\hat{c} \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \ell \mu _{2} (\hat{d}^{\ell }-\hat{c}^{\ell } )^{\sigma _{2}}}{8k_{1}k_{2} ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }}W_{2} (H_{1}+H_{2} )+ \frac{\mu \nu \ell \mu _{1} (\hat{b}^{\ell }-\hat{a}^{\ell } )^{\sigma _{1}}}{8k_{1}k_{2} ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }}W_{1} (H_{3}+H_{4} ) \\ &\quad \leq \frac{\ell ^{\frac{\mu }{k_{1}} +\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[^{\ell ,\ell }I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{\ell ,\ell }I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{c} ) + {}^{\ell ,\ell }I_{\hat{b}-, \hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{d} ) + {}^{\ell ,\ell }I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell ^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \bigl[ ^{\ell }I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},\hat{c} ) + {}^{\ell }I_{\hat{a}+}^{\mu , k_{1} }\rho ( \hat{b},\hat{d} ) + {}^{\ell }I_{\hat{b}-}^{\mu , k_{1} }\rho ( \hat{a},\hat{c} ) + {}^{\ell }I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{k_{2}}{\nu +k_{2}s_{2}}+B \biggl( \frac{\nu }{k_{2}} ,s_{2}+1 \biggr) \biggr] \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell }-\hat{c}^{\ell } \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell }-\hat{d}^{\ell } \bigr)^{\sigma _{2}} \bigr] (H_{1}+H_{2} ) \\ &\qquad {}+ \frac{\mu \ell ^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \bigl[ ^{\ell }I_{\hat{c}+}^{\nu , k_{2} } \rho (\hat{a},\hat{d} ) + {}^{\ell }I_{\hat{c}+}^{\nu , k_{2} }\rho ( \hat{b},\hat{d} ) + {}^{\ell }I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{a},\hat{c} ) + {}^{\ell }I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{k_{1}}{\mu +k_{1}s_{1}}+B \biggl( \frac{\mu }{k_{1}} ,s_{1}+1 \biggr) \biggr] \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell }-\hat{b}^{\ell } \bigr)^{\sigma _{1}} \bigr] (H_{3}+H_{4} ) \\ &\quad \leq \frac{\mu \nu }{k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{k_{1}k_{2}}{(\mu +k_{1}s_{1})(\nu +k_{2}s_{2})}+ \frac{k_{2}B (\frac{\mu }{k_{1}} ,s_{1}+1 )}{\nu +k_{2}s_{2}} \\ &\qquad {}+ \frac{k_{1}B (\frac{\nu }{k_{2}},s_{2}+1 )}{\mu +k_{1}s_{1}} +B \biggl(\frac{\mu }{k_{1}} ,s_{1}+1 \biggr)B \biggl( \frac{\nu }{k_{2}} ,s_{2}+1 \biggr) \biggr\} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell }-\hat{c}^{\ell } \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell }-\hat{d}^{\ell } \bigr)^{\sigma _{2}} \bigr] (H_{1}+H_{2} ) \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell }-\hat{b}^{\ell } \bigr)^{\sigma _{1}} \bigr] (H_{3}+H_{4} ), \end{aligned}$$

where \(H_{1}\), \(H_{2}\), \(H_{3}\) and \(H_{4}\) are defined in Corollary 3.22.

Corollary 3.25

Taking \(\ell =1\) and \(\sigma _{1}=\sigma _{2}=2\) in Corollary 3.24, we get

$$\begin{aligned} &2^{s_{1}+s_{2}}\rho \biggl(\frac{\hat{a}+\hat{b}}{2}, \frac{\hat{c}+\hat{d}}{2} \biggr) \\ &\quad \leq \frac{2^{s_{2}-1}\Gamma _{k_{1}} ( \mu +k_{1} ) }{ ( \hat{b}-\hat{a} ) ^{\frac{\mu }{k_{1}} }} \biggl[ I_{\hat{a}+}^{\mu , k_{1} } \rho \biggl( \hat{b}, \frac{\hat{c}+\hat{d}}{2} \biggr) + I_{\hat{b}-}^{\mu , k_{1} } \rho \biggl( \hat{a}, \frac{\hat{c}+\hat{d}}{2} \biggr) \biggr] \\ &\qquad {}+ \frac{2^{s_{1}-1}\Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{d}-\hat{c} ) ^{\frac{\nu }{k_{2}} }} \biggl[ I_{\hat{c}+}^{\nu , k_{2} } \rho \biggl( \frac{\hat{a}+\hat{b}}{2}, \hat{d} \biggr) + I_{\hat{d}-}^{\nu , k_{2} } \rho \biggl( \frac{\hat{a}+\hat{b}}{2},\hat{c} \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \mu _{2} (\hat{d}-\hat{c} )^{2}}{8k_{1}k_{2} ( \hat{b}-\hat{a} ) ^{\frac{\mu }{k_{1}} }}W_{2}^{\star } \bigl(H_{1}^{\star }+H_{2}^{\star } \bigr)+ \frac{\mu \nu \mu _{1} (\hat{b}-\hat{a} )^{2}}{8k_{1}k_{2} ( \hat{d}-\hat{c} ) ^{\frac{\nu }{k_{2}} }}W_{1}^{\star } \bigl(H_{3}^{\star }+H_{4}^{\star } \bigr) \\ &\quad \leq \frac{\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{b}-\hat{a} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}-\hat{c} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[{}^{1,1}I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{1,1}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{c} ) \\ &\qquad {}+ {}^{1,1}I_{\hat{b}-,\hat{c}+}^{\mu , \nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{d} ) + {}^{1,1}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}-\hat{a} ) ^{\frac{\mu }{k_{1}} }} \bigl[ I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},\hat{c} ) + I_{\hat{a}+}^{\mu , k_{1} }\rho ( \hat{b},\hat{d} ) + I_{\hat{b}-}^{\mu , k_{1} }\rho ( \hat{a},\hat{c} ) + I_{\hat{b}-}^{\mu , k_{1} }\rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{k_{2}}{\nu +k_{2}s_{2}}+B \biggl( \frac{\nu }{k_{2}} ,s_{2}+1 \biggr) \biggr] \\ &\qquad {}-\mu _{2} \frac{\mu \nu k_{2} (\hat{d}-\hat{c} )^{2}}{k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}-\hat{a} ) ^{\frac{\mu }{k_{1}} }} \bigl(H_{1}^{\star }+H_{2}^{\star } \bigr) \\ &\qquad {}+ \frac{\mu \Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}-\hat{c} ) ^{\frac{\nu }{k_{2}} }} \bigl[ I_{\hat{c}+}^{\nu , k_{2} }\rho ( \hat{a},\hat{d} ) + I_{\hat{c}+}^{\nu , k_{2} }\rho ( \hat{b},\hat{d} ) + I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{a},\hat{c} ) + I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{k_{1}}{\mu +k_{1}s_{1}}+B \biggl( \frac{\mu }{k_{1}} ,s_{1}+1 \biggr) \biggr] \\ &\qquad {}-\mu _{1} \frac{\mu \nu k_{1} (\hat{b}-\hat{a} )^{2}}{k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}-\hat{c} ) ^{\frac{\nu }{k_{2}} }} \bigl(H_{3}^{\star }+H_{4}^{\star } \bigr) \\ &\quad \leq \frac{\mu \nu }{k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{k_{1}k_{2}}{(\mu +k_{1}s_{1})(\nu +k_{2}s_{2})}+ \frac{k_{2}B (\frac{\mu }{k_{1}} ,s_{1}+1 )}{\nu +k_{2}s_{2}} \\ &\qquad {}+ \frac{k_{1}B (\frac{\nu }{k_{2}} ,s_{2}+1 )}{\mu +k_{1}s_{1}} +B \biggl(\frac{\mu }{k_{1}} ,s_{1}+1 \biggr)B \biggl(\frac{\nu }{k_{2}} ,s_{2}+1 \biggr) \biggr\} \\ &\qquad {}-\mu _{2} \frac{\mu \nu k_{2} (\hat{d}-\hat{c} )^{2}}{k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}-\hat{a} ) ^{\frac{\mu }{k_{1}} }} \bigl(H_{1}^{\star }+H_{2}^{\star } \bigr) \\ &\qquad {}-\mu _{1} \frac{\mu \nu k_{1} (\hat{b}-\hat{a} )^{2}}{k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}-\hat{c} ) ^{\frac{\nu }{k_{2}} }} \bigl(H_{3}^{\star }+H_{4}^{\star } \bigr), \end{aligned}$$

where

$$\begin{aligned}& H_{1}^{\star }= \int _{\hat{a}}^{\hat{b}} \frac{dx}{ ( \hat{b}-x ) ^{1-\frac{\mu }{k_{1}} }},\qquad H_{2}^{\star }= \int _{\hat{a}}^{\hat{b}} \frac{dx}{ ( x-\hat{a} ) ^{1-\frac{\mu }{k_{1}} }}, \\& H_{3}^{\star }= \int _{\hat{c}}^{\hat{d}} \frac{dy}{ ( \hat{d}-y ) ^{1-\frac{\nu }{k_{2}} }},\qquad H_{4}^{\star }= \int _{\hat{c}}^{\hat{d}} \frac{dy}{ ( y-\hat{c} ) ^{1-\frac{\nu }{k_{2}} }}, \end{aligned}$$

and

$$ W_{1}^{\star }=\frac{4k_{1}}{\mu +2k_{1}}-\frac{4k_{1}}{\mu +k_{1}}+ \frac{k_{1}}{\mu },\qquad W_{2}^{\star }=\frac{4k_{2}}{\nu +2k_{2}}- \frac{4k_{2}}{\nu +k_{2}}+\frac{k_{2}}{\nu }. $$

Corollary 3.26

Taking \(k_{1}, k_{2}\to 1\) in Corollary 3.25, we have

$$\begin{aligned} &2^{s_{1}+s_{2}}\rho \biggl(\frac{\hat{a}+\hat{b}}{2}, \frac{\hat{c}+\hat{d}}{2} \biggr) \\ &\quad \leq \frac{2^{s_{2}-1}\Gamma ( \mu +1 ) }{ ( \hat{b}-\hat{a} ) ^{\mu }} \biggl[ I_{\hat{a}+}^{\mu } \rho \biggl( \hat{b}, \frac{\hat{c}+\hat{d}}{2} \biggr) + I_{\hat{b}-}^{\mu } \rho \biggl( \hat{a}, \frac{\hat{c}+\hat{d}}{2} \biggr) \biggr] \\ &\qquad {}+ \frac{2^{s_{1}-1}\Gamma ( \nu +1 ) }{ ( \hat{d}-\hat{c} ) ^{\nu }} \biggl[ I_{\hat{c}+}^{\nu } \rho \biggl( \frac{\hat{a}+\hat{b}}{2},\hat{d} \biggr) + I_{\hat{d}-}^{\nu } \rho \biggl( \frac{\hat{a}+\hat{b}}{2},\hat{c} \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \mu _{2} (\hat{d}-\hat{c} )^{2}}{8 ( \hat{b}-\hat{a} ) ^{\mu }}W_{4}^{\star } \bigl(H_{5}^{\star }+H_{6}^{\star } \bigr)+ \frac{\mu \nu \mu _{1} (\hat{b}-\hat{a} )^{2}}{8 ( \hat{d}-\hat{c} ) ^{\nu }}W_{3}^{\star } \bigl(H_{7}^{\star }+H_{8}^{\star } \bigr) \\ &\quad \leq \frac{\Gamma ( \mu +1 ) \Gamma ( \nu +1 ) }{ ( \hat{b}-\hat{a} ) ^{\mu } ( \hat{d}-\hat{c} ) ^{\nu }} \\ &\qquad {}\times \bigl[{}^{1,1}I_{\hat{a}+,\hat{c}+}^{\mu ,\nu } \rho ( \hat{b},\hat{d} ) + {}^{1,1}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu } \rho ( \hat{b},\hat{c} ) + {}^{1,1}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu }\rho ( \hat{a},\hat{d} ) + {}^{1,1}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu }\rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \Gamma ( \mu +1 ) }{2 ( \hat{b}-\hat{a} ) ^{\mu }} \bigl[ I_{\hat{a}+}^{\mu } \rho ( \hat{b},\hat{c} ) + I_{\hat{a}+}^{\mu }\rho ( \hat{b},\hat{d} ) + I_{\hat{b}-}^{\mu }\rho ( \hat{a},\hat{c} ) + I_{\hat{b}-}^{\mu }\rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{1}{\nu +s_{2}}+B (\nu ,s_{2}+1 ) \biggr] \\ &\qquad {}-\mu _{2} \frac{\mu \nu (\hat{d}-\hat{c} )^{2}}{(\nu +1)(\nu +2) ( \hat{b}-\hat{a} ) ^{\mu }} \bigl(H_{5}^{\star }+H_{6}^{\star } \bigr) \\ &\qquad {}+ \frac{\mu \Gamma ( \nu +1 ) }{2 ( \hat{d}-\hat{c} ) ^{\nu }} \bigl[ I_{\hat{c}+}^{\nu } \rho ( \hat{a},\hat{d} ) + I_{\hat{c}+}^{\nu }\rho (\hat{b},\hat{d} ) + I_{\hat{d}-}^{\nu }\rho ( \hat{a},\hat{c} ) + I_{\hat{d}-}^{\nu }\rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{1}{\mu +s_{1}}+B (\mu ,s_{1}+1 ) \biggr] \\ &\qquad {}-\mu _{1} \frac{\mu \nu (\hat{b}-\hat{a} )^{2}}{(\mu +1)(\mu +2) ( \hat{d}-\hat{c} ) ^{\nu }} \bigl(H_{7}^{\star }+H_{8}^{\star } \bigr) \\ &\quad \leq \mu \nu \bigl[ \rho ( \hat{a},\hat{c} ) +\rho ( \hat{a},\hat{d} ) + \rho ( \hat{b},\hat{c} ) + \rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{1}{(\mu +s_{1})(\nu +s_{2})}+ \frac{B (\mu ,s_{1}+1 )}{\nu +s_{2}}+ \frac{B (\nu ,s_{2}+1 )}{\mu +s_{1}} +B (\mu ,s_{1}+1 )B (\nu ,s_{2}+1 ) \biggr\} \\ &\qquad {}-\mu _{2} \frac{\mu \nu (\hat{d}-c )^{2}}{(\nu +1)(\nu +2) ( \hat{b}-\hat{a} ) ^{\mu }} \bigl(H_{5}^{\star }+H_{6}^{\star } \bigr)-\mu _{1} \frac{\mu \nu (\hat{b}-\hat{a} )^{2}}{(\mu +1)(\mu +2) ( \hat{d}-\hat{c} ) ^{\nu }} \bigl(H_{7}^{\star }+H_{8}^{\star } \bigr), \end{aligned}$$

where

$$\begin{aligned} &H_{5}^{\star }= \int _{\hat{a}}^{\hat{b}} \frac{dx}{ ( \hat{b}-x ) ^{1-\mu }},\qquad H_{6}^{\star }= \int _{\hat{a}}^{\hat{b}} \frac{dx}{ ( x-\hat{a} ) ^{1-\mu }}, \\ &H_{7}^{\star }= \int _{\hat{c}}^{\hat{d}} \frac{dy}{ ( \hat{d}-y ) ^{1-\nu }},\qquad H_{8}^{\star }= \int _{\hat{c}}^{\hat{d}} \frac{dy}{ ( y-\hat{c} ) ^{1-\nu }}, \end{aligned}$$

and

$$ W_{3}^{\star }=\frac{4}{\mu +2}-\frac{4}{\mu +1}+ \frac{1}{\mu },\qquad W_{4}^{\star }=\frac{4}{\nu +2}- \frac{4}{\nu +1}+\frac{1}{\nu }. $$

Corollary 3.27

Taking \(\mu _{1}, \mu _{2}\to 0^{+}\) in Corollary 3.26, we obtain

$$\begin{aligned} &2^{s_{1}+s_{2}}\rho \biggl( \frac{\hat{a}+\hat{b}}{2}, \frac{\hat{c}+\hat{d}}{2} \biggr) \\ &\quad \leq \frac{2^{s_{2}-1}\Gamma ( \mu +1 ) }{ ( \hat{b}-\hat{a} )^{\mu }} \biggl[ I_{\hat{a}+}^{\mu }\rho \biggl( \hat{b}, \frac{\hat{c}+\hat{d}}{2} \biggr) +I_{\hat{b}-}^{\mu }\rho \biggl( \hat{a},\frac{\hat{c}+\hat{d}}{2} \biggr) \biggr] \\ &\qquad {}+ \frac{2^{s_{1}-1}\Gamma ( \nu +1 ) }{ ( \hat{d}-\hat{c} ) ^{\nu }} \biggl[ I_{\hat{c}+}^{\nu }\rho \biggl( \frac{\hat{a}+\hat{b}}{2}, \hat{d} \biggr) +I_{\hat{d}-}^{\nu }\rho \biggl( \frac{\hat{a}+\hat{b}}{2},\hat{c} \biggr) \biggr] \\ &\quad \leq \frac{\Gamma ( \mu +1 ) \Gamma ( \nu +1 ) }{ ( \hat{b}-\hat{a} ) ^{\mu } ( \hat{d}-\hat{c} ) ^{\nu }} \bigl[ I_{\hat{a}+,\hat{c}+}^{\mu ,\nu } \rho ( \hat{b},\hat{d} ) + I_{\hat{a}+,\hat{d}-}^{\mu ,\nu }\rho ( \hat{b},\hat{c} ) \\ &\qquad {} + I_{\hat{b}-,\hat{c}+}^{\mu ,\nu }\rho ( \hat{a},\hat{d} ) + I_{\hat{b}-,\hat{d}-}^{\mu ,\nu } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \Gamma ( \mu +1 ) }{2 ( \hat{b}-\hat{a} )^{\mu }} \bigl[ I_{\hat{a}+}^{\mu }\rho ( \hat{b}, \hat{c} ) + I_{\hat{a}+}^{\mu }\rho ( \hat{b}, \hat{d} ) + I_{\hat{b}-}^{\mu }\rho ( \hat{a},\hat{c} ) + I_{\hat{b}-}^{\mu }\rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{1}{\nu +s_{2}}+B(\nu ,s_{2}+1) \biggr\} \\ &\qquad {}+ \frac{\mu \Gamma ( \nu +1 ) }{2 ( \hat{d}-\hat{c} ) ^{\nu }} \bigl[ I_{\hat{c}+}^{\nu } \rho ( \hat{a},\hat{d} ) + I_{\hat{c}+}^{\nu }\rho ( \hat{b},\hat{d} ) + I_{\hat{d}-}^{\nu }\rho ( \hat{a},\hat{c} ) + I_{\hat{d}-}^{\nu }\rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{1}{\mu +s_{1}}+B(\mu ,s_{1}+1) \biggr\} \\ &\quad \leq \mu \nu \bigl[ \rho ( \hat{a},\hat{c} ) +\rho ( \hat{a},\hat{d} ) + \rho ( \hat{b},\hat{c} ) + \rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{1}{(\mu +s_{1})(\nu +s_{2})}+ \frac{B(\mu ,s_{1}+1)}{\nu +s_{2}}+ \frac{B(\nu ,s_{2}+1)}{\mu +s_{1}} +B( \mu ,s_{1}+1)B(\nu ,s_{2}+1) \biggr\} . \end{aligned}$$

Remark 3.28

If \(\mu =\nu =1\) and \(s_{1}=s_{2}=s\), then the inequalities in Corollary 3.27 coincide with Theorem 2.1 of [1].

Conclusion

In this paper two inequalities of trapezium type are presented for the Katugampola \((k_{1}, k_{2})\)-fractional integrals taking coordinated distance-disturbed \((\ell _{1},h_{1})\)-\((\ell _{2},h_{2})\)-convexity of higher orders \((\sigma _{1}, \sigma _{2})\) into account. The special cases are discussed to see the compatibility with the previously known results. It is found that the results are highly compatible and they can be extended for other types of convexities. We omit here their proofs and the details are left to the interested reader working in the same domain. We hope that current work will attract the attention of researchers working in mathematical inequalities, fractional calculus, differential equations, difference equations, applied mathematics and other related fields.

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Acknowledgements

T. Abdeljawad would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

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Correspondence to Thabet Abdeljawad.

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Kashuri, A., Iqbal, S., Liko, R. et al. New trapezium type inequalities of coordinated distance-disturbed convex type functions of higher orders via extended Katugampola fractional integrals. Adv Differ Equ 2021, 120 (2021). https://doi.org/10.1186/s13662-021-03280-5

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MSC

  • 26A51
  • 26A33
  • 26D07
  • 26D10
  • 26D15

Keywords

  • Coordinated convex functions
  • Hermite–Hadamard type inequalities
  • Katugampola fractional integrals