On Some Important Dynamic Inequalities of Hardy–Hilbert-Type on Timescales

Abstract

In this article, by using some algebraic inequalities, nabla Hölder inequalities, and nabla Jensen’s inequalities on timescales, we proved some new nabla Hilbert-type dynamic inequalities on timescales. These inequalities extend some known dynamic inequalities on timescales and unify some continuous inequalities and their corresponding discrete analogues. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

1. Introduction

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

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

(1)

where $p>1$, $q=\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$p-1$}\right.$.

The continuous versions of inequality (1) is given by:

$${\int }_0^{\infty }{\int }_0^{\infty }\frac{f\left(x\right)g\left(y\right)}{x+y}dxdy⩽\frac{\pi }{\sin \frac{\pi }{p}}{\left({\int }_0^{\infty }{f}^p\left(x\right)dx\right)}^{\frac{1}{p}}{\left({\int }_0^{\infty }{g}^{p^{\prime }}\left(x\right)dx\right)}^{\frac{1}{q}},$$

(2)

unless $f\equiv 0$ or $g\equiv 0,$ where f and g are measurable non-negative functions such that ${\int }_0^{\infty }{f}^p\left(x\right)dx<\infty$ and ${\int }_0^{\infty }{g}^p\left(x\right)dx<\infty$. The constant $\frac{\pi }{\sin \frac{\pi }{p}},$ in (1) and (2), is the best possible.

In [2], Pachpatte proved that if $f\in C^1[[0,x],{\mathbb{R}}^{+}],$ $g\in C^1[[0,y],{\mathbb{R}}^{+}]$ with $f\left(0\right)=g\left(0\right)=0$ and $p,$ q are two positive functions defined for $t\in [0,x)$ and $\tau \in [0,y),$ with $P\left(t\right)={\int }_0^{t}p\left(\tau \right)d\tau$ and $Q\left(t\right)={\int }_0^{t}q\left(\tau \right)d\tau$ for $s\in [0,x)$ and $t\in [0,y)$ where $x,$ y are positive real numbers. Let $\Phi$ and $\Psi$ be two real-valued non-negative, convex, and sub-multiplicative functions defined on $[0,\infty ).$ Then

$$\begin{array}{cc}{\int }_0^x{\int }_0^y{\displaystyle \frac{\Phi \left(f\right(s\left)\right)\Psi \left(g\right(t\left)\right)}{s+t}}dsdt⩽\hfill & L(x,y)({\int }_0^x(x-s){\left(p\left(s\right)\Phi {\left(\frac{f^{\prime }\left(s\right)}{p\left(s\right)}\right)}^{2}ds\right)}^{\frac{1}{2}}\hfill \\ \hfill & \times ({\int }_0^y(y-t){\left(q\left(t\right)\Psi {\left(\frac{g^{\prime }\left(t\right)}{q\left(t\right)}\right)}^{2}dt\right)}^{\frac{1}{2}},\hfill \end{array}$$

(3)

where

$$L(x,y)=\frac{1}{2}{\left({\int }_0^x{\left(\frac{\Phi \left(P\right(s\left)\right)}{P\left(s\right)}\right)}^{2}ds\right)}^{\frac{1}{2}}{\left({\int }_0^y{\left(\frac{\Psi \left(Q\right(t\left)\right)}{Q\left(t\right)}\right)}^{2}dt\right)}^{\frac{1}{2}}.$$

In [3], Handley et al. proved that if $\left\{a_{\ell ,m_{\ell }}\right\}$ $(\ell =1,2,\cdots ,n)$ are n sequences of non-negative real numbers defined for $m_{\ell }=1,\cdots ,k_{\ell }$ with $a_{1,0}=a_{2,0}\cdots a_{n,0}=0.$ Let $\left\{p_{\ell ,m_{\ell }}\right\}$ be n sequences of positive real numbers defined for $m_{\ell }=1,\cdots ,k_{\ell },$ where $k_{\ell }$ are natural numbers. Set $P_{\ell ,m_{\ell }}={\sum }_{s_{\ell }}^{m_{\ell }}{p}_{\ell ,s_{\ell }}.$ Let ${\Phi }_{\ell }$ $(\ell =1,2,\cdots ,n)$ be n real valued non-negative convex and sub-multiplicative functions defined on $(0,\infty ).$ Let ${\alpha }_{\ell }\in \left(0.1\right),$ and set ${\alpha }_{\ell }^{\prime }=1-\alpha ,$ $\alpha ={\sum }_{\ell =1}^n{\alpha }_{\ell },$ and ${\alpha }_{\ell }^{\prime }=n-\alpha .$ Then

$$\begin{array}{c}\hfill {\displaystyle \sum _{m_1=1}^{k_1}}\cdots {\displaystyle \sum _{m_n=1}^{k_n}}\frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left(a_{\ell ,m_{\ell }}\right)}{{\left({\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }{m}_{\ell }\right)}^{\alpha {}^{\prime }}}⩽M(k_1,\cdots ,k_n){\displaystyle \prod _{\ell =1}^n}({\displaystyle \sum _{m_{\ell }=1}^{k_{\ell }}}(k_{\ell }-m_{\ell }+1){\left(p_{\ell ,m_{\ell }}{\Phi }_{\ell }{\left(\frac{\nabla a_{\ell ,m_{\ell }}}{p_{\ell ,m_{\ell }}}\right)}^{\frac{1}{{\alpha }_{\ell }}}\right)}^{{\alpha }_{\ell }}\end{array}$$

(4)

where

$$M(k_1,\cdots ,k_n)=\frac{1}{{\left({\alpha }^{\prime }\right)}^{{\alpha }^{\prime }}}{\displaystyle \prod _{\ell =1}^n}{\left({\displaystyle \sum _{m_{\ell }=1}^{k_{\ell }}}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell ,m_{\ell }}\right)}{P_{\ell ,m_{\ell }}}\right)}^{\frac{1}{{\alpha }_{\ell }^{\prime }}}\right)}^{{\alpha }_{\ell }^{\prime }}.$$

Moreover, in the same paper [3], the authors proved that if $f_{\ell }\in C^1([0,k_{\ell }],{\mathbb{R}}_{+}])$ $\ell =1,\cdots ,n,$ with $f_{\ell }\left(0\right)=0.$ Let $p_{\ell }\left({\xi }_{\ell }\right)$ be n positive functions defined for ${\xi }_{\ell }\in [0,x_{\ell }]$ $(\ell =1,\cdots ,n).$ Set $P_{\ell }\left(s_{\ell }\right)={\int }_0^{s_{\ell }}{p}_{\ell }\left({\xi }_{\ell }\right)d{\xi }_{\ell }$ for $s_{\ell }\in [0,x_{\ell }],$ where $x_{\ell }$ are positive real numbers. Let ${\Phi }_{\ell },$ ${\alpha }_{\ell },$ ${\alpha }_{\ell }^{\prime },$ $\alpha ,$ and ${\alpha }^{\prime }$ be as in Equation (4). Then

$$\begin{array}{ccc}\hfill & \hfill & {\int }_0^{x_1}\cdots {\int }_0^{x_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left(f\left(s_{\ell }\right)\right)}{{\left({\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }{s}_{\ell }\right)}^{\alpha {}^{\prime }}}}d{s}_n\cdots d{s}_1\hfill \\ \hfill & \hfill & ⩽L(x_1,\cdots ,x_n){\displaystyle {\prod }_{\ell =1}^n}({\int }_0^{x_{\ell }}(x_{\ell }-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }{\left(\frac{f^{\prime }\left(s_{\ell }\right)}{p\left(s_{\ell }\right)}\right)}^{\frac{1}{{\alpha }_{\ell }}}d{s}_{\ell }\right)}^{{\alpha }_{\ell }},\hfill \end{array}$$

(5)

where

$$L(x_1,\cdots ,x_n)=\frac{1}{{\left({\alpha }^{\prime }\right)}^{{\alpha }^{\prime }}}{\displaystyle \prod _{\ell =1}^n}{\left({\int }_0^{x_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell }\right)}\right)}^{\frac{1}{{\alpha }_{\ell }^{\prime }}}d{s}_{\ell }\right)}^{{\alpha }_{\ell }^{\prime }}.$$

In [4], Pachpatte established the following Hilbert-type integral inequalities under the conditions: If $h⩾1,$ $l⩾1,$ and $f\left(t\right)⩾0,$ $g\left(t\right)⩾0,$ for $t\in (0,x)$ and $\tau \in (0,y),$ where x and y are positive real numbers and define $ϝ\left(s\right)={\int }_0^{s}f\left(t\right)dt$ and $G\left(t\right)={\int }_0^{t}g\left(\tau \right)d\tau ,$ for $s\in (0,x)$ and $t\in (0,y).$ Let P, Q, $\Phi$, $\Psi$ and ${\alpha }^{\prime }$ are defined as in (3), then

$$\begin{array}{cc}{\int }_0^x{\int }_0^y{\displaystyle \frac{{ϝ}^h\left(s\right)G^l\left(t\right)}{s+t}}dsdt⩽\hfill & \frac{1}{2}hl{\left(xy\right)}^{\frac{1}{2}}{\left({\int }_0^x(x-s){\left({ϝ}^{h-1}\left(s\right)f\left(s\right)\right)}^{2}ds\right)}^{\frac{1}{2}}\hfill \\ \hfill & \times {\left({\int }_0^y(y-t){\left(G^{l-1}g\left(t\right)\right)}^{2}dt\right)}^{\frac{1}{2}},\hfill \end{array}$$

(6)

and

$$\begin{array}{cc}{\int }_0^x{\int }_0^y{\displaystyle \frac{\Phi \left(ϝ\right(s\left)\right)\Psi \left(G\right(t\left)\right)}{s+t}}dsdt⩽\hfill & L(x,y){\left({\int }_0^x(x-s){\left(p\left(s\right)\Phi \left(\frac{f\left(s\right)}{p\left(s\right)}\right)\right)}^{2}ds\right)}^{\frac{1}{2}}\hfill \\ \hfill & \times {\left({\int }_0^y(y-t){\left(q\left(t\right)\Psi \left(\frac{g\left(t\right)}{q\left(t\right)}\right)\right)}^{2}dt\right)}^{\frac{1}{2}}\hfill \end{array}$$

(7)

where

$$L(x,y)=\frac{1}{2}{\left({\int }_0^x{\left(\frac{\Phi \left(P\right(s\left)\right)}{P\left(s\right)}\right)}^{2}ds\right)}^{\frac{1}{2}}{\left({\int }_0^y{\left(\frac{\Psi \left(Q\right(t\left)\right)}{Q\left(t\right)}\right)}^{2}dt\right)}^{\frac{1}{2}},$$

and

$$\begin{array}{cc}{\int }_0^x{\int }_0^y{\displaystyle \frac{P\left(s\right)Q\left(t\right)\Phi \left(ϝ\right(s\left)\right)\Psi \left(G\right(t\left)\right)}{s+t}}dsdt⩽\hfill & \frac{1}{2}{\left(xy\right)}^{\frac{1}{2}}{\left({\int }_0^x(x-s){\left(p\left(s\right)\Phi \left(f\left(s\right)\right)\right)}^{2}ds\right)}^{\frac{1}{2}}\hfill \\ \hfill & \times {\left({\int }_0^y(y-t){\left(q\left(t\right)\Psi \left(g\left(t\right)\right)\right)}^{2}dt\right)}^{\frac{1}{2}}.\hfill \end{array}$$

(8)

and

$$\begin{array}{ccc}\hfill & \hfill & {\int }_0^{x_1}{\int }_0^{y_1}\cdots {\int }_0^{x_n}{\int }_0^{y_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({ϝ}_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}\left(s_{\ell }\right)\left(t_{\ell }\right)\right)}^{\frac{1}{\alpha }}}}d{s}_{n}d{t}_n\cdots d{s}_{1}d{t}_1\hfill \\ \hfill & \hfill & ⩾L(x_1{y}_1,\cdots ,x_n{y}_n)\hfill \\ \hfill & \hfill & \times {\displaystyle {\prod }_{\ell =1}^n}{\left({\int }_0^{x_{\ell }}{\int }_0^{y_{\ell }}(x_{\ell }-s_{\ell })(y_{\ell }-t_{\ell }){\left(p_{\ell }(s_{\ell },t_{\ell }){\Phi }_{\ell }\left(\frac{f_{\ell }(s_{\ell },t_{\ell })}{p_{\ell }(s_{\ell },t_{\ell })}\right)\right)}^{{\beta }_{\ell }}d{s}_{\ell }d{t}_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(9)

where

$$L(x_1{y}_1,\cdots ,x_n{y}_n)={\displaystyle \prod _{\ell =1}^n}{\left({\int }_0^{x_{\ell }}{\int }_0^{y_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right)}{P_{\ell }(s_{\ell },t_{\ell })}\right)}^{{\alpha }_{\ell }}d{s}_{\ell }d{t}_{\ell }\right)}^{\frac{1}{{\alpha }_{\ell }}}.$$

Over the past decade, a great number of dynamic Hilbert-type inequalities on timescales has been established by many researchers who were motivated by some applications, see the papers [5,6,7,8,9,10,11,12,13,14,15,16,17].

A timescale $\mathbb{T}$ is an arbitrary, nonempty closed subset of the set of real numbers $\mathbb{R}$. Throughout the article, we assume that $\mathbb{T}$ has the topology that it inherits from the standard topology on $\mathbb{R}$. We define the forward jump operator $\sigma :\mathbb{T}\to \mathbb{T}$ for any $t\in \mathbb{T}$ by

$$\sigma \left(t\right):=\inf \{s\in \mathbb{T}:s>t\},$$

and the backward jump operator $\rho :\mathbb{T}\to \mathbb{T}$ for any $t\in \mathbb{T}$ by

$$\rho \left(t\right):=\sup \{s\in \mathbb{T}:s<t\}.$$

In the preceding two definitions, we set $\inf \varnothing =\sup \mathbb{T}$ (i.e., if t is the maximum of $\mathbb{T}$, then $\sigma \left(t\right)=t$) and $\sup \varnothing =\inf \mathbb{T}$ (i.e., if t is the minimum of $\mathbb{T}$, then $\rho \left(t\right)=t$), where ∅ denotes the empty set. For more details on time scales calculus see [11].

A point $t\in \mathbb{T}$ with $\inf \mathbb{T}<t<\sup \mathbb{T}$ is said to be right-scattered if $\sigma \left(t\right)>t$, right-dense if $\sigma \left(t\right)=t$, left-scattered if $\rho \left(t\right)<t$, and left-dense if $\rho \left(t\right)=t$. Points that are simultaneously right-dense and left-dense are said to be dense points, whereas points that are simultaneously right-scattered and left-scattered are said to be isolated points.

The forward graininess function $\mu :\mathbb{T}\to [0,\infty )$ is defined for any $t\in \mathbb{T}$ by $\mu \left(t\right):=\sigma \left(t\right)-t$.

If $ϝ:\mathbb{T}\to \mathbb{R}$ is a function, then the function ${ϝ}^{\sigma }:\mathbb{T}\to \mathbb{R}$ is defined by ${ϝ}^{\sigma }\left(t\right)=ϝ\left(\sigma \left(t\right)\right),\forall t\in \mathbb{T}$, that is ${ϝ}^{\sigma }=ϝ\circ \sigma$. Similarly, the function ${ϝ}^{\rho }:\mathbb{T}\to \mathbb{R}$ is defined by ${ϝ}^{\rho }\left(t\right)=g\left(\rho \left(t\right)\right),\forall t\in \mathbb{T}$, that is ${ϝ}^{\rho }=ϝ\circ \rho$.

The interval $[a,b]$ in $\mathbb{T}$ is defined by

$${[a,b]}_{\mathbb{T}}=\{t\in \mathbb{T}:a\le t\le b\}.$$

We define open intervals and half-closed intervals similarly.

A function $ϝ:\mathbb{T}\to \mathbb{R}$ is said to be right-dense continuous (rd-continuous) if $ϝ$ is continuous at all right-dense points in $\mathbb{T}$ and its left-sided limits exist at all left-dense points in $\mathbb{T}$.

In a similar manner, a function $ϝ:\mathbb{T}\to \mathbb{R}$ is said to be left-dense continuous (ld-continuous) if $ϝ$ is continuous at all left-dense points in $\mathbb{T}$ and its right-sided limits exist at all right-dense points in $\mathbb{T}$.

The delta integration by parts on timescales is given by the following formula:

$${\int }_a^b{g}^{\Delta }\left(t\right)ϝ\left(t\right)\Delta t=g\left(b\right)ϝ\left(b\right)-g\left(a\right)ϝ\left(a\right)-{\int }_a^b{g}^{\sigma }\left(t\right){ϝ}^{\Delta }\left(t\right)\Delta t,$$

(10)

whereas the nabla integration by parts on timescales is given by

$${\int }_a^b{g}^{\nabla }\left(t\right)ϝ\left(t\right)\nabla t=g\left(b\right)ϝ\left(b\right)-g\left(a\right)ϝ\left(a\right)-{\int }_a^b{g}^{\rho }\left(t\right){ϝ}^{\nabla }\left(t\right)\nabla t.$$

(11)

The following relationships will be used.

(i)

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

$$\begin{array}{c}\sigma \left(t\right)=\rho \left(t\right)=t,\mu \left(t\right)=\nu \left(t\right)=0,{ϝ}^{\Delta }\left(t\right)={ϝ}^{\nabla }\left(t\right)={ϝ}^{\prime }\left(t\right),\\ {\int }_a^bϝ\left(t\right)\Delta t={\int }_a^bϝ\left(t\right)\nabla t={\int }_a^bϝ\left(t\right)dt.\end{array}$$

(12)

(ii)

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

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

(13)

where $\Delta$ and ∇ are the forward and backward difference operators, respectively.

Next, we write Hölder’s inequality and Jensen’s inequality on timescales, where $C{C}_{ld}$ denotes the set of all $ld$-continuous functions $ϝ(x,y)$ in x and y, and $C{C}_{ld}^1$ is the set of all functions in $C{C}_{ld}$ for which both the first partial derivative ${\nabla }_1$ and ${\nabla }_2$ exist in $C{C}_{ld}$. Similarly we can define $C{C}_{ld}^2$.

Lemma 1

(Dynamic Hölder’s Inequality [7]). Suppose $u,$ $v\in \mathbb{T}$ with $u<v.$ Assume $ϝ,$ $g\in C{C}_{ld}^1({[u,v]}_{\mathbb{T}}\times {[u,v]}_{\mathbb{T}},\mathbb{R})$ be integrable functions and $\frac{1}{p}+\frac{1}{q}=1$ with $p>1$ then

$$\begin{array}{cc}{\int }_u^v{\int }_u^v\left|ϝ(r,t)g(r,t)\right|\nabla r\nabla t\hfill & \le {\left[{\int }_u^v{\int }_u^v{\left|ϝ(r,t)\right|}^p\nabla r\nabla t\right]}^{\frac{1}{p}}\hfill \\ \hfill & \times {\left[{\int }_u^v{\int }_u^v{\left|g(r,t)\right|}^q\nabla r\nabla t\right]}^{\frac{1}{q}}.\hfill \end{array}$$

(14)

This inequality is reversed if $0<p<1$ and if $p<0$ or $q<0.$

Lemma 2

(Dynamic Jensen’s inequality [7]). Let r, $t\in R$ and $-\infty ⩽m,n⩽\infty .$ If $ϝ\in C{C}_{ld}^1(\mathbb{R},(m,n))$ and $\Phi :(m,n)⟶\mathbb{R}$ is convex then

$$\varphi \left({\displaystyle \frac{{\int }_u^v{\int }_{\omega }^sϝ(r,t)\nabla r\nabla t}{{\int }_u^v{\int }_{\omega }^s\nabla r\nabla t}}\right)⩽{\displaystyle \frac{{\int }_u^v{\int }_{\omega }^s\varphi \left(ϝ(r,t)\right)\nabla r\nabla t}{{\int }_u^v{\int }_{\omega }^s\nabla r\nabla t}}.$$

(15)

This inequality is reversed if $\varphi \in C_{ld}\left((c,d),\mathbb{R}\right)$ is concave.

Definition 1.

Φ is called a super-multiplicative function on $[0,\infty )$ if

$$\Phi \left(xy\right)⩾\Phi \left(x\right)\Phi \left(y\right),\mathrm{for}\mathrm{all}x,y⩾0.$$

(16)

In this paper, we prove some extensions of the nabla integral Hardy–Hilbert inequality to a general timescale. As special cases of our results, we will recover some dynamic integral and discrete inequalities known in the literature. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

2. Main Results

In this section, we state and prove the main results that extend several results in the literature.

Theorem 1.

Let $\mathbb{T}$ be timescales with $t_0,$ $x_{\ell },$ $y_{\ell },$ $s_{\ell },$ $t_{\ell }\in \mathbb{T},$ $(\ell =1,\cdots ,n).$ Assume ${\omega }_{\ell }(s_{\ell },t_{\ell })\in C_{ld}^2\left({[t_0,x_{\ell })}_{\mathbb{T}}\times {[t_0,y_{\ell })}_{\mathbb{T}},[0,\infty )\right)$ $(\ell =1,\cdots ,n),$ where $x_{\ell }$ and $y_{\ell }$ are positive real numbers. Define ${\Omega }_{\ell }(s_{\ell },t_{\ell })={\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{\omega }_{\ell }({\xi }_{\ell },{\tau }_{\ell })\nabla {\xi }_{\ell }\nabla {\tau }_{\ell }.$ Let $p_{\ell }({\xi }_{\ell },{\tau }_{\ell })$ be n positive, left-dense continuous functions defined for ${\xi }_{\ell }\in {(t_0,s_{\ell })}_{\mathbb{T}},$ ${\tau }_{\ell }\in {(t_0,t_{\ell })}_{\mathbb{T}}$ and define $P_{\ell }(s_{\ell },t_{\ell })={\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{p}_{\ell }({\xi }_{\ell },{\tau }_{\ell })\nabla {\xi }_{\ell }\nabla {\tau }_{\ell }.$ Let ${\Phi }_{\ell }$ $(\ell =1,\cdots ,n)$ be n real-valued, non-negative concave and super-multiplicative functions defined on $(0,\infty ).$ Then for $s_{\ell }\in {[t_0,x_{\ell })}_{\mathbb{T}},$ $t_{\ell }\in {[t_0,y_{\ell })}_{\mathbb{T}},$ $\frac{1}{{\alpha }_{\ell }}+\frac{1}{{\beta }_{\ell }}=1,$ $0<{\beta }_{\ell }<1,$ and ${\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}=\frac{1}{\alpha }$ we have that

$$\begin{array}{c}{\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{\frac{1}{\alpha }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ ⩾L(x_1{y}_1,\cdots ,x_n{y}_n)\hfill \\ \times {\prod }_{\ell =1}^n{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}(\rho \left(x_{\ell }\right)-s_{\ell })(\rho \left(y_{\ell }\right)-t_{\ell }){\left(p_{\ell }(s_{\ell },t_{\ell }){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }(s_{\ell },t_{\ell })}{p_{\ell }(s_{\ell },t_{\ell })}\right)\right)}^{{\beta }_{\ell }}\nabla s_{\ell }\nabla t_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}\hfill \end{array}$$

(17)

where

$$L(x_1{y}_1,\cdots ,x_n{y}_n)={\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right)}{P_{\ell }(s_{\ell },t_{\ell })}\right)}^{{\alpha }_{\ell }}\nabla s_{\ell }\nabla t_{\ell }\right)}^{\frac{1}{{\alpha }_{\ell }}}.$$

Proof. 

From the hypotheses of Theorem 1, it is easy to observe that

$$\begin{array}{ccc}{\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)\hfill & =\hfill & {\Phi }_{\ell }\left({\displaystyle \frac{P_{\ell }(s_{\ell },t_{\ell }){\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{p}_{\ell }(t_{\ell },{\tau }_{\ell })\left(\frac{{\omega }_{\ell }(t_{\ell },{\tau }_{\ell })}{p_{\ell }(t_{\ell },{\tau }_{\ell })}\right)\nabla t_{\ell }\nabla {\tau }_{\ell }}{{\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{p}_{\ell }(t_{\ell },{\tau }_{\ell })\nabla t_{\ell }\nabla {\tau }_{\ell }}}\right)\hfill \\ \hfill & ⩾\hfill & {\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right){\Phi }_{\ell }\left({\displaystyle \frac{{\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{p}_{\ell }(t_{\ell },{\tau }_{\ell })\left(\frac{{\omega }_{\ell }(t_{\ell },{\tau }_{\ell })}{p_{\ell }(t_{\ell },{\tau }_{\ell })}\right)\nabla t_{\ell }\nabla {\tau }_{\ell }}{{\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{p}_{\ell }(t_{\ell },{\tau }_{\ell })\nabla t_{\ell }\nabla {\tau }_{\ell }}}\right).\hfill \end{array}$$

(18)

By using inverse Jensen dynamic inequality, we obtain that

$$\begin{array}{ccc}\hfill {\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)& ⩾& \frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right)}{P_{\ell }(s_{\ell },t_{\ell })}{\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{p}_{\ell }(t_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }(t_{\ell },{\tau }_{\ell })}{p_{\ell }(t_{\ell },{\tau }_{\ell })}\right)\nabla t_{\ell }\nabla {\tau }_{\ell }.\hfill \end{array}$$

(19)

Applying the inverse Hölder’s inequality on the left-hand side of (19) with indices ${\alpha }_{\ell }$ and ${\beta }_{\ell },$ it is easy to observe that

$$\begin{array}{ccc}\hfill {\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)& ⩾& \frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right)}{P_{\ell }(s_{\ell },t_{\ell })}{\left[(s_{\ell }-t_0)(t_{\ell }-t_0)\right]}^{\frac{1}{{\alpha }_{\ell }}}{\left({\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{\left(p_{\ell }(t_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }(t_{\ell },{\tau }_{\ell })}{p_{\ell }(t_{\ell },{\tau }_{\ell })}\right)\right)}^{{\beta }_{\ell }}\nabla t_{\ell }\nabla {\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(20)

By using the following inequality on the term ${\left[(s_{\ell }-t_0)(t_{\ell }-t_0)\right]}^{\frac{1}{{\alpha }_{\ell }}},$

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

(21)

we obtain that

$$\begin{array}{ccc}& & {\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{\frac{1}{\alpha }}}}\\ & & ⩾{\displaystyle \prod _{\ell =1}^n}\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right)}{P_{\ell }(s_{\ell },t_{\ell })}{\left({\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{\left(p_{\ell }(t_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }(t_{\ell },{\tau }_{\ell })}{p_{\ell }(t_{\ell },{\tau }_{\ell })}\right)\right)}^{\frac{1}{{\beta }_{\ell }}}\nabla t_{\ell }\nabla {\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(22)

Integrating both sides of (22) over $s_{\ell },$ $t_{\ell }$ from $t_0$ to $x_{\ell },$ $y_{\ell }$ $(\ell =1,\cdots ,n),$ we obtain that

$$\begin{array}{c}{\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{\frac{1}{\alpha }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ ⩾{\prod }_{\ell =1}^n{\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right)}{P_{\ell }(s_{\ell },t_{\ell })}{\left({\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{\left(p_{\ell }(t_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }(t_{\ell },{\tau }_{\ell })}{p_{\ell }(t_{\ell },{\tau }_{\ell })}\right)\right)}^{{\beta }_{\ell }}\nabla t_{\ell }\nabla {\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}\nabla s_{\ell }\nabla t_{\ell }.\hfill \end{array}$$

(23)

Applying the inverse Hölder’s inequality on the left-hand side of (23) with indices ${\alpha }_{\ell }$ and ${\beta }_{\ell },$ it is easy to observe that

$$\begin{array}{ccc}\hfill & & {\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{\frac{1}{\alpha }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ & & ⩾{\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right)}{P_{\ell }(s_{\ell },t_{\ell })}\right)}^{{\alpha }_{\ell }}\nabla s_{\ell }\nabla t_{\ell }\right)}^{\frac{1}{{\alpha }_{\ell }}}\hfill \\ \hfill & & \times {\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}\left({\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{\left(p_{\ell }(t_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }(t_{\ell },{\tau }_{\ell })}{p_{\ell }(t_{\ell },{\tau }_{\ell })}\right)\right)}^{{\beta }_{\ell }}\nabla t_{\ell }\nabla {\tau }_{\ell }\right)\nabla s_{\ell }\nabla t_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(24)

Using Fubini’s theorem, we observe that

$$\begin{array}{ccc}\hfill & & {\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{\frac{1}{\alpha }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ & & ⩾L(x_1{y}_1,\cdots ,x_n{y}_n)\hfill \\ & & \times {\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}(x_{\ell }-s_{\ell })(y_{\ell }-t_{\ell }){\left(p_{\ell }(s_{\ell },t_{\ell }){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }(s_{\ell },t_{\ell })}{p_{\ell }(s_{\ell },t_{\ell })}\right)\right)}^{{\beta }_{\ell }}\nabla s_{\ell }\nabla t_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

By using the fact $x_{\ell }⩾\rho \left(x_{\ell }\right),$ and $y_{\ell }⩾\rho \left(y_{\ell }\right),$ we obtain that

$$\begin{array}{ccc}& & {\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{\frac{1}{\alpha }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ \hfill & & ⩾L(x_1{y}_1,\cdots ,x_n{y}_n)\hfill \\ \hfill & & \times {\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}(\rho \left(x_{\ell }\right)-s_{\ell })(\rho \left(y_{\ell }\right)-t_{\ell }){\left(p_{\ell }(s_{\ell },t_{\ell }){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }(s_{\ell },t_{\ell })}{p_{\ell }(s_{\ell },t_{\ell })}\right)\right)}^{{\beta }_{\ell }}\nabla s_{\ell }\nabla t_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

This completes the proof. □

Remark 1.

In Theorem 1, if $\mathbb{T}=\mathbb{R}$, we obtain the result due to Zhao et al.’s [9], Theorem 2.

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

Corollary 1.

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

$$\begin{array}{c}{A}_{s_{\ell },t_{\ell },m_{s_{\ell }},m_{t_{\ell }}}={\sum }_{m_{t_{\ell }}}^{m_{s_{\ell }}}{\sum }_{m_{{\eta }_{\ell }}}^{m_{t_{\ell }}}{a}_{s_{\ell },t_{\ell },m_{t_{\ell }},m_{{\eta }_{\ell }}}\\ P_{s_{\ell },t_{\ell },m_{s_{\ell }},m_{t_{\ell }}}={\sum }_{m_{t_{\ell }}}^{m_{s_{\ell }}}{\sum }_{m_{{\eta }_{\ell }}}^{m_{t_{\ell }}}{p}_{s_{\ell },t_{\ell },m_{t_{\ell }},m_{{\eta }_{\ell }}}.\end{array}$$

(25)

Then

$$\begin{array}{ccc}& & {\displaystyle \sum _{m_{s_1}}^{k_{s_1}}}{\displaystyle \sum _{m_{t_1}}^{k_{t_1}}}\cdots {\displaystyle \sum _{m_{s_n}}^{k_{s_n}}}{\displaystyle \sum _{m_{t_n}}^{k_{t_n}}}\frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left(A_{s_{\ell },t_{\ell },m_{s_{\ell }},m_{t_{\ell }}}\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}\left(m_{s_{\ell }}{m}_{t_{\ell }}\right)\right)}^{\frac{1}{\alpha }}}\hfill \\ \hfill & & ⩾C(k_{s_1}{k}_{t_1},\cdots ,k_{s_n}{k}_{t_n})\hfill \\ \hfill & & \times {\displaystyle \prod _{\ell =1}^n}{\left({\displaystyle \sum _{m_{s_{\ell }}}^{k_{s_{\ell }}}}{\displaystyle \sum _{m_{t_{\ell }}}^{k_{t_{\ell }}}}(k_{s_{\ell }}-(m_{s_{\ell }}-1))(k_{t_{\ell }}-(m_{t_{\ell }}-1)){\left(P_{s_{\ell },t_{\ell },m_{s_{\ell }},m_{t_{\ell }}}{\Phi }_{\ell }\left(\frac{a_{s_{\ell },t_{\ell },m_{s_{\ell }},m_{t_{\ell }}}}{P_{s_{\ell },t_{\ell },m_{s_{\ell }},m_{t_{\ell }}}}\right)\right)}^{{\beta }_{\ell }}\right)}^{\frac{1}{{\beta }_{\ell }}}\hfill \end{array}$$

where

$$C(k_{s_1}{k}_{t_1},\cdots ,k_{s_n}{k}_{t_n})={\displaystyle \prod _{\ell =1}^n}{\left({\displaystyle \sum _{m_{s_{\ell }}}^{k_{s_{\ell }}}}{\displaystyle \sum _{m_{t_{\ell }}}^{k_{t_{\ell }}}}\left(\frac{{\Phi }_{\ell }\left(P_{s_{\ell },t_{\ell },m_{s_{\ell }},m_{t_{\ell }}}\right)}{P_{s_{\ell },t_{\ell },m_{s_{\ell }},m_{t_{\ell }}}}\right)\right)}^{{\beta }_{\ell }}{)}^{\frac{1}{{\beta }_{\ell }}}.$$

Remark 2.

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

Corollary 2.

Let ${\omega }_{\ell }\left({\xi }_{\ell }\right)\in C_{ld}^1{[t_0,x_{\ell }]}_{\mathbb{T}},$ $(\ell =1,\cdots ,n)$, where $x_{\ell }$ positive real number, and define ${\Omega }_{\ell }\left(s_{\ell }\right)={\int }_{t_0}^{s_{\ell }}{\omega }_{\ell }\left({\xi }_{\ell }\right)\nabla {\xi }_{\ell },$ then for $s_{\ell }\in {[t_0,x_{\ell })}_{\mathbb{T}},$ $\frac{1}{{\alpha }_{\ell }}+\frac{1}{{\beta }_{\ell }}=1,$ $0<{\beta }_{\ell }<1,$ and ${\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}=\frac{1}{\alpha }$. Let $p_{\ell }\left({\xi }_{\ell }\right)$ be n positive functions defined for ${\xi }_{\ell }\in {(t_0,x_{\ell })}_{\mathbb{T}}$ $(\ell =1,\cdots ,n)$ and define $P_{\ell }\left(s_{\ell }\right)={\int }_{t_0}^{s_{\ell }}{p}_{\ell }\left({\xi }_{\ell }\right)\nabla {\xi }_{\ell },$ where $x_{\ell }$ are positive real number and let ${\Phi }_{\ell }$ be n real-valued non-negative, concave, and super-multiplicative function defined on $(0,\infty ).$ Then

$$\begin{array}{ccc}\hfill & \hfill & {\int }_{t_0}^{x_1}\cdots {\int }_{t_0}^{x_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }({\Omega }_{\ell }\left(s_{\ell }\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-t_0)\right)}^{\frac{1}{\alpha }}}}\nabla s_n\cdots \nabla s_1\hfill \\ \hfill & \hfill & ⩾L^{*}(x_1,\cdots ,x_n){\prod }_{\ell =1}^n{\left({\int }_{t_0}^{x_{\ell }}(\rho \left(x_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }\left(s_{\ell }\right)}{p_{\ell }\left(s_{\ell }\right)}\right)\right)}^{{\beta }_{\ell }}\nabla s_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}},\hfill \end{array}$$

(26)

where

$$L^{*}(x_1,\cdots ,x_n)={\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell }\right)}\right)}^{{\alpha }_{\ell }}\nabla s_{\ell }\right)}^{\frac{1}{{\alpha }_{\ell }}}.$$

Corollary 3.

In Corollary 2, if we take $n=2,$ ${\beta }_{\ell }=\frac{1}{2}$ then the inequality (26) changes to

$$\begin{array}{cc}{\int }_{t_0}^{x_1}{\int }_{t_0}^{x_2}{\displaystyle \frac{{\Phi }_1\left({\Omega }_1\left(s_1\right)\right){\Phi }_2\left({\Omega }_2\left(s_2\right)\right)}{{\left((s_1-t_0)+(s_2-t_0)\right)}^{-2}}}\nabla s_1\nabla s_2\hfill & ⩾L^{**}(x_1,x_2){\left({\int }_{t_0}^{x_1}(\rho \left(x_1\right)-s_1){\left(p_1\left(s_1\right)\Phi \left(\frac{{\omega }_1\left(s_1\right)}{p_1\left(s_1\right)}\right)\right)}^2\nabla s_1\right)}^{\frac{1}{2}}\hfill \\ \hfill & \times {\left({\int }_{t_0}^{x_2}(\rho \left(x_2\right)-s_2){\left(p_2\left(s_2\right)\Psi \left(\frac{{\omega }_2\left(s_2\right)}{p_2\left(s_2\right)}\right)\right)}^2\nabla s_2\right)}^{\frac{1}{2}}\hfill \end{array}$$

(27)

where

$$L^{**}(x_1,x_2)=4{\left({\int }_{t_0}^{x_1}{\left(\frac{{\Phi }_1\left(P_1\left(s_1\right)\right)}{P_1\left(s_1\right)}\right)}^{-1}\nabla s_1\right)}^{-1}{\left({\int }_{t_0}^{x_2}{\left(\frac{{\Phi }_2\left(P_2\left(s_2\right)\right)}{P_2\left(s_2\right)}\right)}^{-1}\nabla s_2\right)}^{-1}$$

Remark 3.

In Corollary 3, if we take $\mathbb{T}=\mathbb{R},$ then the inequality (27) changes to

$$\begin{array}{cc}{\int }_0^{x_1}{\int }_0^{x_2}{\displaystyle \frac{{\Phi }_1\left({\Omega }_1\left(s_1\right)\right){\Phi }_2\left({\Omega }_2\left(s_2\right)\right)}{{(s_1+s_2)}^{-2}}}d{s}_{1}d{s}_2\hfill & ⩾L^{**}(x_1,x_2){\left({\int }_0^{x_1}(x_1-s_1){\left(p_1\left(s_1\right)\Phi \left(\frac{{\omega }_1\left(s_1\right)}{p_1\left(s_1\right)}\right)\right)}^{2}d{s}_1\right)}^{\frac{1}{2}}\hfill \\ \hfill & \times {\left({\int }_0^{x_2}(x_2-s_2){\left(p_2\left(s_2\right)\Psi \left(\frac{{\omega }_2\left(s_2\right)}{p_2\left(s_2\right)}\right)\right)}^{2}d{s}_2\right)}^{\frac{1}{2}}\hfill \end{array}$$

(28)

where

$$L^{**}(x_1,x_2)=4{\left({\int }_0^{x_1}{\left(\frac{{\Phi }_1\left(P_1\left(s_1\right)\right)}{P_1\left(s_1\right)}\right)}^{-1}d{s}_1\right)}^{-1}{\left({\int }_0^{x_2}{\left(\frac{{\Phi }_2\left(P_2\left(s_2\right)\right)}{P_2\left(s_2\right)}\right)}^{-1}d{s}_2\right)}^{-1}$$

This is an inverse of the inequality (7) which was proved by Pachpatte [4].

Corollary 4.

In Corollary 2, if we take ${\beta }_{\ell }=\frac{n-1}{n}$ the inequality (26) becomes

$$\begin{array}{ccc}& & {\int }_{t_0}^{x_1}\cdots {\int }_{t_0}^{x_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }({\Omega }_{\ell }\left(s_{\ell }\right)}{{\left({\sum }_{\ell =1}^n(s_{\ell }-t_0)\right)}^{\frac{-n}{n-1}}}}\nabla s_n\cdots \nabla s_1\hfill \\ \hfill & & ⩾L^{*}(x_1,\cdots ,x_n){\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}(\rho \left(x_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }\left(s_{\ell }\right)}{p_{\ell }\left(s_{\ell }\right)}\right)\right)}^{\frac{n-1}{n}}\nabla s_{\ell }\right)}^{\frac{n}{n-1}}\hfill \end{array}$$

where

$$L^{*}(x_1,\cdots ,x_n)=n^{\frac{n}{n-1}}{\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell }\right)}\right)}^{-(n-1)}\nabla s_{\ell }\right)}^{\frac{-1}{n-1}}.$$

Theorem 2.

Let $\mathbb{T}$ be timescales with $t_0,$ $x_{\ell },$ $y_{\ell },$ $s_{\ell },$ $t_{\ell }\in \mathbb{T},$ $(\ell =1,\cdots ,n).$ Let ${\omega }_{\ell }({\xi }_{\ell },{\tau }_{\ell }),$ $p_{\ell }({\xi }_{\ell },{\tau }_{\ell }),$ $P_{\ell }({\xi }_{\ell },{\tau }_{\ell }),$ ${\alpha }_{\ell },$ and ${\beta }_{\ell }$ be as Theorem 1 and define ${\Omega }_{\ell }(s_{\ell },t_{\ell })=\frac{1}{P_{\ell }({\xi }_{\ell },{\tau }_{\ell })}{\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{p}_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\omega }_{\ell }({\xi }_{\ell },{\tau }_{\ell })\nabla {\xi }_{\ell }\nabla {\tau }_{\ell }$ for ${\xi }_{\ell },$ $s_{\ell }\in {(t_0,x_{\ell })}_{\mathbb{T}},$ ${\tau }_{\ell },$ $t_{\ell }\in {(t_0,y_{\ell })}_{\mathbb{T}},$ where $x_{\ell }$ and $y_{\ell }$ are positive real numbers. Let ${\Phi }_{\ell }$ $(\ell =1,\cdots ,n)$ be n real-valued non-negative concave and super-multiplicative functions defined on $(0,\infty ).$ Then

$$\begin{array}{c}{\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }(s_{\ell },t_{\ell }){\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{\frac{1}{\alpha }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ ⩾{\prod }_{\ell =1}^n{\left[(x_{\ell }-t_0)(y_{\ell }-t_0)\right]}^{\frac{1}{{\alpha }_{\ell }}}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}(\rho \left(x_{\ell }\right)-s_{\ell })(\rho \left(y_{\ell }\right)-t_{\ell }){\left(p_{\ell }(s_{\ell },t_{\ell }){\Phi }_{\ell }\left({\omega }_{\ell }(s_{\ell },t_{\ell })\right)\right)}^{{\beta }_{\ell }}\nabla s_{\ell }\nabla t_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}\hfill \end{array}$$

(29)

Proof. 

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

$$\begin{array}{ccc}{\Phi }_{\ell }\left({\omega }_{\ell }(s_{\ell },t_{\ell })\right)\hfill & =\hfill & {\Phi }_{\ell }\left(\frac{1}{P_{\ell }(s_{\ell },t_{\ell })}{\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{p}_{\ell }(t_{\ell },{\tau }_{\ell }){\Omega }_{\ell }(t_{\ell },{\tau }_{\ell })\nabla t_{\ell }\nabla {\tau }_{\ell }\right)\hfill \\ \hfill & ⩾\hfill & \frac{1}{P_{\ell }(s_{\ell },t_{\ell })}{\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{p}_{\ell }({\sigma }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({\omega }_{\ell }(t_{\ell },{\tau }_{\ell })\right)\nabla t_{\ell }\nabla {\tau }_{\ell }.\hfill \end{array}$$

(30)

Applying the inverse Hölder’s inequality on the left-hand side of (30) with indices ${\alpha }_{\ell }$ and ${\beta }_{\ell },$ it is easy to observe that

$$\begin{array}{c}\hfill {\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)⩾\frac{1}{P_{\ell }(s_{\ell },t_{\ell })}{\left[(s_{\ell }-t_0)(t_{\ell }-t_0)\right]}^{\frac{1}{{\alpha }_{\ell }}}{\left({\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{\left(p_{\ell }(t_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({\omega }_{\ell }(t_{\ell },{\tau }_{\ell })\right)\right)}^{{\beta }_{\ell }}\nabla t_{\ell }\nabla {\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\end{array}$$

By using the inequality (21), on the term ${\left[(s_{\ell }-t_0)(t_{\ell }-t_0)\right]}^{\frac{1}{{\alpha }_{\ell }}}$ we obtain that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{P_{\ell }(s_{\ell },t_{\ell }){\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{\frac{1}{\alpha }}}}& ⩾& {\left({\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{\left(p_{\ell }(t_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({\omega }_{\ell }(t_{\ell },{\tau }_{\ell })\right)\right)}^{{\beta }_{\ell }}\nabla t_{\ell }\nabla {\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}\hfill \\ \hfill \end{array}$$

(31)

Integrating both sides of (31) over $s_{\ell },$ $t_{\ell }$ from $t_0$ to $x_{\ell },$ $y_{\ell }$ $(\ell =1,\cdots ,n),$ we obtain that

$$\begin{array}{ccc}\hfill & & {\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }(s_{\ell },t_{\ell }){\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{\frac{1}{\alpha }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ \hfill & & ⩾{\displaystyle \prod _{\ell =1}^n}{\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}{\left({\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{\left(p_{\ell }(t_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({\omega }_{\ell }(t_{\ell },{\tau }_{\ell })\right)\right)}^{{\beta }_{\ell }}\nabla {\sigma }_{\ell }\nabla {\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

Applying the inverse Hölder’s inequality on the left-hand side of (32) with indices ${\alpha }_{\ell }$ and ${\beta }_{\ell },$ it is easy to observe that

$$\begin{array}{c}{\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }(s_{\ell },t_{\ell }){\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{\frac{1}{\alpha }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ ⩾{\displaystyle \prod _{\ell =1}^n}{\left[(x_{\ell }-t_0)(y_{\ell }-t_0)\right]}^{\frac{1}{{\alpha }_{\ell }}}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}{\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{\left(p_{\ell }(t_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({\omega }_{\ell }(t_{\ell },{\tau }_{\ell })\right)\right)}^{{\beta }_{\ell }}\nabla t_{\ell }\nabla {\tau }_{\ell }\nabla s_{\ell }\nabla t_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(32)

By using Fubini’s theorem, we observe that

$$\begin{array}{ccc}& & {\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }(s_{\ell },t_{\ell }){\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{\frac{1}{\alpha }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ & & ⩾{\displaystyle \prod _{\ell =1}^n}{\left[(x_{\ell }-t_0)(y_{\ell }-t_0)\right]}^{\frac{1}{{\alpha }_{\ell }}}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}(x_{\ell }-s_{\ell })(y_{\ell }-t_{\ell }){\left(p_{\ell }(s_{\ell },t_{\ell }){\Phi }_{\ell }\left({\omega }_{\ell }(s_{\ell },t_{\ell })\right)\right)}^{{\beta }_{\ell }}\nabla s_{\ell }\nabla t_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

By using the fact $x_{\ell }⩾\rho \left(x_{\ell }\right),$ and $y_{\ell }⩾\rho \left(y_{\ell }\right),$ we obtain that

$$\begin{array}{ccc}& & {\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }(s_{\ell },t_{\ell }){\Phi }_{\ell }\left({\Omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{\frac{1}{\alpha }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ \hfill & & ⩾{\displaystyle \prod _{\ell =1}^n}{\left[(x_{\ell }-t_0)(y_{\ell }-t_0)\right]}^{\frac{1}{{\alpha }_{\ell }}}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}(\rho \left(x_{\ell }\right)-s_{\ell })(\rho \left(y_{\ell }\right)-t_{\ell }){\left(p_{\ell }(s_{\ell },t_{\ell }){\Phi }_{\ell }\left({\omega }_{\ell }(s_{\ell },t_{\ell })\right)\right)}^{{\beta }_{\ell }}\nabla s_{\ell }\nabla t_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}\hfill \end{array}$$

This completes the proof. □

Remark 4.

In Theorem 2, if $\mathbb{T}=\mathbb{R}$, we obtain the result due to Zhao et al. [9], Theorem 3.

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

Corollary 5.

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

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

(33)

Then

$$\begin{array}{ccc}& & {\displaystyle \sum _{m_{s_1}}^{k_{s_1}}}{\displaystyle \sum _{m_{t_1}}^{k_{t_1}}}\cdots {\displaystyle \sum _{m_{s_n}}^{k_{s_n}}}{\displaystyle \sum _{m_{t_n}}^{k_{t_n}}}\frac{{\prod }_{\ell =1}^n{P}_{s_{\ell },t_{\ell },m_{s_{\ell }},m_{t_{\ell }}}{\Phi }_{\ell }\left(A_{s_{\ell },t_{\ell },m_{s_{\ell }},m_{t_{\ell }}}\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}\left(m_{s_{\ell }}{m}_{t_{\ell }}\right)\right)}^{\frac{1}{\alpha }}}\hfill \\ \hfill & & ⩾{\displaystyle \prod _{\ell =1}^n}{\left(k_{s_{\ell }}{k}_{t_{\ell }}\right)}^{\frac{1}{{\alpha }_{\ell }}}{\left({\displaystyle \sum _{m_{s_{\ell }}}^{k_{s_{\ell }}}}{\displaystyle \sum _{m_{t_{\ell }}}^{k_{t_{\ell }}}}(k_{s_{\ell }}-(m_{s_{\ell }}-1))(k_{t_{\ell }}-(m_{t_{\ell }}-1)){\left(p_{s_{\ell },t_{\ell },m_{s_{\ell }},m_{t_{\ell }}}{\Phi }_{\ell }\left(a_{s_{\ell },t_{\ell },m_{s_{\ell }},m_{t_{\ell }}}\right)\right)}^{{\beta }_{\ell }}\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

Remark 5.

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

$${\Omega }_{\ell }(s_{\ell },t_{\ell })=\frac{1}{P_{\ell }(s_{\ell },t_{\ell })}{\int }_{t_0}^{s_{\ell }}{\int }_0^{t_{\ell }}{p}_{\ell }(t_{\ell },{\tau }_{\ell }){\omega }_{\ell }(t_{\ell },{\tau }_{\ell })\nabla t_{\ell }\nabla {\tau }_{\ell }$$

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

$${\Omega }_{\ell }\left(s_{\ell }\right)=\frac{1}{P_{\ell }\left(s_{\ell }\right)}{\int }_{t_0}^{s_{\ell }}{p}_{\ell }\left(t_{\ell }\right){\omega }_{\ell }\left(t_{\ell }\right)\nabla t_{\ell }.$$

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

Corollary 6.

Let ${\omega }_{\ell }\left({\xi }_{\ell }\right),$ $p_{\ell }\left({\xi }_{\ell }\right),$ $P_{\ell }\left({\xi }_{\ell }\right),$ ${\alpha }_{\ell }$ and ${\beta }_{\ell }$ be as Corollary 2 and define ${\Omega }_{\ell }\left(s_{\ell }\right)=\frac{1}{P\left(s_{\ell }\right)}{\int }_{t_0}^{s_{\ell }}{p}_{\ell }\left({\xi }_{\ell }\right){\omega }_{\ell }\left({\xi }_{\ell }\right)\nabla {\xi }_{\ell }$ for ${\xi }_{\ell }$ $s_{\ell }\in {(t_0,x_{\ell })}_{\mathbb{T}}$, where $x_{\ell }$ are positive real numbers. Let ${\Phi }_{\ell }$ be n real-valued, non-negative, concave, and super-multiplicative function defined on $(0,\infty ).$ Then

$$\begin{array}{c}{\int }_{t_0}^{x_1}\cdots {\int }_{t_0}^{x_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }({\Omega }_{\ell }\left(s_{\ell }\right)}{{\left(\alpha {\sum }_{\ell =1}^n\frac{1}{{\alpha }_{\ell }}(s_{\ell }-t_0)\right)}^{\frac{1}{\alpha }}}}\nabla s_n\cdots \nabla s_1\hfill \\ ⩾{\prod }_{\ell =1}^n{(x_{\ell }-t_0)}^{\frac{1}{{\alpha }_{\ell }}}{\left({\int }_{t_0}^{x_{\ell }}(\rho \left(x_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left({\omega }_{\ell }\left(s_{\ell }\right)\right)\right)}^{{\beta }_{\ell }}\nabla s_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(34)

Corollary 7.

In Corollary 6, if we take $n=2,$ ${\beta }_{\ell }=\frac{1}{2}$ then the inequality (26) changes to

$$\begin{array}{c}{\int }_{t_0}^{x_1}{\int }_{t_0}^{x_2}{\displaystyle \frac{P_1\left(s_1\right)P_2\left(s_2\right){\Phi }_1\left({\Omega }_1\left(s_1\right)\right){\Phi }_2\left({\Omega }_2\left(s_2\right)\right)}{{\left((s_1-t_0)+(s_2-t_0)\right)}^{-2}}}\nabla s_1\nabla s_2⩾4{\left[(x_1-t_0)(x_2-t_0)\right]}^{-1}\hfill \\ \times {\left({\int }_{t_0}^{x_1}(\rho \left(x_1\right)-s_1){\left(p_1\left(s_1\right){\Phi }_1\left({\omega }_1\left(s_1\right)\right)\right)}^2\nabla s_1\right)}^{\frac{1}{2}}{\left({\int }_{t_0}^{x_2}(\rho \left(x_2\right)-s_2){\left(p_2\left(s_2\right){\Phi }_2\left({\omega }_2\left(s_2\right)\right)\right)}^2\nabla s_2\right)}^{\frac{1}{2}}.\hfill \end{array}$$

(35)

Remark 6.

In Corollary 7, if we take $\mathbb{T}=\mathbb{R},$ then the inequality (35) changes to

$$\begin{array}{c}{\int }_0^{x_1}{\int }_0^{x_2}{\displaystyle \frac{P_1\left(s_1\right)P_2\left(s_2\right){\Phi }_1\left({\Omega }_1\left(s_1\right)\right){\Phi }_2\left({\Omega }_2\left(s_2\right)\right)}{{(s_1+s_2)}^{-2}}}d{s}_{1}d{s}_2⩾4{\left[x_1{x}_2\right]}^{-1}\hfill \\ \times {\left({\int }_0^{x_1}(x_1-s_1){\left(p_1\left(s_1\right){\Phi }_1\left({\omega }_1\left(s_1\right)\right)\right)}^{2}d{s}_1\right)}^{\frac{1}{2}}{\left({\int }_0^{x_2}(x_2-s_2){\left(p_2\left(s_2\right){\Phi }_2\left({\omega }_2\left(s_2\right)\right)\right)}^{2}d{s}_2\right)}^{\frac{1}{2}}.\hfill \end{array}$$

(36)

This is an inverse of the inequality (8) which was proved by Pachpatte [4].

Corollary 8.

In Corollary 7, 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, the inequality (35) changes to

$$\begin{array}{c}{\int }_{t_0}^{x_1}{\int }_{t_0}^{x_2}{\displaystyle \frac{{\Phi }_1\left({\Omega }_1\left(s_1\right)\right){\Phi }_2\left({\Omega }_2\left(s_2\right)\right)}{{\left(s_1{s}_2\right)}^{-1}{\left((s_1-t_0)+(s_2-t_0)\right)}^{-2}}}\nabla s_1\nabla s_2⩾4{\left[(x_1-t_0)(x_2-t_0)\right]}^{-1}\hfill \\ \times {\left({\int }_{t_0}^{x_1}(\rho \left(x_1\right)-s_1){\left({\Phi }_1\left({\omega }_1\left(s_1\right)\right)\right)}^2\nabla s_1\right)}^{\frac{1}{2}}{\left({\int }_{t_0}^{x_2}(\rho \left(x_2\right)-s_2){\left({\Phi }_2\left({\omega }_2\left(s_2\right)\right)\right)}^2\nabla s_2\right)}^{\frac{1}{2}}.\hfill \end{array}$$

(37)

Remark 7.

In Corollary 8, if we take $\mathbb{T}=\mathbb{R},$ then the inequality (37) change to

$$\begin{array}{ccc}& & {\int }_0^{x_1}{\int }_0^{x_2}{\displaystyle \frac{{\Phi }_1\left({\Omega }_1\left(s_1\right)\right){\Phi }_2\left({\Omega }_2\left(s_2\right)\right)}{{\left(s_1{s}_2\right)}^{-1}{\left(s_1+s_2\right)}^{-2}}}d{s}_{1}d{s}_2⩾4{\left[x_1{x}_2\right]}^{-1}\hfill \\ \hfill & & \times {\left({\int }_0^{x_1}(x_1-s_1){\left({\Phi }_1\left({\omega }_1\left(s_1\right)\right)\right)}^{2}d{s}_1\right)}^{\frac{1}{2}}{\left({\int }_0^{x_2}(x_2-s_2){\left({\Phi }_2\left({\omega }_2\left(s_2\right)\right)\right)}^{2}d{s}_2\right)}^{\frac{1}{2}}.\hfill \end{array}$$

This is an inverse inequality of the following inequality which was proved by Pachpatte [9].

$$\begin{array}{ccc}& & {\int }_0^x{\int }_0^y{\displaystyle \frac{\Phi (\Omega (s\left)\right)\Psi \left(G\right(t\left)\right)}{{\left(st\right)}^{-1}\left(s+t\right)}}dsdt⩽\frac{1}{2}{\left[xy\right]}^{\frac{1}{2}}\hfill \\ \hfill & & \times {\left({\int }_0^x(x-s_1){\left(\Phi \left(\omega \left(s\right)\right)\right)}^{2}ds\right)}^{\frac{1}{2}}{\left({\int }_0^y(y-t){\left(\Psi \left(g\left(t\right)\right)\right)}^{2}dt\right)}^{\frac{1}{2}}.\hfill \end{array}$$

Corollary 9.

In Corollary 6, if we take ${\beta }_{\ell }=\frac{n-1}{n}$ $(\ell =1,\cdots ,n)$ the inequality (34) becomes.

$$\begin{array}{ccc}& & {\int }_{t_0}^{x_1}\cdots {\int }_{t_0}^{x_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }({\Omega }_{\ell }\left(s_{\ell }\right)}{{\left({\sum }_{\ell =1}^n(s_{\ell }-t_0)\right)}^{\frac{-n}{n-1}}}}\nabla s_n\cdots \nabla s_1\hfill \\ \hfill & & ⩾n^{\frac{n}{n-1}}{\displaystyle \prod _{\ell =1}^n}{(x_{\ell }-t_0)}^{\frac{-1}{n-1}}{\left({\int }_{t_0}^{x_{\ell }}(\rho \left(x_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left({\omega }_{\ell }\left(s_{\ell }\right)\right)\right)}^{\frac{n-1}{n}}\nabla s_{\ell }\right)}^{\frac{n}{n-1}}.\hfill \end{array}$$

Theorem 3.

Let $\mathbb{T}$ be timescales with $t_0,$ $x_{\ell },$ $y_{\ell },$ $s_{\ell },$ $t_{\ell }\in \mathbb{T},$ $(\ell =1,\cdots ,n).$ Assume ${\omega }_{\ell }(s_{\ell },t_{\ell })$ $(\ell =1,\cdots ,n)$ are non-negative, left-dense continuous functions defined on ${[t_0,x_{\ell })}_{\mathbb{T}}\times {[t_0,y_{\ell })}_{\mathbb{T}},$ where $x_{\ell }$ and $y_{\ell }$ are positive real numbers and with ${\omega }_{\ell }(t_0,t_{\ell })={\omega }_{\ell }(s_{\ell },t_0)=0,$ $(\ell =1,\cdots ,n).$ Let $p_{\ell }\left({\xi }_{\ell }\right)$ and $q_{\ell }\left({\tau }_{\ell }\right)$ be positive left-dense, continuous functions defined for ${\xi }_{\ell }\in {(t_0,s_{\ell })}_{\mathbb{T}},$ ${\tau }_{\ell }\in {(t_0,t_{\ell })}_{\mathbb{T}}.$ Set

$$P_{\ell }(s_{\ell },t_{\ell })={\int }_{t_0}^{t_{\ell }}{\int }_{t_0}^{s_{\ell }}{p}_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)\nabla {\xi }_{\ell }\nabla {\tau }_{\ell }.$$

(38)

The functions ${\omega }_{\ell }(s_{\ell },t_{\ell })$ have partial ∇- derivatives ${\omega }_{\ell }^{{\nabla }_1}(s_{\ell },t_{\ell })$ and ${\omega }_{\ell }^{{\nabla }_2}(s_{\ell },t_{\ell })$ with respect $s_{\ell }$ and $t_{\ell }$ respectively and ${\omega }_{\ell }^{{\nabla }_1{\nabla }_2}(s_{\ell },t_{\ell })={\omega }_{\ell }^{{\nabla }_2{\nabla }_1}(s_{\ell },t_{\ell }).$ Let ${\Phi }_{\ell }$ $(\ell =1,\cdots ,n)$ be n real-valued non-negative concave and super-multiplicative functions defined on $(0,\infty ).$ Let ${\alpha }_{\ell }\in (1,\infty ).$ Set ${\alpha }_{\ell }^{\prime }=1-{\alpha }_{\ell }$ $(\ell =1,\cdots ,n),$ $\alpha ={\sum }_{\ell =1}^n{\alpha }_{\ell },$ and ${\alpha }^{\prime }={\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }=n-\alpha .$ Then for $s_{\ell }\in {[t_0,x_{\ell })}_{\mathbb{T}}$ and $t_{\ell }\in {[t_0,y_{\ell })}_{\mathbb{T}},$ we have that

$$\begin{array}{c}{\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\prod }_{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({\omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{{\alpha }^{\prime }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ ⩾G(x_1{y}_1,\cdots ,x_n{y}_n)\hfill \\ \times {\displaystyle {\prod }_{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}(\rho \left(x_{\ell }\right)-s_{\ell })(\rho \left(y_{\ell }\right)-t_{\ell }){\left(p_{\ell }\left(s_{\ell }\right)q_{\ell }\left(t_{\ell }\right){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }^{{\nabla }_2{\nabla }_1}(s_{\ell },t_{\ell })}{p_{\ell }\left(s_{\ell }\right)q_{\ell }\left(t_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}\nabla s_{\ell }\nabla t_{\ell }\right)}^{{\alpha }_{\ell }}\hfill \end{array}$$

(39)

where

$$G(x_1{y}_1,\cdots ,x_n{y}_n)={\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right)}{P_{\ell }(s_{\ell },t_{\ell })}\right)}^{\frac{1}{{\alpha }_{\ell }^{\prime }}}\nabla s_{\ell }\nabla t_{\ell }\right)}^{{\alpha }_{\ell }^{\prime }}.$$

Proof. 

From the hypotheses of Theorem 3, we obtain

$${\omega }_{\ell }(s_{\ell },t_{\ell })={\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{\omega }_{\ell }^{{\nabla }_2{\nabla }_1}(t_{\ell },{\tau }_{\ell })\nabla t_{\ell }\nabla {\tau }_{\ell }.$$

(40)

From (40) and $S_8$, it is easy to observe that

$$\begin{array}{ccc}{\Phi }_{\ell }\left({\omega }_{\ell }(s_{\ell },t_{\ell })\right)\hfill & =\hfill & {\Phi }_{\ell }\left({\displaystyle \frac{P_{\ell }(s_{\ell },t_{\ell }){\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{p}_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)\left(\frac{{\omega }_{\ell }^{{\nabla }_2{\nabla }_1}(t_{\ell },{\tau }_{\ell })}{p_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\nabla t_{\ell }\nabla {\tau }_{\ell }}{{\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{p}_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)\nabla t_{\ell }\nabla {\tau }_{\ell }}}\right)\hfill \\ \hfill & ⩾\hfill & {\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right){\Phi }_{\ell }\left({\displaystyle \frac{{\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{p}_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)\left(\frac{{\omega }_{\ell }^{{\nabla }_2{\nabla }_1}(t_{\ell },{\tau }_{\ell })}{p_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\nabla t_{\ell }\nabla {\tau }_{\ell }}{{\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{p}_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)\nabla t_{\ell }\nabla {\tau }_{\ell }}}\right).\hfill \end{array}$$

(41)

By using an inverse Jensen’s dynamic inequality, we obtain that

$$\begin{array}{ccc}\hfill {\Phi }_{\ell }\left({\omega }_{\ell }(s_{\ell },t_{\ell })\right)& ⩾& \frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right)}{P_{\ell }(s_{\ell },t_{\ell })}{\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{p}_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }^{{\nabla }_2{\nabla }_1}(t_{\ell },{\tau }_{\ell })}{p_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\nabla t_{\ell }\nabla {\tau }_{\ell }\hfill \\ \hfill \end{array}$$

(42)

Applying the inverse Hölder’s inequality on the left-hand side of (42) with indices $1/{\alpha }_{\ell }$ and $1/{\alpha }_{\ell }^{\prime },$ we obtain

$$\begin{array}{ccc}{\Phi }_{\ell }\left({\omega }_{\ell }(s_{\ell },t_{\ell })\right)\hfill & ⩾\hfill & \frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right)}{P_{\ell }(s_{\ell },t_{\ell })}{\left[(s_{\ell }-t_0)(t_{\ell }-t_0)\right]}^{{\alpha }_{\ell }^{\prime }}\hfill \\ \hfill & \hfill & \times {\left({\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{\left(p_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }^{{\nabla }_2{\nabla }_1}(t_{\ell },{\tau }_{\ell })}{p_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}\nabla t_{\ell }\nabla {\tau }_{\ell }\right)}^{{\alpha }_{\ell }}.\hfill \end{array}$$

(43)

Using the following inequality on the term ${\left[(s_{\ell }-t_0)(t_{\ell }-t_0)\right]}^{{\alpha }_{\ell }^{\prime }}$, where ${\alpha }_{\ell }^{\prime }<0$ and ${\lambda }_{\ell }>0.$

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

(44)

we obtain that

$$\begin{array}{ccc}{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\omega }_{\ell }(s_{\ell },t_{\ell })\right)\hfill & ⩾\hfill & {\prod }_{\ell =1}^n\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right)}{P_{\ell }(s_{\ell },t_{\ell })}{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{{\alpha }^{\prime }}\hfill \\ \hfill & \hfill & \times {\left({\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{\left(p_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }^{{\nabla }_2{\nabla }_1}(t_{\ell },{\tau }_{\ell })}{p_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}\nabla t_{\ell }\nabla {\tau }_{\ell }\right)}^{{\alpha }_{\ell }}.\hfill \end{array}$$

(45)

From (45), we have that

$$\begin{array}{c}{\prod }_{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({\omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{{\alpha }^{\prime }}}}\hfill \\ ⩾{\prod }_{\ell =1}^n\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right)}{P_{\ell }(s_{\ell },t_{\ell })}{\left({\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{\left(p_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }^{{\nabla }_2{\nabla }_1}(t_{\ell },{\tau }_{\ell })}{p_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}\nabla t_{\ell }\nabla {\tau }_{\ell }\right)}^{{\alpha }_{\ell }}.\hfill \end{array}$$

(46)

Integrating both sides of (46) over $s_{\ell },$ $t_{\ell }$ from $t_0$ to $x_{\ell },$ $y_{\ell }$ $(\ell =1,\cdots ,n),$ we obtain that

$$\begin{array}{c}{\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\prod }_{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({\omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{{\alpha }^{\prime }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ ⩾{\prod }_{\ell =1}^n{\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right)}{P_{\ell }(s_{\ell },t_{\ell })}{\left({\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{\left(p_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }^{{\nabla }_2{\nabla }_1}(t_{\ell },{\tau }_{\ell })}{p_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}\nabla t_{\ell }\nabla {\tau }_{\ell }\right)}^{{\alpha }_{\ell }}\nabla s_{\ell }\nabla t_{\ell }\hfill \end{array}$$

(47)

Applying the inverse Hölder’s inequality on the left-hand side of (47) with indices $1/{\alpha }_{\ell }$ and $1/{\alpha }_{\ell }^{\prime },$ we obtain

$$\begin{array}{ccc}\hfill & & {\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\displaystyle \prod _{\ell =1}^n}{\displaystyle \frac{{\Phi }_{\ell }\left({\omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{{\alpha }^{\prime }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ & & ⩾{\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right)}{P_{\ell }(s_{\ell },t_{\ell })}\right)}^{\frac{1}{{\alpha }_{\ell }^{\prime }}}\nabla s_{\ell }\nabla t_{\ell }\right)}^{{\alpha }_{\ell }^{\prime }}\hfill \\ \hfill & & \times {\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}\left({\int }_{t_0}^{s_{\ell }}{\int }_{t_0}^{t_{\ell }}{\left(p_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }^{{\nabla }_2{\nabla }_1}(t_{\ell },{\tau }_{\ell })}{p_{\ell }\left(t_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}\nabla t_{\ell }\nabla {\tau }_{\ell }\right)\nabla s_{\ell }\nabla t_{\ell }\right)}^{{\alpha }_{\ell }}\hfill \\ \hfill \end{array}$$

(48)

By using Fubini’s theorem, we observe that

$$\begin{array}{ccc}\hfill & & {\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\displaystyle \prod _{\ell =1}^n}{\displaystyle \frac{{\Phi }_{\ell }\left({\omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{{\alpha }^{\prime }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ & & ⩾G(x_1{y}_1,\cdots ,x_n{y}_n)\hfill \\ \hfill & & \times {\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}(x_{\ell }-s_{\ell })(y_{\ell }-t_{\ell }){\left(p_{\ell }\left(s_{\ell }\right)q_{\ell }\left(t_{\ell }\right){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }^{{\nabla }_2{\nabla }_1}(s_{\ell },t_{\ell })}{p_{\ell }\left(s_{\ell }\right)q_{\ell }\left(t_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}\nabla s_{\ell }\nabla t_{\ell }\right)}^{{\alpha }_{\ell }}.\hfill \end{array}$$

(49)

By using the fact $x_{\ell }⩾\rho \left(x_{\ell }\right),$ and $y_{\ell }⩾\rho \left(y_{\ell }\right),$ we obtain that

$$\begin{array}{ccc}& & {\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\displaystyle \prod _{\ell =1}^n}{\displaystyle \frac{{\Phi }_{\ell }\left({\omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{{\alpha }^{\prime }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ \hfill & & ⩾G(x_1{y}_1,\cdots ,x_n{y}_n)\hfill \\ \hfill & & \times {\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}(\rho \left(x_{\ell }\right)-s_{\ell })(\rho \left(y_{\ell }\right)-t_{\ell }){\left(p_{\ell }\left(s_{\ell }\right)q_{\ell }\left(t_{\ell }\right){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }^{{\nabla }_2{\nabla }_1}(s_{\ell },t_{\ell })}{p_{\ell }\left(s_{\ell }\right)q_{\ell }\left(t_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}\nabla s_{\ell }\nabla t_{\ell }\right)}^{{\alpha }_{\ell }}.\hfill \end{array}$$

This completes the proof. □

Remark 8.

In Theorem 3, if $\mathbb{T}=\mathbb{Z}$, we obtain the result due to Zhao et al.’s [10], Theorem 1.5.

Remark 9.

In Theorem 3, if we take $\mathbb{T}=\mathbb{R}$, we obtain the result due to Zhao et al.’s [10], Theorem 1.6.

Remark 10.

Let ${\omega }_{\ell }(s_{\ell },t_{\ell }),$ $x_{\ell },$ $y_{\ell },$ ${\omega }_{\ell }(t_0,t_{\ell }),$ ${\omega }_{\ell }(s_{\ell },t_0),$ ${\omega }_{\ell }^{{\nabla }_1}(s_{\ell },t_{\ell }),$ ${\omega }_{\ell }^{{\nabla }_2}(s_{\ell },t_{\ell }),$ ${\omega }_{\ell }^{{\nabla }_2{\nabla }_1}(s_{\ell },t_{\ell }),$ $p_{\ell }\left({\xi }_{\ell }\right),$ $q_{\ell }\left({\tau }_{\ell }\right)$ and $P(s_{\ell },t_{\ell })$ be as in Theorem 3. Let ${\Phi }_{\ell },$ ${\alpha }_{\ell },$ ${\alpha }_{\ell }^{\prime },$ $\alpha ,$ and ${\alpha }^{\prime }$ be the same as in Theorem 4. Similar to the proof of Theorem 3, we have

$$\begin{array}{ccc}& & {\int }_{t_0}^{x_1}{\int }_{t_0}^{y_1}\cdots {\int }_{t_0}^{x_n}{\int }_{t_0}^{y_n}{\displaystyle \prod _{\ell =1}^n}{\displaystyle \frac{{\Phi }_{\ell }\left({\omega }_{\ell }(s_{\ell },t_{\ell })\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-t_0)(t_{\ell }-t_0)\right)}^{{\alpha }^{\prime }}}}\nabla s_n\nabla t_n\cdots \nabla s_1\nabla t_1\hfill \\ \hfill & & ⩽G^{*}(x_1{y}_1,\cdots ,x_n{y}_n)\hfill \\ \hfill & & \times {\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}(\sigma \left(x_{\ell }\right)-s_{\ell })(\sigma \left(y_{\ell }\right)-t_{\ell }){\left(p_{\ell }\left(s_{\ell }\right)q_{\ell }\left(t_{\ell }\right){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }^{{\nabla }_2{\nabla }_1}(s_{\ell },t_{\ell })}{p_{\ell }\left(s_{\ell }\right)q_{\ell }\left(t_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}\nabla s_{\ell }\nabla t_{\ell }\right)}^{{\alpha }_{\ell }}.\hfill \end{array}$$

where

$$G^{*}(x_1{y}_1,\cdots ,x_n{y}_n)=\frac{1}{{\left({\alpha }^{\prime }\right)}^{{\alpha }^{\prime }}}{\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\int }_{t_0}^{y_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },t_{\ell })\right)}{P_{\ell }(s_{\ell },t_{\ell })}\right)}^{\frac{1}{{\alpha }_{\ell }^{\prime }}}\nabla s_{\ell }\nabla t_{\ell }\right)}^{{\alpha }_{\ell }^{\prime }}.$$

This is an inverse form of the inequality (39).

Corollary 10. 

Let $\mathbb{T}$ be a timescale with $t_0,$ $x_{\ell },$ $s_{\ell }\in \mathbb{T}.$ Let ${\omega }_{\ell }\in C_{ld}{[t_0,k_{\ell }]}_{\mathbb{T}},$ $(\ell =1\cdots ,n)$ with ${\omega }_{\ell }\left(t_0\right)=0,$ let $p_{\ell }\left({\xi }_{\ell }\right)$ be n positive functions defined for ${\xi }_{\ell }\in {[t_0,s_{\ell }]}_{\mathbb{T}}.$ Set $P_{\ell }\left(s_{\ell }\right)={\int }_{t_0}^{s_{\ell }}{p}_{\ell }\left({\xi }_{\ell }\right)\nabla {\xi }_{\ell }$ for $s_{\ell }\in {[t_0,x_{\ell }]}_{\mathbb{T}},$ where $x_{\ell }$ are positive real numbers. Let ${\Phi }_{\ell },$ ${\alpha }_{\ell },$ ${\alpha }_{\ell }^{\prime },$ $\alpha ,$ and ${\alpha }^{\prime }$ be as in Theorem 3. Then

$$\begin{array}{ccc}\hfill & & {\int }_{t_0}^{x_1}\cdots {\int }_{t_0}^{x_n}{\displaystyle \prod _{\ell =1}^n}{\displaystyle \frac{{\Phi }_{\ell }\left({\omega }_{\ell }\left(s_{\ell }\right)\right)}{{\left(\frac{1}{{\alpha }^{\prime }}{\sum }_{\ell =1}^n{\alpha }_{\ell }^{\prime }(s_{\ell }-t_0)\right)}^{{\alpha }^{\prime }}}}\nabla s_n\cdots \nabla s_1\hfill \\ & & ⩾G^{**}(x_1,\cdots ,x_n)\hfill \\ \hfill & & \times {\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}(\rho \left(x_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left(\frac{{\omega }_{\ell }^{\nabla }\left(s_{\ell }\right)}{p_{\ell }\left(s_{\ell }\right)}\right)\right)}^{\frac{1}{{\alpha }_{\ell }}}\nabla s_{\ell }\right)}^{{\alpha }_{\ell }}.\hfill \end{array}$$

(50)

where

$$G^{**}(x_1,\cdots ,x_n)={\displaystyle \prod _{\ell =1}^n}{\left({\int }_{t_0}^{x_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell }\right)}\right)}^{\frac{1}{{\alpha }_{\ell }^{\prime }}}\nabla s_{\ell }\right)}^{{\alpha }_{\ell }^{\prime }}.$$

Remark 11.

In Corollary 10, if we take $\mathbb{T}=\mathbb{Z},$ we obtain an inverse form of inequality (4), which was given by Handley et al.

Remark 12.

In Corollary 10, if we take $\mathbb{T}=\mathbb{R},$ we obtain an inverse form of inequality (5), which was given by Handley et al.

Remark 13.

In inequality (50) taking $n=2,$ ${\alpha }_1={\alpha }_2=2,$ then ${\alpha }_1^{\prime }={\alpha }_2^{\prime }=-1,$, we have

$$\begin{array}{ccc}\hfill & & {\int }_{t_0}^{x_1}{\int }_{t_0}^{x_2}{\displaystyle \prod _{\ell =1}^n}{\displaystyle \frac{{\Phi }_1\left({\omega }_1\left(s_1\right)\right){\Phi }_1\left({\omega }_2\left(s_2\right)\right)}{{\left((s_1-t_0)+(s_2-t_0)\right)}^{-2}}}\nabla s_1\nabla s_2\hfill \\ & & ⩾D(x_1,x_2){\left({\int }_{t_0}^{x_1}(\rho \left(x_1\right)-s_1){\left(p_1\left(s_1\right){\Phi }_1\left(\frac{{\omega }_1^{\nabla }\left(s_1\right)}{p_1\left(s_1\right)}\right)\right)}^{\frac{1}{2}}\nabla s_1\right)}^2\hfill \\ \hfill & & \times {\left({\int }_{t_0}^{x_2}(\rho \left(x_2\right)-s_2){\left(p_2\left(s_2\right){\Phi }_2\left(\frac{{\omega }_2^{\nabla }\left(s_2\right)}{p_2\left(s_2\right)}\right)\right)}^{\frac{1}{2}}\nabla s_2\right)}^2.\hfill \end{array}$$

(51)

where

$$D(x_1,x_2)=4{\left({\int }_{t_0}^{x_1}{\left(\frac{{\Phi }_1\left(P_1\left(s_1\right)\right)}{P_2\left(s_1\right)}\right)}^{-1}\nabla s_1\right)}^{-1}{\left({\int }_{t_0}^{x_2}{\left(\frac{{\Phi }_2\left(P_2\left(s_2\right)\right)}{P_2\left(s_2\right)}\right)}^{-1}\nabla s_2\right)}^{-1}.$$

Remark 14.

If we take $\mathbb{T}=\mathbb{Z}$, the inequality (51) is an inverse of inequality due to Pachpatte [2].

Remark 15.

If we take $\mathbb{T}=\mathbb{R}$, the inequality (51) is an inverse of inequality due to Pachpatte [2].

3. Conclusions

In this article, we introduced some investigations of the nabla Hilbert inequality on a general timescale, some dynamic integral and discrete inequalities, known in the literature, are extended as special cases of our results. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.