(γ,a)-Nabla Reverse Hardy–Hilbert-Type Inequalities on Time Scales

Abstract

In this article, using a ($\gamma$,a)-nabla conformable integral on time scales, we study several novel Hilbert-type dynamic inequalities via nabla time scales calculus. Our results generalize various inequalities on time scales, unifying and extending several discrete inequalities and their corresponding continuous analogues. We say that symmetry plays an essential role in determining the correct methods with which to solve dynamic inequalities.

1. Introduction

Hardy [1] proved the classical discrete double series inequality of the Hilbert type: If $\left\{a_m\right\},\left\{b_n\right\}⩾0$ and ${\sum }_{n=1}^{\infty }{a}_n^p<\infty$ and ${\sum }_{n=1}^{\infty }{b}_n^q<\infty ,$ then we have

$$\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{\displaystyle \frac{a_n{b}_m}{m+n}}⩽\frac{\pi }{sin\frac{\pi }{p}}{\left(\sum _{n=1}^{\infty }{a}_n^p\right)}^{\frac{1}{p}}{\left(\sum _{m=1}^{\infty }{b}_m^q\right)}^{\frac{1}{q}},$$

(1)

Unless all the sequence $\left\{a_m\right\}$ or $\left\{b_n\right\}$ is null. Hardy also established the following integral analogous of the inequality (1):

$${\int }_0^{\infty }{\int }_0^{\infty }\frac{\vartheta \left(\xi \right)g\left(\zeta \right)}{\xi +\zeta }dxdy⩽\frac{\pi }{sin\frac{\pi }{p}}{\left({\int }_0^{\infty }{\vartheta }^p\left(\xi \right)dx\right)}^{\frac{1}{p}}{\left({\int }_0^{\infty }{g}^{p^{\prime }}\left(\xi \right)dx\right)}^{\frac{1}{p^{\prime }}},$$

(2)

Unless $\vartheta \equiv 0$ or $g\equiv 0,$ where $\vartheta$ and g are measurable non-negative functions, such that ${\int }_0^{\infty }{\vartheta }^p\left(\xi \right)dx<\infty$ and ${\int }_0^{\infty }{g}^{p^{\prime }}\left(\xi \right)dx<\infty$ and $p>1,$ $p^{\prime }=p(p-1)$, with the best constant $\frac{\mathsf{\Pi }}{sin\frac{\pi }{p}}$, in (1) and (2).

In [2], Pachpatte established the following Hilbert-type integral inequalities under the following conditions: If $h⩾1,$ $l⩾1,$ and $f\left(\varsigma \right)⩾0,$ $g\left(\varsigma \right)⩾0,$ for $\varsigma \in (0,\xi )$ and $\tau \in (0,\zeta ),$ where $\xi$ and $\zeta$ are positive real numbers and define $\vartheta \left(s\right)={\int }_0^{s}f\left(\varsigma \right)d\varsigma$ and $G\left(\varsigma \right)={\int }_0^{\varsigma }g\left(\tau \right)d\tau ,$ for $s\in (0,\xi )$ and $\varsigma \in (0,\zeta ),$ and $\Phi$, $\Psi$ are two real-valued non-negative, convex, and submultiplicative functions defined on $(0,\infty ]$, then

$$\begin{array}{c}\hfill {\int }_0^{\xi }{\int }_0^{\zeta }{\displaystyle \frac{{\vartheta }^h\left(s\right)G^l\left(\chi \right)}{s+\chi }}dsd\chi ⩽\frac{1}{2}hl{\left(\xi \zeta \right)}^{\frac{1}{2}}{\left({\int }_0^{\xi }(\xi -s){\left({\vartheta }^{h-1}\left(s\right)\vartheta \left(s\right)\right)}^{2}ds\right)}^{\frac{1}{2}}\\ \hfill \times {\left({\int }_0^{\zeta }(\zeta -\chi ){\left(G^{l-1}g\left(\chi \right)\right)}^{2}d\chi \right)}^{\frac{1}{2}},\end{array}$$

(3)

and

$$\begin{array}{c}\hfill {\int }_0^{\xi }{\int }_0^{\zeta }{\displaystyle \frac{\Phi \left(\vartheta \right(s\left)\right)\Psi \left(G\right(\chi \left)\right)}{s+\chi }}dsd\chi ⩽L(\xi ,\zeta ){\left({\int }_0^{\xi }(\xi -s){\left(p\left(s\right)\Phi \left(\frac{\vartheta \left(s\right)}{p\left(s\right)}\right)\right)}^{2}ds\right)}^{\frac{1}{2}}\\ \hfill \times {\left({\int }_0^{\zeta }(\zeta -\chi ){\left(q\left(\chi \right)\Psi \left(\frac{g\left(\chi \right)}{q\left(\chi \right)}\right)\right)}^{2}d\chi \right)}^{\frac{1}{2}}\end{array}$$

(4)

where

$$L(\xi ,\zeta )=\frac{1}{2}{\left({\int }_0^{\xi }{\left(\frac{\Phi \left(P\right(s\left)\right)}{P\left(s\right)}\right)}^{2}ds\right)}^{\frac{1}{2}}{\left({\int }_0^{\zeta }{\left(\frac{\Psi \left(Q\right(\chi \left)\right)}{Q\left(\chi \right)}\right)}^{2}d\chi \right)}^{\frac{1}{2}},$$

and

$$\begin{array}{c}\hfill {\int }_0^{\xi }{\int }_0^{\zeta }{\displaystyle \frac{P\left(s\right)Q\left(\chi \right)\Phi \left(\vartheta \right(s\left)\right)\Psi \left(G\right(\chi \left)\right)}{s+\chi }}dsd\chi ⩽\frac{1}{2}{\left(\xi \zeta \right)}^{\frac{1}{2}}{\left({\int }_0^{\xi }(\xi -s){\left(p\left(s\right)\Phi \left(\vartheta \left(s\right)\right)\right)}^{2}ds\right)}^{\frac{1}{2}}\\ \hfill \times {\left({\int }_0^{\zeta }(\zeta -\chi ){\left(q\left(\chi \right)\Psi \left(g\left(\chi \right)\right)\right)}^{2}d\chi \right)}^{\frac{1}{2}}.\end{array}$$

(5)

For more results on Hilbert-type inequalities and other, please see [3,4,5,6]. See also [7,8,9,10,11,12,13]. As we know that time scale $\mathbb{T}$ is an arbitrary, non-empty closed subset of the set of real numbers $\mathbb{R}$, and the jump operators, forward and backward, are defined by $\sigma \left(\varsigma \right):=inf\{\iota \in \mathbb{T}:\iota >\varsigma \}$, $\rho \left(\varsigma \right):=sup\{\iota \in \mathbb{T}:\iota <\varsigma \}$. For more details, see [7]. The following relations are used.

(i)

For $\mathbb{T}=\mathbb{R}$, then

$$\begin{array}{c}\sigma \left(\varsigma \right)=\rho \left(\varsigma \right)=\varsigma ,\mu \left(\varsigma \right)=\nu \left(\varsigma \right)=0,{\vartheta }^{\Delta }\left(\varsigma \right)={\vartheta }^{\nabla }\left(\varsigma \right)={\vartheta }^{\prime }\left(\varsigma \right),\\ {\int }_a^b\vartheta \left(\varsigma \right)\Delta \varsigma ={\int }_a^b\vartheta \left(\varsigma \right)\nabla \varsigma ={\int }_a^b\vartheta \left(\varsigma \right)d\varsigma .\end{array}$$

(6)

(ii)

For $\mathbb{T}=\mathbb{Z}$, then

$$\begin{array}{c}\sigma \left(\varsigma \right)=\varsigma +1,\rho \left(\varsigma \right)=\varsigma -1,\mu \left(\varsigma \right)=\nu \left(\varsigma \right)=1,\\ {\vartheta }^{\Delta }\left(\varsigma \right)=\Delta \vartheta \left(\varsigma \right),{\vartheta }^{\nabla }\left(\varsigma \right)=\nabla \vartheta \left(\varsigma \right),\\ {\int }_a^b\vartheta \left(\varsigma \right)\Delta \varsigma =\sum _{\varsigma =a}^{b-1}\vartheta \left(\varsigma \right),{\int }_a^b\vartheta \left(\varsigma \right)\nabla \varsigma =\sum _{\varsigma =a+1}^b\vartheta \left(\varsigma \right).\end{array}$$

(7)

For more details on the conformable nabla calculus, see [3].

We suppose that $C{C}_{ld}$ denotes the set of all $ld$-continuous functions $\vartheta (\xi ,\zeta )$ in $\xi$ and $\zeta$ and $C{C}_{ld}^1$ is the set of all functions in $C{C}_{ld}$ for which both the first partial derivative ${\nabla }_{a_1}^{\gamma }$ and the first partial derivative ${\nabla }_{a_2}^{\gamma }$ exist in $C{C}_{ld}$. Similarly, we can define $C{C}_{ld}^2$.

In order to obtain our result in this paper, we need the following lemmas.

Lemma 1

(Reversed Dynamic Hölder’s Inequality). Suppose $u,$ $v\in \mathbb{T}$ with $u<v.$ Assume $\vartheta ,$ $g\in C{C}_{ld}^1({[u,v]}_{\mathbb{T}}\times {[u,v]}_{\mathbb{T}},\mathbb{R})$ be integrable functions and $\frac{1}{p}+\frac{1}{q}=1$ with $p<1$, then

$$\begin{array}{ccc}\hfill {\int }_u^v{\int }_u^v\left|\vartheta (r,\chi )g(r,\chi )\right|{\nabla }_a^{\gamma }r{\nabla }_a^{\gamma }\chi & \ge & {\left[{\int }_u^v{\int }_u^v{\left|\vartheta (r,\chi )\right|}^p{\nabla }_a^{\gamma }r{\nabla }_a^{\gamma }\chi \right]}^{\frac{1}{p}}\hfill \\ & & \times {\left[{\int }_u^v{\int }_u^v{\left|g(r,\chi )\right|}^q{\nabla }_a^{\gamma }r{\nabla }_a^{\gamma }\chi \right]}^{\frac{1}{q}}.\hfill \end{array}$$

(8)

Proof. 

This lemma is a direct extension of the (Lemma 9, [14]). □

Lemma 2

(Reversed Dynamic Jensen’s inequality). Let r, $\chi \in \mathbb{R}$ and $-\infty ⩽m,n⩽\infty .$ If $\vartheta \in C{C}_{ld}^1(\mathbb{R},(m,n))$ and $\Phi :(m,n)⟶\mathbb{R}$ is concave, then

$$\varphi \left({\displaystyle \frac{{\int }_u^v{\int }_{\omega }^s\vartheta (r,\chi ){\nabla }_{a_1}^{\gamma }r{\nabla }_{a_2}^{\gamma }\chi }{{\int }_u^v{\int }_{\omega }^s{\nabla }_{a_1}^{\gamma }r{\nabla }_{a_2}^{\gamma }\chi }}\right)\ge {\displaystyle \frac{{\int }_u^v{\int }_{\omega }^s\varphi \left(\vartheta (r,\chi )\right){\nabla }_{a_1}^{\gamma }r{\nabla }_{a_2}^{\gamma }\chi }{{\int }_u^v{\int }_{\omega }^s{\nabla }_{a_1}^{\gamma }r{\nabla }_{a_2}^{\gamma }\chi }}.$$

(9)

Proof. 

This lemma is a direct extension of the (Lemma 10, [14]). □

Definition 1.

Φ is called a supermultiplicative function on $[0,\infty )$ if

$$\Phi \left(\xi \zeta \right)⩾\Phi \left(\xi \right)\Phi \left(\zeta \right),\mathrm{for}\mathrm{all}\xi ,\zeta ⩾0.$$

(10)

In this paper, we establish a ($\gamma$,a)-nabla conformable integral inequality of Hardy–Hilbert type on a time scale. In special cases, we will recover some dynamic continuous and discrete inequalities known in the literature. Symmetry plays an essential role in determining the correct methods with which to solve dynamic inequalities.

Now, our main results will be presented.

2. Main Results

First, we suppose the following assumptions:

$\left(S_1\right)$

$\mathbb{T}$ be time scales with ${\chi }_0,$ ${\xi }_{\ell },$ ${\zeta }_{\ell },$ $s_{\ell },$ ${\chi }_{\ell }\in \mathbb{T},$ $(\ell =1,\dots ,n).$

$\left(S_2\right)$

${\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })$ are non-negative, nabla integrable functions defined as ${[{\chi }_0,{\xi }_{\ell })}_{\mathbb{T}}\times {[{\chi }_0,{\zeta }_{\ell })}_{\mathbb{T}}$ $(\ell =1,\dots ,n).$

$\left(S_3\right)$

${\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })$ have partial ${\nabla }_a^{\gamma }$- derivatives ${\vartheta }_{\ell }^{{\nabla }_{a_1}^{\gamma }}(s_{\ell },{\chi }_{\ell })$ and ${\vartheta }_{\ell }^{{\nabla }_{a_2}^{\gamma }}(s_{\ell },{\chi }_{\ell })$ with respect to $s_{\ell }$ and ${\chi }_{\ell }$, respectively.

$\left(S_4\right)$

All functions used in this section are integrable according to ${\nabla }_a^{\gamma }$ sense.

$\left(S_5\right)$

${\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\in C_{ld}^2\left({[{\chi }_0,{\xi }_{\ell })}_{\mathbb{T}}\times {[{\chi }_0,{\zeta }_{\ell })}_{\mathbb{T}},[0,\infty )\right)$$(\ell =1,\dots ,n).$

$\left(S_6\right)$

$p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })$ are n positive nabla-integrable functions defined for ${\varsigma }_{\ell }\in {({\chi }_0,s_{\ell })}_{\mathbb{T}},$ ${\tau }_{\ell }\in {({\chi }_0,{\chi }_{\ell })}_{\mathbb{T}}.$

$\left(S_7\right)$

$p_{\ell }\left({\varsigma }_{\ell }\right)$ and $q_{\ell }\left({\tau }_{\ell }\right)$ are positive nabla-integrable functions defined for ${\varsigma }_{\ell }\in {({\chi }_0,s_{\ell })}_{\mathbb{T}},$ ${\tau }_{\ell }\in {({\chi }_0,{\chi }_{\ell })}_{\mathbb{T}}.$

$\left(S_8\right)$

${\Phi }_{\ell }$ and ${\Psi }_{\ell }$, $(\ell =1,\dots ,n)$ are n real-valued, non-negative concave and supermultiplicative functions defined on $(0,\infty ).$

$\left(S_9\right)$

${\xi }_{\ell }$ and ${\zeta }_{\ell }$ are positive real numbers.

$\left(S_{10}\right)$

$s_{\ell }\in {[{\chi }_0,{\xi }_{\ell })}_{\mathbb{T}}$ and ${\chi }_{\ell }\in {[{\chi }_0,{\zeta }_{\ell })}_{\mathbb{T}}.$

$\left(S_{11}\right)$

${\vartheta }_{\ell }({\chi }_0,{\chi }_{\ell })={\vartheta }_{\ell }(s_{\ell },{\chi }_0)=0,$$(\ell =1,\dots ,n).$

$\left(S_{12}\right)$

${\vartheta }_{\ell }^{{\nabla }_{a_1}^{\gamma }{\nabla }_{a_2}^{\gamma }}(s_{\ell },{\chi }_{\ell })={\vartheta }_{\ell }^{{\nabla }_{a_2}^{\gamma }{\nabla }_{a_1}^{\gamma }}(s_{\ell },{\chi }_{\ell }).$

$\left(S_{13}\right)$

$P_{\ell }(s_{\ell },{\chi }_{\ell })={\int }_{{\chi }_0}^{{\chi }_{\ell }}{\int }_{{\chi }_0}^{s_{\ell }}{p}_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }.$

$\left(S_{14}\right)$

${\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })={\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }.$

$\left(S_{15}\right)$

$P_{\ell }(s_{\ell },{\chi }_{\ell })={\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{p}_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }.$

$\left(S_{16}\right)$

${\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })=\frac{1}{P_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })}{\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{p}_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }.$

$\left(S_{17}\right)$

${\alpha }_{\ell }\in (1,\infty ),$${\alpha }_{\ell }^{\prime }=1-{\alpha }_{\ell },$$\alpha ={\sum }_{\ell =1}^n{\alpha }_{\ell },$ and ${\alpha }^{\prime }={\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }=n-\alpha ,$ $(\ell =1,\dots ,n).$

$\left(S_{18}\right)$

$0<{\beta }_{\ell }<1.$

$\left(S_{19}\right)$

$h_{\ell }⩾2.$

$\left(S_{20}\right)$

${\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}=\frac{1}{\alpha }.$

$\left(S_{21}\right)$

$h_{\ell }⩾1.$

$\left(S_{22}\right)$

${\vartheta }_{\ell }\left({\varsigma }_{\ell }\right)\in C_{ld}^1{[{\chi }_0,{\xi }_{\ell }]}_{\mathbb{T}},$$(\ell =1,\dots ,n).$

$\left(S_{23}\right)$

${\xi }_{\ell }$ is a positive real number.

$\left(S_{24}\right)$

${\Theta }_{\ell }\left(s_{\ell }\right)={\int }_{{\chi }_0}^{s_{\ell }}{\vartheta }_{\ell }\left({\varsigma }_{\ell }\right){\nabla }_a^{\gamma }{\varsigma }_{\ell }.$

$\left(S_{25}\right)$

$s_{\ell }\in {[{\chi }_0,{\xi }_{\ell })}_{\mathbb{T}}.$

$\left(S_{26}\right)$

$p_{\ell }\left({\varsigma }_{\ell }\right)$ are n positive functions.

$\left(S_{27}\right)$

$P_{\ell }\left(s_{\ell }\right)={\int }_{{\chi }_0}^{s_{\ell }}{p}_{\ell }\left({\varsigma }_{\ell }\right){\nabla }_a^{\gamma }{\varsigma }_{\ell }.$

$\left(S_{28}\right)$

${\Theta }_{\ell }\left(s_{\ell }\right)=\frac{1}{P_{\ell }\left(s_{\ell }\right)}{\int }_{{\chi }_0}^{s_{\ell }}p\left({\varsigma }_{\ell }\right)\vartheta \left({\varsigma }_{\ell }\right){\nabla }_a^{\gamma }{\varsigma }_{\ell }.$

$\left(S_{29}\right)$

${\vartheta }_{\ell }\left({\chi }_0\right)=0.$

The first important inequality is stated in the following theorem:

Theorem 1.

Let $S_1,$ $S_4$, $S_5,$ $S_{14},$ $S_6,$ $S_{15}$ and $S_8$ be satisfied. Then, for $S_{10},$ $S_{18}$ and $S_{20}$, we find that

$$\begin{array}{ccc}& & {\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{\frac{1}{\alpha }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1\hfill \\ \hfill & & ⩾L({\xi }_1{\zeta }_1,\dots ,{\xi }_n{y}_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}(\rho \left({\xi }_{\ell }\right)-s_{\ell })(\rho \left({\zeta }_{\ell }\right)-{\chi }_{\ell }){\left(p_{\ell }(s_{\ell },{\chi }_{\ell }){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })}{p_{\ell }(s_{\ell },{\chi }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}\hfill \end{array}$$

(11)

where

$$L({\xi }_1{\zeta }_1,\dots ,{\xi }_n{y}_n)=\prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{P_{\ell }(s_{\ell },{\chi }_{\ell })}\right)}^{{\alpha }_{\ell }}{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }\right)}^{\frac{1}{{\alpha }_{\ell }}}.$$

Proof. 

From the hypotheses of Theorem 1, $S_{14},$ $S_{15},$ and $S_8,$ it is easy to observe that

$$\begin{array}{ccc}\hfill {\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)& =& {\Phi }_{\ell }\left({\displaystyle \frac{P_{\ell }(s_{\ell },{\chi }_{\ell }){\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{p}_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })\left(\frac{{\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })}{p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })}\right){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }}{{\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{p}_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }}}\right)\hfill \\ & ⩾& {\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right){\Phi }_{\ell }\left({\displaystyle \frac{{\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{p}_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })\left(\frac{{\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })}{p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })}\right){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }}{{\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{p}_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }}}\right).\hfill \end{array}$$

(12)

Using inverse Jensen dynamic inequality, we obtain that

$$\begin{array}{ccc}\hfill {\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)& ⩾& \frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{P_{\ell }(s_{\ell },{\chi }_{\ell })}{\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{p}_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })}{p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })}\right){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }.\hfill \end{array}$$

(13)

Applying inverse Hölder’s inequality on the right-hand side of (13) with indices ${\alpha }_{\ell }$ and ${\beta }_{\ell },$ it is easy to observe that

$${\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)⩾\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{P_{\ell }(s_{\ell },{\chi }_{\ell })}{\left[(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right]}^{\frac{1}{{\alpha }_{\ell }}}{\left({\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{\left(p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })}{p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.$$

(14)

Using the following inequality on the term ${\left[(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right]}^{\frac{1}{{\alpha }_{\ell }}},$

$$\prod _{\ell =1}^n{m}_{\ell }^{\frac{1}{{\alpha }_{\ell }}}⩾{\left(\alpha \sum _{\ell =1}^n\frac{1}{{\alpha }_{\ell }}{m}_{\ell }\right)}^{\frac{1}{\alpha }},$$

(15)

we get that

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{\frac{1}{\alpha }}}}\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{P_{\ell }(s_{\ell },{\chi }_{\ell })}{\left({\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{\left(p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })}{p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })}\right)\right)}^{\frac{1}{{\beta }_{\ell }}}{\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(16)

Integrating both sides of (16) over $s_{\ell },$ ${\chi }_{\ell }$ from ${\chi }_0$ to ${\xi }_{\ell },$ ${\zeta }_{\ell }$ $(\ell =1,\dots ,n),$ we obtain that

$${\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{\frac{1}{\alpha }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1$$

$$⩾\prod _{\ell =1}^n{\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{P_{\ell }(s_{\ell },{\chi }_{\ell })}{\left({\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{\left(p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })}{p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }.$$

(17)

Applying inverse Hölder’s inequality on the right-hand side of (17) with indices ${\alpha }_{\ell }$ and ${\beta }_{\ell },$ it is easy to observe that

$${\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{\frac{1}{\alpha }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1$$

$$⩾\prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{P_{\ell }(s_{\ell },{\chi }_{\ell })}\right)}^{{\alpha }_{\ell }}{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }\right)}^{\frac{1}{{\alpha }_{\ell }}}$$

(18)

$$\times \prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}\left({\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{\left(p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })}{p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }\right){\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.$$

Using Fubini’s theorem, we observe that

$$\begin{array}{ccc}\hfill & & {\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{\frac{1}{\alpha }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1\hfill \\ & & ⩾L({\xi }_1{\zeta }_1,\dots ,{\xi }_n{y}_n)\hfill \\ & & \times \prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}({\xi }_{\ell }-s_{\ell })({\zeta }_{\ell }-{\chi }_{\ell }){\left(p_{\ell }(s_{\ell },{\chi }_{\ell }){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })}{p_{\ell }(s_{\ell },{\chi }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

Using the fact ${\xi }_{\ell }⩾\rho \left({\xi }_{\ell }\right),$ and ${\zeta }_{\ell }⩾\rho \left({\zeta }_{\ell }\right),$, we get that

$$\begin{array}{ccc}& & {\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{\frac{1}{\alpha }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1\hfill \\ \hfill & & ⩾L({\xi }_1{\zeta }_1,\dots ,{\xi }_n{y}_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}(\rho \left({\xi }_{\ell }\right)-s_{\ell })(\rho \left({\zeta }_{\ell }\right)-{\chi }_{\ell }){\left(p_{\ell }(s_{\ell },{\chi }_{\ell }){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })}{p_{\ell }(s_{\ell },{\chi }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

This completes the proof. □

Remark 1.

In Theorem 1, if $\mathbb{T}=\mathbb{R}$, $\gamma =1$, we get the result due to Zhao et al. [8] (Theorem 2).

As a special case of Theorem 1, when $\mathbb{T}=\mathbb{Z}$, $\gamma =1$, we have $\rho \left(n\right)=n-1,$ and we get the following result.

Corollary 1.

Let $\left\{a_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}\right\}$ and $\left\{p_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}\right\},$ $(\ell =1,\dots ,n)$ be n sequences of non-negative numbers defined for $m_{s_{\ell }}=1,\dots ,k_{s_{\ell }},$ and $m_{{\chi }_{\ell }}=1,\dots ,k_{{\chi }_{\ell }},$ and define

$$\begin{array}{c}\hfill A_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}=\sum _{m_{{\varsigma }_{\ell }}}^{m_{s_{\ell }}}\sum _{m_{{\eta }_{\ell }}}^{m_{{\chi }_{\ell }}}{a}_{s_{\ell },{\chi }_{\ell },m_{{\varsigma }_{\ell }},m_{{\eta }_{\ell }}}\\ \hfill P_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}=\sum _{m_{{\varsigma }_{\ell }}}^{m_{s_{\ell }}}\sum _{m_{{\eta }_{\ell }}}^{m_{{\chi }_{\ell }}}{p}_{s_{\ell },{\chi }_{\ell },m_{{\varsigma }_{\ell }},m_{{\eta }_{\ell }}}.\end{array}$$

(19)

Then,

$$\begin{array}{ccc}& & \sum _{m_{s_1}}^{k_{s_1}}\sum _{m_{{\chi }_1}}^{k_{{\chi }_1}}\dots \sum _{m_{s_n}}^{k_{s_n}}\sum _{m_{{\chi }_n}}^{k_{{\chi }_n}}\frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left(A_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}\left(m_{s_{\ell }}{m}_{{\chi }_{\ell }}\right)\right)}^{\frac{1}{\alpha }}}\hfill \\ \hfill & & ⩾C(k_{s_1}{k}_{{\chi }_1},\dots ,k_{s_n}{k}_{{\chi }_n})\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left(\sum _{m_{s_{\ell }}}^{k_{s_{\ell }}}\sum _{m_{{\chi }_{\ell }}}^{k_{{\chi }_{\ell }}}(k_{s_{\ell }}-(m_{s_{\ell }}-1))(k_{{\chi }_{\ell }}-(m_{{\chi }_{\ell }}-1)){\left(P_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}{\Phi }_{\ell }\left(\frac{a_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}}{P_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}}\right)\right)}^{{\beta }_{\ell }}\right)}^{\frac{1}{{\beta }_{\ell }}}\hfill \end{array}$$

where

$$C(k_{s_1}{k}_{{\chi }_1},\dots ,k_{s_n}{k}_{{\chi }_n})=\prod _{\ell =1}^n{\left(\sum _{m_{s_{\ell }}}^{k_{s_{\ell }}}\sum _{m_{{\chi }_{\ell }}}^{k_{{\chi }_{\ell }}}\left(\frac{{\Phi }_{\ell }\left(P_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}\right)}{P_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}}\right)\right)}^{{\beta }_{\ell }}{)}^{\frac{1}{{\beta }_{\ell }}}.$$

Remark 2.

Let ${\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }),$ $p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }),$ $P_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }),$ and ${\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })$ change to ${\vartheta }_{\ell }\left({\varsigma }_{\ell }\right),$ $p_{\ell }\left({\varsigma }_{\ell }\right),$ $P_{\ell }\left(s_{\ell }\right)$ and ${\vartheta }_{\ell }\left(s_{\ell }\right),$ respectively, and with suitable changes, we have the following new corollary:

Corollary 2.

Let $S_{22},$ $S_{23},$ $S_{24},$ $S_{26},$ $S_{27}$ be satisfied. Then, for $S_{18},$ $S_{20}$ and $S_{25}$ we find that

$$\begin{array}{ccc}\hfill & & {\int }_{{\chi }_0}^{{\xi }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }({\Theta }_{\ell }\left(s_{\ell }\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-{\chi }_0)\right)}^{\frac{1}{\alpha }}}}{\nabla }_a^{\gamma }{s}_1\dots {\nabla }_a^{\gamma }{s}_n\hfill \\ \hfill & & ⩾L^{*}({\xi }_1,\dots ,{\xi }_n)\prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}(\rho \left({\xi }_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }\left(s_{\ell }\right)}{p_{\ell }\left(s_{\ell }\right)}\right)\right)}^{{\beta }_{\ell }}{\nabla }_a^{\gamma }{s}_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}\hfill \end{array}$$

(20)

where

$$L^{*}({\xi }_1,\dots ,{\xi }_n)=\prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell }\right)}\right)}^{{\alpha }_{\ell }}{\nabla }_a^{\gamma }{s}_{\ell }\right)}^{\frac{1}{{\alpha }_{\ell }}}.$$

Remark 3.

In Corollary 2, if we take $n=2$, ${\beta }_{\ell }=\frac{1}{2}$, then the Inequality (21) changes to

$$\begin{array}{c}{\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\xi }_1}{\displaystyle \frac{{\Phi }_1\left({\Theta }_1\left(s_1\right)\right){\Phi }_2\left({\Theta }_2\left(s_2\right)\right)}{{\left((s_1-{\chi }_0)+(s_2-{\chi }_0)\right)}^{-2}}}{\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{s}_2⩾L^{**}({\xi }_1,{\xi }_2)({\int }_{{\chi }_0}^{{\xi }_1}(\rho \left({\xi }_1\right)-s_1)\\ {\left(p_1\left(s_1\right)\Phi \left(\frac{{\vartheta }_1\left(s_1\right)}{p_1\left(s_1\right)}\right)\right)}^2{\nabla }_a^{\gamma }{s}_1{)}^{\frac{1}{2}}\\ \times {\left({\int }_{{\chi }_0}^{{\xi }_2}(\rho \left({\xi }_2\right)-s_2){\left(p_2\left(s_2\right)\Psi \left(\frac{{\vartheta }_2\left(s_2\right)}{p_2\left(s_2\right)}\right)\right)}^2{\nabla }_a^{\gamma }{s}_2\right)}^{\frac{1}{2}}\end{array}$$

(21)

where

$$L^{**}({\xi }_1,{\xi }_2)=4{\left({\int }_{{\chi }_0}^{{\xi }_1}{\left(\frac{{\Phi }_1\left(P_1\left(s_1\right)\right)}{P_1\left(s_1\right)}\right)}^{-1}{\nabla }_a^{\gamma }{s}_1\right)}^{-1}{\left({\int }_{{\chi }_0}^{{\xi }_2}{\left(\frac{{\Phi }_2\left(P_2\left(s_2\right)\right)}{P_2\left(s_2\right)}\right)}^{-1}{\nabla }_a^{\gamma }{s}_2\right)}^{-1}$$

.

Remark 4.

In Corollary 3, if we take $\mathbb{T}=\mathbb{R}$, $\gamma =1$, then the inequality (22) changes to

$$\begin{array}{c}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_1}{\displaystyle \frac{{\Phi }_1\left({\Theta }_1\left(s_1\right)\right){\Phi }_2\left({\Theta }_2\left(s_2\right)\right)}{{(s_1+s_2)}^{-2}}}d{s}_{1}d{s}_2\\ ⩾L^{**}({\xi }_1,{\xi }_2){\left({\int }_0^{{\xi }_1}({\xi }_1-s_1){\left(p_1\left(s_1\right)\Phi \left(\frac{{\vartheta }_1\left(s_1\right)}{p_1\left(s_1\right)}\right)\right)}^{2}d{s}_1\right)}^{\frac{1}{2}}\\ \times {\left({\int }_0^{{\xi }_2}({\xi }_2-s_2){\left(p_2\left(s_2\right)\Psi \left(\frac{{\vartheta }_2\left(s_2\right)}{p_2\left(s_2\right)}\right)\right)}^{2}d{s}_2\right)}^{\frac{1}{2}}\end{array}$$

(22)

where

$$L^{**}({\xi }_1,{\xi }_2)=4{\left({\int }_0^{{\xi }_1}{\left(\frac{{\Phi }_1\left(P_1\left(s_1\right)\right)}{P_1\left(s_1\right)}\right)}^{-1}d{s}_1\right)}^{-1}{\left({\int }_0^{{\xi }_2}{\left(\frac{{\Phi }_2\left(P_2\left(s_2\right)\right)}{P_2\left(s_2\right)}\right)}^{-1}d{s}_2\right)}^{-1}.$$

This is an inverse of the Inequality (4), which was proved by Pachpatte [15].

Corollary 3.

In Corollary 2, if we take ${\beta }_{\ell }=\frac{n-1}{n}$ the Inequality (21) becomes

$$\begin{array}{ccc}& & {\int }_{{\chi }_0}^{{\xi }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }({\Theta }_{\ell }\left(s_{\ell }\right)}{{\left({\sum }_{\ell =1}^n(s_{\ell }-{\chi }_0)\right)}^{\frac{-n}{n-1}}}}{\nabla }_a^{\gamma }{s}_n\dots {\nabla }_a^{\gamma }{s}_1\hfill \\ \hfill & & ⩾L^{*}({\xi }_1,\dots ,{\xi }_n)\prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}(\rho \left({\xi }_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }\left(s_{\ell }\right)}{p_{\ell }\left(s_{\ell }\right)}\right)\right)}^{\frac{n-1}{n}}{\nabla }_a^{\gamma }{s}_{\ell }\right)}^{\frac{n}{n-1}}\hfill \end{array}$$

where

$$L^{*}({\xi }_1,\dots ,{\xi }_n)=n^{\frac{n}{n-1}}\prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell }\right)}\right)}^{-(n-1)}{\nabla }_a^{\gamma }{s}_{\ell }\right)}^{\frac{-1}{n-1}}.$$

Theorem 2.

Let $S_1,$ $S_4$, $S_5,$ $S_6,$ $S_9,$ $S_{15},$ and $S_{16}$ be satisfied. Then for $S_{10},$ $S_{18}$ and $S_{20}$, we have that

$$\begin{array}{ccc}& & {\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }(s_{\ell },{\chi }_{\ell }){\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{\frac{1}{\alpha }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n{\left[({\xi }_{\ell }-{\chi }_0)({\zeta }_{\ell }-{\chi }_0)\right]}^{\frac{1}{{\alpha }_{\ell }}}{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}(\rho \left({\xi }_{\ell }\right)-s_{\ell })(\rho \left({\zeta }_{\ell }\right)-{\chi }_{\ell }){\left(p_{\ell }(s_{\ell },{\chi }_{\ell }){\Phi }_{\ell }\left({\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(23)

Proof. 

From the hypotheses of Theorem 2, and using inverse Jensen dynamic inequality, we have

$$\begin{array}{ccc}\hfill {\Phi }_{\ell }\left({\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)& =& {\Phi }_{\ell }\left(\frac{1}{P_{\ell }(s_{\ell },{\chi }_{\ell })}{\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{p}_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\Theta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }\right)\hfill \\ & & ⩾\frac{1}{P_{\ell }(s_{\ell },{\chi }_{\ell })}{\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{p}_{\ell }({\sigma }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })\right){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }.\hfill \end{array}$$

(24)

Applying inverse Hölder’s inequality on the right-hand side of (24) with indices ${\alpha }_{\ell }$ and ${\beta }_{\ell },$, it is easy to observe that

$$\begin{array}{c}{\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)⩾\frac{1}{P_{\ell }(s_{\ell },{\chi }_{\ell })}{\left[(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right]}^{\frac{1}{{\alpha }_{\ell }}}\left({\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}\right(p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\\ \left({\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })\right){)}^{{\beta }_{\ell }}{\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }{)}^{\frac{1}{{\beta }_{\ell }}}.\end{array}$$

Using the Inequality (15), on the term ${\left[(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right]}^{\frac{1}{{\alpha }_{\ell }}}$, we get that

$$\begin{array}{c}{\displaystyle \frac{P_{\ell }(s_{\ell },{\chi }_{\ell }){\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{\frac{1}{\alpha }}}}⩾\\ {\left({\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{\left(p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}\end{array}$$

(25)

Integrating both sides of (25) over $s_{\ell },$ ${\chi }_{\ell }$ from ${\chi }_0$ to ${\xi }_{\ell },$ ${\zeta }_{\ell }$ $(\ell =1,\dots ,n),$ we get that

$$\begin{array}{ccc}\hfill & & {\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }(s_{\ell },{\chi }_{\ell }){\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{\frac{1}{\alpha }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n{\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}{\left({\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{\left(p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\nabla }_a^{\gamma }{\sigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

Applying inverse Hölder’s inequality on the right hand side of (17) with indices ${\alpha }_{\ell }$ and ${\beta }_{\ell },$ it is easy to observe that

$$\begin{array}{c}{\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }(s_{\ell },{\chi }_{\ell }){\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{\frac{1}{\alpha }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1\\ ⩾\prod _{\ell =1}^n{\left[({\xi }_{\ell }-{\chi }_0)({\zeta }_{\ell }-{\chi }_0)\right]}^{\frac{1}{{\alpha }_{\ell }}}({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}{\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{\left(p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })\right)\right)}^{{\beta }_{\ell }}\\ {\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }{)}^{\frac{1}{{\beta }_{\ell }}}.\end{array}$$

Using Fubini’s theorem, we observe that

$$\begin{array}{c}{\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }(s_{\ell },{\chi }_{\ell }){\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{\frac{1}{\alpha }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1\\ ⩾\prod _{\ell =1}^n{\left[({\xi }_{\ell }-{\chi }_0)({\zeta }_{\ell }-{\chi }_0)\right]}^{\frac{1}{{\alpha }_{\ell }}}({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}({\xi }_{\ell }-s_{\ell })({\zeta }_{\ell }-{\chi }_{\ell }){\left(p_{\ell }(s_{\ell },{\chi }_{\ell }){\Phi }_{\ell }\left({\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)\right)}^{{\beta }_{\ell }}\\ {\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }{)}^{\frac{1}{{\beta }_{\ell }}}.\end{array}$$

By using the fact ${\xi }_{\ell }⩾\rho \left({\xi }_{\ell }\right),$ and ${\zeta }_{\ell }⩾\rho \left({\zeta }_{\ell }\right),$ we get that

$$\begin{array}{c}{\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }(s_{\ell },{\chi }_{\ell }){\Phi }_{\ell }\left({\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{\frac{1}{\alpha }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1\\ ⩾\prod _{\ell =1}^n{\left[({\xi }_{\ell }-{\chi }_0)({\zeta }_{\ell }-{\chi }_0)\right]}^{\frac{1}{{\alpha }_{\ell }}}({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}(\rho \left({\xi }_{\ell }\right)-s_{\ell })\\ (\rho \left({\zeta }_{\ell }\right)-{\chi }_{\ell }){\left(p_{\ell }(s_{\ell },{\chi }_{\ell }){\Phi }_{\ell }\left({\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }{)}^{\frac{1}{{\beta }_{\ell }}}\end{array}$$

This completes the proof. □

Remark 5.

In Theorem 2, if $\mathbb{T}=\mathbb{R}$, we get the result due to Zhao et al. [8] (Theorem 3).

As a special case of Theorem 2, when $\mathbb{T}=\mathbb{Z}$, $\gamma =1$, we have $\rho \left(n\right)=n-1,$ we get the following result.

Corollary 4.

Let $\left\{a_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}\right\}$ and $\left\{p_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}\right\},$ $(\ell =1,\dots ,n)$ be n sequences of non-negative numbers defined for $m_{s_{\ell }}=1,\dots ,k_{s_{\ell }},$ and $m_{{\chi }_{\ell }}=1,\dots ,k_{{\chi }_{\ell }},$ and define

$$\begin{array}{ccc}\hfill A_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}& =& \frac{1}{P_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}}\sum _{m_{{\varsigma }_{\ell }}}^{m_{s_{\ell }}}\sum _{m_{{\eta }_{\ell }}}^{m_{{\chi }_{\ell }}}{a}_{s_{\ell },{\chi }_{\ell },m_{{\varsigma }_{\ell }},m_{{\eta }_{\ell }}}{p}_{s_{\ell },{\chi }_{\ell },m_{{\varsigma }_{\ell }},m_{{\eta }_{\ell }}},\hfill \\ \hfill P_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}& =& \sum _{m_{{\varsigma }_{\ell }}}^{m_{s_{\ell }}}\sum _{m_{{\eta }_{\ell }}}^{m_{{\chi }_{\ell }}}{p}_{s_{\ell },{\chi }_{\ell },m_{{\varsigma }_{\ell }},m_{{\eta }_{\ell }}}.\hfill \end{array}$$

(26)

Then,

$$\begin{array}{c}\sum _{m_{s_1}}^{k_{s_1}}\sum _{m_{{\chi }_1}}^{k_{{\chi }_1}}\dots \sum _{m_{s_n}}^{k_{s_n}}\sum _{m_{{\chi }_n}}^{k_{{\chi }_n}}\frac{{\prod }_{\ell =1}^n{P}_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}{\Phi }_{\ell }\left(A_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}\left(m_{s_{\ell }}{m}_{{\chi }_{\ell }}\right)\right)}^{\frac{1}{\alpha }}}\\ ⩾\prod _{\ell =1}^n{\left(k_{s_{\ell }}{k}_{{\chi }_{\ell }}\right)}^{\frac{1}{{\alpha }_{\ell }}}(\sum _{m_{s_{\ell }}}^{k_{s_{\ell }}}\sum _{m_{{\chi }_{\ell }}}^{k_{{\chi }_{\ell }}}(k_{s_{\ell }}-(m_{s_{\ell }}-1))(k_{{\chi }_{\ell }}-(m_{{\chi }_{\ell }}-1))\\ {\left(p_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}{\Phi }_{\ell }\left(a_{s_{\ell },{\chi }_{\ell },m_{s_{\ell }},m_{{\chi }_{\ell }}}\right)\right)}^{{\beta }_{\ell }}{)}^{\frac{1}{{\beta }_{\ell }}}.\end{array}$$

Remark 6.

Let ${\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }),$ $p_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }),$ $P_{\ell }({\varsigma }_{\ell },{\tau }_{\ell })$ be defined as above and

$${\Theta }_{\ell }(s_{\ell },{\chi }_{\ell })=\frac{1}{P_{\ell }(s_{\ell },{\chi }_{\ell })}{\int }_{{\chi }_0}^{s_{\ell }}{\int }_0^{{\chi }_{\ell }}{p}_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\vartheta }_{\ell }({\varsigma }_{\ell },{\tau }_{\ell }){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }$$

changes to ${\vartheta }_{\ell }\left({\varsigma }_{\ell }\right),$ $p_{\ell }\left({\varsigma }_{\ell }\right),$ $P_{\ell }\left(s_{\ell }\right),$ and

$${\Theta }_{\ell }\left(s_{\ell }\right)=\frac{1}{P_{\ell }\left(s_{\ell }\right)}{\int }_{{\chi }_0}^{s_{\ell }}{p}_{\ell }\left({\varsigma }_{\ell }\right){\vartheta }_{\ell }\left({\varsigma }_{\ell }\right){\nabla }_a^{\gamma }{\varsigma }_{\ell }.$$

respectively, and with suitable changes, we have the following new corollary:

Corollary 5.

Let $S_{22},$ $S_{23},$ $S_{26},$ $S_{27}$ and $S_{28}$ be satisfied. Then, for $S_{18},$ $S_{20}$ and $S_{25}$, we find that

$$\begin{array}{ccc}\hfill & & {\int }_{{\chi }_0}^{{\xi }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }({\Theta }_{\ell }\left(s_{\ell }\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-{\chi }_0)\right)}^{\frac{1}{\alpha }}}}{\nabla }_a^{\gamma }{s}_n\dots {\nabla }_a^{\gamma }{s}_1\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n{({\xi }_{\ell }-{\chi }_0)}^{\frac{1}{{\alpha }_{\ell }}}{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}(\rho \left({\xi }_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left({\vartheta }_{\ell }\left(s_{\ell }\right)\right)\right)}^{{\beta }_{\ell }}{\nabla }_a^{\gamma }{s}_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(27)

Corollary 6.

In Corollary 5, if we take $n=2,$ ${\beta }_{\ell }=\frac{1}{2}$, then the inequality (21) changes to

$$\begin{array}{c}{\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\xi }_1}{\displaystyle \frac{P_1\left(s_1\right)P_2\left(s_2\right){\Phi }_1\left({\Theta }_1\left(s_1\right)\right){\Phi }_2\left({\Theta }_2\left(s_2\right)\right)}{{\left((s_1-{\chi }_0)+(s_2-{\chi }_0)\right)}^{-2}}}{\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{s}_2⩾4{\left[({\xi }_1-{\chi }_0)({\xi }_2-{\chi }_0)\right]}^{-1}\\ \times {\left({\int }_{{\chi }_0}^{{\xi }_1}(\rho \left({\xi }_1\right)-s_1){\left(p_1\left(s_1\right){\Phi }_1\left({\vartheta }_1\left(s_1\right)\right)\right)}^2{\nabla }_a^{\gamma }{s}_1\right)}^{\frac{1}{2}}\left({\int }_{{\chi }_0}^{{\xi }_2}(\rho \left({\xi }_2\right)-s_2)\right(p_2\left(s_2\right)\\ {\Phi }_2\left({\vartheta }_2\left(s_2\right)\right){)}^2{\nabla }_a^{\gamma }{s}_2{)}^{\frac{1}{2}}.\end{array}$$

(28)

Remark 7.

In Corollary 6, if we take $\mathbb{T}=\mathbb{R},$ $\gamma =1$, then Inequality (28) changes to

$$\begin{array}{c}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_1}{\displaystyle \frac{P_1\left(s_1\right)P_2\left(s_2\right){\Phi }_1\left({\Theta }_1\left(s_1\right)\right){\Phi }_2\left({\Theta }_2\left(s_2\right)\right)}{{(s_1+s_2)}^{-2}}}d{s}_{1}d{s}_2⩾4{\left[{\xi }_1{\xi }_2\right]}^{-1}\\ \times {\left({\int }_0^{{\xi }_1}({\xi }_1-s_1){\left(p_1\left(s_1\right){\Phi }_1\left({\vartheta }_1\left(s_1\right)\right)\right)}^{2}d{s}_1\right)}^{\frac{1}{2}}\\ {\left({\int }_0^{{\xi }_2}({\xi }_2-s_2){\left(p_2\left(s_2\right){\Phi }_2\left({\vartheta }_2\left(s_2\right)\right)\right)}^{2}d{s}_2\right)}^{\frac{1}{2}}.\end{array}$$

(29)

This is an inverse of Inequality (5), which was proved by Pachpatte [15].

Corollary 7.

In Corollary 6, let $p_1\left(s_1\right)=p_2\left(s_2\right)=1,$ then $P_1\left(s_1\right)=s_1,$ $P_2\left(s_2\right)=s_2.$ Therefore, Inequality (28) changes to

$$\begin{array}{c}{\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\xi }_1}{\displaystyle \frac{{\Phi }_1\left({\Theta }_1\left(s_1\right)\right){\Phi }_2\left({\Theta }_2\left(s_2\right)\right)}{{\left(s_1{s}_2\right)}^{-1}{\left((s_1-{\chi }_0)+(s_2-{\chi }_0)\right)}^{-2}}}{\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{s}_2⩾4{\left[({\xi }_1-{\chi }_0)({\xi }_2-{\chi }_0)\right]}^{-1}\\ \times {\left({\int }_{{\chi }_0}^{{\xi }_1}(\rho \left({\xi }_1\right)-s_1){\left({\Phi }_1\left({\vartheta }_1\left(s_1\right)\right)\right)}^2{\nabla }_a^{\gamma }{s}_1\right)}^{\frac{1}{2}}\\ {\left({\int }_{{\chi }_0}^{{\xi }_2}(\rho \left({\xi }_2\right)-s_2){\left({\Phi }_2\left({\vartheta }_2\left(s_2\right)\right)\right)}^2{\nabla }_a^{\gamma }{s}_2\right)}^{\frac{1}{2}}.\end{array}$$

(30)

Remark 8.

In Corollary 7, if we take $\mathbb{T}=\mathbb{R},$ then, the Inequality (30) changes to

$$\begin{array}{ccc}& & {\int }_0^{{\xi }_1}{\int }_0^{{\xi }_1}{\displaystyle \frac{{\Phi }_1\left({\Theta }_1\left(s_1\right)\right){\Phi }_2\left({\Theta }_2\left(s_2\right)\right)}{{\left(s_1{s}_2\right)}^{-1}{\left(s_1+s_2\right)}^{-2}}}d{s}_{1}d{s}_2⩾4{\left[{\xi }_1{\xi }_2\right]}^{-1}\hfill \\ \hfill & & \times {\left({\int }_0^{{\xi }_1}({\xi }_1-s_1){\left({\Phi }_1\left({\vartheta }_1\left(s_1\right)\right)\right)}^{2}d{s}_1\right)}^{\frac{1}{2}}{\left({\int }_0^{{\xi }_2}({\xi }_2-s_2){\left({\Phi }_2\left({\vartheta }_2\left(s_2\right)\right)\right)}^{2}d{s}_2\right)}^{\frac{1}{2}}.\hfill \end{array}$$

This is an inverse inequality of the following inequality, which was proved by Pachpatte [8].

$$\begin{array}{ccc}& & {\int }_0^{\xi }{\int }_0^{\zeta }{\displaystyle \frac{\Phi (\Theta (s\left)\right)\Psi \left(G\right(\chi \left)\right)}{{\left(s\chi \right)}^{-1}\left(s+\chi \right)}}dsd\chi ⩽\frac{1}{2}{\left[\xi \zeta \right]}^{\frac{1}{2}}\hfill \\ \hfill & & \times {\left({\int }_0^{\xi }(\xi -s_1){\left(\Phi \left(\vartheta \left(s\right)\right)\right)}^{2}ds\right)}^{\frac{1}{2}}{\left({\int }_0^{\zeta }(\zeta -\chi ){\left(\Psi \left(g\left(\chi \right)\right)\right)}^{2}d\chi \right)}^{\frac{1}{2}}.\hfill \end{array}$$

Corollary 8.

In Corollary 5, if we take ${\beta }_{\ell }=\frac{n-1}{n}$ $(\ell =1,\dots ,n)$ Inequality (27)

$$\begin{array}{c}\hfill {\int }_{{\chi }_0}^{{\xi }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }({\Theta }_{\ell }\left(s_{\ell }\right)}{{\left({\sum }_{\ell =1}^n(s_{\ell }-{\chi }_0)\right)}^{\frac{-n}{n-1}}}}{\nabla }_a^{\gamma }{s}_n\dots {\nabla }_a^{\gamma }{s}_1\\ \hfill ⩾n^{\frac{n}{n-1}}\prod _{\ell =1}^n{({\xi }_{\ell }-{\chi }_0)}^{\frac{-1}{n-1}}{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}(\rho \left({\xi }_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left({\vartheta }_{\ell }\left(s_{\ell }\right)\right)\right)}^{\frac{n-1}{n}}{\nabla }_a^{\gamma }{s}_{\ell }\right)}^{\frac{n}{n-1}}.\end{array}$$

Theorem 3.

Let $S_1,$ $S_4$, $S_2,$ $S_9,$ $S_{11},$ $S_7,$ $S_{13},$ $S_3,$ $S_{12},$ $S_8$ and $S_{17}$ be satisfied. Then, for $S_{10}$ we have that

$$\begin{array}{c}{\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{{\alpha }^{\prime }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1\\ ⩾G({\xi }_1{\zeta }_1,\dots ,{\xi }_n{y}_n)\\ \times \prod _{\ell =1}^n({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}(\rho \left({\xi }_{\ell }\right)-s_{\ell })(\rho \left({\zeta }_{\ell }\right)-{\chi }_{\ell }){\left(p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\chi }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }^{{\nabla }_{a_2}^{\gamma }{\nabla }_{a_1}^{\gamma }}(s_{\ell },{\chi }_{\ell })}{p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\chi }_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}\\ {\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }{)}^{{\alpha }_{\ell }},\end{array}$$

(31)

where

$$G({\xi }_1{\zeta }_1,\dots ,{\xi }_n{y}_n)=\prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{P_{\ell }(s_{\ell },{\chi }_{\ell })}\right)}^{\frac{1}{{\alpha }_{\ell }^{\prime }}}{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }\right)}^{{\alpha }_{\ell }^{\prime }}.$$

Proof. 

From the hypotheses of Theorem 3, we obtain

$${\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })={\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{\vartheta }_{\ell }^{{\nabla }_{a_2}^{\gamma }{\nabla }_{a_1}^{\gamma }}({\varsigma }_{\ell },{\tau }_{\ell }){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }.$$

(32)

From (32) and $S_8$, it is easy to observe that

$${\Phi }_{\ell }\left({\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)={\Phi }_{\ell }\left({\displaystyle \frac{P_{\ell }(s_{\ell },{\chi }_{\ell }){\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{p}_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)\left(\frac{{\vartheta }_{\ell }^{{\nabla }_{a_2}^{\gamma }{\nabla }_{a_1}^{\gamma }}({\varsigma }_{\ell },{\tau }_{\ell })}{p_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }}{{\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{p}_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }}}\right)$$

$$⩾{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right){\Phi }_{\ell }\left({\displaystyle \frac{{\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{p}_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)\left(\frac{{\vartheta }_{\ell }^{{\nabla }_{a_2}^{\gamma }{\nabla }_{a_1}^{\gamma }}({\varsigma }_{\ell },{\tau }_{\ell })}{p_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }}{{\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{p}_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }}}\right).$$

(33)

Using inverse Jensen’s dynamic inequality, we get that

$$\begin{array}{c}{\Phi }_{\ell }\left({\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)⩾\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{P_{\ell }(s_{\ell },{\chi }_{\ell })}{\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{p}_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)\\ {\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }^{{\nabla }_{a_2}^{\gamma }{\nabla }_{a_1}^{\gamma }}({\varsigma }_{\ell },{\tau }_{\ell })}{p_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right){\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }.\end{array}$$

(34)

Applying inverse Hölder’s inequality on the right-hand side of (34) with indices $1/{\alpha }_{\ell }$ and $1/{\alpha }_{\ell }^{\prime },$ we obtain

$${\Phi }_{\ell }\left({\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)⩾\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{P_{\ell }(s_{\ell },{\chi }_{\ell })}{\left[(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right]}^{{\alpha }_{\ell }^{\prime }}$$

$$\times {\left({\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{\left(p_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }^{{\nabla }_{a_2}^{\gamma }{\nabla }_{a_1}^{\gamma }}({\varsigma }_{\ell },{\tau }_{\ell })}{p_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}{\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }\right)}^{{\alpha }_{\ell }}.$$

(35)

Using the following inequality on the term ${\left[(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right]}^{{\alpha }_{\ell }^{\prime }}$, where ${\alpha }_{\ell }^{\prime }<0$ and ${\lambda }_{\ell }>0.$

$$\prod _{\ell =1}^n{\lambda }_{\ell }^{{\alpha }_{\ell }^{\prime }}⩾{\left(\frac{1}{{\alpha }^{\prime }}\left(\sum _{\ell =1}^n{\alpha }_{\ell }^{\prime }{\lambda }_{\ell }\right)\right)}^{{\alpha }^{\prime }}.$$

(36)

We obtain that

$$\prod _{\ell =1}^n{\Phi }_{\ell }\left({\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)⩾\prod _{\ell =1}^n\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{P_{\ell }(s_{\ell },{\chi }_{\ell })}{\left(\frac{1}{{\alpha }^{\prime }}\sum _{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{{\alpha }^{\prime }}$$

$$\times {\left({\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{\left(p_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }^{{\nabla }_{a_2}^{\gamma }{\nabla }_{a_1}^{\gamma }}({\varsigma }_{\ell },{\tau }_{\ell })}{p_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}{\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }\right)}^{{\alpha }_{\ell }}.$$

(37)

From (37), we find that

$$\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{{\alpha }^{\prime }}}}$$

$$⩾\prod _{\ell =1}^n\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{P_{\ell }(s_{\ell },{\chi }_{\ell })}{\left({\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{\left(p_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }^{{\nabla }_{a_2}^{\gamma }{\nabla }_{a_1}^{\gamma }}({\varsigma }_{\ell },{\tau }_{\ell })}{p_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}{\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }\right)}^{{\alpha }_{\ell }}.$$

(38)

Integrating both sides of (38) over $s_{\ell },$ ${\chi }_{\ell }$ from ${\chi }_0$ to ${\xi }_{\ell },$ ${\zeta }_{\ell }$ $(\ell =1,\dots ,n),$ we find that

$${\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{{\alpha }^{\prime }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1$$

$$⩾\prod _{\ell =1}^n{\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{P_{\ell }(s_{\ell },{\chi }_{\ell })}{\left({\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{\left(p_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }^{{\nabla }_{a_2}^{\gamma }{\nabla }_{a_1}^{\gamma }}({\varsigma }_{\ell },{\tau }_{\ell })}{p_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}{\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }\right)}^{{\alpha }_{\ell }}{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }$$

(39)

Applying inverse Hölder’s inequality on the right-hand side of (39) with indices $1/{\alpha }_{\ell }$ and $1/{\alpha }_{\ell }^{\prime },$, we obtain

$$\begin{array}{ccc}\hfill & & {\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{{\alpha }^{\prime }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1\hfill \\ & & ⩾\prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{P_{\ell }(s_{\ell },{\chi }_{\ell })}\right)}^{\frac{1}{{\alpha }_{\ell }^{\prime }}}{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }\right)}^{{\alpha }_{\ell }^{\prime }}\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}\left({\int }_{{\chi }_0}^{s_{\ell }}{\int }_{{\chi }_0}^{{\chi }_{\ell }}{\left(p_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }^{{\nabla }_{a_2}^{\gamma }{\nabla }_{a_1}^{\gamma }}({\varsigma }_{\ell },{\tau }_{\ell })}{p_{\ell }\left({\varsigma }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}{\nabla }_a^{\gamma }{\varsigma }_{\ell }{\nabla }_a^{\gamma }{\tau }_{\ell }\right){\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }\right)}^{{\alpha }_{\ell }}\hfill \\ \hfill \end{array}$$

(40)

By using Fubini’s theorem, we observe that

$$\begin{array}{ccc}& & {\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{{\alpha }^{\prime }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1\hfill \\ \hfill & & ⩾G({\xi }_1{\zeta }_1,\dots ,{\xi }_n{y}_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}({\xi }_{\ell }-s_{\ell })({\zeta }_{\ell }-{\chi }_{\ell }){\left(p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\chi }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }^{{\nabla }_{a_2}^{\gamma }{\nabla }_{a_1}^{\gamma }}(s_{\ell },{\chi }_{\ell })}{p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\chi }_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }\right)}^{{\alpha }_{\ell }}.\hfill \end{array}$$

(41)

Using the fact ${\xi }_{\ell }⩾\rho \left({\xi }_{\ell }\right),$ and ${\zeta }_{\ell }⩾\rho \left({\zeta }_{\ell }\right),$ we get that

$$\begin{array}{ccc}& & {\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{{\alpha }^{\prime }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1\hfill \\ \hfill & & ⩾G({\xi }_1{\zeta }_1,\dots ,{\xi }_n{y}_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}(\rho \left({\xi }_{\ell }\right)-s_{\ell })(\rho \left({\zeta }_{\ell }\right)-{\chi }_{\ell }){\left(p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\chi }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }^{{\nabla }_{a_2}^{\gamma }{\nabla }_{a_1}^{\gamma }}(s_{\ell },{\chi }_{\ell })}{p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\chi }_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }\right)}^{{\alpha }_{\ell }}.\hfill \end{array}$$

This completes the proof. □

Remark 9.

In Theorem 3, if $\mathbb{T}=\mathbb{Z}$, $\gamma =1$, we get the result due to Zhao et al. [9] (Theorem 1.5).

Remark 10.

In Theorem 3, if we take $\mathbb{T}=\mathbb{R}$, $\gamma =1$, we get the result due to Zhao et al. [9] (Theorem 1.6).

Remark 11.

Let $S_1,$ $S_2,$ $S_9,$ $S_{11},$ $S_7,$ $S_{13},$ $S_3$ and $S_{12}$ be satisfied and let ${\Phi }_{\ell },$ ${\alpha }_{\ell },$ ${\alpha }_{\ell }^{\prime },$ $\alpha ,$ and ${\alpha }^{\prime }$ be the same as Theorem 3. Similar to proof of Theorem 3, we have

$$\begin{array}{ccc}& & {\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\zeta }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}{\int }_{{\chi }_0}^{{\zeta }_n}\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({\vartheta }_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-{\chi }_0)({\chi }_{\ell }-{\chi }_0)\right)}^{{\alpha }^{\prime }}}}{\nabla }_a^{\gamma }{s}_n{\nabla }_a^{\gamma }{\chi }_n\dots {\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{\chi }_1\hfill \\ \hfill & & ⩽G^{*}({\xi }_1{\zeta }_1,\dots ,{\xi }_n{y}_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}(\sigma \left({\xi }_{\ell }\right)-s_{\ell })(\sigma \left({\zeta }_{\ell }\right)-{\chi }_{\ell }){\left(p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\chi }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }^{{\nabla }_{a_2}^{\gamma }{\nabla }_{a_1}^{\gamma }}(s_{\ell },{\chi }_{\ell })}{p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\chi }_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }\right)}^{{\alpha }_{\ell }}.\hfill \end{array}$$

where

$$G^{*}({\xi }_1{\zeta }_1,\dots ,{\xi }_n{y}_n)=\frac{1}{{\left({\alpha }^{\prime }\right)}^{{\alpha }^{\prime }}}\prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\int }_{{\chi }_0}^{{\zeta }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\chi }_{\ell })\right)}{P_{\ell }(s_{\ell },{\chi }_{\ell })}\right)}^{\frac{1}{{\alpha }_{\ell }^{\prime }}}{\nabla }_a^{\gamma }{s}_{\ell }{\nabla }_a^{\gamma }{\chi }_{\ell }\right)}^{{\alpha }_{\ell }^{\prime }}.$$

This is an inverse form of Inequality (31).

Corollary 9.

Let $S_{22},$ $S_{23},$ $S_{25},$ $S_{26},$ $S_{27},$ $S_{29},$ $S_{17}$ and $S_8$ be satisfied. Then, we have that

$$\begin{array}{ccc}\hfill & & {\int }_{{\chi }_0}^{{\xi }_1}\dots {\int }_{{\chi }_0}^{{\xi }_n}\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({\vartheta }_{\ell }\left(s_{\ell }\right)\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-{\chi }_0)\right)}^{{\alpha }^{\prime }}}}{\nabla }_a^{\gamma }{s}_n\dots {\nabla }_a^{\gamma }{s}_1\hfill \\ \hfill & & ⩾G^{**}({\xi }_1,\dots ,{\xi }_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}(\rho \left({\xi }_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left(\frac{{\vartheta }_{\ell }^{{\nabla }_a^{\gamma }}\left(s_{\ell }\right)}{p_{\ell }\left(s_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}{\nabla }_a^{\gamma }{s}_{\ell }\right)}^{{\alpha }_{\ell }}.\hfill \end{array}$$

(42)

where

$$G^{**}({\xi }_1,\dots ,{\xi }_n)=\prod _{\ell =1}^n{\left({\int }_{{\chi }_0}^{{\xi }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell }\right)}\right)}^{\frac{1}{{\alpha }_{\ell }^{\prime }}}{\nabla }_a^{\gamma }{s}_{\ell }\right)}^{{\alpha }_{\ell }^{\prime }}.$$

Remark 12.

In Corollary 9, if we take $\mathbb{T}=\mathbb{Z},$ $\gamma =1$, we get an inverse form of inequality due to Handley et al. [16].

Remark 13.

In Corollary 9, if we take $\mathbb{T}=\mathbb{R},$ $\gamma =1$, we get an inverse form of inequality due to Handley et al. [16].

Remark 14.

In Inequality (42), taking $n=2,$ ${\alpha }_1={\alpha }_2=2,$ then ${\alpha }_1^{\prime }={\alpha }_2^{\prime }=-1$, we have

$$\begin{array}{ccc}\hfill & & {\int }_{{\chi }_0}^{{\xi }_1}{\int }_{{\chi }_0}^{{\xi }_2}\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_1\left({\vartheta }_1\left(s_1\right)\right){\Phi }_1\left({\vartheta }_2\left(s_2\right)\right)}{{\left((s_1-{\chi }_0)+(s_2-{\chi }_0)\right)}^{-2}}}{\nabla }_a^{\gamma }{s}_1{\nabla }_a^{\gamma }{s}_2\hfill \\ \hfill & & ⩾D({\xi }_1,{\xi }_2){\left({\int }_{{\chi }_0}^{{\xi }_1}(\rho \left({\xi }_1\right)-s_1){\left(p_1\left(s_1\right){\Phi }_1\left(\frac{{\vartheta }_1^{{\nabla }_a^{\gamma }}\left(s_1\right)}{p_1\left(s_1\right)}\right)\right)}^{\frac{1}{2}}{\nabla }_a^{\gamma }{s}_1\right)}^2\hfill \\ \hfill & & \times {\left({\int }_{{\chi }_0}^{{\xi }_2}(\rho \left({\xi }_2\right)-s_2){\left(p_2\left(s_2\right){\Phi }_2\left(\frac{{\vartheta }_2^{{\nabla }_a^{\gamma }}\left(s_2\right)}{p_2\left(s_2\right)}\right)\right)}^{\frac{1}{2}}{\nabla }_a^{\gamma }{s}_2\right)}^2.\hfill \end{array}$$

(43)

where

$$D({\xi }_1,{\xi }_2)=4{\left({\int }_{{\chi }_0}^{{\xi }_1}{\left(\frac{{\Phi }_1\left(P_1\left(s_1\right)\right)}{P_2\left(s_1\right)}\right)}^{-1}{\nabla }_a^{\gamma }{s}_1\right)}^{-1}{\left({\int }_{{\chi }_0}^{{\xi }_2}{\left(\frac{{\Phi }_2\left(P_2\left(s_2\right)\right)}{P_2\left(s_2\right)}\right)}^{-1}{\nabla }_a^{\gamma }{s}_2\right)}^{-1}.$$

Remark 15.

If we take $\mathbb{T}=\mathbb{Z}$, $\gamma =1$, Inequality (43) is an inverse of inequality due to Pachpatte [2].

Remark 16.

If we take $\mathbb{T}=\mathbb{R}$, $\gamma =1$, Inequality (43) is an inverse of inequality due to Pachpatte [2].

3. Conclusions

In this article, we presented some investigations of the $(\gamma ,a)$-nabla Hilbert inequality on time scales. Some dynamic integral and discrete inequalities, known in the literature, are generalized as special cases of our results. We obtained the discrete and the continuous inequalities as special cases of our main results. In future work, I will ask if it is possible to generalize these results using a q-difference operator.