On Some New Dynamic Hilbert-Type Inequalities across Time Scales

Abstract

In this article, we present some novel dynamic Hilbert-type inequalities within the framework of time scales $\mathbb{T}$. We achieve this by utilizing Hölder’s inequality, the chain rule, and the mean inequality. As specific instances of our findings (when $\mathbb{T}=\mathbb{N}$ and $\mathbb{T}=\mathbb{R}$), we obtain the discrete and continuous analogues of previously established inequalities. Additionally, we derive other inequalities for different time scales, such as $\mathbb{T}=q^{{\mathbb{N}}_0}$ for $q>1,$ which, to the best of the authors’ knowledge, is a largely novel conclusion.

1. Introduction

In the early 1900s, the renowned mathematician David Hilbert formulated his famous inequality, known as the double series Hilbert inequality (see [1]), wherein he established that if ${\left\{G_m\right\}}_{m=1}^{\infty },$ ${\left\{Z_n\right\}}_{n=1}^{inf}$ are two real sequences, such that $0<{\sum }_{m=1}^{\infty }{G}_m^2<\infty$ and $0<{\sum }_{n=1}^{\infty }{Z}_n^2<\infty ,$ then

$$\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }\frac{G_m{Z}_n}{m+n}\le 2\pi {\left(\sum _{m=1}^{\infty }{G}_m^2\right)}^{\frac{1}{2}}{\left(\sum _{n=1}^{\infty }{Z}_n^2\right)}^{\frac{1}{2}}.$$

(1)

In 1911, Schur demonstrated in his paper [2] that the constant $\pi$ in (1) is optimal. Furthermore, he established the integral analogue of (1), which is now recognized as the Hilbert integral inequality in the form

$${\int }_0^{\infty }{\int }_0^{\infty }\frac{S\left(x\right)T\left(y\right)}{x+y}dxdy\le \pi {\left({\int }_0^{\infty }{S}^2\left(x\right)dx\right)}^{\frac{1}{2}}{\left({\int }_0^{\infty }{T}^2\left(y\right)dy\right)}^{\frac{1}{2}},$$

(2)

where S and T are real functions such that $0<{\int }_0^{inf}{S}^2\left(x\right)dx<\infty ,$ $0<{\int }_0^{\infty }{T}^2\left(y\right)dy<\infty$, and $\pi$ in (2) is still the best possible constant factor.

The inequalities expressed as (1) and (2) are crucial in the theory and application of integral inequalities, especially in analyzing both the qualitative and quantitative aspects of solutions to differential and integral equations. Recently, there has been rapid development in fractal theory, which has found widespread use in science and engineering. Some researchers have utilized fractal theory and weight function methods to generalize classical inequalities effectively. For instance, Liu [3] established a Hilbert-type integral inequality and its equivalent form on a fractal set. Hilbert-type inequalities play a significant role in mathematics, particularly in complex and numerical analysis. Over the years, these inequalities have seen numerous refinements, generalizations, extensions, and applications in the literature (see [4,5,6,7]).

In 1925, Hardy [8] extended (1) by introducing a pair of conjugate exponents $(\eta ,\lambda )$, where $\eta ,\lambda >1$ and satisfying $1/\eta +1/\lambda =1,$ as follows. If $G_m,$ $Z_n\ge 0$, such that $0<{\sum }_{m=1}^{\infty }{G}_m^{\eta }<\infty$, and $0<{\sum }_{n=1}^{\infty }{Z}_n^{\lambda }<\infty ,$ then

$$\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }\frac{G_m{Z}_n}{m+n}\le \frac{\pi }{\sin \frac{\pi }{\eta }}{\left(\sum _{m=1}^{\infty }{G}_m^{\eta }\right)}^{\frac{1}{\eta }}{\left(\sum _{n=1}^{\infty }{Z}_n^{\lambda }\right)}^{\frac{1}{\lambda }}.$$

(3)

In [9], the authors established the equivalent integral form of (3) as

$${\int }_0^{\infty }{\int }_0^{\infty }\frac{S\left(x\right)T\left(y\right)}{x+y}dxdy\le \frac{\pi }{sin\frac{\pi }{\eta }}{\left({\int }_0^{\infty }{S}^{\eta }\left(x\right)dx\right)}^{\frac{1}{\eta }}{\left({\int }_0^{\infty }{T}^{\lambda }\left(y\right)dy\right)}^{\frac{1}{\lambda }},$$

(4)

where $S,$ $T\ge 0$, such that $0<{\int }_0^{\infty }{S}^p\left(x\right)dx<\infty$ and $0<{\int }_0^{\infty }{T}^{\lambda }\left(y\right)dy<\infty .$ The constant factor $\pi /sin(\pi /\eta )$ in (3) and (4) is optimal.

In 1998, Pachpatte [10] presented a new inequality akin to the Hilbert inequality as follows: let $G\left(s\right):\left\{0,1,2,\dots ,p\right\}\subset \mathbb{N}\to \mathbb{R}$ and $Z\left(\vartheta \right):\left\{0,1,2,\dots ,q\right\}\subset \mathbb{N}\to \mathbb{R}$ with $G\left(0\right)=Z\left(0\right)=0.$ Define the operators as $\nabla G_s=G_s-G_{s-1},$ $\nabla Z_{\vartheta }=Z_{\vartheta }-Z_{\vartheta -1}.$ Then

$$\begin{array}{ccc}\hfill \sum _{s=1}^p\sum _{\vartheta =1}^q\frac{\left|G_s\right|\left|Z_{\vartheta }\right|}{s+\vartheta }& \le & \frac{1}{2}\sqrt{pq}{\left(\sum _{s=1}^p(p-s+1){\left|\nabla G_s\right|}^2\right)}^{\frac{1}{2}}\hfill \\ & & \times {\left(\sum _{\vartheta =1}^q(q-\vartheta +1){\left|\nabla Z_{\vartheta }\right|}^2\right)}^{\frac{1}{2}}.\hfill \end{array}$$

(5)

In 2000, Pachpatte [11] generalized (5) by introducing one pair of conjugate exponents $(\eta ,\mu ):\eta ,\mu >1$ with $1/\eta +1/\mu =1$ and proved that if $G\left(s\right):\left\{0,1,2,\dots ,p\right\}\subset \mathbb{N}\to \mathbb{R}$ and $Z\left(\vartheta \right):\left\{0,1,2,\dots ,q\right\}\subset \mathbb{N}\to \mathbb{R}$ with $G\left(0\right)=Z\left(0\right)=0,$ then

$$\begin{array}{ccc}\hfill \sum _{s=1}^p\sum _{\vartheta =1}^q\frac{\left|G_s\right|\left|Z_{\vartheta }\right|}{\mu s^{\eta -1}+\eta {\vartheta }^{\mu -1}}& \le & \frac{1}{\eta \mu }{p}^{\frac{\eta -1}{\eta }}{q}^{\frac{\mu -1}{\mu }}{\left(\sum _{s=1}^p(p-s+1){\left|\nabla G_s\right|}^{\eta }\right)}^{\frac{1}{\eta }}\hfill \\ & & \times {\left(\sum _{\vartheta =1}^q(q-\vartheta +1){\left|\nabla Z_{\vartheta }\right|}^{\mu }\right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(6)

In 2002, Kim et al. [12] extended (6) and demonstrated that if $\eta ,\mu >1,$ $G\left(s\right):\left\{0,1,2,\dots ,p\right\}\subset \mathbb{N}\to \mathbb{R}$ and $Z\left(\vartheta \right):\left\{0,1,2,\dots ,q\right\}\subset \mathbb{N}\to \mathbb{R}$ with $G\left(0\right)=Z\left(0\right)=0,$ then

$$\begin{array}{ccc}\hfill \sum _{s=1}^p\sum _{\vartheta =1}^q\frac{\left|G_s\right|\left|Z_{\vartheta }\right|}{\mu s^{\frac{\left(\eta -1\right)\left(\eta +\mu \right)}{\eta \mu }}+\eta {\vartheta }^{\frac{\left(\mu -1\right)\left(\eta +\mu \right)}{\eta \mu }}}& \le & \frac{1}{\eta +\mu }{p}^{\frac{\eta -1}{\eta }}{q}^{\frac{\mu -1}{\mu }}{\left(\sum _{s=1}^p(p-s+1){\left|\nabla G_s\right|}^{\eta }\right)}^{\frac{1}{\eta }}\hfill \\ & & \times {\left(\sum _{\vartheta =1}^q(q-\vartheta +1){\left|\nabla Z_{\vartheta }\right|}^{\mu }\right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(7)

Also, the authors [12] established the continuous analogue of (7) as follows: if $\eta ,\mu >1$ and $S\left(\theta \right),$ and $T\left(\vartheta \right)$ are real continuous functions on $\left(0,x\right),$ $\left(0,y\right),$ respectively, with $S\left(0\right)=T\left(0\right)=0,$ then for $x,y\in \left(0,\infty \right),$ we have

$$\begin{array}{ccc}& & {\int }_0^x{\int }_0^y\frac{\left|S\left(\theta \right)\right|\left|T\left(\vartheta \right)\right|}{\mu {\theta }^{\frac{\left(\eta -1\right)\left(\eta +\mu \right)}{\eta \mu }}+\eta {\vartheta }^{\frac{\left(\mu -1\right)\left(\eta +\mu \right)}{\eta \mu }}}d\theta d\vartheta \hfill \\ & \le & \frac{1}{\eta +\mu }{x}^{\frac{\eta -1}{\eta }}{y}^{\frac{\mu -1}{\mu }}{\left({\int }_0^x\left(x-\theta \right){\left|S^{\prime }\left(\theta \right)\right|}^{\eta }d\theta \right)}^{\frac{1}{\eta }}{\left({\int }_0^y\left(y-\vartheta \right){\left|T^{\prime }\left(\vartheta \right)\right|}^{\mu }d\vartheta \right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

In 2011, Chang-Jian et al. [13] generalized (5) and demonstrated that if ${\lambda }_i>1$, such that $1/{\lambda }_i+1/q_i=1$, $G_i\left(s_i\right)$ is a real sequence defined for $s_i=0,1,2,\dots ,m_i,$ where $m_i$ is a natural number and $G_i\left(0\right)=0,$ $i=1,2,\dots ,n.$. Define the operator ∇ by $\nabla G_i\left(s_i\right)=G_i\left(s_i\right)-G_i(s_i-1)$ for any function $G_i\left(s_i\right),$ $i=1,2,\dots ,n.$ Then

$$\begin{array}{ccc}& & \sum _{s_1=1}^{m_1}\sum _{s_2=1}^{m_2}\dots \sum _{s_n=1}^{m_n}\frac{{\prod }_{i=1}^n\left|G_i\left(s_i\right)\right|}{{\left({\displaystyle \sum _{i=1}^n}{s}_i/q_i\right)}^{{\displaystyle \sum _{i=1}^n}1/q_i}}\hfill \\ & \le & M\prod _{i=1}^n{\left({\displaystyle \sum _{s_i=1}^{m_i}}\left(m_i-s_i+1\right){\left|\nabla G_i\left(s_i\right)\right|}^{{\lambda }_i}\right)}^{\frac{1}{{\lambda }_i}},\hfill \end{array}$$

(8)

where

$$M={\left(n-\sum _{i=1}^n\frac{1}{{\lambda }_i}\right)}^{{\sum }_{i=1}^{n}1/{\lambda }_i-n}.\prod _{i=1}^n{m}_i^{1/q_i}.$$

Also, the authors of [13] proved that if $h_i\ge 1$ and ${\lambda }_i>1$ are constants such that $1/{\lambda }_i+1/q_i=1$, $T_i\left(s_i\right)$ is a real valued differentiable function defined on $[0,x_i),$ where $x_i\in \left(0,\infty \right).$ Assume $T_i\left(0\right)=0$ for $i=1,2,\dots ,n.$ Then

$$\begin{array}{ccc}& & {\int }_0^{x_1}\dots {\int }_0^{x_n}\frac{{\prod }_{i=1}^n\left|T_i^{h_i}\left(s_i\right)\right|}{{\left({\displaystyle \sum _{i=1}^n}{s}_i/q_i\right)}^{{\displaystyle \sum _{i=1}^n}1/q_i}}d{s}_n\dots d{s}_1\hfill \\ & \le & K\prod _{i=1}^n{\left({\int }_0^{x_i}\left(x_i-s_i\right){\left|T_i^{h_i-1}\left(s_i\right).T_i^{\prime }\left(s_i\right)\right|}^{{\lambda }_i}d{s}_i\right)}^{\frac{1}{{\lambda }_i}},\hfill \end{array}$$

(9)

where

$$K={\left(n-\sum _{i=1}^n\frac{1}{{\lambda }_i}\right)}^{{\sum }_{i=1}^{n}1/{\lambda }_i-n}.\prod _{i=1}^n{h}_i{x}_i^{1/q_i}.$$

Also, they demonstrated that if ${\lambda }_i,q_i>1$, such that $1/{\lambda }_i+1/q_i=1$, $G_i(s_i,t_i)$ is a real sequence defined for $\left(s_i,t_i\right),$ where $s_i=0,1,2,\dots ,m_i,$ $t_i=0,1,2,\dots ,n_i$ and $m_i,n_i$ $\left(i=1,2,\dots ,n\right)$ are natural numbers and assuming that $G_i(0,t_i)=G_i(s_i,0)=0$ for all $i=1,2,\dots ,n.$ Define the operator ${\nabla }_1$ and ${\nabla }_2$ as

$$H{\nabla }_1{G}_i(s_i,t_i)=G_i(s_i,t_i)-G_i(s_i-1,t_i),$$

$${\nabla }_2{G}_i(s_i,t_i)=G_i(s_i,t_i)-G_i(s_i,t_i-1).$$

Then

$$\begin{array}{ccc}& & \sum _{s_1=1}^{m_1}\sum _{t_1=1}^{n_1}\dots \sum _{s_n=1}^{m_n}\sum _{t_n=1}^{n_n}\frac{{\prod }_{i=1}^n\left|G_i(s_i,t_i)\right|}{{\left({\displaystyle \sum _{i=1}^n}{s}_i{t}_i/q_i\right)}^{{\displaystyle \sum _{i=1}^n}1/q_i}}\hfill \\ & \le & L\prod _{i=1}^n{\left(\sum _{s_i=1}^{m_i}\sum _{t_i=1}^{n_i}\left(m_i-s_i+1\right)\left(n_i-t_i+1\right){\left|{\nabla }_2{\nabla }_1{G}_i(s_i,t_i)\right|}^{{\lambda }_i}\right)}^{\frac{1}{{\lambda }_i}},\hfill \end{array}$$

(10)

where

$$L={\left(n-\sum _{i=1}^n\frac{1}{{\lambda }_i}\right)}^{{\sum }_{i=1}^{n}1/{\lambda }_i-n}.\prod _{i=1}^n{\left(m_i{n}_i\right)}^{1/q_i}.$$

In the last few decades, much attention has been devoted to establishing discrete analogues of the corresponding continuous results in various fields of analysis. This appears along with establishing a dynamic inequality in this paper by using a general domain called a time scale $\mathbb{T}$. A time scale $\mathbb{T}$ is an arbitrary non-empty closed subset of the real numbers $\mathbb{R}$. For more details about dynamic inequalities and applications on time scales, see [14,15,16,17,18,19].

The aim of this paper is to prove similar analogues of the inequalities (8), (9) on time scales, and we can also generalize (10) on time scale delta calculus for an increasing function by establishing some new dynamic Hilbert-type inequalities on time scale delta calculus.

The remainder of this paper is organized as follows. In Section 2, we show some basics of the time scale theory and some lemmas on time scales needed in Section 3, where we prove our results. These results as special cases when $\mathbb{T}=\mathbb{N}$ and $\mathbb{T}=\mathbb{R}$ give the inequalities ((8) and (10)), (9), respectively. Also, we can obtain other inequalities on different time scales, like $\mathbb{T}=q^{\mathbb{N}}$ for $q>1.$

2. Preliminaries and Basic Lemmas

In 2001, Bohner and Peterson [20] defined the forward jump operator by $\sigma \left(\tau \right):=inf\{s\in \mathbb{T}:s>\tau \}$. For any function $S:\mathbb{T}\to \mathbb{R}$, the notation $S^{\sigma }\left(\tau \right)$ denotes $S\left(\sigma \right(\tau \left)\right)$. We define the time scale interval ${[a,b]}_{\mathbb{T}}$ by ${[a,b]}_{\mathbb{T}}:=[a,b]\cap \mathbb{T}.$

In the following, we state the definition of rd−continuous and Δ−derivative function.

Definition 1

([20]). A function $S:\mathbb{T}\to \mathbb{R}$ is called rd−continuous provided it is continuous at right-dense points in $\mathbb{T}$ and its left-sided limits exist (finite) at left-dense points in $\mathbb{T}$. The set of rd−continuous functions $S:\mathbb{T}\to \mathbb{R}$ is denoted by ${\mathrm{C}}_{rd}(\mathbb{T},\mathbb{R})$.

Definition 2

([20]). Assume that $S:\mathbb{T}\to \mathbb{R}$ is a function and let $t\in \mathbb{T}.$ We define $S^{\Delta }\left(t\right)$ to be the number, provided it exists, as follows: for any $\epsilon >0$, there is a neighborhood U of $t,$ $U=(t-\delta ,$ $t+\delta )\cap \mathbb{T}$ for some $\delta >0$, such that

$$|S\left(\sigma \left(t\right)\right)-S\left(\gamma \right)-S^{\Delta }\left(t\right)(\sigma \left(t\right)-\gamma )|\le \epsilon |\sigma \left(t\right)-\gamma |\mathit{for}\mathit{all}\gamma \in U,\gamma \ne \sigma \left(t\right).$$

In this case, we say $S^{\Delta }\left(t\right)$ is the delta or Hilger derivative of S at t.

In the following, we state several values of $\Delta$—differentiable function at a point $t\in \mathbb{T}$.

Theorem 1

([20]). Assume $S:\mathbb{T}\to \mathbb{R}$ is a function and let $t\in {\mathbb{T}}^k$. Then we have the following.

1.

If S is differentiable at t, then f is continuous at t.

2.

If S is continuous at t and t is right-scattered, then S is differentiable at t with

$$S^{\Delta }\left(t\right)=\frac{S\left(\sigma \right(t\left)\right)-S\left(t\right)}{\sigma \left(t\right)-t}.$$

3.

If t is right-dense, then S is differentiable if the limit

$$\underset{\gamma \to t}{lim}\frac{S\left(t\right)-S\left(\gamma \right)}{t-\gamma },$$

exists as a finite number. In this case,

$$S^{\Delta }\left(t\right)=\underset{\gamma \to t}{lim}\frac{S\left(t\right)-S\left(\gamma \right)}{t-\gamma }.$$

Example 1.

1. If $\mathbb{T}=\mathbb{R},$ then for $S:\mathbb{R}\to \mathbb{R},$ we obtain

$$S^{\Delta }\left(t\right)=\underset{\gamma \to t}{\lim }\frac{S\left(t\right)-S\left(\gamma \right)}{t-\gamma }=S^{\prime }\left(t\right),$$

where $S^{\prime }$ is the usual derivative.

2. If $\mathbb{T}=\mathbb{N},$ then $\sigma \left(t\right)=t+1$, and for $S:\mathbb{N}\to \mathbb{R},$ we have

$$S^{\Delta }\left(t\right)=\frac{S\left(\sigma \right(t\left)\right)-S\left(t\right)}{\sigma \left(t\right)-t}=\frac{S(t+1)-S\left(t\right)}{1}=\Delta S\left(t\right),$$

where Δ is the usual forward difference operator.

3. If $\mathbb{T}=\{t:t=q^k,$ $k\in {\mathbb{N}}_0,$ $q>1\}$, then we have $\sigma \left(t\right)=qt$ and

$$S^{\Delta }\left(t\right)={\Delta }_{q}S\left(t\right)=\frac{S\left(qt\right)-S\left(t\right)}{(q-1)t}.$$

The following theorem is about the chain rule formula on time scales.

Theorem 2

(Chain Rule [20] Theorem 1.87). Assume $T:\mathbb{R}\to \mathbb{R}$ is continuous, $T:\mathbb{T}\to \mathbb{R}$ is delta-differentiable on $\mathbb{T}$ and $S:\mathbb{R}\to \mathbb{R}$ is continuously differentiable. Then γ exists in the real interval $[t,\sigma (t\left)\right]$ with

$${\left(S\circ T\right)}^{\Delta }\left(t\right)=S^{\prime }\left(T\left(\gamma \right)\right)T^{\Delta }\left(t\right).$$

(11)

Definition 3

([20]). A function $S:\mathbb{T}\to \mathbb{R}$ is called an antiderivative of $s:\mathbb{T}\to \mathbb{R}$, provided that

$$S^{\Delta }\left(t\right)=s\left(t\right)\mathit{holds}\mathit{for}\mathit{all}t\in {\mathbb{T}}^k.$$

In this case, the Cauchy integral of s is defined by

$${\int }_r^{\alpha }s\left(t\right)\Delta t=S\left(\alpha \right)-S\left(r\right),\mathit{for}\mathit{all}r,\alpha \in \mathbb{T}.$$

Theorem 3

([20]). Every rd–continuous function $S:\mathbb{T}\to \mathbb{R}$ has an antiderivative. In particular, if $t_0\in \mathbb{T}$, then

$${\left({\int }_{t_0}^{t}S\left(\tau \right)\Delta \tau \right)}^{\Delta }=S\left(t\right),\mathit{for}t\in \mathbb{T}.$$

In the following, we present the properties of integration on time scales.

Theorem 4

([20]). If $a,$ $b,$ $c\in \mathbb{T},$ $\alpha ,$ $\beta \in \mathbb{R}$ and $S,$ $T\in {\mathrm{C}}_{rd}\left(\right[a,$ ${b]}_{\mathbb{T}},\mathbb{R})$, then

1.

${\int }_a^b\left[\alpha S\left(t\right)+\beta T\left(t\right)\right]\Delta t=\alpha {\int }_a^{b}S\left(t\right)\Delta t+\beta {\int }_a^{b}T\left(t\right)\Delta t$.

2.

${\int }_a^{b}S\left(t\right)\Delta t=-{\int }_b^{a}S\left(t\right)\Delta t$.

3.

${\int }_a^{b}S\left(t\right)\Delta t={\int }_a^{c}S\left(t\right)\Delta t+{\int }_c^{b}S\left(t\right)\Delta t$.

4.

${\int }_a^{a}S\left(t\right)\Delta t=0.$

5.

$\left|{\int }_a^{b}S\left(t\right)\Delta t\right|\le {\int }_a^b\left|S\left(t\right)\right|\Delta t.$

6.

If $S\left(t\right)\ge 0$ for all $t\in {[a,b]}_{\mathbb{T}},$ then ${\int }_a^{b}S\left(t\right)\Delta t\ge 0.$

Theorem 5

([21]). Let $a,$ $b\in \mathbb{T}$ and $S\in C_{rd}(\mathbb{T},\mathbb{R}).$ Then, the following properties hold:

$\left(i\right)$ If $\mathbb{T}=\mathbb{R}$, then

$${\int }_a^{b}S\left(t\right)\Delta t={\int }_a^{b}S\left(t\right)dt.$$

$\left(ii\right)$ If $\mathbb{T}={\mathbb{N}}_0\cup \left\{0\right\}$, then

$${\int }_a^{b}S\left(t\right)\Delta t=\sum _{t=a}^{b-1}S\left(t\right).$$

$\left(iii\right)$ If $\mathbb{T}=\{t:t=q^k,k\in {\mathbb{N}}_0,$ $q>1\}$, then

$${\int }_{t_0}^{\infty }S\left(t\right)\Delta t=\sum _{k=n_0}^{\infty }S\left(q^k\right)\mu \left(q^k\right).$$

In the following, we present some auxiliary lemmas that we need to prove our results.

Lemma 1

(Integration by Parts [21]). If $a,b\in \mathbb{T}$ and $S,$ $T\in {\mathrm{C}}_{rd}{([a,b]}_{\mathbb{T}},$ $\mathbb{R}),$ then

$${\int }_a^{b}S\left(t\right)T^{\Delta }\left(t\right)\Delta t={\left[S\left(t\right)T\left(t\right)\right]}_a^b-{\int }_a^b{S}^{\Delta }\left(t\right)T^{\sigma }\left(t\right)\Delta t.$$

(12)

Lemma 2

(lHölder’s Inequality [21,22]). If $a,$ $b\in \mathbb{T}$ and $S,$ $T\in {\mathrm{C}}_{rd}{([a,b]}_{\mathbb{T}},$ $\mathbb{R}),$ then

$${\int }_a^b\left|S\left(t\right)T\left(t\right)\right|\Delta t\le {\left[{\int }_a^b{\left|S\left(t\right)\right|}^{\gamma }\Delta t\right]}^{\frac{1}{\gamma }}{\left[{\int }_a^b{\left|T\left(t\right)\right|}^{\nu }\Delta t\right]}^{\frac{1}{\nu }},$$

(13)

where $\gamma >1$ and $1/\gamma +1/\nu =1.$ The inequality (13) is reversed for $0<\gamma <1$ or $\gamma <0.$

Let ${\mathbb{T}}_1$ and ${\mathbb{T}}_2$ be time scales. Assume that $C{C}_{rd}$ denotes the set of functions $S\left(t_1,t_2\right)$ on ${\mathbb{T}}_1\times {\mathbb{T}}_2,$ where S is $rd-$ continuous in $t_1$ and $t_2.$ Let $C{\mathrm{C}}_{rd}^{\prime }$ denote the set of all functions $C{C}_{rd}$ for which both the ${\Delta }_1$ partial derivative with respect to $t_1$ and the ${\Delta }_2$ partial derivative with respect to $t_2$ exist and are in $C{C}_{rd}.$

Lemma 3

(Two dimensional Hölder’s inequality [23] Theorem 3.3). Assume that $a,$ $b\in \mathbb{T}$ with $a<b,$ $S,$ $T\in C{C}_{rd}({\left[a,b\right]}_{\mathbb{T}}\times {\left[a,b\right]}_{\mathbb{T}},\mathbb{R})$ and $\gamma ,$ $\nu >1$ such that $1/\gamma +1/\nu =1.$ Then

$${\int }_a^b{\int }_a^b\left|S(\tau ,\xi )T(\tau ,\xi )\right|{\Delta }_1\tau {\Delta }_2\xi \le {\left[{\int }_a^b{\int }_a^b{\left|S(\tau ,\xi )\right|}^{\gamma }{\Delta }_1\tau {\Delta }_2\xi \right]}^{\frac{1}{\gamma }}{\left[{\int }_a^b{\int }_a^b{\left|T(\tau ,\xi )\right|}^{\nu }{\Delta }_1\tau {\Delta }_2\xi \right]}^{\frac{1}{\nu }}.$$

(14)

Lemma 4

(Fubini’s theorem [24]). If $a,b,c,d\in \mathbb{T}$ and $S\in C{C}_{rd}\left({\left[a,b\right]}_{\mathbb{T}}\times {\left[c,d\right]}_{\mathbb{T}},\mathbb{R}\right)$ is Δ−integrable, then

$${\int }_a^b\left({\int }_c^{d}S\left(x,y\right){\Delta }_{2}y\right){\Delta }_{1}x={\int }_c^d\left({\int }_a^{b}S\left(x,y\right){\Delta }_{1}x\right){\Delta }_{2}y.$$

Lemma 5

(Mean inequality [9]). If ${\alpha }_i,{\beta }_i>0$ for $i=1,2,\dots ,n,$ then

$$\prod _{i=1}^n{\alpha }_i^{{\beta }_i}\le \frac{{\left({\sum }_{i=1}^n{\alpha }_i{\beta }_i\right)}^{{\sum }_{i=1}^n{\beta }_i}}{{\left({\sum }_{i=1}^n{\beta }_i\right)}^{{\sum }_{i=1}^n{\beta }_i}}.$$

(15)

Lemma 6.

Let $p_i,r_i>1$ with $1/p_i+1/r_i=1$ and $s_i>0,$ where $i=1,2,\dots ,n.$ Then

$$\prod _{i=1}^n{s}_i^{1/r_i}\le \frac{{\left({\sum }_{i=1}^n{s}_i/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}{{\left(n-{\sum }_{i=1}^{n}1/p_i\right)}^{\left(n-{\sum }_{i=1}^{n}1/p_i\right)}}.$$

(16)

Proof. 

Applying Lemma 5 with ${\alpha }_i=s_i$ and ${\beta }_i=1/r_i,$ we observe that

$$\prod _{i=1}^n{s}_i^{1/r_i}\le \frac{{\left({\sum }_{i=1}^n{s}_i/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}{{\left({\sum }_{i=1}^{n}1/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}.$$

(17)

Since $1/r_i=1-1/p_i,$ we can obtain that

$$\sum _{i=1}^{n}1/r_i=\sum _{i=1}^n\left(1-1/p_i\right)=n-\sum _{i=1}^{n}1/p_i,$$

and then the inequality (17) becomes

$$\prod _{i=1}^n{s}_i^{1/r_i}\le \frac{{\left({\sum }_{i=1}^n{s}_i/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}{{\left(n-{\sum }_{i=1}^{n}1/p_i\right)}^{n-{\sum }_{i=1}^{n}1/p_i}},$$

which is (16). □

3. Main Results

In this section, we present the key results of our study. Firstly, we establish the time scale version of (8).

Theorem 6.

Let $a_i,{\epsilon }_i\in \mathbb{T},$ $p_i,r_i>1$, such that $1/p_i+1/r_i=1$ and let ${\lambda }_i\in {\mathrm{C}}_{rd}({\left[a_i,{\epsilon }_i\right]}_{\mathbb{T}},$ $\mathbb{R})$ be a delta-differentiable function with ${\lambda }_i\left(a_i\right)=0;$ $i=1,2,\dots ,n.$ Then

$$\begin{array}{ccc}& & {\int }_{a_n}^{{\epsilon }_n}\dots {\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{i=1}^n\left|{\lambda }_i\left({\xi }_i\right)\right|}{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}\Delta {\xi }_1\dots \Delta {\xi }_n\hfill \\ & \le & M\prod _{i=1}^n{\left({\int }_{a_i}^{{\epsilon }_i}\left[{\epsilon }_i-\sigma \left({\xi }_i\right)\right]{\left|{\lambda }_i^{\Delta }\left({\xi }_i\right)\right|}^{p_i}\Delta {\xi }_i\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

(18)

where

$$M={\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{\left({\epsilon }_i-a_i\right)}^{\frac{1}{q_i}}.$$

(19)

Proof. 

Applying the property (5) of Theorem 4 and the hypothesis ${\lambda }_i\left(a_i\right)=0$, we obtain

$$\begin{array}{ccc}\hfill {\int }_{a_i}^{{\xi }_i}\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|\Delta t_i& \ge & \left|{\int }_{a_i}^{{\xi }_i}{\lambda }_i^{\Delta }\left(t_i\right)\Delta t_i\right|\hfill \\ & =& \left|{\lambda }_i\left({\xi }_i\right)-{\lambda }_i\left(a_i\right)\right|=\left|{\lambda }_i\left({\xi }_i\right)\right|,\hfill \end{array}$$

and then

$$\prod _{i=1}^n\left|{\lambda }_i\left({\xi }_i\right)\right|\le \prod _{i=1}^n{\int }_{a_i}^{{\xi }_i}\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|\Delta t_i.$$

(20)

Applying (13) on ${\int }_{a_i}^{{\xi }_i}\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|\Delta t_i$ with $f\left(t_i\right)=\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|$ and $g\left(t_i\right)=1,$ we observe that

$$\begin{array}{ccc}\hfill {\int }_{a_i}^{{\xi }_i}\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|\Delta t_i& \le & {\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}{\left({\int }_{a_i}^{{\xi }_i}\Delta t_i\right)}^{\frac{1}{r_i}}\hfill \\ & =& {\left({\xi }_i-a_i\right)}^{\frac{1}{r_i}}{\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

and then

$$\begin{array}{ccc}\hfill \prod _{i=1}^n{\int }_{a_i}^{{\xi }_i}\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|\Delta t_i& \le & \prod _{i=1}^n{\left({\xi }_i-a_i\right)}^{\frac{1}{r_i}}{\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}\hfill \\ & =& \prod _{i=1}^n{\left({\xi }_i-a_i\right)}^{\frac{1}{r_i}}\prod _{i=1}^n{\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}.\hfill \end{array}$$

(21)

Substituting (21) into (20), we obtain

$$\prod _{i=1}^n\left|{\lambda }_i\left({\xi }_i\right)\right|\le \prod _{i=1}^n{\left({\xi }_i-a_i\right)}^{\frac{1}{r_i}}\prod _{i=1}^n{\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}.$$

(22)

Applying (16) with $s_i={\xi }_i-a_i,$ we have

$$\prod _{i=1}^n{\left({\xi }_i-a_i\right)}^{1/r_i}\le \frac{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}{{\left(n-{\sum }_{i=1}^{n}1/p_i\right)}^{n-{\sum }_{i=1}^{n}1/p_i}}.$$

(23)

Substituting (23) into (22), we obtain

$$\prod _{i=1}^n\left|{\lambda }_i\left({\xi }_i\right)\right|\le \frac{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}{{\left(n-{\sum }_{i=1}^{n}1/p_i\right)}^{n-{\sum }_{i=1}^{n}1/p_i}}\prod _{i=1}^n{\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}.$$

(24)

Dividing (24) by ${\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}$ and integrating over ${\xi }_i$ from $a_i$ to ${\epsilon }_i,$ $i=1,2,\dots ,n,$ we observe that

$$\begin{array}{cc}\hfill & {\int }_{a_n}^{{\epsilon }_n}\dots {\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{i=1}^n\left|{\lambda }_i\left({\xi }_i\right)\right|}{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}\Delta {\xi }_1\dots \Delta {\xi }_n\hfill \\ \hfill & \le {\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}{\int }_{a_n}^{{\epsilon }_n}\dots {\int }_{a_1}^{{\epsilon }_1}\prod _{i=1}^n{\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}\Delta {\xi }_1\dots \Delta {\xi }_n\hfill \\ \hfill & ={\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{\int }_{a_i}^{{\epsilon }_i}{\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}\Delta {\xi }_i.\hfill \end{array}$$

(25)

Applying (13) on ${\int }_{a_i}^{{\epsilon }_i}{\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}\Delta {\xi }_i$ with $f\left({\xi }_i\right)={\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}$ and $g\left({\xi }_i\right)=1,$ we have

$$\begin{array}{ccc}\hfill {\int }_{a_i}^{{\epsilon }_i}{\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}\Delta {\xi }_i& \le & {\left({\int }_{a_i}^{{\epsilon }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\Delta {\xi }_i\right)}^{\frac{1}{p_i}}{\left({\int }_{a_i}^{{\epsilon }_i}\Delta {\xi }_i\right)}^{\frac{1}{r_i}}\hfill \\ & =& {\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}{\left({\int }_{a_i}^{{\epsilon }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\Delta {\xi }_i\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

and then

$$\begin{array}{ccc}& & \prod _{i=1}^n{\int }_{a_i}^{{\epsilon }_i}{\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}\Delta {\xi }_i\hfill \\ & \le & \prod _{i=1}^n{\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}{\left({\int }_{a_i}^{{\epsilon }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\Delta {\xi }_i\right)}^{\frac{1}{p_i}}\hfill \\ & =& \prod _{i=1}^n{\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}\prod _{i=1}^n{\left({\int }_{a_i}^{{\epsilon }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\Delta {\xi }_i\right)}^{\frac{1}{p_i}}.\hfill \end{array}$$

(26)

Substituting (26) into (25), we have

$$\begin{array}{cc}\hfill & {\int }_{a_n}^{{\epsilon }_n}\dots {\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{i=1}^n\left|{\lambda }_i\left({\xi }_i\right)\right|}{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}\Delta {\xi }_1\dots \Delta {\xi }_n\hfill \\ \hfill & \le {\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}\prod _{i=1}^n{\left({\int }_{a_i}^{{\epsilon }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\Delta {\xi }_i\right)}^{\frac{1}{p_i}}.\hfill \end{array}$$

(27)

Applying (12) on ${\int }_{a_i}^{{\epsilon }_i}\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)\Delta {\xi }_i$ with $f\left({\xi }_i\right)={\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i$ and $g^{\Delta }\left({\xi }_i\right)=1,$ we obtain

$$\begin{array}{ccc}& & {\int }_{a_i}^{{\epsilon }_i}\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)\Delta {\xi }_i\hfill \\ & =& {\left.\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)g\left({\xi }_i\right)\right|}_{a_i}^{{\epsilon }_i}-{\int }_{a_i}^{{\epsilon }_i}{\left|{\lambda }_i^{\Delta }\left({\xi }_i\right)\right|}^{p_i}{g}^{\sigma }\left({\xi }_i\right)\Delta {\xi }_i,\hfill \end{array}$$

(28)

where $g\left({\xi }_i\right)={\xi }_i-{\epsilon }_i.$ Since $g\left({\epsilon }_i\right)=0,$ we can find from (28) that

$${\int }_{a_i}^{{\epsilon }_i}\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{\Delta }\left(t_i\right)\right|}^{p_i}\Delta t_i\right)\Delta {\xi }_i={\int }_{a_i}^{{\epsilon }_i}{\left|{\lambda }_i^{\Delta }\left({\xi }_i\right)\right|}^{p_i}\left[{\epsilon }_i-\sigma \left({\xi }_i\right)\right]\Delta {\xi }_i.$$

(29)

Substituting (29) into (27), we obtain

$$\begin{array}{ccc}& & {\int }_{a_n}^{{\epsilon }_n}\dots {\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{i=1}^n\left|{\lambda }_i\left({\xi }_i\right)\right|}{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}\Delta {\xi }_1\dots \Delta {\xi }_n\hfill \\ & \le & {\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}\hfill \\ & & \times \prod _{i=1}^n{\left({\int }_{a_i}^{{\epsilon }_i}\left[{\epsilon }_i-\sigma \left({\xi }_i\right)\right]{\left|{\lambda }_i^{\Delta }\left({\xi }_i\right)\right|}^{p_i}\Delta {\xi }_i\right)}^{\frac{1}{p_i}}.\hfill \end{array}$$

(30)

Substituting (19) into (30), we obtain

$$\begin{array}{ccc}& & {\int }_{a_n}^{{\epsilon }_n}\dots {\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{i=1}^n\left|{\lambda }_i\left({\xi }_i\right)\right|}{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}\Delta {\xi }_1\dots \Delta {\xi }_n\hfill \\ & \le & M\prod _{i=1}^n{\left({\int }_{a_i}^{{\epsilon }_i}\left[{\epsilon }_i-\sigma \left({\xi }_i\right)\right]{\left|{\lambda }_i^{\Delta }\left({\xi }_i\right)\right|}^{p_i}\Delta {\xi }_i\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

which is (18). □

Corollary 1.

If we put $\mathbb{T}={\mathbb{N}}_0$ and $a_i=0$ for $i=1,2,\dots ,n,$ into Theorem 6, then $\sigma \left({\xi }_i\right)={\xi }_i+1$, and we obtain the analogue of inequality (8) as follows

$$\sum _{{\xi }_1=0}^{{\epsilon }_1-1}\dots \sum _{{\xi }_n=0}^{{\epsilon }_n-1}\frac{{\prod }_{i=1}^n\left|{\lambda }_i\left({\xi }_i\right)\right|}{{\left({\sum }_{i=1}^n{\xi }_i/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}\le M\prod _{i=1}^n{\left(\sum _{{\xi }_i=0}^{{\epsilon }_i-1}\left[{\epsilon }_i-{\xi }_i-1\right]{\left|\Delta {\lambda }_i\left({\xi }_i\right)\right|}^{p_i}\right)}^{\frac{1}{p_i}},$$

(31)

where

$$M={\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{\epsilon }_i^{\frac{1}{r_i}}.$$

In the following, we present some special cases in (the continuous and quantum) calculie, i.e., when $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=q^{{\mathbb{N}}_0}$ for $q>1$. These cases are new and interesting for the reader.

Corollary 2.

In Theorem 6, if $\mathbb{T}=\mathbb{R},$ $a_i=0,$ $p_i,r_i>1$, such that $1/p_i+1/r_i=1$ and ${\lambda }_i\in \mathrm{C}(\left[0,{\epsilon }_i\right],$ $\mathbb{R})$ is a differentiable function with ${\lambda }_i\left(0\right)=0;$ $i=1,2,\dots ,n,$ then $\sigma \left({\xi }_i\right)={\xi }_i$ and we obtain

$${\int }_0^{{\epsilon }_n}\dots {\int }_0^{{\epsilon }_1}\frac{{\prod }_{i=1}^n\left|{\lambda }_i\left({\xi }_i\right)\right|}{{\left({\sum }_{i=1}^n{\xi }_i/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}d{\xi }_1\dots d{\xi }_n\le M\prod _{i=1}^n{\left({\int }_0^{{\epsilon }_i}\left[{\epsilon }_i-{\xi }_i\right]{\left|{\lambda }_i^{\prime }\left({\xi }_i\right)\right|}^{p_i}d{\xi }_i\right)}^{\frac{1}{p_i}},$$

where

$$M={\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{\epsilon }_i^{\frac{1}{q_i}}.$$

Corollary 3.

In Theorem 6, if $\mathbb{T}=q^{{\mathbb{N}}_0}$ for $q>1,$ $p_i,r_i>1$, such that $1/p_i+1/r_i=1$ and ${\lambda }_i\in \mathrm{C}({\left[a_i,{\epsilon }_i\right]}_{\mathbb{T}},$ $\mathbb{R})$ with ${\lambda }_i\left(a_i\right)=0;$ $i=1,2,\dots ,n,$ then $\sigma \left({\xi }_i\right)=q{\xi }_i$ and we obtain

$$\sum _{{\xi }_1=a_1}^{{\epsilon }_1/q}\dots \sum _{{\xi }_n=a_n}^{{\epsilon }_n/q}\frac{{\prod }_{i=1}^n\left(q-1\right){\xi }_i\left|{\lambda }_i\left({\xi }_i\right)\right|}{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}\le M\prod _{i=1}^n{\left(\sum _{{\xi }_i=a_i}^{{\epsilon }_i/q}\left(q-1\right){\xi }_i\left[{\epsilon }_i-q{\xi }_i\right]{\left|{\Delta }_q{\lambda }_i\left({\xi }_i\right)\right|}^{p_i}\right)}^{\frac{1}{p_i}},$$

where

$${\Delta }_q{\lambda }_i\left({\xi }_i\right)=\frac{{\lambda }_i\left(q{\xi }_i\right)-{\lambda }_i\left({\xi }_i\right)}{\left(q-1\right){\xi }_i},$$

and

$$M={\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}.$$

In the following theorem, we generalize the previous results for two variables.

Theorem 7.

Assume that $a_i,{\epsilon }_i,{ϵ}_i\in \mathbb{T},$ $p_i,r_i>1$, such that $1/p_i+1/r_i=1$ and ${\lambda }_i\in C{\mathrm{C}}_{rd}^{\prime }({\left[a_i,{\epsilon }_i\right]}_{\mathbb{T}}\times {\left[a_i,{ϵ}_i\right]}_{\mathbb{T}},$ $\mathbb{R})$ with ${\lambda }_i(a_i,{\xi }_i)={\lambda }_i({\tau }_i,a_i)=0$ for ${\xi }_i\in {\left[a_i,{\epsilon }_i\right]}_{\mathbb{T}}$ and ${\tau }_i\in {\left[a_i,{ϵ}_i\right]}_{\mathbb{T}},$ $i=1,2,\dots ,n.$ Then

$$\begin{array}{cc}\hfill & {\int }_{a_n}^{{ϵ}_n}{\int }_{a_1}^{{ϵ}_1}\dots {\int }_{a_n}^{{\epsilon }_n}{\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{i=1}^n\left|{\lambda }_i({\tau }_i,{\xi }_i)\right|}{{\left({\sum }_{i=1}^n\left({\tau }_i-a_i\right)\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}{\Delta }_2{\xi }_1\dots {\Delta }_2{\xi }_n{\Delta }_1{\tau }_1\dots {\Delta }_1{\tau }_n\hfill \\ \hfill & \le N\prod _{i=1}^n{\left({\int }_{a_i}^{{\epsilon }_i}{\int }_{a_i}^{{ϵ}_i}\left({ϵ}_i-\sigma \left({\tau }_i\right)\right)\left({\epsilon }_i-\sigma \left({\xi }_i\right)\right){\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\xi }_i)\right|}^{p_i}{\Delta }_1{\tau }_i{\Delta }_2{\xi }_i\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

(32)

where

$$N={\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{\left({ϵ}_i-a_i\right)}^{\frac{1}{r_i}}{\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}.$$

(33)

Here, the ${\Delta }_1$—derivative of the function $\lambda (\tau ,\xi )$ is the $\Delta$—derivative with respect to the first variable τ and the ${\Delta }_2$—derivative of the function $\lambda (\tau ,\xi )$ is the $\Delta$—derivative with respect to the second variable $\xi .$

Proof. 

Applying the property (5) of Theorem 4, Fubini’s theorem and the hypothesis ${\lambda }_i(a_i,{\xi }_i)={\lambda }_i({\tau }_i,a_i)=0$, we obtain

$$\begin{array}{ccc}& & {\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i\hfill \\ & \ge & \left|{\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i){\Delta }_2{\vartheta }_i{\Delta }_1{t}_i\right|=\left|{\int }_{a_i}^{{\xi }_i}\left({\int }_{a_i}^{{\tau }_i}{\left[{\lambda }_i^{{\Delta }_2}(t_i,{\vartheta }_i)\right]}^{{\Delta }_1}{\Delta }_1{t}_i\right){\Delta }_2{\vartheta }_i\right|\hfill \\ & =& \left|{\lambda }_i({\tau }_i,{\xi }_i)-{\lambda }_i({\tau }_i,a_i)-{\lambda }_i(a_i,{\xi }_i)+{\lambda }_i(a_i,a_i)\right|=\left|{\lambda }_i({\tau }_i,{\xi }_i)\right|,\hfill \end{array}$$

and then

$$\prod _{i=1}^n\left|{\lambda }_i({\tau }_i,{\xi }_i)\right|\le \prod _{i=1}^n{\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i.$$

(34)

Applying (14) on ${\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i$ with $h(t_i,{\vartheta }_i)=1,$ $f(t_i,{\vartheta }_i)=1$ and $g(t_i,{\vartheta }_i)=\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|,$ we observe

$$\begin{array}{ccc}& & {\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i\hfill \\ & \le & {\left({\tau }_i-a_i\right)}^{\frac{1}{r_i}}{\left({\xi }_i-a_i\right)}^{\frac{1}{r_i}}{\left({\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

and then

$$\begin{array}{ccc}& & \prod _{i=1}^n{\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i\hfill \\ & \le & \prod _{i=1}^n{\left({\tau }_i-a_i\right)}^{\frac{1}{r_i}}{\left({\xi }_i-a_i\right)}^{\frac{1}{r_i}}\prod _{i=1}^n{\left({\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i\right)}^{\frac{1}{p_i}}.\hfill \end{array}$$

(35)

Substituting (35) into (34), we observe that

$$\prod _{i=1}^n\left|{\lambda }_i({\tau }_i,{\xi }_i)\right|\le \prod _{i=1}^n{\left({\tau }_i-a_i\right)}^{\frac{1}{r_i}}{\left({\xi }_i-a_i\right)}^{\frac{1}{r_i}}\prod _{i=1}^n{\left({\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i\right)}^{\frac{1}{p_i}}.$$

(36)

Applying (16) with $s_i=\left({\tau }_i-a_i\right)\left({\xi }_i-a_i\right),$ we have

$$\prod _{i=1}^n{\left({\tau }_i-a_i\right)}^{\frac{1}{r_i}}{\left({\xi }_i-a_i\right)}^{\frac{1}{r_i}}\le \frac{{\left({\sum }_{i=1}^n\left({\tau }_i-a_i\right)\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}{{\left(n-{\sum }_{i=1}^{n}1/p_i\right)}^{n-{\sum }_{i=1}^{n}1/p_i}}.$$

(37)

Substituting (37) into (36), we obtain

$$\begin{array}{ccc}\hfill \prod _{i=1}^n\left|{\lambda }_i({\tau }_i,{\xi }_i)\right|& \le & \frac{{\left({\sum }_{i=1}^n\left({\tau }_i-a_i\right)\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}{{\left(n-{\sum }_{i=1}^{n}1/p_i\right)}^{n-{\sum }_{i=1}^{n}1/p_i}}\hfill \\ & & \times \prod _{i=1}^n{\left({\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i\right)}^{\frac{1}{p_i}}.\hfill \end{array}$$

(38)

Dividing (38) by ${\left({\sum }_{i=1}^n\left({\tau }_i-a_i\right)\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}$ and integrating over ${\xi }_i$ and ${\tau }_i$ from $a_i$ to ${\epsilon }_i$ and ${ϵ}_i$ for $i=1,2,\dots ,n,$ respectively, we observe that

$$\begin{array}{cc}\hfill & {\int }_{a_n}^{{ϵ}_n}{\int }_{a_1}^{{ϵ}_1}\dots {\int }_{a_n}^{{\epsilon }_n}{\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{i=1}^n\left|{\lambda }_i({\tau }_i,{\xi }_i)\right|}{{\left({\sum }_{i=1}^n\left({\tau }_i-a_i\right)\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}{\Delta }_2{\xi }_1\dots {\Delta }_2{\xi }_n{\Delta }_1{\tau }_1\dots {\Delta }_1{\tau }_n\hfill \\ \hfill & \le {\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\hfill \\ \hfill & \times {\int }_{a_n}^{{ϵ}_n}{\int }_{a_1}^{{ϵ}_1}\dots {\int }_{a_n}^{{\epsilon }_n}{\int }_{a_1}^{{\epsilon }_1}\prod _{i=1}^n{\left({\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i\right)}^{\frac{1}{p_i}}{\Delta }_2{\xi }_1\dots {\Delta }_2{\xi }_n{\Delta }_1{\tau }_1\dots {\Delta }_1{\tau }_n\hfill \\ \hfill & ={\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\hfill \\ \hfill & \times \prod _{i=1}^n{\int }_{a_i}^{{ϵ}_i}{\int }_{a_i}^{{\epsilon }_i}{\left({\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i\right)}^{\frac{1}{p_i}}{\Delta }_2{\xi }_i{\Delta }_1{\tau }_i.\hfill \end{array}$$

(39)

Applying (14) on ${\int }_{a_i}^{{ϵ}_i}{\int }_{a_i}^{{\epsilon }_i}{\left({\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i\right)}^{\frac{1}{p_i}}{\Delta }_2{\xi }_i{\Delta }_1{\tau }_i$ with $h({\xi }_i,{\tau }_i)=1,$ $f({\xi }_i,{\tau }_i)=1$ and

$$g({\xi }_i,{\tau }_i)={\left({\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i\right)}^{\frac{1}{p_i}},$$

we have

$$\begin{array}{cc}\hfill & {\int }_{a_i}^{{ϵ}_i}{\int }_{a_i}^{{\epsilon }_i}{\left({\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i\right)}^{\frac{1}{p_i}}{\Delta }_2{\xi }_i{\Delta }_1{\tau }_i\hfill \\ \hfill & \le {\left({ϵ}_i-a_i\right)}^{\frac{1}{r_i}}{\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}{\left({\int }_{a_i}^{{ϵ}_i}{\int }_{a_i}^{{\epsilon }_i}{\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i{\Delta }_2{\xi }_i{\Delta }_1{\tau }_i\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

and then

$$\begin{array}{cc}\hfill & \prod _{i=1}^n{\int }_{a_i}^{{ϵ}_i}{\int }_{a_i}^{{\epsilon }_i}{\left({\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i\right)}^{\frac{1}{p_i}}{\Delta }_2{\xi }_i{\Delta }_1{\tau }_i\hfill \\ \hfill & \le \prod _{i=1}^n{\left({ϵ}_i-a_i\right)}^{\frac{1}{r_i}}{\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}\prod _{i=1}^n{\left({\int }_{a_i}^{{ϵ}_i}{\int }_{a_i}^{{\epsilon }_i}{\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i{\Delta }_2{\xi }_i{\Delta }_1{\tau }_i\right)}^{\frac{1}{p_i}}.\hfill \end{array}$$

(40)

Substituting (40) into (39) and applying the Fubini theorem, we obtain

$$\begin{array}{cc}\hfill & {\int }_{a_n}^{{ϵ}_n}{\int }_{a_1}^{{ϵ}_1}\dots {\int }_{a_n}^{{\epsilon }_n}{\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{i=1}^n\left|{\lambda }_i({\tau }_i,{\xi }_i)\right|}{{\left({\sum }_{i=1}^n\left({\tau }_i-a_i\right)\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}{\Delta }_2{\xi }_1\dots {\Delta }_2{\xi }_n{\Delta }_1{\tau }_1\dots {\Delta }_1{\tau }_n\hfill \\ \hfill & \le {\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{\left({ϵ}_i-a_i\right)}^{\frac{1}{r_i}}{\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}\hfill \\ \hfill & \times \prod _{i=1}^n{\left({\int }_{a_i}^{{ϵ}_i}{\int }_{a_i}^{{\epsilon }_i}{\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i{\Delta }_2{\xi }_i{\Delta }_1{\tau }_i\right)}^{\frac{1}{p_i}}\hfill \\ \hfill & ={\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{\left({ϵ}_i-a_i\right)}^{\frac{1}{r_i}}{\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}\hfill \\ \hfill & \times \prod _{i=1}^n{\left({\int }_{a_i}^{{\epsilon }_i}\left({\int }_{a_i}^{{ϵ}_i}\left[{\int }_{a_i}^{{\tau }_i}{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i{\Delta }_1{t}_i\right]{\Delta }_1{\tau }_i\right){\Delta }_2{\xi }_i\right)}^{\frac{1}{p_i}}.\hfill \end{array}$$

(41)

Applying (12) on ${\int }_{a_i}^{{ϵ}_i}\left({\int }_{a_i}^{{\tau }_i}\left[{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right]{\Delta }_1{t}_i\right){\Delta }_1{\tau }_i,$ with

$$f\left({\tau }_i\right)={\int }_{a_i}^{{\tau }_i}\left[{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right]{\Delta }_1{t}_i\mathrm{and}{g}^{\Delta }\left({\tau }_i\right)=1,$$

we obtain

$$\begin{array}{ccc}& & {\int }_{a_i}^{{ϵ}_i}\left({\int }_{a_i}^{{\tau }_i}\left[{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right]{\Delta }_1{t}_i\right){\Delta }_1{\tau }_i\hfill \\ & =& {\left.g\left({\tau }_i\right){\int }_{a_i}^{{\tau }_i}\left[{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right]{\Delta }_1{t}_i\right|}_{a_i}^{{ϵ}_i}\hfill \\ & & -{\int }_{a_i}^{{ϵ}_i}{g}^{\sigma }\left({\tau }_i\right)\left[{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right]{\Delta }_1{\tau }_i,\hfill \end{array}$$

(42)

where $g\left({\tau }_i\right)={\tau }_i-{ϵ}_i.$ Since $g\left({ϵ}_i\right)=0,$ we know from (42) that

$$\begin{array}{ccc}& & {\int }_{a_i}^{{ϵ}_i}\left({\int }_{a_i}^{{\tau }_i}\left[{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right]{\Delta }_1{t}_i\right){\Delta }_1{\tau }_i\hfill \\ & =& {\int }_{a_i}^{{ϵ}_i}\left[{ϵ}_i-\sigma \left({\tau }_i\right)\right]\left[{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right]{\Delta }_1{\tau }_i.\hfill \end{array}$$

(43)

Integrating (43) over ${\xi }_i$ from $a_i$ to ${\epsilon }_i$ and then applying Fubini’s theorem, we obtain

$$\begin{array}{ccc}& & {\int }_{a_i}^{{\epsilon }_i}{\int }_{a_i}^{{ϵ}_i}\left({\int }_{a_i}^{{\tau }_i}\left[{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right]{\Delta }_1{t}_i\right){\Delta }_1{\tau }_i{\Delta }_2{\xi }_i\hfill \\ & =& {\int }_{a_i}^{{\epsilon }_i}{\int }_{a_i}^{{ϵ}_i}\left[{ϵ}_i-\sigma \left({\tau }_i\right)\right]\left[{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right]{\Delta }_1{\tau }_i{\Delta }_2{\xi }_i\hfill \\ & =& {\int }_{a_i}^{{ϵ}_i}{\int }_{a_i}^{{\epsilon }_i}\left[{ϵ}_i-\sigma \left({\tau }_i\right)\right]\left[{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right]{\Delta }_2{\xi }_i{\Delta }_1{\tau }_i\hfill \\ & =& {\int }_{a_i}^{{ϵ}_i}\left[{ϵ}_i-\sigma \left({\tau }_i\right)\right]\left({\int }_{a_i}^{{\epsilon }_i}\left[{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right]{\Delta }_2{\xi }_i\right){\Delta }_1{\tau }_i.\hfill \end{array}$$

(44)

Applying (12) on ${\int }_{a_i}^{{\epsilon }_i}\left[{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right]{\Delta }_2{\xi }_i$ with $f\left({\xi }_i\right)={\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i$ and $g^{\Delta }\left({\xi }_i\right)=1,$ we see

$$\begin{array}{ccc}& & {\int }_{a_i}^{{\epsilon }_i}\left[{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right]{\Delta }_2{\xi }_i\hfill \\ & =& {\left.g\left({\xi }_i\right)\left({\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right)\right|}_{a_i}^{{\epsilon }_i}\hfill \\ & & -{\int }_{a_i}^{{\epsilon }_i}{g}^{\sigma }\left({\xi }_i\right){\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\xi }_i)\right|}^{p_i}{\Delta }_2{\xi }_i,\hfill \end{array}$$

(45)

where $g\left({\xi }_i\right)={\xi }_i-{\epsilon }_i.$ Since $g\left({\epsilon }_i\right)=0,$ we know from (45) that

$${\int }_{a_i}^{{\epsilon }_i}\left[{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right]{\Delta }_2{\xi }_i={\int }_{a_i}^{{\epsilon }_i}\left({\epsilon }_i-\sigma \left({\xi }_i\right)\right){\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\xi }_i)\right|}^{p_i}{\Delta }_2{\xi }_i.$$

(46)

Substituting (46) into (44) and applying Fubini’s theorem, we obtain

$$\begin{array}{ccc}& & {\int }_{a_i}^{{\epsilon }_i}{\int }_{a_i}^{{ϵ}_i}\left({\int }_{a_i}^{{\tau }_i}\left[{\int }_{a_i}^{{\xi }_i}{\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}(t_i,{\vartheta }_i)\right|}^{p_i}{\Delta }_2{\vartheta }_i\right]{\Delta }_1{t}_i\right){\Delta }_1{\tau }_i{\Delta }_2{\xi }_i\hfill \\ & =& {\int }_{a_i}^{{ϵ}_i}\left({ϵ}_i-\sigma \left({\tau }_i\right)\right)\left({\int }_{a_i}^{{\epsilon }_i}\left({\epsilon }_i-\sigma \left({\xi }_i\right)\right){\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\xi }_i)\right|}^{p_i}{\Delta }_2{\xi }_i\right){\Delta }_1{\tau }_i\hfill \\ & =& {\int }_{a_i}^{{\epsilon }_i}{\int }_{a_i}^{{ϵ}_i}\left({ϵ}_i-\sigma \left({\tau }_i\right)\right)\left({\epsilon }_i-\sigma \left({\xi }_i\right)\right){\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\xi }_i)\right|}^{p_i}{\Delta }_1{\tau }_i{\Delta }_2{\xi }_i.\hfill \end{array}$$

(47)

Substituting (47) into (41), we see that

$$\begin{array}{cc}\hfill & {\int }_{a_n}^{{ϵ}_n}{\int }_{a_1}^{{ϵ}_1}\dots {\int }_{a_n}^{{\epsilon }_n}{\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{i=1}^n\left|{\lambda }_i({\tau }_i,{\xi }_i)\right|}{{\left({\sum }_{i=1}^n\left({\tau }_i-a_i\right)\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}{\Delta }_2{\xi }_1\dots {\Delta }_2{\xi }_n{\Delta }_1{\tau }_1\dots {\Delta }_1{\tau }_n\hfill \\ \hfill & \le {\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{\left({ϵ}_i-a_i\right)}^{\frac{1}{r_i}}{\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}\hfill \\ \hfill & \times \prod _{i=1}^n{\left({\int }_{a_i}^{{\epsilon }_i}{\int }_{a_i}^{{ϵ}_i}\left({ϵ}_i-\sigma \left({\tau }_i\right)\right)\left({\epsilon }_i-\sigma \left({\xi }_i\right)\right){\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\xi }_i)\right|}^{p_i}{\Delta }_1{\tau }_i{\Delta }_2{\xi }_i\right)}^{\frac{1}{p_i}}.\hfill \end{array}$$

(48)

Substituting (33) into (48), we have

$$\begin{array}{cc}\hfill & {\int }_{a_n}^{{ϵ}_n}{\int }_{a_1}^{{ϵ}_1}\dots {\int }_{a_n}^{{\epsilon }_n}{\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{i=1}^n\left|{\lambda }_i({\tau }_i,{\xi }_i)\right|}{{\left({\sum }_{i=1}^n\left({\tau }_i-a_i\right)\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}{\Delta }_2{\xi }_1\dots {\Delta }_2{\xi }_n{\Delta }_1{\tau }_1\dots {\Delta }_1{\tau }_n\hfill \\ \hfill & \le N\prod _{i=1}^n{\left({\int }_{a_i}^{{\epsilon }_i}{\int }_{a_i}^{{ϵ}_i}\left({ϵ}_i-\sigma \left({\tau }_i\right)\right)\left({\epsilon }_i-\sigma \left({\xi }_i\right)\right){\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\xi }_i)\right|}^{p_i}{\Delta }_1{\tau }_i{\Delta }_2{\xi }_i\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

which is (32). □

Corollary 4.

If $\mathbb{T}={\mathbb{N}}_0$ and $a_i=0,$ then $\sigma \left({\xi }_i\right)={\xi }_i+1$ and we obtain the analogue of inequality (10) as follows:

$$\begin{array}{cc}\hfill & \sum _{{\tau }_1=0}^{{ϵ}_1-1}\sum _{{\tau }_n=0}^{{ϵ}_n-1}\dots \sum _{{\xi }_1=0}^{{\epsilon }_1-1}\sum _{{\xi }_n=0}^{{\epsilon }_n-1}\frac{{\prod }_{i=1}^n\left|{\lambda }_i({\tau }_i,{\xi }_i)\right|}{{\left({\sum }_{i=1}^n{\tau }_i{\xi }_i/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}\hfill \\ \hfill & \le N\prod _{i=1}^n{\left(\sum _{{\xi }_i=0}^{{\epsilon }_i-1}\sum _{{\tau }_i=0}^{{ϵ}_i-1}\left({ϵ}_i-{\tau }_i-1\right)\left({\epsilon }_i-{\xi }_i-1\right){\left|{\Delta }_2{\Delta }_1{\lambda }_i({\tau }_i,{\xi }_i)\right|}^{p_i}\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

where ${\Delta }_1{\lambda }_i({\tau }_i,{\xi }_i)={\lambda }_i({\tau }_i+1,{\xi }_i)-{\lambda }_i({\tau }_i,{\xi }_i),$ ${\Delta }_2{\lambda }_i({\tau }_i,{\xi }_i)={\lambda }_i({\tau }_i,{\xi }_i+1)-{\lambda }_i({\tau }_i,{\xi }_i)$ and

$$N={\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{\left({\epsilon }_i{ϵ}_i\right)}^{\frac{1}{r_i}}.$$

In the following corollaries, we show some particular cases in (the continuous and quantum) calculie, i.e., when $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=q^{{\mathbb{N}}_0}$ for $q>1$, which are original.

Corollary 5.

If $\mathbb{T}=\mathbb{R},$ $a_i=0,$ $p_i,r_i>1$, such that $1/p_i+1/r_i=1,$ ${\lambda }_i\in \mathrm{C}(\left[0,{\epsilon }_i\right]\times [0,{ϵ}_i],$ $\mathbb{R})$ with ${\lambda }_i(0,{\xi }_i)={\lambda }_i({\tau }_i,0)=0$ for ${\xi }_i\in \left[0,{\epsilon }_i\right]$ and ${\tau }_i\in \left[0,{ϵ}_i\right],$ $i=1,2,\dots ,n,$ then $\sigma \left({\xi }_i\right)={\xi }_i$ and we obtain

$$\begin{array}{cc}\hfill & {\int }_0^{{ϵ}_n}{\int }_0^{{ϵ}_1}\dots {\int }_0^{{\epsilon }_n}{\int }_0^{{\epsilon }_1}\frac{{\prod }_{i=1}^n\left|{\lambda }_i({\tau }_i,{\xi }_i)\right|}{{\left({\sum }_{i=1}^n{\tau }_i{\xi }_i/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}d{\xi }_1\dots d{\xi }_{n}d{\tau }_1\dots d{\tau }_n\hfill \\ \hfill & \le N\prod _{i=1}^n{\left({\int }_0^{{\epsilon }_i}{\int }_0^{{ϵ}_i}\left({ϵ}_i-{\tau }_i\right)\left({\epsilon }_i-{\xi }_i\right){\left|\frac{{\partial }^2}{\partial {\xi }_i\partial {\tau }_i}{\lambda }_i({\tau }_i,{\xi }_i)\right|}^{p_i}d{\tau }_{i}d{\xi }_i\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

where

$$N={\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{\left({ϵ}_i{\epsilon }_i\right)}^{\frac{1}{r_i}}.$$

Corollary 6.

If $\mathbb{T}=q^{{\mathbb{N}}_0}$ for $q>1,$ $p_i,r_i>1$ such that $1/p_i+1/r_i=1$ and ${\lambda }_i:{\left[a_i,{\epsilon }_i\right]}_{\mathbb{T}}\times {\left[a_i,{ϵ}_i\right]}_{\mathbb{T}}\to \mathbb{R}$ with ${\lambda }_i(a_i,{\xi }_i)={\lambda }_i({\tau }_i,a_i)=0$ for ${\xi }_i\in {\left[a_i,{\epsilon }_i\right]}_{\mathbb{T}}$ and ${\tau }_i\in {\left[a_i,{ϵ}_i\right]}_{\mathbb{T}},$ $i=1,2,\dots ,n,$ then $\sigma \left(t\right)=qt$ and we obtain

$$\begin{array}{cc}\hfill & \sum _{{\tau }_1=a_1}^{{ϵ}_1/q}\sum _{{\tau }_n=a_n}^{{ϵ}_n/q}\dots \sum _{{\xi }_1=a_1}^{{\epsilon }_1/q}\sum _{{\xi }_n=a_n}^{{\epsilon }_n/q}\frac{{\prod }_{i=1}^n{\left(q-1\right)}^2{\tau }_i{\xi }_i\left|{\lambda }_i({\tau }_i,{\xi }_i)\right|}{{\left({\sum }_{i=1}^n\left({\tau }_i-a_i\right)\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}\hfill \\ \hfill & \le N\prod _{i=1}^n{\left(\sum _{{\xi }_i=a_i}^{{\epsilon }_i/q}\sum _{{\tau }_i=a_i}^{{ϵ}_i/q}\left({ϵ}_i-q{\tau }_i\right)\left({\epsilon }_i-q{\xi }_i\right){\left|{\lambda }_i^{{\Delta }_2{\Delta }_1}({\tau }_i,{\xi }_i)\right|}^{p_i}\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

where

$$N={\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{\left({ϵ}_i-a_i\right)}^{\frac{1}{q_i}}{\left({\epsilon }_i-a_i\right)}^{\frac{1}{q_i}}.$$

Theorem 8.

Let $a_i,{\epsilon }_i\in \mathbb{T},$ $h_i\ge 1,$ $p_i,r_i>1$ such that $1/p_i+1/r_i=1$ and ${\lambda }_i\in {\mathrm{C}}_{rd}({\left[a_i,{\epsilon }_i\right]}_{\mathbb{T}},$ ${\mathbb{R}}^{+}\cup \left\{0\right\})$ is a delta-differentiable function and an increasing function with ${\lambda }_i\left(a_i\right)=0;$ $i=1,2,\dots ,n.$ Then

$$\begin{array}{ccc}& & {\int }_{a_1}^{{\epsilon }_1}\dots {\int }_{a_n}^{{\epsilon }_n}\frac{{\prod }_{i=1}^n{\lambda }_i^{h_i}\left({\xi }_i\right)}{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}\Delta {\xi }_1\dots \Delta {\xi }_n\hfill \\ & \le & Q\prod _{i=1}^n{\left({\int }_{a_i}^{{\epsilon }_i}\left({\epsilon }_i-\sigma \left({\xi }_i\right)\right){\left({\left[{\lambda }_i^{\sigma }\left({\xi }_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left({\xi }_i\right)\right)}^{p_i}\Delta {\xi }_i\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

(49)

where

$$Q={\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{h}_i{\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}.$$

(50)

Proof. 

Applying the chain rule formula (11) on the term ${\lambda }_i^{h_i}\left(t_i\right),$ $h_i\ge 1,$ we obtain

$${\left[{\lambda }_i^{h_i}\left(t_i\right)\right]}^{\Delta }=h_i{\lambda }_i^{h_i-1}\left({\zeta }_i\right){\lambda }_i^{\Delta }\left(t_i\right),$$

(51)

where ${\zeta }_i\in \left[t_i,\sigma \left(t_i\right)\right].$ Since ${\lambda }_i$ is an increasing function, $h_i\ge 1$ and ${\zeta }_i\le \sigma \left(t_i\right),$ we know from (51) that

$${\left[{\lambda }_i^{h_i}\left(t_i\right)\right]}^{\Delta }\le h_i{\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right),$$

and then (where ${\lambda }_i\left(a_i\right)=0$), we observe that

$$h_i{\int }_{a_i}^{{\xi }_i}{\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\Delta t_i\ge {\int }_{a_i}^{{\xi }_i}{\left[{\lambda }_i^{h_i}\left(t_i\right)\right]}^{\Delta }\Delta t_i={\lambda }_i^{h_i}\left({\xi }_i\right)-{\lambda }_i^{h_i}\left(a_i\right)={\lambda }_i^{h_i}\left({\xi }_i\right).$$

Thus,

$$\prod _{i=1}^n{h}_i{\int }_{a_i}^{{\xi }_i}{\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\Delta t_i\ge \prod _{i=1}^n{\lambda }_i^{h_i}\left({\xi }_i\right).$$

(52)

Applying (13) on ${\int }_{a_i}^{{\xi }_i}{\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\Delta t_i$ with $f\left(t_i\right)={\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)$ and $g\left(t_i\right)=1,$ we have

$$\begin{array}{ccc}\hfill {\int }_{a_i}^{{\xi }_i}{\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\Delta t_i& \le & {\left({\int }_{a_i}^{{\xi }_i}\Delta t_i\right)}^{\frac{1}{r_i}}{\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}\hfill \\ & =& {\left({\xi }_i-a_i\right)}^{\frac{1}{r_i}}{\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

and then

$$\prod _{i=1}^n{\int }_{a_i}^{{\xi }_i}{\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\Delta t_i\le \prod _{i=1}^n{\left({\xi }_i-a_i\right)}^{\frac{1}{r_i}}\prod _{i=1}^n{\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}.$$

(53)

Substituting (53) into (52), we get

$$\prod _{i=1}^n{\lambda }_i^{h_i}\left({\xi }_i\right)\le \prod _{i=1}^n{h}_i{\left({\xi }_i-a_i\right)}^{\frac{1}{r_i}}\prod _{i=1}^n{\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}.$$

(54)

Applying (16) with $s_i={\xi }_i-a_i,$ we have

$$\prod _{i=1}^n{\left({\xi }_i-a_i\right)}^{1/r_i}\le \frac{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}{{\left(n-{\sum }_{i=1}^{n}1/p_i\right)}^{n-{\sum }_{i=1}^{n}1/p_i}}.$$

(55)

Substituting (55) into (54), we obtain

$$\begin{array}{ccc}\hfill \prod _{i=1}^n{\lambda }_i^{h_i}\left({\xi }_i\right)& \le & \frac{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}{{\left(n-{\sum }_{i=1}^{n}1/p_i\right)}^{n-{\sum }_{i=1}^{n}1/p_i}}\hfill \\ & & \times \prod _{i=1}^n{h}_i{\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}.\hfill \end{array}$$

(56)

Dividing (56) by ${\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}$ and integrating over ${\xi }_i$ from $a_i$ to ${\epsilon }_i,$ $i=1,2,\dots ,n,$ we observe that

$$\begin{array}{cc}\hfill & {\int }_{a_n}^{{\epsilon }_n}\dots {\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{i=1}^n{\lambda }_i^{h_i}\left({\xi }_i\right)}{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}\Delta {\xi }_1\dots \Delta {\xi }_n\hfill \\ \hfill & \le {\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\hfill \\ \hfill & \times {\int }_{a_n}^{{\epsilon }_n}\dots {\int }_{a_1}^{{\epsilon }_1}\prod _{i=1}^n{h}_i{\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}\Delta {\xi }_1\dots \Delta {\xi }_n\hfill \\ \hfill & ={\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\hfill \\ \hfill & \times \prod _{i=1}^n{h}_i{\int }_{a_i}^{{\epsilon }_i}{\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}\Delta {\xi }_i.\hfill \end{array}$$

(57)

Applying (13) on ${\int }_{a_i}^{{\epsilon }_i}{\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}\Delta {\xi }_i$ with $f\left({\xi }_i\right)=1$ and

$$g\left({\xi }_i\right)={\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}},$$

we have

$$\begin{array}{ccc}& & {\int }_{a_i}^{{\epsilon }_i}{\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}\Delta {\xi }_i\hfill \\ & \le & {\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}{\left({\int }_{a_i}^{{\epsilon }_i}\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)\Delta {\xi }_i\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

and then

$$\begin{array}{cc}\hfill & \prod _{i=1}^n{\int }_{a_i}^{{\epsilon }_i}{\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)}^{\frac{1}{p_i}}\Delta {\xi }_i\hfill \\ \hfill & \le \prod _{i=1}^n{\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}\prod _{i=1}^n{\left({\int }_{a_i}^{{\epsilon }_i}\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)\Delta {\xi }_i\right)}^{\frac{1}{p_i}}.\hfill \end{array}$$

(58)

Substituting (58) into (57), we have

$$\begin{array}{ccc}& & {\int }_{a_n}^{{\epsilon }_n}\dots {\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{i=1}^n{\lambda }_i^{h_i}\left({\xi }_i\right)}{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}\Delta {\xi }_1\dots \Delta {\xi }_n\hfill \\ & \le & {\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{h}_i{\left({\epsilon }_i-a_i\right)}^{\frac{1}{q_i}}\hfill \\ & & \times \prod _{i=1}^n{\left({\int }_{a_i}^{{\epsilon }_i}\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)\Delta {\xi }_i\right)}^{\frac{1}{p_i}}.\hfill \end{array}$$

(59)

Applying (12) on ${\int }_{a_i}^{{\epsilon }_i}\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)\Delta {\xi }_i$ with $f\left({\xi }_i\right)={\int }_{a_i}^{{\xi }_i}({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}$ ${\lambda }_i^{\Delta }\left(t_i\right){)}^{p_i}\Delta t_i$ and $g^{\Delta }\left({\xi }_i\right)=1,$ we obtain

$$\begin{array}{ccc}& & {\int }_{a_i}^{{\epsilon }_i}\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)\Delta {\xi }_i\hfill \\ & =& {\left.g\left({\xi }_i\right)\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)\right|}_{a_i}^{{\epsilon }_i}\hfill \\ & & -{\int }_{a_i}^{{\epsilon }_i}{g}^{\sigma }\left({\xi }_i\right){\left({\left[{\lambda }_i^{\sigma }\left({\xi }_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left({\xi }_i\right)\right)}^{p_i}\Delta {\xi }_i,\hfill \end{array}$$

(60)

where $g\left({\xi }_i\right)={\xi }_i-{\epsilon }_i.$ Since $g\left({\epsilon }_i\right)=0,$ we know from (60) that

$${\int }_{a_i}^{{\epsilon }_i}\left({\int }_{a_i}^{{\xi }_i}{\left({\left[{\lambda }_i^{\sigma }\left(t_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left(t_i\right)\right)}^{p_i}\Delta t_i\right)\Delta {\xi }_i={\int }_{a_i}^{{\epsilon }_i}\left({\epsilon }_i-\sigma \left({\xi }_i\right)\right){\left({\left[{\lambda }_i^{\sigma }\left({\xi }_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left({\xi }_i\right)\right)}^{p_i}\Delta {\xi }_i,$$

(61)

Substituting (61) into (59), we observe that

$$\begin{array}{ccc}& & {\int }_{a_n}^{{\epsilon }_n}\dots {\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{i=1}^n{\lambda }_i^{h_i}\left({\xi }_i\right)}{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}\Delta {\xi }_1\dots \Delta {\xi }_n\hfill \\ & \le & {\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{h}_i{\left({\epsilon }_i-a_i\right)}^{\frac{1}{q_i}}\hfill \\ & & \times \prod _{i=1}^n{\left({\int }_{a_i}^{{\epsilon }_i}\left({\epsilon }_i-\sigma \left({\xi }_i\right)\right){\left({\left[{\lambda }_i^{\sigma }\left({\xi }_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left({\xi }_i\right)\right)}^{p_i}\Delta {\xi }_i\right)}^{\frac{1}{p_i}}.\hfill \end{array}$$

(62)

From (50), the inequality (62) becomes

$$\begin{array}{ccc}& & {\int }_{a_n}^{{\epsilon }_n}\dots {\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{i=1}^n{\lambda }_i^{h_i}\left({\xi }_i\right)}{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}\Delta {\xi }_1\dots \Delta {\xi }_n\hfill \\ & \le & Q\prod _{i=1}^n{\left({\int }_{a_i}^{{\epsilon }_i}\left({\epsilon }_i-\sigma \left({\xi }_i\right)\right){\left({\left[{\lambda }_i^{\sigma }\left({\xi }_i\right)\right]}^{h_i-1}{\lambda }_i^{\Delta }\left({\xi }_i\right)\right)}^{p_i}\Delta {\xi }_i\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

which is (49). □

Remark 1.

If $\mathbb{T}=\mathbb{R}$ and $a_i=0$ for $i=1,2,\dots ,n,$ then we obtain the inequality (9) for the non-negative increasing function λ with ${\lambda }_i\left(0\right)=0,$ $i=1,2,\dots ,n.$

In the following remark, we present the discrete analogue of (9), i.e., when $\mathbb{T}=\mathbb{N}$, which is new and interesting for the reader.

Corollary 7.

If $\mathbb{T}=\mathbb{N}$, $h_i\ge 1,$ $p_i,r_i>1$ such that $1/p_i+1/r_i=1$ and ${\lambda }_i$ is a non-negative and increasing sequence with ${\lambda }_i\left(a_i\right)=0;$ $i=1,2,\dots ,n,$ then

$$\begin{array}{cc}\hfill & \sum _{{\xi }_1=a_1}^{{\epsilon }_1-1}\dots \sum _{{\xi }_n=a_n}^{{\epsilon }_n-1}\frac{{\prod }_{i=1}^n{\lambda }_i^{h_i}\left({\xi }_i\right)}{{\left({\sum }_{i=1}^n\left({\xi }_i-a_i\right)/r_i\right)}^{{\sum }_{i=1}^{n}1/r_i}}\hfill \\ \hfill & \le Q\prod _{i=1}^n{\left(\sum _{{\xi }_i=a_i}^{{\epsilon }_i-1}\left({\epsilon }_i-{\xi }_i-1\right){\left({\left[{\lambda }_i({\xi }_i+1)\right]}^{h_i-1}\Delta {\lambda }_i\left({\xi }_i\right)\right)}^{p_i}\right)}^{\frac{1}{p_i}},\hfill \end{array}$$

where

$$Q={\left(n-\sum _{i=1}^{n}1/p_i\right)}^{{\sum }_{i=1}^{n}1/p_i-n}\prod _{i=1}^n{h}_i{\left({\epsilon }_i-a_i\right)}^{\frac{1}{r_i}}.$$

4. Conclusions and Future Work

In this paper, we establish some new dynamic Hilbert-type inequalities on time scale delta calculus by applying Hölder’s inequality, the chain rule and the mean inequality. In the future, we will prove Hilbert-type inequalities on diamond—α calculus and fractional conformable calculus.