Diamond Alpha Hilbert-Type Inequalities on Time Scales

Abstract

In this article, we will prove some new diamond alpha Hilbert-type dynamic inequalities on time scales which are defined as a linear combination of the nabla and delta integrals. These inequalities extend some known dynamic inequalities on time scales, and unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proven by using some algebraic inequalities, diamond alpha Hölder inequality, and diamond alpha Jensen’s inequality on time scales.

1. Introduction

Over the past decade, a great number of dynamic Hilbert-type inequalities on time scales has been established by many researchers who were motivated by various applications; see the papers [1,2,3,4].

For example, Pachpatte [5] proved that if $\left\{a_m\right\},$$\left\{b_n\right\}$ are two non-negative sequences of real numbers defined for $m=1,\cdots ,k,$ and $n=1,\cdots ,r$ with $a_0=b_0=0,$ and $\left\{p_m\right\},$$\left\{q_n\right\},$ are two positive sequences of real numbers defined for $m=1,\cdots ,k$ and $n=1,\cdots ,r$ where $k,$r are natural numbers. Further $P_m={\displaystyle \sum _{s=1}^m}{p}_s$ and $Q_n={\displaystyle \sum _{t=1}^n}{q}_t,$ and $\Phi$ and $\mathsf{\Psi }$ are two real-valued non-negative, convex, and submultiplicative functions defined on $[0,\infty ),$ then

$$\begin{array}{c}\hfill {\displaystyle \sum _{m=1}^k}{\displaystyle \sum _{n=1}^r}{\displaystyle \frac{\Phi \left(a_m\right)\mathsf{\Psi }\left(b_n\right)}{m+n}}⩽M(k,r)({\displaystyle \sum _{m=1}^k}(k-m+1){\left(p_m\Phi {\left(\frac{\nabla a_m}{p_m}\right)}^2\right)}^{\frac{1}{2}}\\ \hfill \times ({\displaystyle \sum _{n=1}^r}(r-n+1){\left(q_n\mathsf{\Psi }{\left(\frac{\nabla b_n}{q_n}\right)}^2\right)}^{\frac{1}{2}},\end{array}$$

(1)

where

$$M(k,r)=\frac{1}{2}{\left({\displaystyle \sum _{m=1}^k}{\left(\frac{\Phi \left(P_m\right)}{p_m}\right)}^2\right)}^{\frac{1}{2}}{\left({\displaystyle \sum _{n=1}^r}{\left(\frac{\mathsf{\Psi }\left(Q_n\right)}{Q_n}\right)}^2\right)}^{\frac{1}{2}}.$$

Additionally, in the same paper [5], Pachpatte proved that if $ϝ\in C^1[[0,\vartheta ],{\mathbb{R}}^{+}],$$g\in C^1[[0,\varsigma ],{\mathbb{R}}^{+}]$ with $ϝ\left(0\right)=g\left(0\right)=0$ and $p\left(\xi \right),$$q\left(\xi \right)$ are two positive functions defined for $\xi \in [0,\vartheta )$ and $\tau \in [0,\varsigma ),$$P\left(\xi \right)={\int }_0^{\xi }p\left(\xi \right)d\xi$ and $Q\left(\xi \right)={\int }_0^{\xi }p\left(\tau \right)d\tau$ for $s\in [0,\vartheta )$ and $t\in [0,\varsigma )$ where $\vartheta ,$$\varsigma$ are positive real numbers; thus

$$\begin{array}{c}\hfill {\int }_0^{\vartheta }{\int }_0^{\zeta }\frac{\Phi \left(ϝ\right(s\left)\right)\mathsf{\Psi }\left(g\right(\Im \left)\right)}{s+\Im }dsd\Im ⩽L(\vartheta ,\varsigma )\left({\int }_0^{\vartheta }(\vartheta -s){\left(p\left(s\right)\Phi {\left(\frac{{ϝ}^{\prime }\left(s\right)}{p\left(s\right)}\right)}^{2}ds\right)}^{\frac{1}{2}}\right.\\ \hfill \times \left({\int }_0^{\zeta }(\zeta -\Im ){\left(q(\Im )\mathsf{\Psi }{\left(\frac{g^{\prime }(\Im )}{q(\Im )}\right)}^{2}d\Im \right)}^{\frac{1}{2}},\right.\end{array}$$

(2)

where

$$L(\vartheta ,\varsigma )=\frac{1}{2}{\left({\int }_0^{\vartheta }{\left(\frac{\Phi \left(P\right(s\left)\right)}{P\left(s\right)}\right)}^{2}ds\right)}^{\frac{1}{2}}{\left({\int }_0^{\varsigma }{\left(\frac{\mathsf{\Psi }\left(Q\right(\Im \left)\right)}{Q(\Im )}\right)}^{2}d\Im \right)}^{\frac{1}{2}}.$$

Under the same conditions as seen above, with few modifications, Handley et al. [6] extended (1) and (2) as follows:

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

(3)

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

and

$$\begin{array}{c}{\int }_0^{{\vartheta }_1}\dots {\int }_0^{{\vartheta }_n}\frac{{\displaystyle \prod _{\ell =1}^n}{\Phi }_{\ell }\left(ϝ\left(s_{\ell }\right)\right)}{{\left({\displaystyle \sum _{\ell =1}^n}{\gamma }_{\ell }^{\prime }{s}_{\ell }\right)}^{{\gamma }^{\prime }}}d{s}_1\dots d{s}_n\hfill \\ ⩽L\left({\vartheta }_1,\dots ,{\vartheta }_n\right){\displaystyle \prod _{\ell =1}^n}\left({\int }_0^{{\vartheta }_{\ell }}\left({\vartheta }_{\ell }-s_{\ell }\right){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }{\left(\frac{F^{\prime }\left(s_{\ell }\right)}{p\left(s_{\ell }\right)}\right)}^{\frac{1}{{\gamma }_{\ell }}}d{s}_{\ell }\right)}^{{\gamma }_{\ell }},\right.\hfill \\ L\left({\vartheta }_1,\dots ,{\vartheta }_n\right)=\frac{1}{{\left({\gamma }^{\prime }\right)}^{{\gamma }^{\prime }}}{\displaystyle \prod _{\ell =1}^n}{\left({\int }_0^{{\vartheta }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell }\right)}\right)}^{\frac{1}{{\gamma }_{\ell }^{\prime }}}d{s}_{\ell }\right)}^{{\gamma }_{\ell }^{\prime }},\hfill \end{array}$$

(4)

$$L({\vartheta }_1,\cdots ,{\vartheta }_n)=\frac{1}{{\left({\gamma }^{\prime }\right)}^{{\gamma }^{\prime }}}\prod _{\ell =1}^n{\left({\int }_0^{{\vartheta }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell }\right)}\right)}^{\frac{1}{{\gamma }_{\ell }^{\prime }}}d{s}_{\ell }\right)}^{{\gamma }_{\ell }^{\prime }},$$

where ${\gamma }_i\in \left(0.1\right),$${\gamma }_i^{\prime }=1-\gamma ,$$\gamma ={\displaystyle \sum _{i=1}^n}{\gamma }_i,$ and ${\gamma }_i^{\prime }=n-\gamma .$

In [7], Pachpatte established the following Hilbert-type integral inequalities under the following conditions: If $h⩾1,$$l⩾1,$ and $f\left(\xi \right)⩾0,$$g\left(\xi \right)⩾0,$ for $\xi \in (0,\vartheta )$ and $\tau \in (0,\varsigma ),$ where $\vartheta$ and $\varsigma$ are positive real numbers and define $ϝ\left(s\right)={\int }_0^{s}f\left(\xi \right)d\xi$ and $G\left(t\right)={\int }_0^{t}g\left(\tau \right)d\tau ,$ for $s\in (0,\vartheta )$ and $t\in (0,\varsigma ),$ then

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

(5)

and

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

(6)

where

$$L(\vartheta ,\varsigma )=\frac{1}{2}{\left({\int }_0^{\vartheta }{\left(\frac{\Phi \left(P\right(s\left)\right)}{P\left(s\right)}\right)}^{2}ds\right)}^{\frac{1}{2}}{\left({\int }_0^{\varsigma }{\left(\frac{\mathsf{\Psi }\left(Q\right(\Im \left)\right)}{Q(\Im )}\right)}^{2}d\Im \right)}^{\frac{1}{2}},$$

and

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

(7)

$$\begin{array}{ccc}& & {\int }_0^{{\vartheta }_1}{\int }_0^{{\varsigma }_1}\cdots {\int }_0^{{\vartheta }_n}{\int }_0^{{\varsigma }_n}{\displaystyle \frac{{\displaystyle \prod _{\ell =1}^n}{\Phi }_{\ell }\left({ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}\left(s_{\ell }\right)\left({\Im }_{\ell }\right)\right)}^{\frac{1}{\gamma }}}}d{s}_{1}d{\Im }_1\cdots d{s}_{n}d{\Im }_n\hfill \\ \hfill & & ⩾L({\vartheta }_1{\varsigma }_1,\cdots ,{\vartheta }_n{\varsigma }_n)\hfill \\ \hfill & & \times {\displaystyle \prod _{\ell =1}^n}{\left({\int }_0^{{\vartheta }_{\ell }}{\int }_0^{{\varsigma }_{\ell }}({\vartheta }_{\ell }-s_{\ell })({\varsigma }_{\ell }-{\Im }_{\ell }){\left(p_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })}{p_{\ell }(s_{\ell },{\Im }_{\ell })}\right)\right)}^{{\beta }_{\ell }}d{s}_{\ell }d{\Im }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(8)

where

$$L({\vartheta }_1{\varsigma }_1,\cdots ,{\vartheta }_n{\varsigma }_n)=\prod _{\ell =1}^n{\left({\int }_0^{{\vartheta }_{\ell }}{\int }_0^{{\varsigma }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{P_{\ell }(s_{\ell },{\Im }_{\ell })}\right)}^{{\gamma }_{\ell }}d{s}_{\ell }d{\Im }_{\ell }\right)}^{\frac{1}{{\gamma }_{\ell }}}.$$

A time scale $\mathbb{T}$ is an arbitrary, non-empty, 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 $\zeta \in \mathbb{T}$ by

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

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

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

In the preceding two definitions, we set $inf\varnothing =sup\mathbb{T}$ (i.e., if $\zeta$ is the maximum of $\mathbb{T}$, then $\sigma \left(\zeta \right)=\zeta$) and $sup\varnothing =inf\mathbb{T}$ (i.e., if $\zeta$ is the minimum of $\mathbb{T}$, then $\rho \left(\zeta \right)=\zeta$), where ∅ denotes the empty set.

A point $\zeta \in \mathbb{T}$ with $inf\mathbb{T}<\zeta <sup\mathbb{T}$ is said to be right-scattered if $\sigma \left(\zeta \right)>\zeta$, right-dense if $\sigma \left(\zeta \right)=\zeta$, left-scattered if $\rho \left(\zeta \right)<\zeta$, and left-dense if $\rho \left(\zeta \right)=\zeta$. 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 $\zeta \in \mathbb{T}$ by $\mu \left(\zeta \right):=\sigma \left(\zeta \right)-\zeta$.

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

The sets ${\mathbb{T}}^{\kappa }$, ${\mathbb{T}}_{\kappa }$ and ${\mathbb{T}}_{\kappa }^{\kappa }$ are introduced as follows: if $\mathbb{T}$ has a left-scattered maximum ${\zeta }_1$, then ${\mathbb{T}}^{\kappa }=\mathbb{T}-\left\{{\zeta }_1\right\}$, otherwise ${\mathbb{T}}^{\kappa }=\mathbb{T}$. If $\mathbb{T}$ has a right-scattered minimum ${\zeta }_2$, then ${\mathbb{T}}^{\kappa }=\mathbb{T}-\left\{{\zeta }_2\right\}$, otherwise ${\mathbb{T}}_{\kappa }=\mathbb{T}$. Finally, we have ${\mathbb{T}}_{\kappa }^{\kappa }={\mathbb{T}}^{\kappa }\cap {\mathbb{T}}_{\kappa }$.

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

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

We define the open intervals and half-closed intervals similarly.

Assume $ϝ:\mathbb{T}\to \mathbb{R}$ is a function and $\zeta \in {\mathbb{T}}^{\kappa }$. Then ${ϝ}^{\Delta }\left(\zeta \right)\in \mathbb{R}$ is said to be the delta derivative of $ϝ$ at $\zeta$ if for any $\epsilon >0$ there exists a neighborhood U of $\zeta$ such that, for every $s\in U$, we have

$$|ϝ\left(\sigma \left(\zeta \right)\right)-ϝ\left(s\right)]-{ϝ}^{\Delta }\left(\zeta \right)[\sigma \left(\zeta \right)-s]|\le \epsilon |\sigma \left(\zeta \right)-s|.$$

Moreover, $ϝ$ is said to be delta differentiable on ${\mathbb{T}}^{\kappa }$ if it is delta differentiable at every $\zeta \in {\mathbb{T}}^{\kappa }$.

Similarly, we say that ${ϝ}^{\nabla }\left(\zeta \right)\in \mathbb{R}$ is the nabla derivative of $ϝ$ at $\zeta$ if, for any $\epsilon >0$, there is a neighborhood V of $\zeta$, such that for all $s\in V$

$$|[ϝ\left(\rho \left(\zeta \right)\right)-ϝ\left(s\right)]-{ϝ}^{\nabla }\left(\zeta \right)[\rho \left(\zeta \right)-s]|\le \epsilon |\rho \left(\zeta \right)-s|.$$

(9)

Furthermore, $ϝ$ is said to be nabla differentiable on ${\mathbb{T}}_{\kappa }$ if it is nabla differentiable at each $\zeta \in {\mathbb{T}}_{\kappa }$.

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 time scales is given by the following formula

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

(10)

whereas the nabla integration by parts on time scales is given by

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

(11)

The following relations will be used.

(i)

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

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

(12)

(ii)

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

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

(13)

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

Now we will introduce the diamond-$\alpha$ calculus on time scales, and we refer the interested reader to [8,9] for further details on the definitions of nabla and delta integrals and derivatives.

If $\mathbb{T}$ is a time scale, and $ϝ$ is a function that is delta and nabla differentiable on $\mathbb{T}$,then, for any $\zeta \in \mathbb{T}$, the diamond-$\alpha$ dynamic derivative of $ϝ$ at $\zeta$, denoted by ${ϝ}^{{\diamond }_{\alpha }}\left(\zeta \right)$, is defined by

$${ϝ}^{{\diamond }_{\alpha }}\left(\zeta \right)=\alpha {ϝ}^{\Delta }\left(\zeta \right)+(1-\alpha ){ϝ}^{\nabla }\left(\zeta \right),0\le \alpha \le 1.$$

(14)

We conclude from the last relation that a function $ϝ$ is diamond-$\alpha$ differentiable if and only if it is both delta and nabla differentiable. For $\alpha =1$, the diamond-$\alpha$ derivative boils down to a delta derivative, and for $\alpha =0$ it boils down to a nabla derivative.

Assume $ϝ$, $g:\mathbb{T}\to \mathbb{R}$ are diamond-$\alpha$ differentiable functions at $\zeta \in \mathbb{T}$, and let $k\in \mathbb{R}$. Then

(i)

${(ϝ+g)}^{{\diamond }_{\alpha }}\left(\zeta \right)={ϝ}^{{\diamond }_{\alpha }}\left(\zeta \right)+g^{{\diamond }_{\alpha }}\left(\zeta \right)$;

(ii)

${\left(kf\right)}^{{\diamond }_{\alpha }}\left(\zeta \right)=k{f}^{{\diamond }_{\alpha }}\left(\zeta \right)$;

(iii)

${\left(fg\right)}^{{\diamond }_{\alpha }}\left(\zeta \right)={ϝ}^{{\diamond }_{\alpha }}\left(\zeta \right)g\left(\zeta \right)+\alpha {ϝ}^{\sigma }\left(\zeta \right)g^{\Delta }\left(\zeta \right)+(1-\alpha ){ϝ}^{\rho }\left(\zeta \right)g^{\nabla }\left(\zeta \right)$.

Let $ϝ:\mathbb{T}\to \mathbb{R}$ be a continuous function. Then the definite diamond-$\alpha$ integral of $ϝ$ is defined by

$${\int }_a^bϝ\left(\zeta \right){\diamond }_{\alpha }\zeta =\alpha {\int }_a^bϝ\left(\zeta \right)\Delta \zeta +(1-\alpha ){\int }_a^bϝ\left(\zeta \right)\nabla \zeta ,0\le \alpha \le 1,a,b\in \mathbb{T}.$$

(15)

Let a, b, $c\in \mathbb{T}$, $k\in \mathbb{R}$. Then,

(i)

${\int }_a^b\left[ϝ\left(\zeta \right)+g\left(\zeta \right)\right]{\diamond }_{\alpha }\zeta ={\int }_a^bϝ\left(\zeta \right){\diamond }_{\alpha }\zeta +{\int }_a^{b}g\left(\zeta \right){\diamond }_{\alpha }\zeta$;

(ii)

${\int }_a^{b}kf\left(\zeta \right){\diamond }_{\alpha }\zeta =k{\int }_a^bϝ\left(\zeta \right){\diamond }_{\alpha }\zeta$;

(iii)

${\int }_a^bϝ\left(\zeta \right){\diamond }_{\alpha }\zeta ={\int }_a^cϝ\left(\zeta \right){\diamond }_{\alpha }\zeta +{\int }_c^bϝ\left(\zeta \right){\diamond }_{\alpha }\zeta$;

(iv)

${\int }_a^bϝ\left(\zeta \right){\diamond }_{\alpha }\zeta =-{\int }_b^aϝ\left(\zeta \right){\diamond }_{\alpha }\zeta$;

(v)

${\int }_a^aϝ\left(\zeta \right){\diamond }_{\alpha }\zeta =0$;

(vi)

if $ϝ\left(\zeta \right)\ge 0$ on ${[a,b]}_{\mathbb{T}}$, then ${\int }_a^bϝ\left(\zeta \right){\diamond }_{\alpha }\zeta \ge 0$;

(vii)

if $ϝ\left(\zeta \right)\ge g\left(\zeta \right)$ on ${[a,b]}_{\mathbb{T}}$, then ${\int }_a^bϝ\left(\zeta \right){\diamond }_{\alpha }\zeta \ge {\int }_a^{b}g\left(\zeta \right){\diamond }_{\alpha }\zeta$;

(viii)

$\left|{\int }_a^bϝ\left(\zeta \right){\diamond }_{\alpha }\zeta \right|\le {\int }_a^b\left|ϝ\left(\zeta \right)\right|{\diamond }_{\alpha }\zeta$.

Let $ϝ$ be a $diamond-\alpha$ differentiable function on ${[a,b]}_{\mathbb{T}}$. Then $ϝ$ is increasing if ${ϝ}^{{\diamond }_{\alpha }}\left(\zeta \right)>0$, non-decreasing if ${ϝ}^{{\diamond }_{\alpha }}\left(\zeta \right)\ge 0$, decreasing if ${ϝ}^{{\diamond }_{\alpha }}\left(\zeta \right)<0$, and non-increasing if ${ϝ}^{{\diamond }_{\alpha }}\left(\zeta \right)\le 0$ on ${[a,b]}_{\mathbb{T}}$.

Next, we write Hölder’s inequality and Jensen’s inequality on time scales.

Lemma 1

(Dynamic Hölder’s Inequality [3]). Suppose $u,$$v\in \mathbb{T}$ with $u<v.$ Assume ${ϝ}^{*},$$g^{*}\in C{C}^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

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

(16)

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

Lemma 2

(Dynamic Jensen’s inequality [3]). Let $r^{*}$, ${\Im }^{*}\in R$ and $-\infty ⩽m^{*},n^{*}⩽\infty .$ If ${ϝ}^{*}\in C{C}^1(\mathbb{R},(m^{*},n^{*}))$ and $\Phi :(m^{*},n^{*})⟶\mathbb{R}$ is convex then

$$\varphi \left({\displaystyle \frac{{\int }_u^v{\int }_{\omega }^s{ϝ}^{*}(r^{*},{\Im }^{*}){\diamond }_{{\alpha }_1}{r}^{*}{\diamond }_{{\alpha }_2}{\Im }^{*}}{{\int }_u^v{\int }_{\omega }^s{\diamond }_{{\alpha }_1}{r}^{*}{\diamond }_{{\alpha }_2}{\Im }^{*}}}\right)⩽{\displaystyle \frac{{\int }_u^v{\int }_{\omega }^s\varphi \left({ϝ}^{*}(r^{*},{\Im }^{*})\right){\diamond }_{{\alpha }_1}{r}^{*}{\diamond }_{{\alpha }_2}{\Im }^{*}}{{\int }_u^v{\int }_{\omega }^s{\diamond }_{{\alpha }_1}{r}^{*}{\diamond }_{{\alpha }_2}{\Im }^{*}}}.$$

(17)

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

Definition 1.

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

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

(18)

In this paper, we extend some generalizations of the integral Hardy–Hilbert inequality to a general time scale using diamond alpha calculus. As special cases of our results, we will recover some dynamic integral and discrete inequalities known in the literature.

Now we are ready to state and prove our main results.

2. Main Results

First, we enlist the following assumptions for the proof of our main results:

(S₁)

$\mathbb{T}$ be time scales with ${\Im }_0,$${\vartheta }_{\ell },$${\varsigma }_{\ell },$$s_{\ell },$${\Im }_{\ell }\in \mathbb{T},$$(\ell =1,\cdots ,n).$

(S₂)

${ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })$ are non-negative, diamond-Alpha integrable functions defined on ${[{\Im }_0,{\vartheta }_{\ell })}_{\mathbb{T}}\times {[{\Im }_0,{\varsigma }_{\ell })}_{\mathbb{T}}$$(\ell =1,\cdots ,n).$

(S₃)

${ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })$ have partial ${\diamond }_{\alpha }$-derivatives ${ϝ}_{\ell }^{{\diamond }_{{\alpha }_1}}(s_{\ell },{\Im }_{\ell })$ and ${ϝ}_{\ell }^{{\diamond }_{{\alpha }_2}}(s_{\ell },{\Im }_{\ell })$ with respect $s_{\ell }$ and ${\Im }_{\ell }$, respectively.

(S₄)

All functions used in this section are integrable according to ${\diamond }_{\alpha }$ sense.

(S₅)

${ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })\in C^2\left({[{\Im }_0,{\vartheta }_{\ell })}_{\mathbb{T}}\times {[{\Im }_0,{\varsigma }_{\ell })}_{\mathbb{T}},[0,\infty )\right)$$(\ell =1,\cdots ,n).$

(S₆)

$p_{\ell }({\xi }_{\ell },{\tau }_{\ell })$ are n positive diamond-Alpha integrable functions defined for ${\xi }_{\ell }\in {({\Im }_0,s_{\ell })}_{\mathbb{T}},$${\tau }_{\ell }\in {({\Im }_0,{\Im }_{\ell })}_{\mathbb{T}}.$

(S₇)

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

(S₈)

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

(S₉)

${\vartheta }_{\ell }$ and ${\varsigma }_{\ell }$ are positive real numbers.

(S₁₀)

$s_{\ell }\in {[{\Im }_0,{\vartheta }_{\ell })}_{\mathbb{T}}$ and ${\Im }_{\ell }\in {[{\Im }_0,{\varsigma }_{\ell })}_{\mathbb{T}}.$

(S₁₁)

${ϝ}_{\ell }({\Im }_0,{\Im }_{\ell })={ϝ}_{\ell }(s_{\ell },{\Im }_0)=0,$$(\ell =1,\cdots ,n).$

(S₁₂)

${ϝ}_{\ell }^{{\diamond }_{{\alpha }_1}{\diamond }_{{\alpha }_2}}(s_{\ell },{\Im }_{\ell })={ϝ}_{\ell }^{{\diamond }_{{\alpha }_2}{\diamond }_{{\alpha }_1}}(s_{\ell },{\Im }_{\ell }).$

(S₁₃)

$P_{\ell }(s_{\ell },{\Im }_{\ell })={\int }_{{\Im }_0}^{{\Im }_{\ell }}{\int }_{{\Im }_0}^{s_{\ell }}{p}_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }.$

(S₁₄)

${\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })={\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{ϝ}_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }.$

(S₁₅)

$P_{\ell }(s_{\ell },{\Im }_{\ell })={\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{p}_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }.$

(S₁₆)

${\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })=\frac{1}{P_{\ell }({\xi }_{\ell },{\tau }_{\ell })}{\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{p}_{\ell }({\xi }_{\ell },{\tau }_{\ell }){ϝ}_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }.$

(S₁₇)

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

(S₁₈)

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

(S₁₉)

$h_{\ell }⩾2.$

(S₂₀)

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

(S₂₁)

$h_{\ell }⩾1.$

(S₂₂)

${ϝ}_{\ell }\left({\xi }_{\ell }\right)\in C^1{[{\Im }_0,{\vartheta }_{\ell }]}_{\mathbb{T}},$$(\ell =1,\cdots ,n).$

(S₂₃)

${\vartheta }_{\ell }$ is positive real number.

(S₂₄)

${\widetilde{ϝ}}_{\ell }\left(s_{\ell }\right)={\int }_{{\Im }_0}^{s_{\ell }}{ϝ}_{\ell }\left({\xi }_{\ell }\right){\diamond }_{\alpha }{\xi }_{\ell }.$

(S₂₅)

$s_{\ell }\in {[{\Im }_0,{\vartheta }_{\ell })}_{\mathbb{T}}.$

(S₂₆)

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

(S₂₇)

$P_{\ell }\left(s_{\ell }\right)={\int }_{{\Im }_0}^{s_{\ell }}{p}_{\ell }\left({\xi }_{\ell }\right){\diamond }_{\alpha }{\xi }_{\ell }.$

(S₂₈)

${\widetilde{ϝ}}_{\ell }\left(s_{\ell }\right)=\frac{1}{P_{\ell }\left(s_{\ell }\right)}{\int }_{{\Im }_0}^{s_{\ell }}p\left({\xi }_{\ell }\right)ϝ\left({\xi }_{\ell }\right){\diamond }_{\alpha }{\xi }_{\ell }.$

(S₂₉)

${ϝ}_{\ell }\left({\Im }_0\right)=0.$

Now, we are ready to state and prove the main results that extend several results in the literature.

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

$$\begin{array}{ccc}& & {\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\varsigma }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\displaystyle \prod _{\ell =1}^n}{\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\cdots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ \hfill & & ⩾L({\vartheta }_1{\varsigma }_1,\cdots ,{\vartheta }_n{\varsigma }_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}(\rho \left({\vartheta }_{\ell }\right)-s_{\ell })(\rho \left({\varsigma }_{\ell }\right)-{\Im }_{\ell }){\left(p_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })}{p_{\ell }(s_{\ell },{\Im }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}\hfill \end{array}$$

(19)

where

$$L({\vartheta }_1{\varsigma }_1,\cdots ,{\vartheta }_n{\varsigma }_n)=\prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{P_{\ell }(s_{\ell },{\Im }_{\ell })}\right)}^{{\gamma }_{\ell }}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{\frac{1}{{\gamma }_{\ell }}}.$$

Proof. 

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

$$\begin{array}{cc}\hfill {\Phi }_{\ell }\left({\tilde{F}}_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)\right)& ={\Phi }_{\ell }\left(\frac{p_{\ell }\left(s_{\ell },{\Im }_{\ell }\right){\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{p}_{\ell }\left({\xi }_{\ell },{\tau }_{\ell }\right)\left(\frac{{ϝ}_{\ell }\left({\xi }_{\ell },{\tau }_{\ell }\right)}{p_{\ell }\left({\xi }_{\ell },{\tau }_{\ell }\right)}\right){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }}{{\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{s_{\ell }}{p}_{\ell }\left({\xi }_{\ell },{\tau }_{\ell }\right){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }}\right)\hfill \\ \hfill & ⩾{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)\right){\Phi }_{\ell }\left(\frac{{\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{p}_{\ell }\left({\xi }_{\ell },{\tau }_{\ell }\right)\left(\frac{{ϝ}_{\ell }\left({\xi }_{\ell },{\tau }_{\ell }\right)}{p_{\ell }\left({\xi }_{\ell },{\tau }_{\ell }\right)}\right){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }}{{\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{p}_{\ell }\left({\xi }_{\ell },{\tau }_{\ell }\right){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }}\right).\hfill \end{array}$$

(20)

By using inverse Jensen dynamic inequality, we obtain that

$$\begin{array}{ccc}\hfill {\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })\right)& ⩾& \frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{P_{\ell }(s_{\ell },{\Im }_{\ell })}{\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{p}_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }({\xi }_{\ell },{\tau }_{\ell })}\right){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }.\hfill \end{array}$$

(21)

Applying inverse Hölder’s inequality on the right hand side of (21) with indices ${\gamma }_{\ell }$ and ${\beta }_{\ell },$ it is easy to observe that

$${\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })\right)⩾\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{P_{\ell }(s_{\ell },{\Im }_{\ell })}{\left[(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right]}^{\frac{1}{{\gamma }_{\ell }}}{\left({\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{\left(p_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }({\xi }_{\ell },{\tau }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.$$

(22)

By using the following inequality on the term ${\left[(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right]}^{\frac{1}{{\gamma }_{\ell }}},$

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

(23)

we get that

$$\begin{array}{c}\frac{{\displaystyle \prod _{\ell =1}^n}{\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)\right)}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}\left(s_{\ell }-{\Im }_0\right)\left({\Im }_{\ell }-{\Im }_0\right)\right)}^{\frac{1}{\gamma }}}\hfill \\ ⩾{\displaystyle \prod _{\ell =1}^n}\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)}{\left({\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{\left(p_{\ell }\left({\xi }_{\ell },{\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }\left({\tilde{\xi }}_{\ell },{\tau }_{\ell }\right)}{p_{\ell }\left({\tilde{\xi }}_{\ell },{\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\beta }_{\ell }}}{\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}\hfill \end{array}$$

(24)

Integrating both sides of (24) over $s_{\ell },$${\Im }_{\ell }$ from ${\Im }_0$ to ${\vartheta }_{\ell },$${\varsigma }_{\ell }$$(\ell =1,\cdots ,n),$ we obtain that

$$\begin{array}{ccc}\hfill & & {\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\varsigma }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\displaystyle \prod _{\ell =1}^n}{\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\cdots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n{\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{P_{\ell }(s_{\ell },{\Im }_{\ell })}{\left({\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{\left(p_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }({\xi }_{\ell },{\tau }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }.\hfill \end{array}$$

(25)

Applying inverse Hölder’s inequality on the right hand side of (25) with indices ${\gamma }_{\ell }$ and ${\beta }_{\ell },$ it is easy to observe that

$$\begin{array}{c}{\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\zeta }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\zeta }_n}\frac{{\displaystyle \prod _{\ell =1}^n}{\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)\right)}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}\left(s_{\ell }-{\Im }_0\right)\left({\Im }_{\ell }-{\Im }_0\right)\right)}^{\frac{1}{\gamma }}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\dots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ ⩾{\displaystyle \prod _{\ell =1}^n}{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\zeta }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)}\right)}^{{\gamma }_{\ell }}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{\frac{1}{{\gamma }_{\ell }}}\hfill \\ \times {\displaystyle \prod _{\ell =1}^n}{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\zeta }_{\ell }}\left({\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{\left(p_{\ell }\left({\tilde{\xi }}_{\ell },{\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }\left({\xi }_{\ell },{\tau }_{\ell }\right)}{p_{\ell }\left({\tilde{\zeta }}_{\ell },{\tau }_{\ell }\right)}\right)\right)}^{{\beta }_{\ell }}{\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }\right){\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(26)

Using Fubini’s theorem, we observe that

$$\begin{array}{ccc}\hfill & & {\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\varsigma }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\displaystyle \prod _{\ell =1}^n}{\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\cdots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ & & ⩾L({\vartheta }_1{\varsigma }_1,\cdots ,{\vartheta }_n{\varsigma }_n)\hfill \\ & & \times \prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}({\vartheta }_{\ell }-s_{\ell })({\varsigma }_{\ell }-{\Im }_{\ell }){\left(p_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })}{p_{\ell }(s_{\ell },{\Im }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

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

$$\begin{array}{ccc}& & {\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\varsigma }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\displaystyle \prod _{\ell =1}^n}{\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\cdots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ \hfill & & ⩾L({\vartheta }_1{\varsigma }_1,\cdots ,{\vartheta }_n{\varsigma }_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}(\rho \left({\vartheta }_{\ell }\right)-s_{\ell })(\rho \left({\varsigma }_{\ell }\right)-{\Im }_{\ell }){\left(p_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })}{p_{\ell }(s_{\ell },{\Im }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

This completes the proof. □

Remark 1.

In Theorem 1, if $\mathbb{T}=\mathbb{R}$, $\alpha =1$ we get the result due to Zhao et al. ([10], Theorem 2).

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

Corollary 1.

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

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

(27)

Then

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

where

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

Remark 2.

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

Corollary 2.

Let $S_{22},$$S_{23},$$S_{24},$$S_{26},$$S_{27}$ and $S_8$ be satisfied. Then for $S_{18},$$S_{20}$ and $S_{25}$ we have that

$$\begin{array}{ccc}\hfill & & {\int }_{{\Im }_0}^{{\vartheta }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\displaystyle \frac{{\displaystyle \prod _{\ell =1}^n}{\Phi }_{\ell }({\widetilde{ϝ}}_{\ell }\left(s_{\ell }\right))}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\diamond }_{\alpha }{s}_1\cdots {\diamond }_{\alpha }{s}_n\hfill \\ \hfill & & ⩾L^{*}({\vartheta }_1,\cdots ,{\vartheta }_n)\prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}(\rho \left({\vartheta }_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }\left(s_{\ell }\right)}{p_{\ell }\left(s_{\ell }\right)}\right)\right)}^{{\beta }_{\ell }}{\diamond }_{\alpha }{s}_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}\hfill \end{array}$$

(28)

where

$$L^{*}({\vartheta }_1,\cdots ,{\vartheta }_n)=\prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell }\right)}\right)}^{{\gamma }_{\ell }}{\diamond }_{\alpha }{s}_{\ell }\right)}^{\frac{1}{{\gamma }_{\ell }}}.$$

Corollary 3.

In Corollary 2, if we take $n=2,$${\beta }_{\ell }=\frac{1}{2}$ then the inequality (28) changes to

$$\begin{array}{c}\hfill {\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\vartheta }_2}{\displaystyle \frac{{\Phi }_1\left({\widetilde{ϝ}}_1\left(s_1\right)\right){\Phi }_2\left({\widetilde{ϝ}}_2\left(s_2\right)\right)}{{\left((s_1-{\Im }_0)+(s_2-{\Im }_0)\right)}^{-2}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{s}_2⩾L^{**}({\vartheta }_1,{\vartheta }_2){\left({\int }_{{\Im }_0}^{{\vartheta }_1}(\rho \left({\vartheta }_1\right)-s_1){\left(p_1\left(s_1\right)\Phi \left(\frac{{ϝ}_1\left(s_1\right)}{p_1\left(s_1\right)}\right)\right)}^2{\diamond }_{\alpha }{s}_1\right)}^{\frac{1}{2}}\\ \hfill \times {\left({\int }_{{\Im }_0}^{{\vartheta }_2}(\rho \left({\vartheta }_2\right)-s_2){\left(p_2\left(s_2\right)\mathsf{\Psi }\left(\frac{{ϝ}_2\left(s_2\right)}{p_2\left(s_2\right)}\right)\right)}^2{\diamond }_{\alpha }{s}_2\right)}^{\frac{1}{2}}\end{array}$$

(29)

where

$$L^{**}({\vartheta }_1,{\vartheta }_2)=4{\left({\int }_{{\Im }_0}^{{\vartheta }_2}{\left(\frac{{\Phi }_1\left(P_1\left(s_1\right)\right)}{P_1\left(s_1\right)}\right)}^{-1}{\diamond }_{\alpha }{s}_1\right)}^{-1}{\left({\int }_{{\Im }_0}^{{\vartheta }_2}{\left(\frac{{\Phi }_2\left(P_2\left(s_2\right)\right)}{P_2\left(s_2\right)}\right)}^{-1}{\diamond }_{\alpha }{s}_2\right)}^{-1}$$

Remark 3.

In Corollary 3, if we take $\mathbb{T}=\mathbb{R},$ then the inequality (29) changes to

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

(30)

where

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

This is an inverse of the inequality (6) which was proved by Pachpatte [7].

Corollary 4.

In Corollary 2, if we take ${\beta }_{\ell }=\frac{n-1}{n}$ the inequality (28) becomes

$$\begin{array}{ccc}& & {\int }_{{\Im }_0}^{{\vartheta }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\displaystyle \frac{{\displaystyle \prod _{\ell =1}^n}{\Phi }_{\ell }({\widetilde{ϝ}}_{\ell }\left(s_{\ell }\right))}{{\left({\displaystyle \sum _{\ell =1}^n}(s_{\ell }-{\Im }_0)\right)}^{\frac{-n}{n-1}}}}{\diamond }_{\alpha }{s}_1\cdots {\diamond }_{\alpha }{s}_n\hfill \\ \hfill & & ⩾L^{*}({\vartheta }_1,\cdots ,{\vartheta }_n)\prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}(\rho \left({\vartheta }_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }\left(s_{\ell }\right)}{p_{\ell }\left(s_{\ell }\right)}\right)\right)}^{\frac{n-1}{n}}{\diamond }_{\alpha }{s}_{\ell }\right)}^{\frac{n}{n-1}}\hfill \end{array}$$

where

$$L^{*}({\vartheta }_1,\cdots ,{\vartheta }_n)=n^{\frac{n}{n-1}}\prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell }\right)}\right)}^{-(n-1)}{\diamond }_{\alpha }{s}_{\ell }\right)}^{\frac{-1}{n-1}}.$$

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

$$\begin{array}{ccc}& & {\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\varsigma }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\displaystyle \prod _{\ell =1}^n}{P}_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\cdots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n{\left[({\vartheta }_{\ell }-{\Im }_0)({\varsigma }_{\ell }-{\Im }_0)\right]}^{\frac{1}{{\gamma }_{\ell }}}{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}(\rho \left({\vartheta }_{\ell }\right)-s_{\ell })(\rho \left({\varsigma }_{\ell }\right)-{\Im }_{\ell }){\left(p_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left({ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}\hfill \end{array}$$

(31)

Proof. 

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

$$\begin{array}{cc}\hfill {\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)\right)=& {\Phi }_{\ell }\left(\frac{1}{P_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)}{\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{p}_{\ell }\left({\xi }_{\ell },{\tau }_{\ell }\right){ϝ}_{\ell }\left({\xi }_{\ell },{\tau }_{\ell }\right){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }\right)\hfill \\ \hfill & ⩾\frac{1}{P_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)}{\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{p}_{\ell }\left({\sigma }_{\ell },{\tau }_{\ell }\right){\Phi }_{\ell }\left({ϝ}_{\ell }\left({\xi }_{\ell },{\tau }_{\ell }\right)\right){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }.\hfill \end{array}$$

(32)

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

$$\begin{array}{c}\hfill {\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })\right)⩾\frac{1}{P_{\ell }(s_{\ell },{\Im }_{\ell })}{\left[(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right]}^{\frac{1}{{\gamma }_{\ell }}}{\left({\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{\left(p_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({ϝ}_{\ell }({\xi }_{\ell },{\tau }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\end{array}$$

By using the inequality (23), on the term ${\left[(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right]}^{\frac{1}{{\gamma }_{\ell }}}$ we get that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{P_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}& ⩾& {\left({\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{\left(p_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({ϝ}_{\ell }({\xi }_{\ell },{\tau }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \\ \hfill \end{array}$$

(33)

Integrating both sides of (33) over $s_{\ell },$${\Im }_{\ell }$ from ${\Im }_0$ to ${\vartheta }_{\ell },$${\varsigma }_{\ell }$$(\ell =1,\cdots ,n),$ we get that

$$\begin{array}{c}{\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\varsigma }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\displaystyle \prod _{\ell =1}^n}{P}_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\cdots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ ⩾\prod _{\ell =1}^n{\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}{\left({\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{\left(p_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({ϝ}_{\ell }({\xi }_{\ell },{\tau }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\diamond }_{\alpha }{\sigma }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

Applying inverse Hölder’s inequality on the right hand side of (25) with indices ${\gamma }_{\ell }$ and ${\beta }_{\ell },$ it is easy to observe that

$$\begin{array}{ccc}& & {\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\varsigma }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\displaystyle \prod _{\ell =1}^n}{P}_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\cdots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n{\left[({\vartheta }_{\ell }-{\Im }_0)({\varsigma }_{\ell }-{\Im }_0)\right]}^{\frac{1}{{\gamma }_{\ell }}}{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}{\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{\left(p_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({ϝ}_{\ell }({\xi }_{\ell },{\tau }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(34)

By using Fubini’s theorem, we observe that

$$\begin{array}{ccc}& & {\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\varsigma }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\displaystyle \prod _{\ell =1}^n}{P}_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\cdots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ & & ⩾\prod _{\ell =1}^n{\left[({\vartheta }_{\ell }-{\Im }_0)({\varsigma }_{\ell }-{\Im }_0)\right]}^{\frac{1}{{\gamma }_{\ell }}}{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}({\vartheta }_{\ell }-s_{\ell })({\varsigma }_{\ell }-{\Im }_{\ell }){\left(p_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left({ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

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

$$\begin{array}{ccc}& & {\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\varsigma }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\displaystyle \prod _{\ell =1}^n}{P}_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\cdots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n{\left[({\vartheta }_{\ell }-{\Im }_0)({\varsigma }_{\ell }-{\Im }_0)\right]}^{\frac{1}{{\gamma }_{\ell }}}{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}(\rho \left({\vartheta }_{\ell }\right)-s_{\ell })(\rho \left({\varsigma }_{\ell }\right)-{\Im }_{\ell }){\left(p_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left({ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

This completes the proof. □

Remark 4.

In Theorem 2, if $\mathbb{T}=\mathbb{R}$, $\alpha =1$ we get the result due to Zhao et al. ([10], Theorem 3).

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

Corollary 5.

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

$$\begin{array}{c}{A}_{s_{\ell },{\Im }_{\ell },m_{s_{\ell }},m_{{\Im }_{\ell }}}=\frac{1}{P_{s_{\ell },{\Im }_{\ell },m_{s_{\ell }},m_{{\Im }_{\ell }}}}{\displaystyle \sum _{m_{\tilde{\zeta }\ell }}^{m_{s_{\ell }}}}{\displaystyle \sum _{m_{{\eta }_{\ell }}}^{m_{{\Im }_{\ell }}}}{a}_{s_{\ell },{\Im }_{\ell },m_{\tilde{\zeta }\ell },m_{{\eta }_{\ell }}}{p}_{s_{\ell },{\Im }_{\ell },m_{\tilde{\zeta }{\ell }^{\prime }},m_{{\eta }_{\ell }}}\hfill \\ P_{s_{\ell },{\Im }_{\ell },m_{s_{\ell }},m_{{\Im }_{\ell }}}={\displaystyle \sum _{m_{\tilde{\zeta }\ell }}^{m_{s_{\ell }}}}{\displaystyle \sum _{m_{\eta \ell }}^{m_{{\mathcal{S}}_{\ell }}}}{p}_{s_{\ell },{\Im }_{\ell },m_{{\tilde{\zeta }}_{\ell }},m_{{\eta }_{\ell }}}.\hfill \end{array}$$

(35)

Then

$$\begin{array}{ccc}& & {\displaystyle \sum _{m_{s_1}}^{k_{s_1}}}{\displaystyle \sum _{m_{{\Im }_1}}^{k_{{\Im }_1}}}\cdots {\displaystyle \sum _{m_{s_n}}^{k_{s_n}}}{\displaystyle \sum _{m_{{\Im }_n}}^{k_{{\Im }_n}}}\frac{{\displaystyle \prod _{\ell =1}^n}{P}_{s_{\ell },{\Im }_{\ell },m_{s_{\ell }},m_{{\Im }_{\ell }}}{\Phi }_{\ell }\left(A_{s_{\ell },{\Im }_{\ell },m_{s_{\ell }},m_{{\Im }_{\ell }}}\right)}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}\left(m_{s_{\ell }}{m}_{{\Im }_{\ell }}\right)\right)}^{\frac{1}{\gamma }}}\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n{\left(k_{s_{\ell }}{k}_{{\Im }_{\ell }}\right)}^{\frac{1}{{\gamma }_{\ell }}}{\left({\displaystyle \sum _{m_{s_{\ell }}}^{k_{s_{\ell }}}}{\displaystyle \sum _{m_{{\Im }_{\ell }}}^{k_{{\Im }_{\ell }}}}(k_{s_{\ell }}-(m_{s_{\ell }}-1))(k_{{\Im }_{\ell }}-(m_{{\Im }_{\ell }}-1)){\left(p_{s_{\ell },{\Im }_{\ell },m_{s_{\ell }},m_{{\Im }_{\ell }}}{\Phi }_{\ell }\left(a_{s_{\ell },{\Im }_{\ell },m_{s_{\ell }},m_{{\Im }_{\ell }}}\right)\right)}^{{\beta }_{\ell }}\right)}^{\frac{1}{{\beta }_{\ell }}},\hfill \end{array}$$

Remark 5.

Let ${ϝ}_{\ell }({\xi }_{\ell },{\tau }_{\ell }),$$p_{\ell }({\xi }_{\ell },{\tau }_{\ell }),$$P_{\ell }({\xi }_{\ell },{\tau }_{\ell })$ and

$${\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })=\frac{1}{P_{\ell }(s_{\ell },{\Im }_{\ell })}{\int }_{{\Im }_0}^{s_{\ell }}{\int }_0^{{\Im }_{\ell }}{p}_{\ell }({\xi }_{\ell },{\tau }_{\ell }){ϝ}_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }$$

changes to ${\widetilde{ϝ}}_{\ell }\left({\xi }_{\ell }\right),$$p_{\ell }\left({\xi }_{\ell }\right),$$P_{\ell }\left(s_{\ell }\right),$ and

$${\widetilde{ϝ}}_{\ell }\left(s_{\ell }\right)=\frac{1}{P_{\ell }\left(s_{\ell }\right)}{\int }_{{\Im }_0}^{s_{\ell }}{p}_{\ell }\left({\xi }_{\ell }\right){ϝ}_{\ell }\left({\xi }_{\ell }\right){\diamond }_{\alpha }{\xi }_{\ell },$$

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

Corollary 6.

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

$$\begin{array}{ccc}\hfill & & {\int }_{{\Im }_0}^{{\vartheta }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\displaystyle \frac{{\displaystyle \prod _{\ell =1}^n}{P}_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }({\widetilde{ϝ}}_{\ell }\left(s_{\ell }\right)}{{\left(\gamma {\displaystyle \sum _{\ell =1}^n}\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\diamond }_{\alpha }{s}_1\cdots {\diamond }_{\alpha }{s}_n\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n{({\vartheta }_{\ell }-{\Im }_0)}^{\frac{1}{{\gamma }_{\ell }}}{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}(\rho \left({\vartheta }_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left({ϝ}_{\ell }\left(s_{\ell }\right)\right)\right)}^{{\beta }_{\ell }}{\diamond }_{\alpha }{s}_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(36)

Corollary 7.

In Corollary 6, if we take $n=2,$${\beta }_{\ell }=\frac{1}{2}$ then the inequality (28) changes to

$$\begin{array}{ccc}\hfill & & {\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\vartheta }_2}{\displaystyle \frac{P_1\left(s_1\right)P_2\left(s_2\right){\Phi }_1\left({\widetilde{ϝ}}_1\left(s_1\right)\right){\Phi }_2\left({\widetilde{ϝ}}_2\left(s_2\right)\right)}{{\left((s_1-{\Im }_0)+(s_2-{\Im }_0)\right)}^{-2}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{s}_2⩾4{\left[({\vartheta }_1-{\Im }_0)({\vartheta }_1-{\Im }_0)\right]}^{-1}\hfill \\ \hfill & & \times {\left({\int }_{{\Im }_0}^{{\vartheta }_1}(\rho \left({\vartheta }_1\right)-s_1){\left(p_1\left(s_1\right){\Phi }_1\left({ϝ}_1\left(s_1\right)\right)\right)}^2{\diamond }_{\alpha }{s}_1\right)}^{\frac{1}{2}}{\left({\int }_{{\Im }_0}^{{\vartheta }_2}(\rho \left({\vartheta }_2\right)-s_2){\left(p_2\left(s_2\right){\Phi }_2\left({ϝ}_2\left(s_2\right)\right)\right)}^2{\diamond }_{\alpha }{s}_2\right)}^{\frac{1}{2}}.\hfill \end{array}$$

(37)

Remark 6.

In Corollary 7, if we take $\mathbb{T}=\mathbb{R},$ then the inequality (37) changes to

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

(38)

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

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, inequality (37) changes to

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

(39)

Remark 7.

In Corollary 8, if we take $\mathbb{T}=\mathbb{R},$ then inequality (39) changes to

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

This is an inverse inequality of the following inequality which was proven by Pachpatte [10].

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

Corollary 9.

In Corollary 6, if we take ${\beta }_{\ell }=\frac{n-1}{n}$$(\ell =1,\cdots ,n)$ the inequality (30)

$$\begin{array}{ccc}& & {\int }_{{\Im }_0}^{{\vartheta }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\displaystyle \frac{{\displaystyle \prod _{\ell =1}^n}{P}_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }({\widetilde{ϝ}}_{\ell }\left(s_{\ell }\right)}{{\left({\displaystyle \sum _{\ell =1}^n}(s_{\ell }-{\Im }_0)\right)}^{\frac{-n}{n-1}}}}{\diamond }_{\alpha }{s}_1\cdots {\diamond }_{\alpha }{s}_n\hfill \\ \hfill & & ⩾n^{\frac{n}{n-1}}\prod _{\ell =1}^n{({\vartheta }_{\ell }-{\Im }_0)}^{\frac{-1}{n-1}}{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}(\rho \left({\vartheta }_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left({ϝ}_{\ell }\left(s_{\ell }\right)\right)\right)}^{\frac{n-1}{n}}{\diamond }_{\alpha }{s}_{\ell }\right)}^{\frac{n}{n-1}}.\hfill \end{array}$$

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

$$\begin{array}{ccc}\hfill & & {\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\varsigma }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({\widetilde{ϝ}}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\frac{1}{{\gamma }^{\prime }}{\displaystyle \sum _{\ell =1}^n}{\gamma }_{\ell }^{\prime }(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{{\gamma }^{\prime }}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\cdots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ \hfill & & ⩾G({\vartheta }_1{\varsigma }_1,\cdots ,{\vartheta }_n{\varsigma }_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}(\rho \left({\vartheta }_{\ell }\right)-s_{\ell })(\rho \left({\varsigma }_{\ell }\right)-{\Im }_{\ell }){\left(p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\Im }_{\ell }\right){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }^{{\diamond }_{{\alpha }_2}{\diamond }_{{\alpha }_1}}(s_{\ell },{\Im }_{\ell })}{p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\Im }_{\ell }\right)}\right)\right)}^{\frac{1}{{\gamma }_{\ell }}}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{{\gamma }_{\ell }}\hfill \end{array}$$

(40)

where

$$G({\vartheta }_1{\varsigma }_1,\cdots ,{\vartheta }_n{\varsigma }_n)=\prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{P_{\ell }(s_{\ell },{\Im }_{\ell })}\right)}^{\frac{1}{{\gamma }_{\ell }^{\prime }}}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{{\gamma }_{\ell }^{\prime }}.$$

Proof. 

From the hypotheses of Theorem 3, we obtain

$${ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })={\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{ϝ}_{\ell }^{{\diamond }_{{\alpha }_2}{\diamond }_{{\alpha }_1}}({\xi }_{\ell },{\tau }_{\ell }){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }.$$

(41)

From (41) and $S_8$, it is easy to observe that

$$\begin{array}{c}{\Phi }_{\ell }\left({ϝ}_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)\right)={\Phi }_{\ell }\left(\frac{p_{\ell }\left(s_{\ell },{\Im }_{\ell }\right){\int }_{{\Im }_0}^{s_{\ell }}{\int }_{3_0}^{3_{\ell }}{p}_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)\left(\frac{{ϝ}_{\ell }{\diamond }_{a_2}{\diamond }_{a_1}\left({\overline{\zeta }}_{\ell },{\tau }_{\ell }\right)}{p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }}{{\int }_{3_0}^{s_{\ell }}{\int }_{3_0}^{3_{\ell }}{p}_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\diamond }_a{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }}\right)\hfill \\ \hfill ⩾{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)\right){\Phi }_{\ell }\left(\frac{{\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{3_{\ell }}{p}_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)\left(\frac{{ϝ}_{\ell }^{{\diamond }_{a_2}{\diamond }_{a_1}}\left({\xi }_{\ell },{\tau }_{\ell }\right)}{p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_a{\tau }_{\ell }}{{\int }_{3_0}^{s_{\ell }}{\int }_{{\Im }_0}^{3_{\ell }}{p}_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }}\right).\end{array}$$

(42)

By using inverse Jensen’s dynamic inequality, we get that

$$\begin{array}{ccc}\hfill {\Phi }_{\ell }\left({ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })\right)& ⩾& \frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{P_{\ell }(s_{\ell },{\Im }_{\ell })}{\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{p}_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }^{{\diamond }_{{\alpha }_2}{\diamond }_{{\alpha }_1}}({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right){\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }.\hfill \\ \hfill \end{array}$$

(43)

Applying inverse Hölder’s inequality on the right hand side of (43) with indices $1/{\gamma }_{\ell }$ and $1/{\gamma }_{\ell }^{\prime },$ we obtain

$$\begin{array}{cc}\hfill {\Phi }_{\ell }\left({ϝ}_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)\right)⩾& \frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)}{\left[\left(s_{\ell }-{\Im }_0\right)\left({\Im }_{\ell }-{\Im }_0\right)\right]}^{{\gamma }_{\ell }^{\prime }}\hfill \\ \hfill & \times {\left({\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{\left(p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{F_{\ell }^{{\diamond }_{a_2}{\diamond }_{{\alpha }_1}}\left({\xi }_{\ell },{\tau }_{\ell }\right)}{p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\gamma }_{\ell }}}{\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }\right)}^{{\gamma }_{\ell }}.\hfill \end{array}$$

(44)

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

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

(45)

We obtain that

$$\begin{array}{c}{\displaystyle \prod _{\ell =1}^n}{\Phi }_{\ell }\left({ϝ}_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)\right)⩾{\displaystyle \prod _{\ell =1}^n}\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell },{\Im }_{\ell }\right)}{\left(\frac{1}{{\gamma }^{\prime }}{\displaystyle \sum _{\ell =1}^n}{\gamma }_{\ell }^{\prime }\left(s_{\ell }-{\Im }_0\right)\left({\Im }_{\ell }-{\Im }_0\right)\right)}^{{\gamma }^{\prime }}\hfill \\ \hfill \times {\left({\int }_{{\Im }_0}^{s_{\ell }}{\int }_{3_0}^{3_{\ell }}{\left(p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{F_{\ell }^{{\diamond }_{{\alpha }_2}{\diamond }_{{\alpha }_1}}\left({\xi }_{\ell },{\tau }_{\ell }\right)}{p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\tau }_{\ell }}}{\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }\right)}^{{\gamma }_{\ell }}.\end{array}$$

(46)

From (46), we have that

$$\begin{array}{c}\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\frac{1}{{\gamma }^{\prime }}{\displaystyle \sum _{\ell =1}^n}{\gamma }_{\ell }^{\prime }(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{{\gamma }^{\prime }}}}\hfill \\ ⩾\prod _{\ell =1}^n\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{P_{\ell }(s_{\ell },{\Im }_{\ell })}{\left({\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{\left(p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }^{{\diamond }_{{\alpha }_2}{\diamond }_{{\alpha }_1}}({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\gamma }_{\ell }}}{\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }\right)}^{{\gamma }_{\ell }}.\hfill \end{array}$$

(47)

Integrating both sides of (47) over $s_{\ell },$${\Im }_{\ell }$ from ${\Im }_0$ to ${\vartheta }_{\ell },$${\varsigma }_{\ell }$$(\ell =1,\cdots ,n),$ we get that

$$\begin{array}{ccc}\hfill & \hfill & {\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\varsigma }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \prod _{\ell =1}^n}{\displaystyle \frac{{\Phi }_{\ell }\left({ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\frac{1}{{\gamma }^{\prime }}{\displaystyle \sum _{\ell =1}^n}{\gamma }_{\ell }^{\prime }(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{{\gamma }^{\prime }}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\cdots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ \hfill & \hfill & ⩾{\displaystyle \prod _{\ell =1}^n}{\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{P_{\ell }(s_{\ell },{\Im }_{\ell })}{\left({\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{\left(p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }^{{\diamond }_{{\alpha }_2}{\diamond }_{{\alpha }_1}}({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\gamma }_{\ell }}}{\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }\right)}^{{\gamma }_{\ell }}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }.\hfill \end{array}$$

(48)

Applying inverse Hölder’s inequality on the right hand side of (48) with indices $1/{\gamma }_{\ell }$ and $1/{\gamma }_{\ell }^{\prime },$ we obtain

$$\begin{array}{c}{\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\varsigma }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\frac{1}{{\gamma }^{\prime }}{\displaystyle \sum _{\ell =1}^n}{\gamma }_{\ell }^{\prime }(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{{\gamma }^{\prime }}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\cdots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ ⩾\prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{P_{\ell }(s_{\ell },{\Im }_{\ell })}\right)}^{\frac{1}{{\gamma }_{\ell }^{\prime }}}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{{\gamma }_{\ell }^{\prime }}\hfill \\ \times \prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}\left({\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{\left(p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }^{{\diamond }_{{\alpha }_2}{\diamond }_{{\alpha }_1}}({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\gamma }_{\ell }}}{\diamond }_{\alpha }{\xi }_{\ell }{\diamond }_{\alpha }{\tau }_{\ell }\right){\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{{\gamma }_{\ell }}.\hfill \\ \hfill \end{array}$$

(49)

By using Fubini’s theorem, we observe that

$$\begin{array}{ccc}& & {\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\varsigma }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\frac{1}{{\gamma }^{\prime }}{\displaystyle \sum _{\ell =1}^n}{\gamma }_{\ell }^{\prime }(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{{\gamma }^{\prime }}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\cdots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ \hfill & & ⩾G({\vartheta }_1{\varsigma }_1,\cdots ,{\vartheta }_n{\varsigma }_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}({\vartheta }_{\ell }-s_{\ell })({\varsigma }_{\ell }-{\Im }_{\ell }){\left(p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\Im }_{\ell }\right){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }^{{\diamond }_{{\alpha }_2}{\diamond }_{{\alpha }_1}}(s_{\ell },{\Im }_{\ell })}{p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\Im }_{\ell }\right)}\right)\right)}^{\frac{1}{{\gamma }_{\ell }}}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{{\gamma }_{\ell }}.\hfill \end{array}$$

(50)

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

$$\begin{array}{ccc}& & {\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\varsigma }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\frac{1}{{\gamma }^{\prime }}{\displaystyle \sum _{\ell =1}^n}{\gamma }_{\ell }^{\prime }(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{{\gamma }^{\prime }}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\cdots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ \hfill & & ⩾G({\vartheta }_1{\varsigma }_1,\cdots ,{\vartheta }_n{\varsigma }_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}(\rho \left({\vartheta }_{\ell }\right)-s_{\ell })(\rho \left({\varsigma }_{\ell }\right)-{\Im }_{\ell }){\left(p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\Im }_{\ell }\right){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }^{{\diamond }_{{\alpha }_2}{\diamond }_{{\alpha }_1}}(s_{\ell },{\Im }_{\ell })}{p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\Im }_{\ell }\right)}\right)\right)}^{\frac{1}{{\gamma }_{\ell }}}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{{\gamma }_{\ell }}.\hfill \end{array}$$

This completes the proof. □

Remark 8.

In Theorem 3, if $\mathbb{T}=\mathbb{Z}$, $\alpha =1$ we get the result due to Zhao et al. ([11], Theorem 1.5).

Remark 9.

In Theorem 3, if we take $\mathbb{T}=\mathbb{R}$, $\alpha =1$ we get the result due to Zhao et al. ([11], Theorem 1.6).

Remark 10.

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

$$\begin{array}{ccc}& & {\int }_{{\Im }_0}^{{\vartheta }_1}{\int }_{{\Im }_0}^{{\varsigma }_1}\cdots {\int }_{{\Im }_0}^{{\vartheta }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\frac{1}{{\gamma }^{\prime }}{\displaystyle \sum _{\ell =1}^n}{\gamma }_{\ell }^{\prime }(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{{\gamma }^{\prime }}}}{\diamond }_{\alpha }{s}_1{\diamond }_{\alpha }{\Im }_1\cdots {\diamond }_{\alpha }{s}_n{\diamond }_{\alpha }{\Im }_n\hfill \\ \hfill & & ⩽G^{*}({\vartheta }_1{\varsigma }_1,\cdots ,{\vartheta }_n{\varsigma }_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}(\sigma \left({\vartheta }_{\ell }\right)-s_{\ell })(\sigma \left({\varsigma }_{\ell }\right)-{\Im }_{\ell }){\left(p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\Im }_{\ell }\right){\Phi }_{\ell }\left(\frac{{ϝ}_{\ell }^{{\diamond }_{{\alpha }_2}{\diamond }_{{\alpha }_1}}(s_{\ell },{\Im }_{\ell })}{p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\Im }_{\ell }\right)}\right)\right)}^{\frac{1}{{\gamma }_{\ell }}}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{{\gamma }_{\ell }}.\hfill \end{array}$$

where

$$G^{*}({\vartheta }_1{\varsigma }_1,\cdots ,{\vartheta }_n{\varsigma }_n)=\frac{1}{{\left({\gamma }^{\prime }\right)}^{{\gamma }^{\prime }}}\prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{P_{\ell }(s_{\ell },{\Im }_{\ell })}\right)}^{\frac{1}{{\gamma }_{\ell }^{\prime }}}{\diamond }_{\alpha }{s}_{\ell }{\diamond }_{\alpha }{\Im }_{\ell }\right)}^{{\gamma }_{\ell }^{\prime }}.$$

This is an inverse form of the inequality (40).

Corollary 10.

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

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

(51)

where

$$G^{**}({\vartheta }_1,\cdots ,{\vartheta }_n)=\prod _{\ell =1}^n{\left({\int }_{{\Im }_0}^{{\vartheta }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell }\right)}\right)}^{\frac{1}{{\gamma }_{\ell }^{\prime }}}{\diamond }_{\alpha }{s}_{\ell }\right)}^{{\gamma }_{\ell }^{\prime }}.$$

Remark 11.

In Corollary 10, if we take $\mathbb{T}=\mathbb{Z},$$\alpha =1$ we get an inverse form of inequality (3), which was given by Handley et al.

Remark 12.

In Corollary 10, if we take $\mathbb{T}=\mathbb{R},$$\alpha =1$ we get an inverse form of inequality (4), which was given by Handley et al.

Remark 13.

In inequality (51) taking $n=2,$${\gamma }_1={\gamma }_2=2,$ then ${\gamma }_1^{\prime }={\gamma }_2^{\prime }=-1$, we have

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

(52)

where

$$D({\vartheta }_1,{\vartheta }_2)=4{\left({\int }_{{\Im }_0}^{{\vartheta }_1}{\left(\frac{{\Phi }_1\left(P_1\left(s_1\right)\right)}{P_2\left(s_1\right)}\right)}^{-1}{\diamond }_{\alpha }{s}_1\right)}^{-1}{\left({\int }_{{\Im }_0}^{{\vartheta }_2}{\left(\frac{{\Phi }_2\left(P_2\left(s_2\right)\right)}{P_2\left(s_2\right)}\right)}^{-1}{\diamond }_{\alpha }{s}_2\right)}^{-1}.$$

Remark 14.

If we take $\mathbb{T}=\mathbb{Z}$, $\alpha =1$ the inequality (52) is an inverse of inequality of (1), which was given by Pachpatte.

Remark 15.

If we take $\mathbb{T}=\mathbb{R}$, $\alpha =1$ the inequality (52) is an inverse of inequality of (2), which was given by Pachpatte.

3. Conclusions

In this work, by applying ${\diamond }_{\alpha }$ calculus, defined as a linear combination of the nabla and delta integrals, we introduced some novel results of Hardy–Hilbert-type inequalities on a general time-scale. Furthermore, we gave the multidimensional generalization for these inequalities to time scales. We also applied our inequalities to discrete and continuous calculus to obtain some new inequalities as special cases.