On Some Important Class of Dynamic Hilbert’s-Type Inequalities on Time Scales

Abstract

In this important work, we discuss some novel Hilbert-type dynamic inequalities on time scales. The inequalities investigated here generalize several known dynamic inequalities on time scales and unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proved by using some algebraic inequalities, Hölder inequality, and Jensen’s inequality on time scales.

1. Introduction

The celebrated Hardy-Hilbert’s integral inequality [1] is

$${\int }_0^{+\infty }{\int }_0^{+\infty }\frac{ϝ\left(\vartheta \right)g\left(\varsigma \right)}{\vartheta +\varsigma }\mathrm{d}\vartheta \mathrm{d}\varsigma \le \frac{\pi }{sin\frac{\pi }{p}}{\left[{\int }_0^{+\infty }{ϝ}^p\left(\vartheta \right)\mathrm{d}\vartheta \right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}{\left[{\int }_0^{+\infty }{g}^q\left(\varsigma \right)\mathrm{d}\varsigma \right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$q$}\right.},$$

(1)

where $p>1$, $q=p/p-1$. Putting $p=q=2$, we get:

$${\int }_0^{+\infty }{\int }_0^{+\infty }\frac{ϝ\left(\vartheta \right)g\left(\varsigma \right)}{\vartheta +\varsigma }\mathrm{d}\vartheta \mathrm{d}\varsigma \le \pi {\left[{\int }_0^{+\infty }{ϝ}^2\left(\vartheta \right)\mathrm{d}\vartheta \right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\left[{\int }_0^{+\infty }{g}^2\left(\varsigma \right)\mathrm{d}\varsigma \right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},$$

(2)

where the constants $\pi$ and $\frac{\pi }{sin\frac{\pi }{p}}$ are the best possible.

Over the past decade, a great number of dynamic Hilbert-type inequalities on time scales have been established by many researchers who were motivated by some applications; see the papers [2,3,4,5,6,7,8,9,10,11,12,13]. For more details on time scales calculus, see [14].

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

A time scale $\mathbb{T}$ is an arbitrary non-empty closed subset of the real number. We define forward and backward jump operators $\sigma :\mathbb{T}\to \mathbb{T}$ and $\rho :\mathbb{T}:\to \mathbb{T}$ respectively by

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

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

We will need the following important relations between calculus on time scales $\mathbb{T}$ and either continuous calculus on $\mathbb{R}$ or discrete calculus on $\mathbb{Z}$. Note that:

(i) 

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

$$\sigma \left(\iota \right)=\iota ,\mu \left(\iota \right)=0,{ϝ}^{\Delta }\left(\iota \right)={ϝ}^{\prime }\left(\iota \right),{\int }_a^bϝ\left(\iota \right)\Delta \iota ={\int }_a^bϝ\left(\iota \right)d\iota .$$

(3)

(ii) 

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

$$\sigma \left(\iota \right)=\iota +1,\mu \left(\iota \right)=1,{ϝ}^{\Delta }\left(\iota \right)=ϝ(\iota +1)-ϝ\left(\iota \right),{\int }_a^bϝ\left(\iota \right)\Delta \iota =\sum _{\iota =a}^{b-1}ϝ\left(\iota \right).$$

(4)

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

Lemma 1

([3]). Let $u,$ $v\in \mathbb{T}$ with $u<v.$ Assume ${ϝ}^{*}$, $g^{*}\in C{C}_{rd}^1({[u,v]}_{\mathbb{T}}\times {[u,v]}_{\mathbb{T}},\mathbb{R})$ be integrable functions and $\frac{1}{p}+\frac{1}{q}=1$ with $p>1$. Then

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

(5)

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

Lemma 2

([15]). Let $r^{*}$, ${\iota }^{*}\in \mathbb{R}$ and $-+\infty ⩽m^{*},n^{*}⩽+\infty .$ If ${ϝ}^{*}\in C{C}_{rd}^1(\mathbb{R},(m^{*},n^{*}))$, and $\varphi :(m^{*},n^{*})⟶\mathbb{R}$ is convex, then

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

(6)

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

Theorem 1

(Chain rule on time scales [14]). Assume $g:\mathbb{R}\to \mathbb{R}$ is continuous, $g:\mathbb{T}\to \mathbb{R}$ is $\Delta$-differentiable on ${\mathbb{T}}^{\kappa }$, and $ϝ:\mathbb{R}\to \mathbb{R}$ is continuously differentiable. Then there exists $c\in {[\iota ,\sigma \left(\iota \right)]}_{\mathbb{R}}$ with

$${(ϝ\circ g)}^{{\Delta }^{}}\left(\iota \right)={ϝ}^{\prime }\left(g\left(c\right)\right){\left(g\right)}^{{\Delta }^{}}\left(\iota \right).$$

(7)

Definition 1.

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

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

(8)

Next, we write Fubini’s theorem on time scales.

Lemma 3

(Fubini’s Thoerem, see [16]). Assume that $(\vartheta ,{\Sigma }_1,{\mu }_{\Delta })$ and $(\varsigma ,{\Sigma }_2,{\nu }_{\Delta })$ are two finite-dimensional time scales measure spaces. Moreover, suppose that $ϝ:\vartheta \times \varsigma \to \mathbb{R}$ is a delta integrable function and define the functions

$$\varphi \left(\varsigma \right)={\int }_{\vartheta }ϝ(\vartheta ,\varsigma )d{\mu }_{\Delta }\left(\vartheta \right),\varsigma \in \varsigma ,$$

and

$$\psi \left(\vartheta \right)={\int }_{\varsigma }ϝ(\vartheta ,\varsigma )d{\nu }_{\Delta }\left(\varsigma \right),\vartheta \in \vartheta .$$

Then ϕ is delta integrable on ς, and ψ is delta integrable on ϑ and

$${\int }_{\vartheta }d{\mu }_{\Delta }\left(\vartheta \right){\int }_{\varsigma }ϝ(\vartheta ,\varsigma )d{\nu }_{\Delta }\left(\varsigma \right)={\int }_{\varsigma }d{\nu }_{\Delta }\left(\varsigma \right){\int }_{\vartheta }ϝ(\vartheta ,\varsigma )d{\mu }_{\Delta }\left(\vartheta \right).$$

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

2. Main Results

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

$\left(S_1\right)$

$\mathbb{T}$ be time scales with ${\iota }_0,$ ${\upsilon }_{\ell },$ ${\varsigma }_{\ell },$ $s_{\ell },$ ${\iota }_{\ell }\in \mathbb{T},$$(\ell =1,\dots ,n).$

$\left(S_2\right)$

${\lambda }_{\ell }(s_{\ell },{\iota }_{\ell })$ are nonnegative, delta integrable functions defined on ${[{\iota }_0,{\upsilon }_{\ell })}_{\mathbb{T}}\times {[{\iota }_0,{\varsigma }_{\ell })}_{\mathbb{T}}$ $(\ell =1,\dots ,n).$

$\left(S_3\right)$

${\lambda }_{\ell }(s_{\ell },{\iota }_{\ell })$ have a partial $\Delta$- derivatives ${\lambda }_{\ell }^{{\Delta }_1}(s_{\ell },{\iota }_{\ell })$ and ${\lambda }_{\ell }^{{\Delta }_2}(s_{\ell },{\iota }_{\ell })$ with respect $s_{\ell }$ and ${\iota }_{\ell }$ respectively.

$\left(S_4\right)$

All functions used in this section are integrable according to $\Delta$ sense.

$\left(S_5\right)$

${\lambda }_{\ell }(s_{\ell },{\iota }_{\ell })\in C_{rd}^2\left({[{\iota }_0,{\upsilon }_{\ell })}_{\mathbb{T}}\times {[{\iota }_0,{\varsigma }_{\ell })}_{\mathbb{T}},[0,\infty )\right)$$(\ell =1,\dots ,n).$

$\left(S_6\right)$

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

$\left(S_7\right)$

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

$\left(S_8\right)$

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

$\left(S_9\right)$

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

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

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

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

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

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

${\lambda }_{\ell }^{{\Delta }_1{\Delta }_2}(s_{\ell },{\iota }_{\ell })={\lambda }_{\ell }^{{\Delta }_2{\Delta }_1}(s_{\ell },{\iota }_{\ell }).$

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

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

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

${\Lambda }_{\ell }(s_{\ell },{\iota }_{\ell })={\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\iota }_{\ell }}{\lambda }_{\ell }({\xi }_{\ell },{\tau }_{\ell })\Delta {\xi }_{\ell }\Delta {\tau }_{\ell }.$

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

$P_{\ell }(s_{\ell },{\iota }_{\ell })={\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\iota }_{\ell }}{p}_{\ell }({\xi }_{\ell },{\tau }_{\ell })\Delta {\xi }_{\ell }\Delta {\tau }_{\ell }.$

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

${\Lambda }_{\ell }(s_{\ell },{\iota }_{\ell })=\frac{1}{P_{\ell }({\xi }_{\ell },{\tau }_{\ell })}{\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\iota }_{\ell }}{p}_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\lambda }_{\ell }({\xi }_{\ell },{\tau }_{\ell })\Delta {\xi }_{\ell }\Delta {\tau }_{\ell }.$

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

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

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

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

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

$h_{\ell }⩾2.$

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

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

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

$h_{\ell }⩾1.$

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

${\lambda }_{\ell }\left({\xi }_{\ell }\right)\in C_{rd}^1{[{\iota }_0,{\upsilon }_{\ell }]}_{\mathbb{T}},$$(\ell =1,\dots ,n).$

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

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

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

${\Lambda }_{\ell }\left(s_{\ell }\right)={\int }_{{\iota }_0}^{s_{\ell }}{\lambda }_{\ell }\left({\xi }_{\ell }\right)\Delta {\xi }_{\ell }.$

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

$s_{\ell }\in {[{\iota }_0,{\upsilon }_{\ell })}_{\mathbb{T}}.$

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

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

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

$P_{\ell }\left(s_{\ell }\right)={\int }_{{\iota }_0}^{s_{\ell }}{p}_{\ell }\left({\xi }_{\ell }\right)\Delta {\xi }_{\ell }.$

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

${\Lambda }_{\ell }\left(s_{\ell }\right)=\frac{1}{P_{\ell }\left(s_{\ell }\right)}{\int }_{{\iota }_0}^{s_{\ell }}p\left({\xi }_{\ell }\right)\lambda \left({\xi }_{\ell }\right)\Delta {\xi }_{\ell }.$

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

${\lambda }_{\ell }\left({\iota }_0\right)=0.$

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

Theorem 2.

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

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

(9)

where

$$G({\upsilon }_1{\varsigma }_1,\dots ,{\upsilon }_n{\varsigma }_n)=\prod _{\ell =1}^n{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\int }_{{\iota }_0}^{{\varsigma }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\iota }_{\ell })\right)}{P_{\ell }(s_{\ell },{\iota }_{\ell })}\right)}^{\frac{1}{{\gamma }_{\ell }^{\prime }}}{\Delta }^{}{s}_{\ell }{\Delta }^{}{\iota }_{\ell }\right)}^{{\gamma }_{\ell }^{\prime }}.$$

Proof. 

From the hypotheses of Theorem 2, we obtain

$${\lambda }_{\ell }(s_{\ell },{\iota }_{\ell })={\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\Im }_{\ell }}{\lambda }_{\ell }^{{\Delta }_2^{}{\Delta }_1^{}}({\xi }_{\ell },{\tau }_{\ell }){\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }.$$

(10)

From (10) and $S_8$, it is easy to observe that

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

(11)

By using inverse Jensen’s dynamic inequality, we get

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

(12)

Applying inverse Hölder’s inequality on the right hand side of (12) with indices $1/{\gamma }_{\ell }$ and $1/{\gamma }_{\ell }^{\prime },$ we obtain

$$\begin{array}{ccc}\hfill {\Phi }_{\ell }\left({\lambda }_{\ell }(s_{\ell },{\iota }_{\ell })\right)& ⩾& \frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\iota }_{\ell })\right)}{P_{\ell }(s_{\ell },{\iota }_{\ell })}{\left[(s_{\ell }-{\iota }_0)({\iota }_{\ell }-{\iota }_0)\right]}^{{\gamma }_{\ell }^{\prime }}\hfill \\ & & \times {\left({\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\iota }_{\ell }}{\left(p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\lambda }_{\ell }^{{\Delta }_2^{}{\Delta }_1^{}}({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\gamma }_{\ell }}}{\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }\right)}^{{\gamma }_{\ell }}.\hfill \end{array}$$

(13)

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

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

(14)

we obtain that

$$\begin{array}{ccc}\hfill \prod _{\ell =1}^n{\Phi }_{\ell }\left({\lambda }_{\ell }(s_{\ell },{\iota }_{\ell })\right)& ⩾& \prod _{\ell =1}^n\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\iota }_{\ell })\right)}{P_{\ell }(s_{\ell },{\iota }_{\ell })}{\left(\frac{1}{{\gamma }^{\prime }}\sum _{\ell =1}^n{\gamma }_{\ell }^{\prime }(s_{\ell }-{\iota }_0)({\iota }_{\ell }-{\iota }_0)\right)}^{{\gamma }^{\prime }}\hfill \\ & & \times {\left({\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\iota }_{\ell }}{\left(p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\lambda }_{\ell }^{{\Delta }_2^{}{\Delta }_1^{}}({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\gamma }_{\ell }}}{\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }\right)}^{{\gamma }_{\ell }}.\hfill \end{array}$$

(15)

From (15), we obtain that

$$\begin{array}{ccc}\hfill & & \prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({\lambda }_{\ell }(s_{\ell },{\iota }_{\ell })\right)}{{\left(\frac{1}{{\gamma }^{\prime }}{\sum }_{\ell =1}^n{\gamma }_{\ell }^{\prime }(s_{\ell }-{\iota }_0)({\iota }_{\ell }-{\iota }_0)\right)}^{{\gamma }^{\prime }}}}\hfill \\ & & ⩾\prod _{\ell =1}^n\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\iota }_{\ell })\right)}{P_{\ell }(s_{\ell },{\iota }_{\ell })}{\left({\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\iota }_{\ell }}{\left(p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right){\Phi }_{\ell }\left(\frac{{\lambda }_{\ell }^{{\Delta }_2^{}{\Delta }_1^{}}({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }\left({\xi }_{\ell }\right)q_{\ell }\left({\tau }_{\ell }\right)}\right)\right)}^{\frac{1}{{\gamma }_{\ell }}}{\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }\right)}^{{\gamma }_{\ell }}.\hfill \end{array}$$

(16)

Integrating both sides of (16) over $s_{\ell },$ ${\iota }_{\ell }$ from ${\Im }_0$ to ${\upsilon }_{\ell },$ ${\varsigma }_{\ell }$ $(\ell =1,\dots ,n),$ we get

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

(17)

Applying inverse Hölder’s inequality on the right hand side of (17) with indices $1/{\gamma }_{\ell }$ and $1/{\gamma }_{\ell }^{\prime },$ we obtain

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

(18)

By using Fubini’s theorem, we observe that

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

(19)

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

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

This completes the proof. □

Remark 1.

In Theorem 2, if $\mathbb{T}=\mathbb{Z}$, we get the result due to Zhao et al. ([17], Theorem 1.5).

Remark 2.

In Theorem 2, if we take $\mathbb{T}=\mathbb{R}$, we get inequality due to Zhao et al. [17].

Remark 3.

Let $S_1,$ $S_2,$ $S_9,$ $S_{11},$ $S_7,$ $S_{13},$ $S_3$ and $S_{12}$ be satisfied and let ${\Phi }_{\ell },$ ${\gamma }_{\ell },$ ${\gamma }_{\ell }^{\prime },$ γ, and ${\gamma }^{\prime }$ be as in Theorem 2. Similar to proof of Theorem 2, we have

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

where

$$G^{*}({\upsilon }_1{\varsigma }_1,\dots ,{\upsilon }_n{\varsigma }_n)=\frac{1}{{\left({\gamma }^{\prime }\right)}^{{\gamma }^{\prime }}}\prod _{\ell =1}^n{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\int }_{{\iota }_0}^{{\varsigma }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\iota }_{\ell })\right)}{P_{\ell }(s_{\ell },{\iota }_{\ell })}\right)}^{\frac{1}{{\gamma }_{\ell }^{\prime }}}{\Delta }^{}{s}_{\ell }{\Delta }^{}{\iota }_{\ell }\right)}^{{\gamma }_{\ell }^{\prime }}.$$

This is an inverse form of the inequality (9).

Corollary 1.

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

$$\begin{array}{ccc}\hfill & & {\int }_{{\iota }_0}^{{\upsilon }_1}\dots {\int }_{{\iota }_0}^{{\upsilon }_n}\prod _{\ell =1}^n{\displaystyle \frac{{\Phi }_{\ell }\left({\lambda }_{\ell }\left(s_{\ell }\right)\right)}{{\left(\frac{1}{{\gamma }^{\prime }}{\sum }_{\ell =1}^n{\gamma }_{\ell }^{\prime }(s_{\ell }-{\iota }_0)\right)}^{{\gamma }^{\prime }}}}{\Delta }^{}{s}_n\dots {\Delta }^{}{s}_1\hfill \\ \hfill & & ⩾G^{**}({\upsilon }_1,\dots ,{\upsilon }_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}(\rho \left({\upsilon }_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left(\frac{{\lambda }_{\ell }^{{\Delta }^{}}\left(s_{\ell }\right)}{p_{\ell }\left(s_{\ell }\right)}\right)\right)}^{\frac{1}{{\gamma }_{\ell }}}{\Delta }^{}{s}_{\ell }\right)}^{{\gamma }_{\ell }},\hfill \end{array}$$

(20)

where

$$G^{**}({\upsilon }_1,\dots ,{\upsilon }_n)=\prod _{\ell =1}^n{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell }\right)}\right)}^{\frac{1}{{\gamma }_{\ell }^{\prime }}}{\Delta }^{}{s}_{\ell }\right)}^{{\gamma }_{\ell }^{\prime }}.$$

Remark 4.

In Corollary 1, if we take $\mathbb{T}=\mathbb{Z},$ we get an inverse form of inequality due to Handley [18].

Remark 5.

In Corollary 1, if we take $\mathbb{T}=\mathbb{R},$ we get an inverse form of inequality due to Handley [18].

Remark 6.

In inequality (20) taking $n=2,$ ${\gamma }_1={\gamma }_2=2,$ then ${\gamma }_1^{\prime }={\gamma }_2^{\prime }=-1,$ we have

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

(21)

where

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

Remark 7.

If we take $\mathbb{T}=\mathbb{Z}$, the inequality (21) is an inverse of inequality due to Pachpatte [19].

Remark 8.

If we take $\mathbb{T}=\mathbb{R}$, the inequality (21) is an inverse of inequality due to Pachpatte [19].

Theorem 3.

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

$$\begin{array}{ccc}& & {\int }_{{\iota }_0}^{{\upsilon }_1}{\int }_{{\iota }_0}^{{\varsigma }_1}\dots {\int }_{{\Im }_0}^{{\upsilon }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Lambda }_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\sum }_{\ell =1}^n\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\Delta }^{}{s}_n{\Delta }^{}{\Im }_n\dots {\Delta }^{}{s}_1{\Delta }^{}{\Im }_1\hfill \\ \hfill & & ⩾L({\upsilon }_1{\varsigma }_1,\dots ,{\upsilon }_n{\varsigma }_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\int }_{{\iota }_0}^{{\varsigma }_{\ell }}(\rho \left({\upsilon }_{\ell }\right)-s_{\ell })(\rho \left({\varsigma }_{\ell }\right)-{\iota }_{\ell }){\left(p_{\ell }(s_{\ell },{\iota }_{\ell }){\Phi }_{\ell }\left(\frac{{\lambda }_{\ell }(s_{\ell },{\iota }_{\ell })}{p_{\ell }(s_{\ell },{\iota }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\Delta }^{}{s}_{\ell }{\Delta }^{}{\iota }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}\hfill \end{array}$$

(22)

where

$$L({\upsilon }_1{\varsigma }_1,\dots ,{\upsilon }_n{\varsigma }_n)=\prod _{\ell =1}^n{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\int }_{{\iota }_0}^{{\varsigma }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\iota }_{\ell })\right)}{P_{\ell }(s_{\ell },{\iota }_{\ell })}\right)}^{{\gamma }_{\ell }}{\Delta }^{}{s}_{\ell }{\Delta }^{}{\iota }_{\ell }\right)}^{\frac{1}{{\gamma }_{\ell }}}.$$

Proof. 

From the hypotheses of Theorem 3, $S_{14},$ $S_{15},$ and $S_8,$ it is easy to observe that

$$\begin{array}{ccc}\hfill {\Phi }_{\ell }\left({\Lambda }_{\ell }(s_{\ell },{\iota }_{\ell })\right)& =& {\Phi }_{\ell }\left({\displaystyle \frac{P_{\ell }(s_{\ell },{\iota }_{\ell }){\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\iota }_{\ell }}{p}_{\ell }({\xi }_{\ell },{\tau }_{\ell })\left(\frac{{\lambda }_{\ell }({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }({\xi }_{\ell },{\tau }_{\ell })}\right){\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }}{{\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\Im }_{\ell }}{p}_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }}}\right)\hfill \\ & ⩾& {\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\iota }_{\ell })\right){\Phi }_{\ell }\left({\displaystyle \frac{{\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\iota }_{\ell }}{p}_{\ell }({\xi }_{\ell },{\tau }_{\ell })\left(\frac{{\lambda }_{\ell }({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }({\xi }_{\ell },{\tau }_{\ell })}\right){\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }}{{\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\Im }_{\ell }}{p}_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }}}\right).\hfill \end{array}$$

(23)

By using inverse Jensen dynamic inequality, we obtain that

$$\begin{array}{ccc}\hfill {\Phi }_{\ell }\left({\Lambda }_{\ell }(s_{\ell },{\iota }_{\ell })\right)& ⩾& \frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\iota }_{\ell })\right)}{P_{\ell }(s_{\ell },{\iota }_{\ell })}{\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{p}_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{\lambda }_{\ell }({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }({\xi }_{\ell },{\tau }_{\ell })}\right){\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }.\hfill \end{array}$$

(24)

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

$$\begin{array}{ccc}\hfill {\Phi }_{\ell }\left({\Lambda }_{\ell }(s_{\ell },{\iota }_{\ell })\right)& ⩾& \frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\iota }_{\ell })\right)}{P_{\ell }(s_{\ell },{\iota }_{\ell })}{\left[(s_{\ell }-{\iota }_0)({\iota }_{\ell }-{\iota }_0)\right]}^{\frac{1}{{\gamma }_{\ell }}}{\left({\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\iota }_{\ell }}{\left(p_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{\lambda }_{\ell }({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }({\xi }_{\ell },{\tau }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(25)

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

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

(26)

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Lambda }_{\ell }(s_{\ell },{\iota }_{\ell })\right)}{{\left(\gamma {\sum }_{\ell =1}^n\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\iota }_0)({\iota }_{\ell }-{\iota }_0)\right)}^{\frac{1}{\gamma }}}}\hfill \\ & & ⩾\prod _{\ell =1}^n\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\iota }_{\ell })\right)}{P_{\ell }(s_{\ell },{\iota }_{\ell })}{\left({\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\iota }_{\ell }}{\left(p_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{\lambda }_{\ell }({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }({\xi }_{\ell },{\tau }_{\ell })}\right)\right)}^{\frac{1}{{\beta }_{\ell }}}{\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(27)

Integrating both sides of (27) over $s_{\ell },$ ${\iota }_{\ell }$ from ${\Im }_0$ to ${\upsilon }_{\ell },$${\varsigma }_{\ell }$ $(\ell =1,\dots ,n),$ we obtain that

$$\begin{array}{ccc}\hfill & & {\int }_{{\iota }_0}^{{\upsilon }_1}{\int }_{{\iota }_0}^{{\varsigma }_1}\dots {\int }_{{\Im }_0}^{{\upsilon }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Lambda }_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\sum }_{\ell =1}^n\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\Delta }^{}{s}_n{\Delta }^{}{\Im }_n\dots {\Delta }^{}{s}_1{\Delta }^{}{\Im }_1\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n{\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\int }_{{\iota }_0}^{{\varsigma }_{\ell }}\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\iota }_{\ell })\right)}{P_{\ell }(s_{\ell },{\iota }_{\ell })}{\left({\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\iota }_{\ell }}{\left(p_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{\lambda }_{\ell }({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }({\xi }_{\ell },{\tau }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}{\Delta }^{}{s}_{\ell }{\Delta }^{}{\iota }_{\ell }.\hfill \end{array}$$

(28)

Applying inverse Hölder’s inequality on the right hand side of (28) with indices ${\gamma }_{\ell }$ and ${\beta }_{\ell },$ it is easy to observe that

$$\begin{array}{ccc}\hfill & & {\int }_{{\iota }_0}^{{\upsilon }_1}{\int }_{{\iota }_0}^{{\varsigma }_1}\dots {\int }_{{\Im }_0}^{{\upsilon }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Lambda }_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\sum }_{\ell =1}^n\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\Delta }^{}{s}_n{\Delta }^{}{\Im }_n\dots {\Delta }^{}{s}_1{\Delta }^{}{\Im }_1\hfill \\ & & ⩾\prod _{\ell =1}^n{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\int }_{{\iota }_0}^{{\varsigma }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }(s_{\ell },{\iota }_{\ell })\right)}{P_{\ell }(s_{\ell },{\iota }_{\ell })}\right)}^{{\gamma }_{\ell }}{\Delta }^{}{s}_{\ell }{\Delta }^{}{\iota }_{\ell }\right)}^{\frac{1}{{\gamma }_{\ell }}}\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\int }_{{\iota }_0}^{{\varsigma }_{\ell }}\left({\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\iota }_{\ell }}{\left(p_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left(\frac{{\lambda }_{\ell }({\xi }_{\ell },{\tau }_{\ell })}{p_{\ell }({\xi }_{\ell },{\tau }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }\right){\Delta }^{}{s}_{\ell }{\Delta }^{}{\iota }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(29)

Using Fubini’s theorem, we observe that

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

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

$$\begin{array}{ccc}& & {\int }_{{\iota }_0}^{{\upsilon }_1}{\int }_{{\iota }_0}^{{\varsigma }_1}\dots {\int }_{{\Im }_0}^{{\upsilon }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }\left({\Lambda }_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\sum }_{\ell =1}^n\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\Delta }^{}{s}_n{\Delta }^{}{\Im }_n\dots {\Delta }^{}{s}_1{\Delta }^{}{\Im }_1\hfill \\ \hfill & & ⩾L({\upsilon }_1{\varsigma }_1,\dots ,{\upsilon }_n{\varsigma }_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\int }_{{\iota }_0}^{{\varsigma }_{\ell }}(\rho \left({\upsilon }_{\ell }\right)-s_{\ell })(\rho \left({\varsigma }_{\ell }\right)-{\iota }_{\ell }){\left(p_{\ell }(s_{\ell },{\iota }_{\ell }){\Phi }_{\ell }\left(\frac{{\lambda }_{\ell }(s_{\ell },{\iota }_{\ell })}{p_{\ell }(s_{\ell },{\iota }_{\ell })}\right)\right)}^{{\beta }_{\ell }}{\Delta }^{}{s}_{\ell }{\Delta }^{}{\iota }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

This completes the proof. □

Remark 9.

In Theorem 3, if $\mathbb{T}=\mathbb{R}$, we get the result due to Zhao et al. [20] (Theorem 2).

As a special case of Theorem 3, when $\mathbb{T}=\mathbb{Z}$, we have $\rho \left(n\right)=n-1,$ and we get the following result.

Corollary 2.

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

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

(30)

Then

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

where

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

Remark 10.

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

Corollary 3.

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

$$\begin{array}{ccc}& & {\int }_{{\iota }_0}^{{\upsilon }_1}\dots {\int }_{{\iota }_0}^{{\upsilon }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }({\Lambda }_{\ell }\left(s_{\ell }\right)}{{\left(\gamma {\sum }_{\ell =1}^n\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\iota }_0)\right)}^{\frac{1}{\gamma }}}}{\Delta }^{}{s}_n\dots {\Delta }^{}{s}_1\hfill \\ \hfill & & ⩾L^{*}({\upsilon }_1,\dots ,{\upsilon }_n)\prod _{\ell =1}^n{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}(\rho \left({\upsilon }_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left(\frac{{\lambda }_{\ell }\left(s_{\ell }\right)}{p_{\ell }\left(s_{\ell }\right)}\right)\right)}^{{\beta }_{\ell }}{\Delta }^{}{s}_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}\hfill \end{array}$$

(31)

where

$$L^{*}({\upsilon }_1,\dots ,{\upsilon }_n)=\prod _{\ell =1}^n{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell }\right)}\right)}^{{\gamma }_{\ell }}{\Delta }^{}{s}_{\ell }\right)}^{\frac{1}{{\gamma }_{\ell }}}.$$

Corollary 4.

In Corollary 3, if we take $n=2,$ ${\beta }_{\ell }=\frac{1}{2}$, then the inequality (31) changes to

$$\begin{array}{c}\hfill {\int }_{{\iota }_0}^{{\upsilon }_1}{\int }_{{\iota }_0}^{{\upsilon }_2}{\displaystyle \frac{{\Phi }_1\left({\Lambda }_1\left(s_1\right)\right){\Phi }_2\left({\Lambda }_2\left(s_2\right)\right)}{{\left((s_1-{\iota }_0)+(s_2-{\iota }_0)\right)}^{-2}}}{\Delta }^{}{s}_1{\Delta }^{}{s}_2⩾L^{**}({\upsilon }_1,{\upsilon }_2){\left({\int }_{{\iota }_0}^{{\upsilon }_1}(\rho \left({\upsilon }_1\right)-s_1){\left(p_1\left(s_1\right)\Phi \left(\frac{{\lambda }_1\left(s_1\right)}{p_1\left(s_1\right)}\right)\right)}^2{\Delta }^{}{s}_1\right)}^{\frac{1}{2}}\\ \hfill \times {\left({\int }_{{\iota }_0}^{{\upsilon }_2}(\rho \left({\upsilon }_2\right)-s_2){\left(p_2\left(s_2\right)\Psi \left(\frac{{\lambda }_2\left(s_2\right)}{p_2\left(s_2\right)}\right)\right)}^2{\Delta }^{}{s}_2\right)}^{\frac{1}{2}}.\end{array}$$

(32)

where

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

Remark 11.

In Corollary 4, if we take $\mathbb{T}=\mathbb{R},$ then the inequality (32) changes to

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

(33)

where

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

This is an inverse of the inequality due to Pachpatte [21].

Corollary 5.

In Corollary 3, if we take ${\beta }_{\ell }=\frac{n-1}{n}$, the inequality (31) becomes

$$\begin{array}{ccc}& & {\int }_{{\iota }_0}^{{\upsilon }_1}\dots {\int }_{{\iota }_0}^{{\upsilon }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{\Phi }_{\ell }({\lambda }_{\ell }\left(s_{\ell }\right)}{{\left({\sum }_{\ell =1}^n(s_{\ell }-{\iota }_0)\right)}^{\frac{-n}{n-1}}}}{\Delta }^{}{s}_1\dots {\Delta }^{}{s}_n\hfill \\ \hfill & & ⩾L^{*}({\upsilon }_1,\dots ,{\upsilon }_n)\prod _{\ell =1}^n{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}(\rho \left({\upsilon }_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left(\frac{{\lambda }_{\ell }\left(s_{\ell }\right)}{p_{\ell }\left(s_{\ell }\right)}\right)\right)}^{\frac{n-1}{n}}{\Delta }^{}{s}_{\ell }\right)}^{\frac{n}{n-1}}\hfill \end{array}$$

where

$$L^{*}({\upsilon }_1,\dots ,{\upsilon }_n)=n^{\frac{n}{n-1}}\prod _{\ell =1}^n{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\left(\frac{{\Phi }_{\ell }\left(P_{\ell }\left(s_{\ell }\right)\right)}{P_{\ell }\left(s_{\ell }\right)}\right)}^{-(n-1)}{\Delta }^{}{s}_{\ell }\right)}^{\frac{-1}{n-1}}.$$

Theorem 4.

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

$$\begin{array}{ccc}& & {\int }_{{\iota }_0}^{{\upsilon }_1}{\int }_{{\iota }_0}^{{\varsigma }_1}\dots {\int }_{{\Im }_0}^{{\upsilon }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left({\Lambda }_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\sum }_{\ell =1}^n\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\Delta }^{}{s}_n{\Delta }^{}{\Im }_n\dots {\Delta }^{}{s}_1{\Delta }^{}{\Im }_1\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n{\left[({\upsilon }_{\ell }-{\iota }_0)({\varsigma }_{\ell }-{\iota }_0)\right]}^{\frac{1}{{\gamma }_{\ell }}}{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\int }_{{\iota }_0}^{{\varsigma }_{\ell }}(\rho \left({\upsilon }_{\ell }\right)-s_{\ell })(\rho \left({\varsigma }_{\ell }\right)-{\iota }_{\ell }){\left(p_{\ell }(s_{\ell },{\iota }_{\ell }){\Phi }_{\ell }\left({\lambda }_{\ell }(s_{\ell },{\iota }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\Delta }^{}{s}_{\ell }{\Delta }^{}{\iota }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(34)

Proof. 

From the hypotheses of Theorem 4, and by using inverse Jensen dynamic inequality, we have

$$\begin{array}{ccc}\hfill {\Phi }_{\ell }\left({\Lambda }_{\ell }(s_{\ell },{\iota }_{\ell })\right)& =& {\Phi }_{\ell }\left(\frac{1}{P_{\ell }(s_{\ell },{\Im }_{\ell })}{\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{p}_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\lambda }_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }\right)\hfill \\ & & ⩾\frac{1}{P_{\ell }(s_{\ell },{\iota }_{\ell })}{\int }_{{\iota }_0}^{s_{\ell }}{\int }_{{\iota }_0}^{{\iota }_{\ell }}{p}_{\ell }({\sigma }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({\lambda }_{\ell }({\xi }_{\ell },{\tau }_{\ell })\right){\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }.\hfill \end{array}$$

(35)

Applying inverse Hölder’s inequality on the right hand side of (35) with indices ${\gamma }_{\ell }$ and ${\beta }_{\ell },$ it is easy to observe that

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

By using the inequality (26) on the term ${\left[(s_{\ell }-{\iota }_0)({\iota }_{\ell }-{\iota }_0)\right]}^{\frac{1}{{\gamma }_{\ell }}}$, we get

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

(36)

Integrating both sides of (36) over $s_{\ell },$ ${\iota }_{\ell }$ from ${\iota }_0$ to ${\upsilon }_{\ell },$${\varsigma }_{\ell }$ $(\ell =1,\dots ,n),$ we get

$$\begin{array}{ccc}\hfill & & {\int }_{{\iota }_0}^{{\upsilon }_1}{\int }_{{\iota }_0}^{{\varsigma }_1}\dots {\int }_{{\Im }_0}^{{\upsilon }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left({\Lambda }_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\sum }_{\ell =1}^n\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\Delta }^{}{s}_n{\Delta }^{}{\Im }_n\dots {\Delta }^{}{s}_1{\Delta }^{}{\Im }_1\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n{\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}{\left({\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{\left(p_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({\lambda }_{\ell }({\xi }_{\ell },{\tau }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\Delta }^{}{\sigma }_{\ell }{\Delta }^{}{\tau }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

Applying inverse Hölder’s inequality on the right hand side of (37) with indices ${\gamma }_{\ell }$ and ${\beta }_{\ell },$ it is easy to observe that

$$\begin{array}{ccc}& & {\int }_{{\iota }_0}^{{\upsilon }_1}{\int }_{{\iota }_0}^{{\varsigma }_1}\dots {\int }_{{\Im }_0}^{{\upsilon }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left({\Lambda }_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\sum }_{\ell =1}^n\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\Delta }^{}{s}_1{\Delta }^{}{\Im }_1\dots {\Delta }^{}{s}_n{\Delta }^{}{\Im }_n\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n{\left[({\upsilon }_{\ell }-{\iota }_0)({\varsigma }_{\ell }-{\iota }_0)\right]}^{\frac{1}{{\gamma }_{\ell }}}{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\int }_{{\Im }_0}^{{\varsigma }_{\ell }}{\int }_{{\Im }_0}^{s_{\ell }}{\int }_{{\Im }_0}^{{\Im }_{\ell }}{\left(p_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Phi }_{\ell }\left({\lambda }_{\ell }({\xi }_{\ell },{\tau }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }{\Delta }^{}{s}_{\ell }{\Delta }^{}{\Im }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(37)

By using Fubini’s theorem, we observe that

$$\begin{array}{ccc}& & {\int }_{{\iota }_0}^{{\upsilon }_1}{\int }_{{\iota }_0}^{{\varsigma }_1}\dots {\int }_{{\Im }_0}^{{\upsilon }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left({\Lambda }_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\sum }_{\ell =1}^n\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\Delta }^{}{s}_n{\Delta }^{}{\Im }_n\dots {\Delta }^{}{s}_1{\Delta }^{}{\Im }_1\hfill \\ & & ⩾\prod _{\ell =1}^n{\left[({\upsilon }_{\ell }-{\iota }_0)({\varsigma }_{\ell }-{\iota }_0)\right]}^{\frac{1}{{\gamma }_{\ell }}}{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\int }_{{\iota }_0}^{{\varsigma }_{\ell }}({\upsilon }_{\ell }-s_{\ell })({\varsigma }_{\ell }-{\Im }_{\ell }){\left(p_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left({\lambda }_{\ell }(s_{\ell },{\Im }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\Delta }^{}{s}_{\ell }{s}_{\ell }{\Delta }^{}{\Im }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

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

$$\begin{array}{ccc}& & {\int }_{{\iota }_0}^{{\upsilon }_1}{\int }_{{\iota }_0}^{{\varsigma }_1}\dots {\int }_{{\Im }_0}^{{\upsilon }_n}{\int }_{{\Im }_0}^{{\varsigma }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }(s_{\ell },{\Im }_{\ell }){\Phi }_{\ell }\left({\Lambda }_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\gamma {\sum }_{\ell =1}^n\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\Im }_0)({\Im }_{\ell }-{\Im }_0)\right)}^{\frac{1}{\gamma }}}}{\Delta }^{}{s}_n{\Delta }^{}{\Im }_n\dots {\Delta }^{}{s}_1{\Delta }^{}{\Im }_1\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n{\left[({\upsilon }_{\ell }-{\iota }_0)({\varsigma }_{\ell }-{\iota }_0)\right]}^{\frac{1}{{\gamma }_{\ell }}}{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}{\int }_{{\iota }_0}^{{\varsigma }_{\ell }}(\rho \left({\upsilon }_{\ell }\right)-s_{\ell })(\rho \left({\varsigma }_{\ell }\right)-{\iota }_{\ell }){\left(p_{\ell }(s_{\ell },{\iota }_{\ell }){\Phi }_{\ell }\left({\lambda }_{\ell }(s_{\ell },{\iota }_{\ell })\right)\right)}^{{\beta }_{\ell }}{\Delta }^{}{s}_{\ell }{\Delta }^{}{\iota }_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

This completes the proof. □

Remark 12.

In Theorem 4, if $\mathbb{T}=\mathbb{R}$, we get the result due to Zhao et al. [20] (Theorem 3).

As a special case of Theorem 4, when $\mathbb{T}=\mathbb{Z}$, we have $\rho \left(n\right)=n-1,$ and we get the following result.

Corollary 6.

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

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

(38)

Then

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

Remark 13.

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

$${\lambda }_{\ell }(s_{\ell },{\iota }_{\ell })=\frac{1}{P_{\ell }(s_{\ell },{\iota }_{\ell })}{\int }_{{\iota }_0}^{s_{\ell }}{\int }_0^{{\iota }_{\ell }}{p}_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Lambda }_{\ell }({\xi }_{\ell },{\tau }_{\ell }){\Delta }^{}{\xi }_{\ell }{\Delta }^{}{\tau }_{\ell }$$

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

$${\lambda }_{\ell }\left(s_{\ell }\right)=\frac{1}{P_{\ell }\left(s_{\ell }\right)}{\int }_{{\iota }_0}^{s_{\ell }}{p}_{\ell }\left({\xi }_{\ell }\right){\lambda }_{\ell }\left({\xi }_{\ell }\right){\Delta }^{}{\xi }_{\ell }.$$

respectively, and with suitable changes, we have the following result:

Corollary 7.

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

$$\begin{array}{ccc}\hfill & & {\int }_{{\iota }_0}^{{\upsilon }_1}\dots {\int }_{{\iota }_0}^{{\upsilon }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }({\Lambda }_{\ell }\left(s_{\ell }\right)}{{\left(\gamma {\sum }_{\ell =1}^n\frac{1}{{\gamma }_{\ell }}(s_{\ell }-{\iota }_0)\right)}^{\frac{1}{\gamma }}}}{\Delta }^{}{s}_n\dots {\Delta }^{}{s}_1\hfill \\ \hfill & & ⩾\prod _{\ell =1}^n{({\upsilon }_{\ell }-{\iota }_0)}^{\frac{1}{{\gamma }_{\ell }}}{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}(\rho \left({\upsilon }_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left({\lambda }_{\ell }\left(s_{\ell }\right)\right)\right)}^{{\beta }_{\ell }}{\Delta }^{}{s}_{\ell }\right)}^{\frac{1}{{\beta }_{\ell }}}.\hfill \end{array}$$

(39)

Corollary 8.

In Corollary 7, if we take $n=2,$ ${\beta }_{\ell }=\frac{1}{2}$, then the inequality (39) changes to

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

(40)

Remark 14.

In Corollary 8, if we take $\mathbb{T}=\mathbb{R},$ then the inequality (40) changes to

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

(41)

This is an inverse of the inequality due to Pachpatte [21].

Corollary 9.

In Corollary 8, 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 (40) changes to

$$\begin{array}{ccc}\hfill & & {\int }_{{\iota }_0}^{{\upsilon }_1}{\int }_{{\Im }_0}^{{\upsilon }_2}{\displaystyle \frac{{\Phi }_1\left({\Lambda }_1\left(s_1\right)\right){\Phi }_2\left({\Lambda }_2\left(s_2\right)\right)}{{\left(s_1{s}_2\right)}^{-1}{\left((s_1-{\Im }_0)+(s_2-{\Im }_0)\right)}^{-2}}}{\Delta }^{}{s}_1{\Delta }^{}{s}_2⩾4{\left[({\upsilon }_1-{\Im }_0)({\upsilon }_2-{\Im }_0)\right]}^{-1}\hfill \\ \hfill & & \times {\left({\int }_{{\iota }_0}^{{\upsilon }_1}(\rho \left({\upsilon }_1\right)-s_1){\left({\Phi }_1\left({\lambda }_1\left(s_1\right)\right)\right)}^2{\Delta }^{}{s}_1\right)}^{\frac{1}{2}}{\left({\int }_{{\iota }_0}^{{\upsilon }_2}(\rho \left({\upsilon }_2\right)-s_2){\left({\Phi }_2\left({\lambda }_2\left(s_2\right)\right)\right)}^2{\Delta }^{}{s}_2\right)}^{\frac{1}{2}}.\hfill \end{array}$$

(42)

Remark 15.

In Corollary 9, if we take $\mathbb{T}=\mathbb{R},$ then the inequality (42) changes to

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

This is an inverse inequality of the following inequality, which was proved by Pachpatte [20].

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

Corollary 10.

In Corollary 7, if we take ${\beta }_{\ell }=\frac{n-1}{n}$$(\ell =1,\dots ,n)$, the inequality (39)

$$\begin{array}{ccc}& & {\int }_{{\iota }_0}^{{\upsilon }_1}\dots {\int }_{{\iota }_0}^{{\upsilon }_n}{\displaystyle \frac{{\prod }_{\ell =1}^n{P}_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }({\Lambda }_{\ell }\left(s_{\ell }\right)}{{\left({\sum }_{\ell =1}^n(s_{\ell }-{\iota }_0)\right)}^{\frac{-n}{n-1}}}}{\Delta }^{}{s}_1\dots {\Delta }^{}{s}_n\hfill \\ \hfill & & ⩾n^{\frac{n}{n-1}}\prod _{\ell =1}^n{({\upsilon }_{\ell }-{\iota }_0)}^{\frac{-1}{n-1}}{\left({\int }_{{\iota }_0}^{{\upsilon }_{\ell }}(\rho \left({\upsilon }_{\ell }\right)-s_{\ell }){\left(p_{\ell }\left(s_{\ell }\right){\Phi }_{\ell }\left({\lambda }_{\ell }\left(s_{\ell }\right)\right)\right)}^{\frac{n-1}{n}}{\Delta }^{}{s}_{\ell }\right)}^{\frac{n}{n-1}}.\hfill \end{array}$$

3. Conclusions

In this article, we gave some generalizations of the Hardy-Hilbert inequality on a general time scale, and some dynamic integral and discrete inequalities known in the literature were extended as special cases of our results.