Gamma-Nabla Hardy–Hilbert-Type Inequalities on Time Scales

Abstract

We investigated several novel conformable fractional gamma-nabla dynamic Hardy–Hilbert inequalities on time scales in this study. Several continuous inequalities and their corresponding discrete analogues in the literature are combined and expanded by these inequalities. Hölder’s inequality on time scales and a few algebraic inequalities are used to demonstrate our findings.

1. Introduction

This is a statement of the well-known classical extension of Hilbert’s double-series theorem [1]:

Theorem 1.

If $\nu ,$ $\varpi >1$ are such that $\frac{1}{\nu }+\frac{1}{\varpi }⩽1$ and $0<\lambda =2-\frac{1}{\nu }-\frac{1}{\varpi }=\frac{1}{{\nu }^{\prime }}+\frac{1}{{\varpi }^{\prime }}⩽1,$ such that ${\nu }^{\prime }$ and ${\varpi }^{\prime }$ present the exponents’ conjugate; then,

$$\sum _{j=1}^{\infty }\sum _{i=1}^{\infty }\frac{{\vartheta }_j{\pi }_i}{{(j+i)}^{\lambda }}⩽K{\left(\sum _{j=1}^{\infty }{\vartheta }_j^{\nu }\right)}^{\frac{1}{\nu }}(\sum _{i=1}^{\infty }{\pi }_i^{\varpi }{)}^{\frac{1}{\varpi }},$$

(1)

where $K=K(\nu ,\varpi )$ depends on ν and ϖ only.

Readers may find the integral analogue of Theorem 1 in [1].

Theorem 2.

Let $\nu ,$ $\varpi ,$ ${\nu }^{\prime },$ ${\varpi }^{\prime }$ and λ be as in Theorem 1. If $\vartheta \in L^{\nu }(0,\infty )$ and $\theta \in L^{\varpi }(0,\infty ),$ then

$$\begin{array}{c}\hfill {\int }_0^{\infty }{\int }_0^{\infty }\frac{\vartheta \left(\iota \right)\theta \left(\varsigma \right)}{{(\iota +\varsigma )}^{\lambda }}d\iota d\varsigma ⩽K{\left({\int }_0^{\infty }{\vartheta }^{\nu }\left(\iota \right)d\iota \right)}^{\frac{1}{\nu }}{\left({\int }_0^{\infty }{\theta }^{\varpi }\left(\varsigma \right)d\varsigma \right)}^{\frac{1}{\varpi }},\end{array}$$

(2)

where $K=K(\nu ,\varpi )$ depends on ν and ϖ only.

In 2011, Zhao et al. [2] proposed a new inequality similar to Theorem 2.

Theorem 3.

Let $h_i⩾1,$ ${\nu }_i>1$ be constants and $\frac{1}{{\nu }_i}+\frac{1}{{\varpi }_i}=1.$ Let the differentiable fun. ${\vartheta }_i\left({\Im }_i\right)$ on $[0,{\iota }_i),$ where ${\iota }_i\in (0,\infty )$, and we use ${\vartheta }_i^{\prime }$ as a differentiation of ${\vartheta }_i.$ Suppose ${\vartheta }_i\left(0\right)=0$ for $(i=1,\dots ,n).$ Then,

$$\begin{array}{ccc}\hfill & & {\int }_0^{{\iota }_1}{\int }_0^{{\iota }_2}\dots {\int }_0^{{\iota }_n}{\displaystyle \frac{{\prod }_{i=1}^n\left|{\vartheta }_i^{h_i}\left({\Im }_i\right)\right|}{{\left({\sum }_{i=1}^n\frac{{\Im }_i}{{\varpi }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}}}d{\Im }_{n}d{\Im }_{n-1}\dots d{\Im }_1\hfill \\ \hfill & & ⩽K\prod _{i=1}^n{\left({\int }_0^{{\iota }_i}({\iota }_i-{\Im }_i)|{\vartheta }_i^{h_i-1}\left({\Im }_i\right){\vartheta }_i^{\prime }\left({\Im }_i\right){|}^{{\nu }_i}d{\Im }_i\right)}^{\frac{1}{{\nu }_i}},\hfill \end{array}$$

where

$$K={\left(n-\sum _{i=1}^n\frac{1}{{\nu }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\nu }_i}-n}\prod _{i=1}^n{h}_i{\iota }_i^{\frac{1}{{\varpi }_i}}.$$

Moreover, in 2012, Zhoa and Chung [3] proved the following theorem.

Theorem 4.

Let ${\nu }_i>1,$ be constants and $\frac{1}{{\nu }_i}+\frac{1}{{\varpi }_i}=1.$ Let ${\vartheta }_i({\tau }_{1i},\dots ,{\tau }_{ni})$ be real-valued nth differentiable functions defined on $[0,{\iota }_{1i})\times \dots \times [0,{\iota }_{ni}),$ where $0⩽{\iota }_{ji}⩽{\delta }_{ji},$ ${\delta }_{ji}\in (0,\infty )$ and $i,j=1,\dots ,n.$ Suppose

$${\vartheta }_i({\iota }_{1i},\dots ,{\iota }_{ni})={\int }_0^{{\iota }_{1i}}\dots {\int }_0^{{\iota }_{ni}}\frac{{\partial }^n}{\partial {\tau }_{1i}\dots \partial {\tau }_{ni}}{\vartheta }_i({\tau }_{1i},\dots ,{\tau }_{ni})d{\tau }_{ni}\dots d{\tau }_{1i},$$

then

$$\begin{array}{ccc}& & {\int }_0^{{\delta }_{11}}\dots {\int }_0^{{\delta }_{n1}}{\int }_0^{{\delta }_{12}}\dots {\int }_0^{{\delta }_{n2}}\dots {\int }_0^{{\delta }_{1n}}\dots {\int }_0^{{\delta }_{nn}}\hfill \\ \hfill & & {\displaystyle \frac{{\prod }_{i=1}^n{\left({\int }_0^{{\iota }_{1i}}\dots {\int }_0^{{\iota }_{ni}}|\frac{{\partial }^n}{\partial {\tau }_{1i}\dots \partial {\tau }_{ni}}{\vartheta }_i({\tau }_{1i},\dots ,{\tau }_{ni}){|}^{{\nu }_i}d{\tau }_{ni}\dots d{\tau }_{1i}\right)}^{\frac{1}{{\nu }_i}}}{{\left({\sum }_{i=1}^n\frac{\left[{\iota }_{1i}\dots {\iota }_{ni}\right]}{{\varpi }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}}}\hfill \\ \hfill & & d{\iota }_{11}\dots d{\iota }_{n1}d{\iota }_{12}\dots d{\iota }_{n2}\dots d{\iota }_{1n}\dots d{\iota }_{nn}\hfill \\ \hfill & & ⩽N\prod _{i=1}^n{\left({\int }_0^{{\delta }_{1i}}\dots {\int }_0^{{\delta }_{ni}}\prod _{j=1}^n({\delta }_{ji}-{\iota }_{ji})|\frac{{\partial }^n}{\partial {\iota }_{1i}\dots \partial {\iota }_{ni}}{\vartheta }_i({\iota }_{1i},\dots ,{\iota }_{ni}){|}^{{\nu }_i}d{\iota }_{ni}\dots d{\iota }_{1i}\right)}^{\frac{1}{{\nu }_i}},\hfill \end{array}$$

(3)

where

$$N={\left(n-\sum _{i=1}^n\frac{1}{{\nu }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\nu }_i}-n}\prod _{i=1}^n{\left[{\delta }_{1i}\dots {\delta }_{ni}\right]}^{\frac{1}{{\varpi }_i}}.$$

Pachappte [4] proved the following one:

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

(4)

where

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

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

(5)

where

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

Handley et al. [5] extended (4) and (5) as follows:

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

(6)

where

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

and

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

(7)

where

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

In 2006, Zhao and Cheung [6] proved the following reverse inequality.

$$\begin{array}{ccc}& & {\int }_0^{{\vartheta }_1}{\int }_0^{{\varsigma }_1}\dots {\int }_0^{{\vartheta }_n}{\int }_0^{{\varsigma }_n}\prod _{\ell =1}^n{\displaystyle \frac{{\mathsf{\Phi }}_{\ell }\left({ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })\right)}{{\left(\frac{1}{{\gamma }^{\prime }}{\sum }_{\ell =1}^n{\gamma }_{\ell }^{\prime }\left(s_{\ell }{\Im }_{\ell }\right)\right)}^{{\gamma }^{\prime }}}}d{s}_{1}d{\Im }_1\dots d{s}_{n}d{\Im }_n\hfill \\ \hfill & & ⩾G({\vartheta }_1{\varsigma }_1,\dots ,{\vartheta }_n{y}_n)\hfill \\ \hfill & & \times \prod _{\ell =1}^n{\left({\int }_0^{{\vartheta }_{\ell }}{\int }_0^{{\varsigma }_{\ell }}({\vartheta }_{\ell }-s_{\ell })({\varsigma }_{\ell }-{\Im }_{\ell }){\left(p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\Im }_{\ell }\right){\mathsf{\Phi }}_{\ell }\left(\frac{D_2{D}_1{ϝ}_{\ell }(s_{\ell },{\Im }_{\ell })}{p_{\ell }\left(s_{\ell }\right)q_{\ell }\left({\Im }_{\ell }\right)}\right)\right)}^{\frac{1}{{\gamma }_{\ell }}}d{s}_{\ell }d{\Im }_{\ell }\right)}^{{\gamma }_{\ell }}\hfill \end{array}$$

(8)

where

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

and

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

(9)

In [7], Pachpatte studied the Hilbert version inequalities.

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

(10)

and

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

(11)

where

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

and

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

(12)

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

(13)

where

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

In [8,9,10], Yang et al. established some important extensions of a Hardy–Hilbert-type inequality by using the weight coefficient method and techniques of real analysis.

All of the aforementioned findings hold true for both continuous and discrete domains. The purpose of the current research is to provide new, more general conclusions to the time-scale-based disparities previously established. Supreme outcomes, from which many other previous and current results may be taken, would be produced in this way. See the following publications for various dynamic inequalities, integrals of Hilbert’s kind, and other categories of inequalities on time scales [11,12,13,14,15,16,17,18,19,20,21,22,23].

We hope that the reader has a sufficient background on the nabla conformable fractional on time scales. S. Hilger [24] introduced the time scale theory in 1988 as a way to combine continuous and discrete analysis. A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset of the set of real numbers $\mathbb{R}$. In the manuscript, we use the notation ${\nabla }^{(\gamma ,a)}$ for the nabla conformable fractional derivative on time scales instead of ${\nabla }_a^{\gamma }$ for simplification. For more details on nabla conformable fractionals, please see [25].

Definition 1 (Conformable nabla derivative).

Given a function $f:\mathbb{T}⟶\mathbb{R}$ and $a\in \mathbb{T},$ f is $(\gamma ,a)$-nabla differentiable at $\xi >a,$ if it is nabla differentiable at ξ, and its $(\gamma ,a)$-nabla derivative is defined by

$${\nabla }_a^{\gamma }f\left(\xi \right)={\widehat{G}}_{1-\gamma }(\xi ,a)f^{\nabla }\left(\xi \right)\xi >a,$$

(14)

Definition 2 (Conformable nabla integral).

Assume that $0<\gamma ⩽1,$ $a,$ ${\xi }_1,$ ${\xi }_2\in \mathbb{T},$ $a⩽{\xi }_1⩽{\xi }_2$ and $f\in C_{ld}\left(\mathbb{T}\right),$ and the function f is called $(\gamma ,a)$-nabla integrable on $[{\xi }_1,{\xi }_2]$ if

$$\begin{array}{ccc}\hfill {\nabla }_a^{-\gamma }f\left(\xi \right)& =& {\int }_{{\xi }_1}^{{\xi }_2}f\left(\xi \right){\nabla }_a^{\gamma }\xi \hfill \\ & =& {\int }_{{\xi }_1}^{{\xi }_2}f\left(\xi \right){\widehat{G}}_{\gamma -1}({\sigma }^{\gamma -1}\left(\xi \right),a)\nabla \xi ,\hfill \end{array}$$

(15)

exists and is finite.

Lemma 1

(Dynamic Hölder’s Inequality [14]). Let $u,$ $v\in \mathbb{T}$ with $u<v.$ If $\vartheta ,$ $\theta \in C{C}_{rd}^1({[u,v]}_{\mathbb{T}}\times {[u,v]}_{\mathbb{T}},\mathbb{R})$ be integrable functions and $\frac{1}{\nu }+\frac{1}{\varpi }=1$ with $\nu >1.$ Then,

$$\begin{array}{ccc}\hfill {\int }_u^v{\int }_u^v\left|\vartheta (r,\delta )\theta (r,\delta )\right|{\nabla }^{(\gamma ,a)}r{\nabla }^{(\gamma ,a)}\delta & \le & {\left[{\int }_u^v{\int }_u^v{\left|\vartheta (r,\delta )\right|}^{\nu }{\nabla }^{(\gamma ,a)}r{\nabla }^{(\gamma ,a)}\delta \right]}^{\frac{1}{\nu }}\hfill \\ & & \times {\left[{\int }_u^v{\int }_u^v{\left|\theta (r,\delta )\right|}^{\varpi }{\nabla }^{(\gamma ,a)}r{\nabla }^{(\gamma ,a)}\delta \right]}^{\frac{1}{\varpi }}.\hfill \end{array}$$

(16)

This inequality is reversed if $0<\nu <1$ and if $\nu <0$ or $\varpi <0.$

In this study, we prove a few novel conformable fractional dynamic inequalities of the Hardy–Hilbert type on time scales, which are driven by Theorems 3 and 4 given above. We will also extract the discrete counterparts of the continuous Hilbert inequalities that are present in some special situations of our results. We are now prepared to state and support our key findings.

2. Main Results

Theorem 5.

Let $\mathbb{T}$ be a time scale with ${\delta }_0,$ ${\iota }_i,$ ${\Im }_i,$ ${\delta }_i\in \mathbb{T},$ $(i=1,\dots ,n).$ Let $h_i⩾1,$ ${\nu }_i,{\varpi }_i>1$ be constants and $\frac{1}{{\nu }_i}+\frac{1}{{\varpi }_i}=1.$ Let ${\nabla }^{(\gamma ,a)}$- differentiable functions ${\vartheta }_i\left({\Im }_i\right)$ be decreasing on ${[{\delta }_0,{\iota }_i)}_{\mathbb{T}},$ where ${\iota }_i\in (0,\infty ).$ Suppose ${\vartheta }_i\left({\delta }_0\right)=0.$ Then,

$$\begin{array}{ccc}\hfill & & {\int }_{{\delta }_0}^{{\iota }_1}{\int }_{{\delta }_0}^{{\iota }_2}\dots {\int }_{{\delta }_0}^{{\iota }_n}{\displaystyle \frac{{\prod }_{i=1}^n\left|{\vartheta }_i^{h_i}\left({\Im }_i\right)\right|}{{\left({\sum }_{i=1}^n\frac{({\Im }_i-{\delta }_0)}{{\varpi }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}}}{\nabla }^{(\gamma ,a)}{\Im }_n{\nabla }^{(\gamma ,a)}{\Im }_{n-1}\dots {\nabla }^{(\gamma ,a)}{\Im }_1\hfill \\ & & ⩽K\prod _{i=1}^n{\left({\int }_{{\delta }_0}^{{\iota }_i}(\rho \left({\iota }_i\right)-\rho \left({\Im }_i\right))|{\vartheta }_i^{h_i-1}\left({\Im }_i\right){\vartheta }_i^{{\nabla }^{(\gamma ,a)}}\left({\Im }_i\right){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\Im }_i\right)}^{\frac{1}{{\nu }_i}},\hfill \end{array}$$

(17)

where

$$K=K({\iota }_1,\dots ,{\iota }_n)={\left(n-\sum _{i=1}^n\frac{1}{{\nu }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\nu }_i}-n}\prod _{i=1}^n{h}_i{({\iota }_i-{\delta }_0)}^{\frac{1}{{\varpi }_i}}.$$

Proof. 

From Hölder inequality (16), one can see that

$$\begin{array}{ccc}\hfill \prod _{i=1}^n\left|{\vartheta }_i^{h_i}\left({\Im }_i\right)\right|& ⩽& \prod _{i=1}^n{h}_i{\int }_{{\delta }_0}^{{\Im }_i}\left|{\vartheta }_i^{h_i-1}\left({\tau }_i\right){\vartheta }_i^{{\nabla }^{(\gamma ,a)}}\left({\tau }_i\right)\right|{\nabla }^{(\gamma ,a)}{\tau }_i\hfill \\ \hfill & & ⩽\prod _{i=1}^n{h}_i{({\Im }_i-{\delta }_0)}^{\frac{1}{{\varpi }_i}}{\left({\int }_{{\delta }_0}^{{\Im }_i}|{\vartheta }_i^{h_i-1}\left({\tau }_i\right){\vartheta }_i^{{\nabla }^{(\gamma ,a)}}\left({\tau }_i\right){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\tau }_i\right)}^{\frac{1}{{\nu }_i}}.\hfill \end{array}$$

(18)

Using the inequality for the means [26]

$${\left(\prod _{i=1}^n{\lambda }_i^{\frac{1}{{\varpi }_i}}\right)}^{\frac{1}{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}}⩽\frac{1}{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}\sum _{i=1}^n\frac{{\lambda }_i}{{\varpi }_i},{\lambda }_i>0(i=1,\dots ,n),$$

(19)

we have

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{{\prod }_{i=1}^n\left|{\vartheta }_i^{h_i}\left({\Im }_i\right)\right|}{{\left({\sum }_{i=1}^n\frac{({\Im }_i-{\delta }_0)}{{\varpi }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}}}\hfill \\ & & ⩽{\left(n-\sum _{i=1}^n\frac{1}{{\nu }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\nu }_i}-n}\prod _{i=1}^n{h}_i{\left({\int }_{{\delta }_0}^{{\Im }_i}|{\vartheta }_i^{h_i-1}\left({\tau }_i\right){\vartheta }_i^{{\nabla }^{(\gamma ,a)}}\left({\tau }_i\right){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\tau }_i\right)}^{\frac{1}{{\nu }_i}}.\hfill \end{array}$$

(20)

Using the integration of (20) on ${\Im }_i$ from ${\delta }_0$ to ${\iota }_i$ $(i=1,\dots ,n)$, employing the inequality of Hölder’s yields

$$\begin{array}{ccc}\hfill & & {\int }_{{\delta }_0}^{{\iota }_1}{\int }_{{\delta }_0}^{{\iota }_2}\dots {\int }_{{\delta }_0}^{{\iota }_n}{\displaystyle \frac{{\prod }_{i=1}^n\left|{\vartheta }_i^{h_i}\left({\Im }_i\right)\right|}{{\left({\sum }_{i=1}^n\frac{({\Im }_i-{\delta }_0)}{{\varpi }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}}}{\nabla }^{(\gamma ,a)}{\Im }_n{\nabla }^{(\gamma ,a)}{\Im }_{n-1}\dots {\nabla }^{(\gamma ,a)}{\Im }_1\hfill \\ \hfill & & ⩽{\left(n-\sum _{i=1}^n\frac{1}{{\nu }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\nu }_i}-n}\prod _{i=1}^n{h}_i{\int }_{{\delta }_0}^{{\iota }_i}{\left({\int }_{{\delta }_0}^{{\Im }_i}|{\vartheta }_i^{h_i-1}\left({\tau }_i\right){\vartheta }_i^{{\nabla }^{(\gamma ,a)}}\left({\tau }_i\right){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\tau }_i\right)}^{\frac{1}{{\nu }_i}}\hfill \\ \hfill & & ⩽K\prod _{i=1}^n{\left({\int }_{{\delta }_0}^{{\iota }_i}{\int }_{{\delta }_0}^{{\Im }_i}|{\vartheta }_i^{h_i-1}\left({\tau }_i\right){\vartheta }_i^{{\nabla }^{(\gamma ,a)}}\left({\tau }_i\right){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\tau }_i{\nabla }^{(\gamma ,a)}{\Im }_i\right)}^{\frac{1}{{\nu }_i}}\hfill \\ & & =K\prod _{i=1}^n{\left({\int }_{{\delta }_0}^{{\iota }_i}({\iota }_i-{\Im }_i)|{\vartheta }_i^{h_i-1}\left({\Im }_i\right){\vartheta }_i^{{\nabla }^{(\gamma ,a)}}\left({\Im }_i\right){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\Im }_i\right)}^{\frac{1}{{\nu }_i}}.\hfill \end{array}$$

(21)

By exploiting the fact that ${\iota }_i⩽\rho \left({\iota }_i\right),$ we find that

$$\begin{array}{ccc}\hfill & & {\int }_{{\delta }_0}^{{\iota }_1}{\int }_{{\delta }_0}^{{\iota }_2}\dots {\int }_{{\delta }_0}^{{\iota }_n}{\displaystyle \frac{{\prod }_{i=1}^n\left|{\vartheta }_i^{h_i}\left({\Im }_i\right)\right|}{{\left({\sum }_{i=1}^n\frac{({\Im }_i-{\delta }_0)}{{\varpi }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}}}{\nabla }^{(\gamma ,a)}{\Im }_n{\nabla }^{(\gamma ,a)}{\Im }_{n-1}\dots {\nabla }^{(\gamma ,a)}{\Im }_1\hfill \\ \hfill & & ⩽K\prod _{i=1}^n{\left({\int }_{{\delta }_0}^{{\iota }_i}(\rho \left({\iota }_i\right)-\rho \left({\Im }_i\right))|{\vartheta }_i^{h_i-1}\left({\Im }_i\right){\vartheta }_i^{{\nabla }^{(\gamma ,a)}}\left({\Im }_i\right){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\Im }_i\right)}^{\frac{1}{{\nu }_i}}.\hfill \end{array}$$

This concludes the evidence. □

Remark 1.

In Theorem 5, taking $\mathbb{T}=\mathbb{Z},\gamma =1,$ $h_i=1,$ we obtain the results thanks to the authors of ([2], Theorem 1.1).

Remark 2.

In Theorem 5, taking $\mathbb{T}=\mathbb{R},\gamma =1$, we obtain the results thanks to the authors of ([2], Theorem 1.3).

Corollary 1.

In Theorem 5, taking $n=2,$ and $h_1=h_2=1,$ if ${\nu }_1,$ ${\nu }_2>1$ are such that $\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}⩾1$ and $0<\lambda =2-\frac{1}{{\nu }_1}-\frac{1}{{\nu }_2}=\frac{1}{{\varpi }_1}+\frac{1}{{\varpi }_2}⩽1,$ inequality (17) reduces to

$$\begin{array}{ccc}\hfill & & {\int }_{{\delta }_0}^{{\iota }_1}{\int }_{{\delta }_0}^{{\iota }_2}\frac{|{\vartheta }_1\left({\Im }_1\right)\left|\right|{\vartheta }_2\left({\Im }_2\right)|}{{\left({\varpi }_2({\Im }_1-{\delta }_0)+{\varpi }_1({\Im }_2-{\delta }_0)\right)}^{\lambda }}{\nabla }^{(\gamma ,a)}{\Im }_2{\nabla }^{(\gamma ,a)}{\Im }_1⩽\frac{1}{{\left(\lambda {\varpi }_1{\varpi }_2\right)}^{\lambda }}{({\iota }_1-{\delta }_0)}^{\frac{1}{{\varpi }_1}}{({\iota }_2-{\delta }_0)}^{\frac{1}{{\varpi }_2}}\hfill \\ \hfill & & \times {\left({\int }_{{\delta }_0}^{{\iota }_1}(\rho \left({\iota }_1\right)-\rho \left({\Im }_1\right)){\left|{\vartheta }_1^{{\nabla }^{(\gamma ,a)}}\left({\Im }_1\right)\right|}^{{\nu }_1}{\nabla }^{(\gamma ,a)}{\Im }_1\right)}^{\frac{1}{{\nu }_1}}{\left({\int }_{{\delta }_0}^{{\iota }_2}(\rho \left({\iota }_2\right)-\rho \left({\Im }_2\right)){\left|{\vartheta }_2^{{\nabla }^{(\gamma ,a)}}\left({\Im }_2\right)\right|}^{{\nu }_2}{\nabla }^{(\gamma ,a)}{\Im }_2\right)}^{\frac{1}{{\nu }_2}}.\hfill \end{array}$$

(22)

Remark 3.

In a special case, taking $\mathbb{T}=\mathbb{R},\gamma =1$, in (22), we have that

$$\begin{array}{ccc}\hfill & & {\int }_0^{{\iota }_1}{\int }_0^{{\iota }_2}\frac{|{\vartheta }_1\left({\Im }_1\right)\left|\right|{\vartheta }_2\left({\Im }_2\right)|}{{\left({\varpi }_2{\Im }_1+{\varpi }_1{\Im }_2\right)}^{\lambda }}d{\Im }_{2}d{\Im }_1⩽\frac{1}{{\left(\lambda {\varpi }_1{\varpi }_2\right)}^{\lambda }}{\left({\iota }_1\right)}^{\frac{1}{{\varpi }_1}}{\left({\iota }_2\right)}^{\frac{1}{{\varpi }_2}}\hfill \\ & & \times {\left({\int }_0^{{\iota }_1}({\iota }_1-{\Im }_1){\left|{\vartheta }_1^{\prime }\left({\Im }_1\right)\right|}^{{\nu }_1}d{\Im }_1\right)}^{\frac{1}{{\nu }_1}}{\left({\int }_0^{{\iota }_2}({\iota }_2-{\Im }_2){\left|{\vartheta }_2^{\prime }\left({\Im }_2\right)\right|}^{{\nu }_2}d{\Im }_2\right)}^{\frac{1}{{\nu }_2}},\hfill \end{array}$$

(23)

which is an interesting variation of the inequality (2).

Remark 4.

In a special case, taking $\mathbb{T}=\mathbb{Z},\gamma =1$, in (22), we have that

$$\begin{array}{ccc}\hfill & & \sum _{{\Im }_1=1}^{m_1}\sum _{{\Im }_2=1}^{m_2}\frac{|a_1\left({\Im }_1\right)\left|\right|a_2\left({\Im }_2\right)|}{{\left({\varpi }_2{\Im }_1+{\varpi }_1{\Im }_2\right)}^{\lambda }}⩽\frac{1}{{\left(\lambda {\varpi }_1{\varpi }_2\right)}^{\lambda }}{\left(m_1\right)}^{\frac{1}{{\varpi }_1}}{\left(m_2\right)}^{\frac{1}{{\varpi }_2}}\hfill \\ & & \times {\left(\sum _{{\Im }_1=1}^{m_1}(m_1-{\Im }_1+1){\left|{\nabla }^{(\gamma ,a)}{a}_1\left({\Im }_1\right)\right|}^{{\nu }_1}\right)}^{\frac{1}{{\nu }_1}}{\left(\sum _{{\Im }_2=1}^{m_2}(m_2-{\Im }_2+1){\left|{\nabla }^{(\gamma ,a)}{a}_2\left({\Im }_2\right)\right|}^{{\nu }_2}\right)}^{\frac{1}{{\nu }_2}},\hfill \end{array}$$

(24)

which is an interesting variation of the inequality (1).

Corollary 2.

In Corollary 1, if $\lambda =1,$ then $\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}=\frac{1}{{\varpi }_1}+\frac{1}{{\varpi }_2}=1$ and we take ${\nu }_1={\varpi }_2,$ ${\nu }_2={\varpi }_1.$ In this case, inequality (22) reduces to

$$\begin{array}{ccc}\hfill & & {\int }_{{\delta }_0}^{{\iota }_1}{\int }_{{\delta }_0}^{{\iota }_2}\frac{|{\vartheta }_1\left({\Im }_1\right)\left|\right|{\vartheta }_2\left({\Im }_2\right)|}{{\varpi }_2({\Im }_1-{\delta }_0)+{\varpi }_1({\Im }_2-{\delta }_0)}{\nabla }^{(\gamma ,a)}{\Im }_2{\nabla }^{(\gamma ,a)}{\Im }_1⩽\frac{1}{{\nu }_1{\varpi }_1}{({\iota }_1-{\delta }_0)}^{\frac{{\nu }_1-1}{{\nu }_1}}{({\iota }_2-{\delta }_0)}^{\frac{{\varpi }_1-1}{{\varpi }_1}}\hfill \\ \hfill & & \times {\left({\int }_{{\delta }_0}^{{\iota }_1}(\rho \left({\iota }_1\right)-\rho \left({\Im }_1\right)){\left|{\vartheta }_1^{{\nabla }^{(\gamma ,a)}}\left({\Im }_1\right)\right|}^{{\nu }_1}{\nabla }^{(\gamma ,a)}{\Im }_1\right)}^{\frac{1}{{\nu }_1}}{\left({\int }_{{\delta }_0}^{{\iota }_2}(\rho \left({\iota }_2\right)-\rho \left({\Im }_2\right)){\left|{\vartheta }_2^{{\nabla }^{(\gamma ,a)}}\left({\Im }_2\right)\right|}^{{\varpi }_1}{\nabla }^{(\gamma ,a)}{\Im }_2\right)}^{\frac{1}{{\varpi }_1}}.\hfill \end{array}$$

(25)

Remark 5.

In Corollary 2, if $\mathbb{T}=\mathbb{R},\gamma =1$, we obtain an equivalent formulation of the inequality that Pachpatte presented in ([27], Theorem 2).

Remark 6.

In Corollary 2, if $\mathbb{T}=\mathbb{Z},\gamma =1$, we obtain an equivalent formulation of the inequality that Pachpatte presented in ([27], Theorem 1).

Theorem 6.

Let $\mathbb{T}$ be a time scale with ${\delta }_0,$ ${\iota }_i,$ ${\varsigma }_i,$ ${\Im }_i,$ ${\delta }_i\in \mathbb{T},$ $(i=1,\dots ,n).$ Let $h_i⩾1,$ ${\nu }_i,{\varpi }_i>1$ be constants and $\frac{1}{{\nu }_i}+\frac{1}{{\varpi }_i}=1.$ Let the ${\nabla }^{(\gamma ,a)}$-differentiable fun. ${\vartheta }_i({\Im }_i,{\delta }_i)$ be decreasing funs. on ${[{\delta }_0,{\iota }_i)}_{\mathbb{T}}\times {[{\delta }_0,{\varsigma }_i)}_{\mathbb{T}}$ and ${\vartheta }_i({\delta }_0,{\delta }_i)={\vartheta }_i({\Im }_i,{\delta }_0)=0,$ for $(i=1,\dots ,n).$ Partial derivatives of ${\vartheta }_i$ are indicated by ${\vartheta }_i^{{\nabla }_1^{(\gamma ,a)}},$ ${\vartheta }_i^{{\nabla }_2^{(\gamma ,a)}},$ ${\vartheta }_i^{{\nabla }_{12}^{(\gamma ,a)}}={\vartheta }_i^{{\nabla }_{21}^{(\gamma ,a)}}.$ Let

$${\left({\vartheta }_i^{h_i}({\Im }_i,{\delta }_i)\right)}^{{\nabla }_1^{(\gamma ,a)}{\nabla }_2^{(\gamma ,a)}}⩽{\left(h_i{\vartheta }_i^{h_i-1}({\Im }_i,{\delta }_i).{\vartheta }_i^{{\nabla }_1^{(\gamma ,a)}}({\Im }_i,{\delta }_i)\right)}^{{\nabla }_2^{(\gamma ,a)}}={\vartheta }_i^{{\nabla }_{12}^{(\gamma ,a)}}({\Im }_i,{\delta }_i).$$

Then,

$$\begin{array}{ccc}\hfill & & {\int }_{{\delta }_0}^{{\iota }_1}{\int }_{{\delta }_0}^{{\varsigma }_1}\dots {\int }_{{\delta }_0}^{{\iota }_n}{\int }_{{\delta }_0}^{{\varsigma }_n}\frac{{\prod }_{i=1}^n\left|{\vartheta }_i^{h_i}({\Im }_i,{\delta }_i)\right|}{{\left({\sum }_{i=1}^n\frac{({\Im }_i-{\delta }_0)({\delta }_i-{\delta }_0)}{{\varpi }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}}{\nabla }^{(\gamma ,a)}{\delta }_n{\nabla }^{(\gamma ,a)}{\Im }_n\dots {\nabla }^{(\gamma ,a)}{\delta }_1{\nabla }^{(\gamma ,a)}{\Im }_1\hfill \\ \hfill & & ⩽C\prod _{i=1}^n{\left({\int }_{{\delta }_0}^{{\iota }_i}{\int }_{{\delta }_0}^{{\varsigma }_i}(\rho \left({\iota }_i\right)-\rho \left({\Im }_i\right))(\rho \left({\varsigma }_i\right)-\rho \left({\delta }_i\right)){\left|{\vartheta }_i^{{\nabla }_{12}^{(\gamma ,a)}}({\Im }_i,{\delta }_i)\right|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\delta }_i{\nabla }^{(\gamma ,a)}{\Im }_i\right)}^{\frac{1}{{\nu }_i}},\hfill \end{array}$$

(26)

where

$$C=C({\iota }_1{\varsigma }_1,\dots ,{\iota }_n{\varsigma }_n)={\left(n-\sum _{i=1}^n\frac{1}{{\nu }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\nu }_i}-n}\prod _{i=1}^n{\left[({\iota }_i-{\delta }_0)({\varsigma }_i-{\delta }_0)\right]}^{\frac{1}{{\varpi }_i}}.$$

Proof. 

We can write

$$\begin{array}{ccc}\hfill {\vartheta }_i^{h_i}({\Im }_i,{\delta }_i)& =& {\vartheta }_i^{h_i}({\Im }_i,{\delta }_i)-{\vartheta }_i^{h_i}({\delta }_0,{\delta }_i)-{\vartheta }_i^{h_i}({\Im }_i,{\delta }_0)+{\vartheta }_i^{h_i}({\delta }_0,{\delta }_0)\hfill \\ \hfill & =& {\int }_{{\delta }_0}^{{\Im }_i}{\left({\vartheta }_i^{h_i}({\xi }_i,{\delta }_i)\right)}^{{\nabla }_1^{(\gamma ,a)}}{\nabla }_1^{(\gamma ,a)}{\xi }_i-{\int }_{{\delta }_0}^{{\Im }_i}{\left({\vartheta }_i^{h_i}({\xi }_i,{\delta }_0)\right)}^{{\nabla }_1^{(\gamma ,a)}}{\nabla }^{(\gamma ,a)}{\xi }_i\hfill \\ \hfill & =& {\int }_{{\delta }_0}^{{\Im }_i}\left[{\left({\vartheta }_i^{h_i}({\xi }_i,{\delta }_i)\right)}^{{\nabla }_1^{(\gamma ,a)}}-{\left({\vartheta }_i^{h_i}({\xi }_i,{\delta }_0)\right)}^{{\nabla }_1^{(\gamma ,a)}}\right]{\nabla }^{(\gamma ,a)}{\xi }_i\hfill \\ \hfill & ⩽& {\int }_{{\delta }_0}^{{\Im }_i}{\int }_{{\delta }_0}^{{\delta }_i}{\left(h_i{\vartheta }_i^{h_i-1}({\xi }_i,{\eta }_i).{\vartheta }_i^{{\nabla }_1^{(\gamma ,a)}}({\xi }_i,{\eta }_i)\right)}^{{\nabla }_2^{(\gamma ,a)}}{\nabla }^{(\gamma ,a)}{\eta }_i{\nabla }^{(\gamma ,a)}{\xi }_i\hfill \\ & =& {\int }_{{\delta }_0}^{{\Im }_i}{\int }_{{\delta }_0}^{{\delta }_i}{\vartheta }_i^{{\nabla }_{12}^{(\gamma ,a)}}({\xi }_i,{\eta }_i){\nabla }^{(\gamma ,a)}{\eta }_i{\nabla }^{(\gamma ,a)}{\xi }_i.\hfill \end{array}$$

(27)

By (27) applying (16) and (28), we obtain

$$\begin{array}{ccc}\hfill & & \prod _{i=1}^n|{\vartheta }_i^{h_i}({\Im }_i,{\delta }_i)|⩽\prod _{i=1}^n{\int }_{{\delta }_0}^{{\Im }_i}{\int }_{{\delta }_0}^{{\delta }_i}\left|{\vartheta }_i^{{\nabla }_{12}^{(\gamma ,a)}}({\xi }_i,{\eta }_i)\right|{\nabla }_1^{(\gamma ,a)}{\eta }_i{\nabla }_2^{(\gamma ,a)}{\xi }_i\hfill \\ \hfill & & ⩽\prod _{i=1}^n{\left[({\Im }_i-{\delta }_0)({\delta }_i-{\delta }_0)\right]}^{\frac{1}{{\varpi }_i}}{\left({\int }_{{\delta }_0}^{{\Im }_i}{\int }_{{\delta }_0}^{{\delta }_i}{\left|{\vartheta }_i^{{\nabla }_{12}^{(\gamma ,a)}}({\xi }_i,{\eta }_i)\right|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\eta }_i{\nabla }^{(\gamma ,a)}{\xi }_i\right)}^{\frac{1}{{\nu }_i}}.\hfill \end{array}$$

(28)

Using inequality (19), we find that

$$\begin{array}{c}\hfill \frac{{\prod }_{i=1}^n\left|{\vartheta }_i^{h_i}({\Im }_i,{\delta }_i)\right|}{{\left({\sum }_{i=1}^n\frac{({\Im }_i-{\delta }_0)({\delta }_i-{\delta }_0)}{{\varpi }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}}⩽{\left(n-\sum _{i=1}^n\frac{1}{{\nu }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\nu }_i}-n}\prod _{i=1}^n{\left({\int }_{{\delta }_0}^{{\Im }_i}{\int }_{{\delta }_0}^{{\delta }_i}{\left|{\vartheta }_i^{{\nabla }_{12}^{(\gamma ,a)}}({\xi }_i,{\eta }_i)\right|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\eta }_i{\nabla }^{(\gamma ,a)}{\xi }_i\right)}^{\frac{1}{{\nu }_i}}.\end{array}$$

(29)

Integrating (29) with ${\Im }_i$ and ${\delta }_i$, and applying (16) and Fubini’s theorem, yields

$$\begin{array}{ccc}\hfill & & {\int }_{{\delta }_0}^{{\iota }_1}{\int }_{{\delta }_0}^{{\varsigma }_1}\dots {\int }_{{\delta }_0}^{{\iota }_n}{\int }_{{\delta }_0}^{{\varsigma }_n}\frac{{\prod }_{i=1}^n\left|{\vartheta }_i^{h_i}({\Im }_i,{\delta }_i)\right|}{{\left({\sum }_{i=1}^n\frac{({\Im }_i-{\delta }_0)({\delta }_i-{\delta }_0)}{{\varpi }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}}{\nabla }^{(\gamma ,a)}{\delta }_n{\nabla }^{(\gamma ,a)}{\Im }_n\dots {\nabla }^{(\gamma ,a)}{\delta }_1{\nabla }^{(\gamma ,a)}{\Im }_1\hfill \\ \hfill & & ⩽{\left(n-\sum _{i=1}^n\frac{1}{{\nu }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\nu }_i}-n}\hfill \\ \hfill & & \times \prod _{i=1}^n\left({\int }_{{\delta }_0}^{{\iota }_i}{\int }_{{\delta }_0}^{{\varsigma }_i}{\left({\int }_{{\delta }_0}^{{\Im }_i}{\int }_{{\delta }_0}^{{\delta }_i}{\left|{\vartheta }_i^{{\nabla }_{12}^{(\gamma ,a)}}({\xi }_i,{\eta }_i)\right|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\eta }_i{\nabla }^{(\gamma ,a)}{\xi }_i\right)}^{\frac{1}{{\nu }_i}}{\nabla }^{(\gamma ,a)}{\delta }_i{\nabla }^{(\gamma ,a)}{\Im }_i\right)\hfill \\ \hfill & & ⩽{\left(n-\sum _{i=1}^n\frac{1}{{\nu }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\nu }_i}-n}\hfill \\ \hfill & & \times \prod _{i=1}^n{\left[({\iota }_i-{\delta }_0)({\varsigma }_i-{\delta }_0)\right]}^{\frac{1}{{\varpi }_i}}{\left({\int }_{{\delta }_0}^{{\iota }_i}{\int }_{{\delta }_0}^{{\varsigma }_i}\left({\int }_{{\delta }_0}^{{\Im }_i}{\int }_{{\delta }_0}^{{\delta }_i}{\left|{\vartheta }_i^{{\nabla }_{12}^{(\gamma ,a)}}({\xi }_i,{\eta }_i)\right|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\eta }_i{\nabla }^{(\gamma ,a)}{\xi }_i\right){\nabla }^{(\gamma ,a)}{\delta }_i{\nabla }^{(\gamma ,a)}{\Im }_i\right)}^{\frac{1}{{\nu }_i}}\hfill \\ & & =C\prod _{i=1}^n{\left({\int }_{{\delta }_0}^{{\iota }_i}{\int }_{{\delta }_0}^{{\varsigma }_i}({\iota }_i-{\Im }_i)({\varsigma }_i-{\delta }_i){\left|{\vartheta }_i^{{\nabla }_{12}^{(\gamma ,a)}}({\Im }_i,{\delta }_i)\right|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\delta }_i{\nabla }^{(\gamma ,a)}{\Im }_i\right)}^{\frac{1}{{\nu }_i}}.\hfill \end{array}$$

(30)

By exploiting the fact that ${\iota }_i⩽\rho \left({\iota }_i\right),$ we obtain

$$\begin{array}{ccc}& & {\int }_{{\delta }_0}^{{\iota }_1}{\int }_{{\delta }_0}^{{\varsigma }_1}\dots {\int }_{{\delta }_0}^{{\iota }_n}{\int }_{{\delta }_0}^{{\varsigma }_n}\frac{{\prod }_{i=1}^n\left|{\vartheta }_i^{h_i}({\Im }_i,{\delta }_i)\right|}{{\left({\sum }_{i=1}^n\frac{({\Im }_i-{\delta }_0)({\delta }_i-{\delta }_0)}{{\varpi }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}}{\nabla }^{(\gamma ,a)}{\delta }_n{\nabla }^{(\gamma ,a)}{\Im }_n\dots {\nabla }^{(\gamma ,a)}{\delta }_1{\nabla }^{(\gamma ,a)}{\Im }_1\hfill \\ \hfill & & ⩽C\prod _{i=1}^n{\left({\int }_{{\delta }_0}^{{\iota }_i}{\int }_{{\delta }_0}^{{\varsigma }_i}(\rho \left({\iota }_i\right)-\rho \left({\Im }_i\right))(\rho \left({\varsigma }_i\right)-\rho \left({\delta }_i\right)){\left|{\vartheta }_i^{{\nabla }_{12}^{(\gamma ,a)}}({\Im }_i,{\delta }_i)\right|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\delta }_i{\nabla }^{(\gamma ,a)}{\Im }_i\right)}^{\frac{1}{{\nu }_i}}.\hfill \end{array}$$

This concludes the evidence. □

Remark 7.

In Theorem 6, if we take $\mathbb{T}=\mathbb{Z},\gamma =1,$ $h_i=1,$ we obtain the results thanks to the authors of ([2], Theorem 1.2).

Remark 8.

In Theorem 6, supposing that $\mathbb{T}=\mathbb{R},\gamma =1,$ we obtain the results thanks to the authors of ([2], Theorem 1.4).

Corollary 3.

Taking $n=2$ and $h_1=h_2=1$ in Theorem 6, we have

$${\vartheta }_1^{{\nabla }_{12}^{(\gamma ,a)}}({\Im }_1,{\delta }_1)={\vartheta }^{{\nabla }_2^{(\gamma ,a)}{\nabla }_1^{(\gamma ,a)}}({\Im }_1,{\delta }_1),{\vartheta }_2^{{\nabla }_{12}^{(\gamma ,a)}}({\Im }_1,{\delta }_1)={\vartheta }^{{\nabla }_2^{(\gamma ,a)}{\nabla }_1^{(\gamma ,a)}}({\Im }_2,{\delta }_2).$$

Moreover, if ${\nu }_1,$ ${\nu }_2>1$ satisfy $\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}⩾1$ and $0<\lambda =2-\frac{1}{{\nu }_1}-\frac{1}{{\nu }_2}=\frac{1}{{\varpi }_1}+\frac{1}{{\varpi }_2}⩽1,$ inequality (26) reduces to

$$\begin{array}{ccc}\hfill & & {\int }_{{\delta }_0}^{{\iota }_1}{\int }_{{\delta }_0}^{{\varsigma }_1}\left({\int }_{{\delta }_0}^{{\iota }_2}{\int }_{{\delta }_0}^{{\varsigma }_2}\frac{|{\vartheta }_1({\Im }_1,{\delta }_1)\left|\right|{\vartheta }_2({\Im }_2,{\delta }_2)|}{{\left({\nu }_1({\Im }_1-{\delta }_0)({\delta }_1-{\delta }_0)+{\varpi }_1({\Im }_2-{\delta }_0)({\delta }_2-{\delta }_0)\right)}^{\lambda }}{\nabla }^{(\gamma ,a)}{\Im }_2{\nabla }^{(\gamma ,a)}{\delta }_2\right){\nabla }^{(\gamma ,a)}{\Im }_1{\nabla }^{(\gamma ,a)}{\delta }_1\hfill \\ \hfill & & ⩽\frac{1}{{\left(\lambda {\varpi }_1{\varpi }_2\right)}^{\lambda }}{\left[({\iota }_1-{\delta }_0)({\varsigma }_1-{\delta }_0)\right]}^{\frac{1}{{\varpi }_1}}{\left[({\iota }_2-{\delta }_0)({\varsigma }_2-{\delta }_0)\right]}^{\frac{{\varpi }_1-1}{{\varpi }_1}}\hfill \\ & & \times {\left({\int }_{{\delta }_0}^{{\iota }_1}{\int }_{{\delta }_0}^{{\varsigma }_1}(\rho \left({\iota }_1\right)-\rho \left({\Im }_1\right))(\rho \left({\varsigma }_1\right)-\rho \left({\delta }_1\right)){\left|{\vartheta }^{{\nabla }_2^{(\gamma ,a)}{\nabla }_1^{(\gamma ,a)}}({\Im }_1,{\delta }_1)\right|}^{{\nu }_1}{\nabla }^{(\gamma ,a)}{\Im }_1{\nabla }^{(\gamma ,a)}{\delta }_1\right)}^{\frac{1}{{\nu }_1}}\hfill \\ \hfill & & {\left({\int }_{{\delta }_0}^{{\iota }_2}{\int }_{{\delta }_0}^{{\varsigma }_2}(\rho \left({\iota }_2\right)-{\delta }_0)(\rho \left({\varsigma }_2\right)-{\delta }_0){\left|{\vartheta }^{{\nabla }_2^{(\gamma ,a)}{\nabla }_1^{(\gamma ,a)}}({\Im }_2,{\delta }_2)\right|}^{{\nu }_2}{\nabla }^{(\gamma ,a)}{\Im }_2{\nabla }^{(\gamma ,a)}{\delta }_2\right)}^{\frac{1}{{\nu }_2}}.\hfill \end{array}$$

(31)

Remark 9.

In a unique scenario, if we take $\mathbb{T}=\mathbb{R}$ in Corollary 3, the inequality (31) reduces to

$$\begin{array}{ccc}\hfill & & {\int }_0^{{\iota }_1}{\int }_0^{{\varsigma }_1}\left({\int }_0^{{\iota }_2}{\int }_0^{{\varsigma }_2}\frac{|{\vartheta }_1({\Im }_1,{\delta }_1)\left|\right|{\vartheta }_2({\Im }_2,{\delta }_2)|}{{\left({\nu }_1{\Im }_1{\delta }_1+{\varpi }_1{\Im }_2{\delta }_2\right)}^{\lambda }}d{\Im }_{2}d{\delta }_2\right)d{\Im }_{1}d{\delta }_1\hfill \\ \hfill & & ⩽\frac{1}{{\left(\lambda {\varpi }_1{\varpi }_2\right)}^{\lambda }}{\left[{\iota }_1{\varsigma }_1\right]}^{\frac{1}{{\varpi }_1}}{\left[{\iota }_2{\varsigma }_2\right]}^{\frac{{\varpi }_1-1}{{\varpi }_1}}\hfill \\ & & \times {\left({\int }_0^{{\iota }_1}{\int }_0^{{\varsigma }_1}({\iota }_1-{\Im }_1)({\varsigma }_1-{\delta }_1){\left|D_1{D}_2{\vartheta }_1({\Im }_1,{\delta }_1)\right|}^{{\nu }_1}d{\Im }_{1}d{\delta }_1\right)}^{\frac{1}{{\nu }_1}}\hfill \\ \hfill & & \times {\left({\int }_0^{{\iota }_2}{\int }_0^{{\varsigma }_2}({\iota }_2-{\Im }_2)({\varsigma }_2-{\delta }_2){\left|D_1{D}_2{\vartheta }_2({\Im }_2,{\delta }_2)\right|}^{{\nu }_2}d{\Im }_{2}d{\delta }_2\right)}^{\frac{1}{{\nu }_2}},\hfill \end{array}$$

(32)

Remark 10.

In a unique scenario, if we take $\mathbb{T}=\mathbb{Z}$ in Corollary 3, the inequality (31) reduces to

$$\begin{array}{ccc}\hfill & & \sum _{{\Im }_1=1}^{m_1}\sum _{{\delta }_1=1}^{n_1}\left(\sum _{{\Im }_2=1}^{m_2}\sum _{{\delta }_2=1}^{n_2}\frac{|a_1({\Im }_1,{\delta }_1)\left|\right|a_2({\Im }_2,{\delta }_2)|}{{\left({\nu }_1{\Im }_1{\delta }_1+{\varpi }_1{\Im }_2{\delta }_2\right)}^{\lambda }}\right)\hfill \\ \hfill & & ⩽\frac{1}{{\left(\lambda {\varpi }_1{\varpi }_2\right)}^{\lambda }}{\left[m_1{n}_1\right]}^{\frac{1}{{\varpi }_1}}{\left[m_2{n}_2\right]}^{\frac{{\varpi }_1-1}{{\varpi }_1}}\hfill \\ & & \times {\left(\sum _{{\Im }_1=1}^{m_1}\sum _{{\delta }_1=1}^{n_1}(n_1-{\delta }_1)(m_1-{\Im }_1){\left|{\nabla }_1^{(\gamma ,a)}{\nabla }_2^{(\gamma ,a)}{a}_1({\Im }_1,{\delta }_1)\right|}^{{\nu }_1}\right)}^{\frac{1}{{\nu }_1}}\hfill \\ \hfill & & \times {\left(\sum _{{\Im }_2=1}^{m_2}\sum _{{\delta }_2=1}^{n_2}(n_2-{\delta }_2)(m_2-{\Im }_2)\left)\right|{\nabla }_1^{(\gamma ,a)}{\nabla }_2^{(\gamma ,a)}{a}_2({\Im }_2,{\delta }_2){|}^{{\nu }_2}\right)}^{\frac{1}{{\nu }_2}},\hfill \end{array}$$

(33)

Corollary 4.

In Corollary 3, if $\lambda =1,$ then $\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}=\frac{1}{{\varpi }_1}+\frac{1}{{\varpi }_2}=1$, and we take ${\nu }_1={\varpi }_2,$ ${\nu }_2={\varpi }_1.$ In this case, the inequality (31) reduces to

$$\begin{array}{ccc}\hfill & & {\int }_{{\delta }_0}^{{\iota }_1}{\int }_{{\delta }_0}^{{\varsigma }_1}\left({\int }_{{\delta }_0}^{{\iota }_2}{\int }_{{\delta }_0}^{{\varsigma }_2}\frac{|{\vartheta }_1({\Im }_1,{\delta }_1)\left|\right|{\vartheta }_2({\Im }_2,{\delta }_2)|}{\left({\nu }_1({\Im }_1-{\delta }_0)({\delta }_1-{\delta }_0)+{\varpi }_1({\Im }_2-{\delta }_0)({\delta }_2-{\delta }_0)\right)}{\nabla }^{(\gamma ,a)}{\Im }_2{\nabla }^{(\gamma ,a)}{\delta }_2\right){\nabla }^{(\gamma ,a)}{\Im }_1{\nabla }^{(\gamma ,a)}{\delta }_1\hfill \\ \hfill & & ⩽\frac{1}{{\nu }_1{\varpi }_1}{\left[({\iota }_1-{\delta }_0)({\varsigma }_1-{\delta }_0)\right]}^{\frac{{\nu }_1-1}{{\nu }_1}}{\left[({\iota }_2-{\delta }_0)({\varsigma }_2-{\delta }_0)\right]}^{\frac{{\varpi }_1-1}{{\varpi }_1}}\hfill \\ & & \times {\left({\int }_{{\delta }_0}^{{\iota }_1}{\int }_{{\delta }_0}^{{\varsigma }_1}(\rho \left({\iota }_1\right)-\rho \left({\Im }_1\right))(\rho \left({\varsigma }_1\right)-\rho \left({\delta }_1\right)){\left|{\vartheta }^{{\nabla }_2^{(\gamma ,a)}{\nabla }_1^{(\gamma ,a)}}({\Im }_1,{\delta }_1)\right|}^{{\nu }_1}{\nabla }^{(\gamma ,a)}{\Im }_1{\nabla }^{(\gamma ,a)}{\delta }_1\right)}^{\frac{1}{{\nu }_1}}\hfill \\ \hfill & & {\left({\int }_{{\delta }_0}^{{\iota }_2}{\int }_{{\delta }_0}^{{\varsigma }_2}(\rho \left({\iota }_2\right)-{\delta }_0)(\rho \left({\varsigma }_2\right)-{\delta }_0){\left|{\vartheta }^{{\nabla }_2^{(\gamma ,a)}{\nabla }_1^{(\gamma ,a)}}({\Im }_2,{\delta }_2)\right|}^{{\nu }_2}{\nabla }^{(\gamma ,a)}{\Im }_2{\nabla }^{(\gamma ,a)}{\delta }_2\right)}^{\frac{1}{{\nu }_2}}.\hfill \end{array}$$

(34)

Remark 11.

In Corollary 4, if $\mathbb{T}=\mathbb{R},\gamma =1$, we obtain an equivalent formulation of the inequality that Pachpatte presented in ([27], Theorem 4).

Remark 12.

In Corollary 4, if $\mathbb{T}=\mathbb{Z},\gamma =1$, we obtain an equivalent formulation of the inequality that Pachpatte presented in ([27], Theorem 3).

Theorem 7.

Let $\mathbb{T}$ be a time scale with ${\delta }_0,$ ${\iota }_{ij},$ ${\tau }_{ij},$ ${\delta }_{ij}\in \mathbb{T},$ $(i,j=1,\dots ,n).$ Let ${\nu }_i,{\varpi }_i>1,$ be constants and $\frac{1}{{\nu }_i}+\frac{1}{{\varpi }_i}=1.$ Let ${\vartheta }_i({\tau }_{1i},\dots ,{\tau }_{ni})$ be real-valued nth ${\nabla }^{(\gamma ,a)}$-differentiable functions also defined on ${[{\delta }_0,{\iota }_{1i})}_{\mathbb{T}}\times \dots \times {[{\delta }_0,{\iota }_{ni})}_{\mathbb{T}},$ where ${\delta }_0⩽{\iota }_{ji}⩽{\delta }_{ji},$ ${\delta }_{ji}\in (0,\infty )$ and $i,j=1,\dots ,n.$ Suppose

$${\vartheta }_i({\iota }_{1i},\dots ,{\iota }_{ni})={\int }_{{\delta }_0}^{{\iota }_{1i}}\dots {\int }_{{\delta }_0}^{{\iota }_{ni}}\frac{{\partial }^n}{{\nabla }^{(\gamma ,a)}{\tau }_{1i}\dots {\nabla }^{(\gamma ,a)}{\tau }_{ni}}{\vartheta }_i({\tau }_{1i},\dots ,{\tau }_{ni}){\nabla }^{(\gamma ,a)}{\tau }_{ni}\dots {\nabla }^{(\gamma ,a)}{\tau }_{1i},$$

then

$$\begin{array}{ccc}& & {\int }_{{\delta }_0}^{{\delta }_{11}}\dots {\int }_{{\delta }_0}^{{\delta }_{n1}}{\int }_{{\delta }_0}^{{\delta }_{12}}\dots {\int }_{{\delta }_0}^{{\delta }_{n2}}\dots {\int }_{{\delta }_0}^{{\delta }_{1n}}\dots {\int }_{{\delta }_0}^{{\delta }_{nn}}\hfill \\ \hfill & & {\displaystyle \frac{{\prod }_{i=1}^n{\left({\int }_{{\delta }_0}^{{\iota }_{1i}}\dots {\int }_{{\delta }_0}^{{\iota }_{ni}}|\frac{{\partial }^n}{{\nabla }^{(\gamma ,a)}{\tau }_{1i}\dots {\nabla }^{(\gamma ,a)}{\tau }_{ni}}{\vartheta }_i({\tau }_{1i}\dots {\tau }_{ni}){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\tau }_{ni}\dots {\nabla }^{(\gamma ,a)}{\tau }_{1i}\right)}^{\frac{1}{{\nu }_i}}}{{\left({\sum }_{i=1}^n\frac{\left[({\iota }_{1i}-{\delta }_0)\dots ({\iota }_{ni}-{\delta }_0)\right]}{{\varpi }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}}}\hfill \\ \hfill & & {\nabla }^{(\gamma ,a)}{\iota }_{11}\dots {\nabla }^{(\gamma ,a)}{\iota }_{n1}\dots {\nabla }^{(\gamma ,a)}{\iota }_{12}\dots {\nabla }^{(\gamma ,a)}{\iota }_{n2}\dots {\nabla }^{(\gamma ,a)}{\iota }_{1n}\dots {\nabla }^{(\gamma ,a)}{\iota }_{nn}\hfill \\ \hfill & & ⩽N\prod _{i=1}^n{\left({\int }_{{\delta }_0}^{{\delta }_{1i}}\dots {\int }_{{\delta }_0}^{{\delta }_{ni}}\prod _{j=1}^n(\rho \left({\delta }_{ji}\right)-{\iota }_{ji})|\frac{{\partial }^n}{{\nabla }^{(\gamma ,a)}{\iota }_{1i}\dots {\nabla }^{(\gamma ,a)}{\iota }_{ni}}{\vartheta }_i({\iota }_{1i},\dots ,{\iota }_{ni}){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\iota }_{1i}\dots {\nabla }^{(\gamma ,a)}{\iota }_{ni}\right)}^{\frac{1}{{\nu }_i}},\hfill \end{array}$$

(35)

where

$$N=N({\delta }_{1i},\dots ,{\delta }_{ni}){\left(n-\sum _{i=1}^n\frac{1}{{\nu }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\nu }_i}-n}\prod _{i=1}^n{\left[({\delta }_{1i}-{\delta }_0)\dots ({\delta }_{ni}-{\delta }_0)\right]}^{\frac{1}{{\varpi }_i}}.$$

Proof. 

From the hypothesis of Theorem 7, we have

$$|{\vartheta }_i({\iota }_{1i},\dots ,{\iota }_{ni})|⩽{\int }_{{\delta }_0}^{{\iota }_{1i}}\dots {\int }_{{\delta }_0}^{{\iota }_{ni}}\left|\frac{{\partial }^n}{{\nabla }^{(\gamma ,a)}{\tau }_{1i}\dots {\nabla }^{(\gamma ,a)}{\tau }_{ni}}{\vartheta }_i({\tau }_{1i},\dots ,{\tau }_{ni})\right|{\nabla }^{(\gamma ,a)}{\tau }_{ni}\dots {\nabla }^{(\gamma ,a)}{\tau }_{1i}.$$

(36)

On the other hand, by using (19) and Hölder’s dynamic inequality, we obtain

$$\begin{array}{ccc}\hfill & & \prod _{i=1}^n\left|{\vartheta }_i({\iota }_{1i},\dots ,{\iota }_{ni})\right|\hfill \\ \hfill & & ⩽\prod _{i=1}^n{\int }_{{\delta }_0}^{{\iota }_{1i}}\dots {\int }_{{\delta }_0}^{{\iota }_{ni}}\left|\frac{{\partial }^n}{{\nabla }^{(\gamma ,a)}{\tau }_{1i}\dots {\nabla }^{(\gamma ,a)}{\tau }_{ni}}{\vartheta }_i({\tau }_{1i},\dots ,{\tau }_{ni})\right|{\nabla }^{(\gamma ,a)}{\tau }_{ni}\dots {\nabla }^{(\gamma ,a)}{\tau }_{1i}\hfill \\ \hfill & & ⩽\prod _{i=1}^n{\left[({\iota }_{1i}-{\delta }_0)\dots ({\iota }_{ni}-{\delta }_0)\right]}^{\frac{1}{{\varpi }_i}}\hfill \\ \hfill & & \times {\left({\int }_{{\delta }_0}^{{\iota }_{1i}}\dots {\int }_{{\delta }_0}^{{\iota }_{ni}}|\frac{{\partial }^n}{{\nabla }^{(\gamma ,a)}{\tau }_{1i},\dots ,{\nabla }^{(\gamma ,a)}{\tau }_{ni}}{\vartheta }_i({\tau }_{1i},\dots ,{\tau }_{ni}){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\tau }_{ni}\dots {\nabla }^{(\gamma ,a)}{\tau }_{1i}\right)}^{\frac{1}{{\nu }_i}}\hfill \\ \hfill & & ⩽\frac{{\left({\sum }_{i=1}^n\frac{\left[({\iota }_{1i}-{\delta }_0)\dots ({\iota }_{ni}-{\delta }_0)\right]}{{\varpi }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}}{{\left(n-{\sum }_{i=1}^n\frac{1}{{\nu }_i}\right)}^{n-{\sum }_{i=1}^n\frac{1}{{\nu }_i}}}\hfill \\ & & \times \prod _{i=1}^n{\left({\int }_{{\delta }_0}^{{\iota }_{1i}}\dots {\int }_{{\delta }_0}^{{\iota }_{ni}}|\frac{{\partial }^n}{{\nabla }^{(\gamma ,a)}{\tau }_{1i}\dots {\nabla }^{(\gamma ,a)}{\tau }_{ni}}{\vartheta }_i({\tau }_{1i},\dots ,{\tau }_{ni}){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\tau }_{ni}\dots {\nabla }^{(\gamma ,a)}{\tau }_{1i}\right)}^{\frac{1}{{\nu }_i}}.\hfill \end{array}$$

(37)

Divide (37) by ${\left({\sum }_{i=1}^n\frac{\left[({\iota }_{1i}-{\delta }_0)\dots ({\iota }_{ni}-{\delta }_0)\right]}{{\varpi }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}},$ and then integrate it over ${\iota }_{ji}$ from ${\delta }_0$ to ${\delta }_{ji}$ $(i,j=1,\dots ,n),$ respectively; using the dynamic Hölder inequality and using the information $\rho \left(n\right)⩾n,$ we obtain

$$\begin{array}{ccc}\hfill & & {\int }_{{\delta }_0}^{{\delta }_{11}}\dots {\int }_{{\delta }_0}^{{\delta }_{n1}}{\int }_{{\delta }_0}^{{\delta }_{12}}\dots {\int }_{{\delta }_0}^{{\delta }_{n2}}\dots {\int }_{{\delta }_0}^{{\delta }_{1n}}\dots {\int }_{{\delta }_0}^{{\delta }_{nn}}\hfill \\ \hfill & & {\displaystyle \frac{{\prod }_{i=1}^n{\left({\int }_{{\delta }_0}^{{\iota }_{1i}}\dots {\int }_{{\delta }_0}^{{\iota }_{ni}}|\frac{{\partial }^n}{{\nabla }^{(\gamma ,a)}{\tau }_{1i}\dots {\nabla }^{(\gamma ,a)}{\tau }_{ni}}{\vartheta }_i({\tau }_{1i},\dots ,{\tau }_{ni}){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\tau }_{ni}\dots {\nabla }^{(\gamma ,a)}{\tau }_{1i}\right)}^{\frac{1}{{\nu }_i}}}{{\left({\sum }_{i=1}^n\frac{\left[({\iota }_{1i}-{\delta }_0)\dots ({\iota }_{ni}-{\delta }_0)\right]}{{\varpi }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}}}\hfill \\ \hfill & & {\nabla }^{(\gamma ,a)}{\iota }_{11}\dots {\nabla }^{(\gamma ,a)}{\iota }_{n1}{\nabla }^{(\gamma ,a)}{\iota }_{12}\dots {\nabla }^{(\gamma ,a)}{\iota }_{n2}\dots {\nabla }^{(\gamma ,a)}{\iota }_{1n}\dots {\nabla }^{(\gamma ,a)}{\iota }_{nn}\hfill \\ \hfill & & ⩽{\left(n-\sum _{i=1}^n\frac{1}{{\nu }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\nu }_i}-n}\hfill \\ \hfill & & \times \prod _{i=1}^n{\int }_{{\delta }_0}^{{\delta }_{1i}}\dots {\int }_{{\delta }_0}^{{\delta }_{ni}}{\left({\int }_{{\delta }_0}^{{\iota }_{1i}}\dots {\int }_{{\delta }_0}^{{\iota }_{ni}}|\frac{{\partial }^n}{{\nabla }^{(\gamma ,a)}{\tau }_{1i},\dots ,{\nabla }^{(\gamma ,a)}{\tau }_{ni}}{\vartheta }_i({\tau }_{1i},\dots ,{\tau }_{ni}){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\tau }_{ni}\dots {\nabla }^{(\gamma ,a)}{\tau }_{1i}\right)}^{\frac{1}{{\nu }_i}}{\nabla }^{(\gamma ,a)}{\iota }_{ni}\dots {\nabla }^{(\gamma ,a)}{\iota }_{1i}\hfill \\ \hfill & & ⩽{\left(n-\sum _{i=1}^n\frac{1}{{\nu }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\nu }_i}-n}\prod _{i=1}^n{\left[({\delta }_{1i}-{\delta }_0)\dots ({\delta }_{ni}-{\delta }_0)\right]}^{\frac{1}{{\varpi }_i}}\hfill \\ \hfill & & {\left({\int }_{{\delta }_0}^{{\delta }_{1i}}\dots {\int }_{{\delta }_0}^{{\delta }_{ni}}\left({\int }_{{\delta }_0}^{{\iota }_{1i}}\dots {\int }_{{\delta }_0}^{{\iota }_{ni}}|\frac{{\partial }^n}{{\nabla }^{(\gamma ,a)}{\tau }_{1i}\dots {\nabla }^{(\gamma ,a)}{\tau }_{ni}}{\vartheta }_i({\tau }_{1i},\dots ,{\tau }_{ni}){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\tau }_{ni}\dots {\nabla }^{(\gamma ,a)}{\tau }_{1i}\right){\nabla }^{(\gamma ,a)}{\iota }_{ni}\dots {\nabla }^{(\gamma ,a)}{\iota }_{1i}\right)}^{\frac{1}{{\nu }_i}}\hfill \\ \hfill & & =N\prod _{i=1}^n{\left({\int }_{{\delta }_0}^{{\delta }_{1i}}\dots {\int }_{{\delta }_0}^{{\delta }_{ni}}\prod _{j=1}^n({\delta }_{ji}-{\iota }_{ji})|\frac{{\partial }^n}{{\nabla }^{(\gamma ,a)}{\iota }_{1i}\dots {\nabla }^{(\gamma ,a)}{\iota }_{ni}}{\vartheta }_i({\iota }_{1i},\dots ,{\iota }_{ni}){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\iota }_{ni}\dots {\nabla }^{(\gamma ,a)}{\iota }_{1i}\right)}^{\frac{1}{{\nu }_i}}\hfill \\ \hfill & & ⩽N\prod _{i=1}^n{\left({\int }_{{\delta }_0}^{{\delta }_{1i}}\dots {\int }_{{\delta }_0}^{{\delta }_{ni}}\prod _{j=1}^n(\rho \left({\delta }_{ji}\right)-{\iota }_{ji})|\frac{{\partial }^n}{{\nabla }^{(\gamma ,a)}{\iota }_{1i}\dots {\nabla }^{(\gamma ,a)}{\iota }_{ni}}{\vartheta }_i({\iota }_{1i},\dots ,{\iota }_{ni}){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\iota }_{ni}\dots {\nabla }^{(\gamma ,a)}{\iota }_{1i}\right)}^{\frac{1}{{\nu }_i}}.\hfill \end{array}$$

This concludes the evidence. □

Remark 13.

In Theorem 7, supposing $\mathbb{Z}=\mathbb{T},andwith \gamma =1,$ we obtain ([3], Theorem 2.1).

Remark 14.

In Theorem 7, supposing $\mathbb{R}=\mathbb{T},andwith \gamma =1$, we obtain ([3], Theorem 2.2).

Corollary 5.

Let ${\vartheta }_i({\iota }_{1i},\dots ,{\iota }_{ni})$ change to ${\vartheta }_i\left({\Im }_i\right)$ in Theorem 7 and in view of ${\vartheta }_i\left({\delta }_0\right)=0,$ $(i=1,\dots ,n),$ and then

$$\begin{array}{ccc}\hfill & & {\int }_{{\delta }_0}^{{\iota }_1}{\int }_{{\delta }_0}^{{\iota }_2}\dots {\int }_{{\delta }_0}^{{\iota }_n}{\displaystyle \frac{{\prod }_{i=1}^n\left|{\vartheta }_i\left({\Im }_i\right)\right|}{{\left({\sum }_{i=1}^n\frac{({\Im }_i-{\delta }_0)}{{\varpi }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\varpi }_i}}}}{\nabla }^{(\gamma ,a)}{\Im }_n{\nabla }^{(\gamma ,a)}{\Im }_{n-1}\dots {\nabla }^{(\gamma ,a)}{\Im }_1\hfill \\ & & ⩽R\prod _{i=1}^n{\left({\int }_{{\delta }_0}^{{\iota }_i}(\rho \left({\iota }_i\right)-\rho \left({\Im }_i\right))|{\vartheta }_i^{{\nabla }^{(\gamma ,a)}}\left({\Im }_i\right){|}^{{\nu }_i}{\nabla }^{(\gamma ,a)}{\tau }_i{\nabla }^{(\gamma ,a)}{\Im }_i\right)}^{\frac{1}{{\nu }_i}},\hfill \end{array}$$

(38)

where

$$R={\left(n-\sum _{i=1}^n\frac{1}{{\nu }_i}\right)}^{{\sum }_{i=1}^n\frac{1}{{\nu }_i}-n}\prod _{i=1}^n{({\iota }_i-{\delta }_0)}^{\frac{1}{{\varpi }_i}}.$$

Remark 15.

Taking $n=2,$ in Corollary 5, if ${\nu }_1,$ ${\nu }_2>1$ are such that $\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}⩾1$ and $0<\lambda =2-\frac{1}{{\nu }_1}-\frac{1}{{\nu }_2}=\frac{1}{{\varpi }_1}+\frac{1}{{\varpi }_2}⩽1,$ inequality (38) reduces to inequality (22).

3. Conclusions

In this work, we used Holder’s inequality to prove a number of Hilbert’s inequalities on the time scale. Some integer and discrete inequalities were obtained as special cases of the results. This work builds on the multiple inequalities reported by Pachpatte in 1998 and 2000 and by Handley et al. and by Zhao et al. in 2012. Moreover, as a future work, we intend to extend these inequalities by 123 using a-conformable calculus and also by employing alpha-conformable calculus on time scales. Moreover, we will try to obtain the diamond alpha version for these results.