Similar Formulas to Hilbert-Type Inequalities on Time Scales Delta Calculus

1. Introduction

During the early 1900s, Hilbert made the discovery of this inequality (refer to [10])

$$\sum _{n=1}^{\infty }\sum _{s=1}^{\infty }\frac{a_s{c}_n}{s+n}\le \pi {\left(\sum _{s=1}^{\infty }{a}_s^2\right)}^{\frac{1}{2}}{\left(\sum _{n=1}^{\infty }{c}_n^2\right)}^{\frac{1}{2}}.$$

(1)

Here, ${\left\{a_s\right\}}_{s=1}^{\infty }$ and ${\left\{c_n\right\}}_{n=1}^{\infty }$ are real sequences satisfying $0<{\sum }_{s=1}^{\infty }{a}_s^2<\infty$ and $0<{\sum }_{n=1}^{\infty }{c}_n^2<\infty .$ This particular expression is known as Hilbert’s double series inequality.

In [17], Schur demonstrated that $\pi$ in (1) is the most optimal constant achievable. Additionally, he unveiled the integral counterpart of (1), which later became recognized as the Hilbert integral inequality, taking the form

$${\int }_0^{\infty }{\int }_0^{\infty }\frac{f\left(\eta \right)g\left(\tau \right)}{\eta +\tau }d\eta d\tau \le \pi {\left({\int }_0^{\infty }{f}^2\left(\eta \right)d\eta \right)}^{\frac{1}{2}}{\left({\int }_0^{\infty }{g}^2\left(\tau \right)d\tau \right)}^{\frac{1}{2}},$$

(2)

where $f,g$ are measurable functions satisfying$0<{\int }_0^{\infty }{f}^2\left(\eta \right)d\eta <\infty$ and $0<{\int }_0^{\infty }{g}^2\left(\tau \right)d\tau <\infty .$

In [8], an extension of (1) is presented as follows: suppose $l,r>1$ with $1/l+1/r=1,$ ${\left\{a_s\right\}}_{s=1}^{\infty },$ ${\left\{c_n\right\}}_{n=1}^{\infty }$ are real sequences satisfying $0<{\sum }_{s=1}^{\infty }{a}_s^r<\infty$ and $0<{\sum }_{n=1}^{\infty }{c}_n^l<\infty ,$ then

$$\sum _{n=1}^{\infty }\sum _{s=1}^{\infty }\frac{a_s{c}_n}{s+n}\le \frac{\pi }{sin\frac{\pi }{r}}{\left(\sum _{s=1}^{\infty }{a}_s^r\right)}^{\frac{1}{r}}{\left(\sum _{n=1}^{\infty }{c}_n^l\right)}^{\frac{1}{l}}.$$

(3)

Here, $\pi /sin(\pi /r)$ is the optimal constant.

In [9], the authors derived the integral counterpart of (3) as

$${\int }_0^{\infty }{\int }_0^{\infty }\frac{f\left(\eta \right)g\left(\tau \right)}{\eta +\tau }d\eta d\tau \le \frac{\pi }{sin\frac{\pi }{r}}{\left({\int }_0^{\infty }{f}^r\left(\eta \right)d\eta \right)}^{\frac{1}{r}}{\left({\int }_0^{\infty }{g}^l\left(\tau \right)d\tau \right)}^{\frac{1}{l}}.$$

(4)

Here, $f,g\ge 0$ are measurable functions satisfying $0<{\int }_0^{\infty }{f}^r\left(\eta \right)d\eta <\infty$ and $0<{\int }_0^{\infty }{g}^l\left(\tau \right)d\tau <\infty .$

In [14], new inequalities akin to the ones presented in (3) and (4) were established as follows: let $l,r>1$ with $1/l+1/r=1.$ Consider sequences $a_w:\left\{0,1,2,...,s\right\}\subset \mathbb{N}\to \mathbb{R}$ and $c_{\vartheta }:\left\{0,1,2,...,n\right\}\subset \mathbb{N}\to \mathbb{R}$where $a\left(0\right)=c\left(0\right)=0.$ Then

$$\begin{array}{ccc}\hfill \sum _{w=1}^s\sum _{\vartheta =1}^n\frac{\left|a_w\right|\left|c_{\vartheta }\right|}{l{w}^{r-1}+r{\vartheta }^{l-1}}& \le & D(l,r,s,n){\left(\sum _{w=1}^s(s-w+1){\left|\nabla a_w\right|}^r\right)}^{\frac{1}{r}}\hfill \\ & & \times {\left(\sum _{\vartheta =1}^n(n-\vartheta +1){\left|\nabla c_{\vartheta }\right|}^l\right)}^{\frac{1}{l}}.\hfill \end{array}$$

(5)

Here, $\nabla a_w=a_w-a_{w-1},$ $\nabla c_{\vartheta }=c_{\vartheta }-c_{\vartheta -1}$ and

$$D(l,r,s,n)=\frac{1}{lr}{s}^{\frac{r-1}{r}}{n}^{\frac{l-1}{l}}.$$

Moreover, if $l,r>1$ with $1/l+1/r=1,$ $f\left(w\right)$ and $g\left(\vartheta \right)$ are real-valued continuous functions with $f\left(0\right)=g\left(0\right)=0,$ then

$$\begin{array}{ccc}\hfill {\int }_0^{\eta }{\int }_0^{\tau }\frac{\left|f\left(w\right)\right|\left|g\left(\vartheta \right)\right|}{l{w}^{r-1}+r{\vartheta }^{l-1}}dwd\vartheta & \le & M(l,r,\eta ,\tau ){\left({\int }_0^{\eta }\left(\eta -w\right){\left|f^{\prime }\left(w\right)\right|}^{r}dw\right)}^{\frac{1}{r}}\hfill \\ & & \times {\left({\int }_0^{\tau }\left(\tau -\vartheta \right){\left|g^{\prime }\left(\vartheta \right)\right|}^{l}d\vartheta \right)}^{\frac{1}{l}}.\hfill \end{array}$$

(6)

Here,

$$M(l,r,\eta ,\tau )=\frac{1}{lr}{\eta }^{\frac{r-1}{r}}{\tau }^{\frac{l-1}{l}}.$$

In [7], Chang-Jian et al. established a set of new inequalities that are similar to extensions of Hilbert’s double-series inequality and derived their integral analogues. These inequalities are outlined as follows: let $r_j>1$ such that $1/l_j+1/r_j=1$ and $a_j\left(w_j\right)$ are real sequences defined for $w_j=0,1,2,...,s_j$ where $s_j\in \mathbb{N}$ and $a_j\left(0\right)=0;$ $j=1,2,...,n.$ Define the operator ∇ as $\nabla a_j\left(w_j\right)=a_j\left(w_j\right)-a_j(w_j-1).$ Then

$$\sum _{w_1=1}^{s_1}\sum _{w_2=1}^{s_2}...\sum _{w_n=1}^{s_n}\frac{{\prod }_{j=1}^n\left|a_j\left(w_j\right)\right|}{{\left({\displaystyle \sum _{j=1}^n}\frac{w_j}{l_j}\right)}^{{\displaystyle \sum _{j=1}^n}\frac{1}{l_j}}}\le K\prod _{j=1}^n{\left(\sum _{w_j=1}^{s_j}\left(s_j-w_j+1\right){\left|\Delta a_j\left(w_j\right)\right|}^{r_j}\right)}^{\frac{1}{r_j}}.$$

(7)

Here,

$$K={\left(n-\sum _{j=1}^n\frac{1}{r_j}\right)}^{{\sum }_{j=1}^n\frac{1}{r_j}-n}\prod _{j=1}^n{s}_j^{\frac{1}{l_j}}.$$

Also, they proved that if $h_j\ge 1,$ $l_j,r_j>1$ are constants with $1/r_j+1/l_j=1,$ $f_j\left(w_j\right)$ are real valued differentiable functions defined on $[0,{\eta }_j)$ where ${\eta }_j\in \left(0,\infty \right)$ and $f_j\left(0\right)=0;$ $j=1,2,...,n,$ then

$${\int }_0^{{\eta }_1}...{\int }_0^{{\eta }_n}\frac{{\prod }_{j=1}^n\left|f_j^{h_j}\left(w_j\right)\right|}{{\left({\displaystyle \sum _{j=1}^n}\frac{w_j}{l_j}\right)}^{{\displaystyle \sum _{j=1}^n}\frac{1}{l_j}}}d{w}_n...d{w}_1\le L\prod _{j=1}^n{\left({\int }_0^{{\eta }_j}\left({\eta }_j-w_j\right){\left|f_j^{h_j-1}\left(w_j\right).f_j^{\prime }\left(w_j\right)\right|}^{r_j}d{w}_j\right)}^{\frac{1}{r_j}},$$

(8)

where

$$L={\left(n-\sum _{j=1}^n\frac{1}{r_j}\right)}^{{\sum }_{j=1}^n\frac{1}{r_j}-n}\prod _{j=1}^n{h}_j{\eta }_j^{\frac{1}{l_j}}.$$

Furthermore, they established that if $l_j,r_j>1$ such that $1/r_j+1/l_j=1,$ $a_j(w_j,z_j)$ be real sequences defined for $\left(w_j,z_j\right)$ where $w_j=0,1,2,...,s_j,$ $z_j=0,1,2,...,n_j;$ $s_j,n_j\in \mathbb{N}$ and $a_j(0,z_j)=a_j(w_j,0)=0$ $\forall j=1,2,...,n.$ Define the operators ${\nabla }_1$ and ${\nabla }_2$ by

$${\nabla }_1{a}_j(w_j,z_j)=a_j(w_j,z_j)-a_j(w_j-1,z_j),$$

$${\nabla }_2{a}_j(w_j,z_j)=a_j(w_j,z_j)-a_j(w_j,z_j-1).$$

Then

$$\begin{array}{ccc}& & \sum _{w_1=1}^{s_1}\sum _{z_1=1}^{t_1}...\sum _{w_n=1}^{s_n}\sum _{z_n=1}^{t_n}\frac{{\prod }_{j=1}^n\left|a_j(w_j,z_j)\right|}{{\left({\displaystyle \sum _{j=1}^n}{w}_j{z}_j/l_j\right)}^{{\displaystyle \sum _{j=1}^n}\frac{1}{l_j}}}\hfill \\ & \le & R\prod _{j=1}^n{\left(\sum _{w_j=1}^{s_j}\sum _{z_j=1}^{t_j}\left(s_j-w_j+1\right)\left(t_j-z_j+1\right){\left|{\nabla }_2{\nabla }_1{a}_j(w_j,z_j)\right|}^{r_j}\right)}^{\frac{1}{r_j}}.\hfill \end{array}$$

(9)

Here,

$$R={\left(n-\sum _{j=1}^n\frac{1}{r_j}\right)}^{{\sum }_{j=1}^n\frac{1}{r_j}-n}.\prod _{j=1}^n{\left(s_j{n}_j\right)}^{\frac{1}{l_j}}.$$

In recent decades, a novel theory known as time scale theory has emerged, aimed at unifying continuous calculus and discrete calculus. The results presented in this paper encompass classical continuous and discrete inequalities as special cases when $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{N},$ respectively. Moreover, these inequalities can be extended to analogous inequalities on various time scales, such as $\mathbb{T}=q^{\stackrel{-}{\mathbb{Z}}}$ for $q>1$. Many researchers have delved into dynamic inequalities on time scales, and for a more comprehensive understanding of these dynamic inequalities on time scales, readers are referred to the following papers: [3,4,12,15,16,19,20,21].

The primary objective of this paper is to establish analogous formulas for Hilbert-type inequalities (7), (9) and (8) within the framework of time scales in delta calculus. It’s important to note that these formulas are derived under specific conditions, which are $a_j\left(s_j\right)=0$ and$a_j(s_j,z_j)=a_j(w_j,n_j)=0$ $\forall j=1,2,...,n.$ These conditions differ from those utilized in a previous work [7]. The outcomes of our research provide novel insights and estimations for these specific categories of inequalities. In particular, we have introduced multivariate summation inequalities for extensions of the Hilbert inequality, which were previously unproven. Additionally, we have obtained their corresponding integral expressions. The proofs of these results are based on the application of Hölder’s inequality on time scales and the mean inequality.

The paper is structured as follows: After this introductory section, the subsequent section offers an overview of fundamental concepts in time scale calculus, which serve as the basis for our proofs. The final section is dedicated to presenting our main findings.

2. Basic Principles

In what follows, the time scale $\mathbb{T}$ is a nonempty closed subset of $\mathbb{R}$, and it could be an interval, a union of intervals, or even a set of isolated points. The real numbers (continuous case), integers (discrete case), and various amalgamations of the two constitute the most prevalent instances of time scales. Given $\nu \in \mathbb{T}$, we establish $\sigma :\mathbb{T}\to \mathbb{T}$ and $\mu :\mathbb{T}\to \mathbb{R}$ as $\sigma \left(\nu \right):=inf\{\alpha \in \mathbb{T}:\alpha >\nu \}$ and $\mu \left(\nu \right):=\sigma \left(\nu \right)-\nu \ge 0.$ These components are referred to as the forward jump operator and the forward graininess function, correspondingly. Considering a function $\Im :\mathbb{T}\to \mathbb{R}$, we introduce the notation:

$${\Im }^{\sigma }\left(\nu \right)=\Im \left(\sigma \left(\nu \right)\right)\forall \nu \in \mathbb{T}.$$

Additionally, we establish the interval ℓ within the context of $\mathbb{T}$ as:

$${\ell }_{\mathbb{T}}:=\ell \cap \mathbb{T}\ell \subset \mathbb{R}.$$

Now, let’s examine a function $\Im :\mathbb{T}\to \mathbb{R}$ and we introduce the concept of the "delta derivative" or "$\Delta$ differentiable" of ℑ at a point $v\in \mathbb{T}$ as follows.

Definition 1 

([5]). We use the term "Δ differentiable" to describe a function ℑ being differentiable at $v\in \mathbb{T}$, if $\forall \epsilon >0$ there is a neighborhood W of v such that for some β the inequality

$$|\Im (\sigma \left(v\right))-\Im (w)-\beta (\sigma \left(v\right)-w\left)\right|\le \epsilon \left|\sigma \right(v)-w|,w\in W$$

is true and in this case we write ${\Im }^{\Delta }\left(v\right)=\beta$.

Theorem 1 

(Properties of delta-derivatives [5]). Assume ℑ is a function and let $v\in {\mathbb{T}}^k$, then

1. 

If ℑ is differentiable at v, then ℑ is continuous at v.

2. 

If ℑ is continuous at v and v is right-scattered (i.e $\sigma \left(v\right)>v)$, then ℑ is differentiable at v with

$${\Im }^{\Delta }\left(v\right)=\frac{\Im \left(\sigma \right(v\left)\right)-\Im \left(v\right)}{\mu \left(v\right)}.$$

3. 

If v is right-dense (i.e $\sigma \left(v\right)=v)$, then ℑ is differentiable iff the limit

$$\underset{w\to v}{lim}\frac{\Im \left(v\right)-\Im \left(w\right)}{v-w},$$

exists as a finite number. In this case

$${\Im }^{\Delta }\left(v\right)=\underset{w\to v}{lim}\frac{\Im \left(v\right)-\Im \left(w\right)}{v-w}.$$

Example 1. 

1. If $\mathbb{T}=\mathbb{R},$ then $\sigma \left(v\right)=v,$ $\mu \left(v\right)=0$ and

$${\Im }^{\Delta }\left(v\right)=\underset{w\to v}{lim}\frac{\Im \left(v\right)-\Im \left(w\right)}{v-w}={\Im }^{\prime }\left(v\right)\forall v\in \mathbb{T},$$

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

2. If $\mathbb{T}=\mathbb{Z},$ then $\sigma \left(v\right)=v+1,$ $\mu \left(v\right)=1$ and

$${\Im }^{\Delta }\left(v\right)=\frac{\Im \left(\sigma \right(\Im \left)\right)-\Im \left(v\right)}{\mu \left(v\right)}=\Im (v+1)-\Im \left(v\right)=\Delta \Im \left(v\right),$$

where Δ is the usual forward difference operator.

3. If $\mathbb{T}=q^{\stackrel{-}{\mathbb{Z}}}:=\{v:v=q^k,$ $k\in \mathbb{Z},$ $q>1\}\cup \{0\},$ then $\sigma \left(v\right)=qv,$ $\mu \left(v\right)=(q-1)v$ and

$${\Im }^{\Delta }\left(v\right)={\Delta }_q\Im \left(v\right)=\frac{\Im \left(qv\right)-\Im \left(v\right)}{(q-1)v}\forall v\in \mathbb{T}\setminus \left\{0\right\}.$$

Theorem 2 

(Chain Rule [5]). Given that ${\rm Y}:\mathbb{T}\to \mathbb{R}$ is a continuous and Δ differentiable and $\Im :\mathbb{R}\to \mathbb{R}$ is continuously differentiable, then

$${\left(\Im \circ {\rm Y}\right)}^{\Delta }\left(v\right)={\Im }^{\prime }\left({\rm Y}\left(\tau \right)\right){{\rm Y}}^{\Delta }\left(v\right)\mathit{for}\tau \in [v,\sigma \left(v\right)].$$

(10)

Definition 2 

([5]). A function ℑ is characterized as $rd-$continuous when it exhibits continuity at every right-dense point within $\mathbb{T}$ and possesses finite left-sided limits at left-dense points in $\mathbb{T}$. We use the symbol ${\mathrm{C}}_{rd}(\mathbb{T},\mathbb{R})$ to represent the sets of all rd-continuous functions and the symbol $\mathrm{C}(\mathbb{T},\mathbb{R})$ to represent the set of all continuous functions.

Following is a description of the concept of an integral on time scales.

Definition 3 

([5]). ℜ is Δ antiderivative of ℑ if

$${\Re }^{\Delta }\left(v\right)=\Im \left(v\right)\mathit{holds}\forall v\in {\mathbb{T}}^k.$$

As a result, for $a,c\in \mathbb{T}$, we deduce the integral of ℑ as

$${\int }_a^c\Im \left(v\right)\Delta v=\Re \left(c\right)-\Re \left(a\right).$$

It’s widely acknowledged that any rd-continuous function possesses an antiderivative. As a result, we can deduce the following outcomes.

Theorem 3 

([5]). If $v_0,v\in \mathbb{T}$, then

$${\left({\int }_{v_0}^v\Im \left(\vartheta \right)\Delta \vartheta \right)}^{\Delta }=\Im \left(v\right).$$

(11)

Theorem 4 

([5]). If $a,c,\tau \in \mathbb{T},$ $\alpha ,\beta \in \mathbb{R}$ and $\Im ,{\rm Y}\in {\mathrm{C}}_{rd}{([a,c]}_{\mathbb{T}},$ $\mathbb{R})$, then

1. 

${\int }_a^c\left[\alpha \Im \left(\delta \right)+\beta {\rm Y}\left(\delta \right)\right]\Delta \delta =\alpha {\int }_a^c\Im \left(\delta \right)\Delta \delta +\beta {\int }_a^c{\rm Y}\left(\delta \right)\Delta \delta ;$

2. 

${\int }_a^a\Im \left(\delta \right)\Delta \delta =0;$

3. 

${\int }_a^c\Im \left(\delta \right)\Delta \delta ={\int }_a^{\tau }\Im \left(\delta \right)\Delta \delta +{\int }_{\tau }^c\Im \left(\delta \right)\Delta \delta ;$

4. 

If $\Im \left(\delta \right)\ge 0;$ $\forall \delta \in {[a,c]}_{\mathbb{T}},$ then ${\int }_a^c\Im \left(\delta \right)\Delta \delta \ge 0.$

5. 

$\left|{\int }_a^c\Im \left(\delta \right)\Delta \delta \right|\le {\int }_a^c\left|\Im \left(\delta \right)\right|\Delta \delta .$

Lemma 1 

(Integration by Parts [1]). If $a,c\in \mathbb{T}$ and $\omega ,\kappa \in {\mathrm{C}}_{rd}{([a,c]}_{\mathbb{T}},$ $\mathbb{R}),$ then

$${\int }_a^c\omega \left(\delta \right){\kappa }^{\Delta }\left(\delta \right)\Delta \delta ={\left[\omega \left(\delta \right)\kappa \left(\delta \right)\right]}_a^c-{\int }_a^c{\omega }^{\Delta }\left(\delta \right){\kappa }^{\sigma }\left(\delta \right)\Delta \delta .$$

(12)

Theorem 5 

([1]). Let $a,c\in \mathbb{T}$ and $\Im \in {\mathrm{C}}_{rd}(\mathbb{T},\mathbb{R}).$ Then

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

$${\int }_a^c\Im \left(\delta \right)\Delta \delta ={\int }_a^c\Im \left(\delta \right)d\delta .$$

$\left(jj\right)$ If $\mathbb{T}=\mathbb{Z}$, then

$${\int }_a^c\Im \left(\delta \right)\Delta \delta =\sum _{\delta =a}^{c-1}\Im \left(\delta \right).$$

$\left(jjj\right)$ If $\mathbb{T}=q^{\stackrel{-}{\mathbb{Z}}}$, then

$${\int }_a^c\Im \left(\delta \right)\Delta \delta =(q-1)\sum _{k={log}_{q}a}^{{log}_{q}c-1}{q}^k\Im \left(q^k\right).$$

Lemma 2 

(Hölder’s Inequality [1]). If $a,c\in \mathbb{T}$ and $\Im ,{\rm Y}\in {\mathrm{C}}_{rd}{([a,c]}_{\mathbb{T}},$ ${\mathbb{R}}^{+}),$ then

$${\int }_a^c\Im \left(\delta \right){\rm Y}\left(\delta \right)\Delta \delta \le {\left[{\int }_a^c\omega \left(\delta \right){\Im }^{\eta }\left(\delta \right)\Delta \delta \right]}^{\frac{1}{\eta }}{\left[{\int }_a^c\omega \left(\delta \right){{\rm Y}}^{\lambda }\left(\delta \right)\Delta \delta \right]}^{\frac{1}{\lambda }},$$

(13)

where $\eta >1$ and $1/\eta +1/\lambda =1.$

Next, we present the Hölder’s inequality in two dimensions on time scales.

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

Lemma 3 

([18], Theorem 3.3). Let $\eta ,\lambda \in \mathbb{T}$ with $\eta <\lambda ,$ $f,g\in C{C}_{rd}({\left[\eta ,\lambda \right]}_{\mathbb{T}}\times {\left[\eta ,\lambda \right]}_{\mathbb{T}},$ $\mathbb{R})$ and $\gamma ,\nu >1$ such that $1/\gamma +1/\nu =1.$ Then

$$\begin{array}{ccc}& & {\int }_{\eta }^{\lambda }{\int }_{\eta }^{\lambda }\left|f(\tau ,\xi )g(\tau ,\xi )\right|{\Delta }_1\tau {\Delta }_2\xi \hfill \\ & \le & {\left[{\int }_{\eta }^{\lambda }{\int }_{\eta }^{\lambda }{\left|f(\tau ,\xi )\right|}^{\gamma }{\Delta }_1\tau {\Delta }_2\xi \right]}^{\frac{1}{\gamma }}{\left[{\int }_{\eta }^{\lambda }{\int }_{\eta }^{\lambda }{\left|g(\tau ,\xi )\right|}^{\nu }{\Delta }_1\tau {\Delta }_2\xi \right]}^{\frac{1}{\nu }}.\hfill \end{array}$$

(14)

Lemma 4 

(Fubini’s theorem [6]). If $\eta ,\lambda ,c,d\in \mathbb{T}$ and $\Im \in C{C}_{rd}\left({\left[\eta ,\lambda \right]}_{\mathbb{T}}\times {\left[c,d\right]}_{\mathbb{T}},\mathbb{R}\right)$ is $\Delta -$integrable, then

$${\int }_{\eta }^{\lambda }\left({\int }_c^d\Im (\tau ,\xi ){\Delta }_2\xi \right){\Delta }_1\tau ={\int }_c^d\left({\int }_{\eta }^{\lambda }\Im (\tau ,\xi ){\Delta }_1\tau \right){\Delta }_2\xi .$$

Lemma 5 

(Mean inequality [9]). If ${\alpha }_j,{\beta }_j>0$ for $j=1,2,...,s,$ then

$$\prod _{j=1}^s{\alpha }_j^{{\beta }_j}\le \frac{{\left({\sum }_{j=1}^s{\alpha }_j{\beta }_j\right)}^{{\sum }_{j=1}^s{\beta }_j}}{{\left({\sum }_{j=1}^s{\beta }_j\right)}^{{\sum }_{j=1}^s{\beta }_j}}.$$

(15)

3. Main Results

Throughout this paper, we will operate under the assumption that the functions are rd-continuous and we will also consider the existence of the integrals. To substantiate our results, it is necessary to prove the following lemma.

Lemma 6. 

Let $l_j,r_j>1$ with $1/l_j+1/r_j=1$ and $w_j>0,$ where $j=1,2,...,n.$ Then

$$\prod _{j=1}^n{w}_j^{\frac{1}{l_j}}\le \frac{{\left({\sum }_{j=1}^n\frac{w_j}{l_j}\right)}^{{\sum }_{j=1}^m\frac{1}{l_j}}}{{\left(s-{\sum }_{j=1}^s\frac{1}{r_j}\right)}^{\left(s-{\sum }_{j=1}^s\frac{1}{r_j}\right)}}.$$

(16)

Proof. 

By utilizing Lemma 5 with ${\alpha }_j=w_j$ and ${\beta }_j=1/l_j,$ we deduce that

$$\prod _{j=1}^s{w}_j^{\frac{1}{l_j}}\le \frac{{\left({\sum }_{j=1}^s\frac{w_j}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}{{\left({\sum }_{j=1}^s\frac{1}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}.$$

(17)

Since ${\sum }_{j=1}^s(1/l_j)={\sum }_{j=1}^s\left(1-(1/r_j)\right)=s-{\sum }_{j=1}^s(1/r_j),$ then (17) becomes

$$\prod _{j=1}^s{w}_j^{\frac{1}{l_j}}\le \frac{{\left({\sum }_{j=1}^s\frac{w_j}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}{{\left(s-{\sum }_{j=1}^s\frac{1}{r_j}\right)}^{s-{\sum }_{j=1}^s\frac{1}{r_j}}},$$

which is (16). □

Theorem 6. 

Let $a_j,{\epsilon }_j\in \mathbb{T},$ $l_j,r_j>1$ such that $1/l_j+1/r_j=1$ and ${\lambda }_j\in {\mathrm{C}}_{rd}({\left[a_j,{\epsilon }_j\right]}_{\mathbb{T}},\mathbb{R})$ with ${\lambda }_j\left({\epsilon }_j\right)=0;$ $j=1,2,...,s.$ Then

$$\begin{array}{ccc}& & {\int }_{a_s}^{{\epsilon }_s}...{\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{j=1}^s\left|{\lambda }_j\left({\xi }_j\right)\right|}{{\left({\sum }_{j=1}^s\frac{{\epsilon }_j-{\xi }_j}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}\Delta {\xi }_1...\Delta {\xi }_s\hfill \\ & \le & A\prod _{j=1}^s{\left({\int }_{a_j}^{{\epsilon }_j}\left(\sigma \left({\xi }_j\right)-a_j\right){\left|{\lambda }_j^{\Delta }\left({\xi }_j\right)\right|}^{r_j}\Delta {\xi }_j\right)}^{\frac{1}{r_j}},\hfill \end{array}$$

(18)

where

$$A={\left(s-\sum _{j=1}^s\frac{1}{r_j}\right)}^{{\sum }_{j=1}^s\frac{1}{r_j}-s}\prod _{j=1}^s{\left({\epsilon }_j-a_j\right)}^{\frac{1}{l_j}}.$$

(19)

Proof. 

By utilizing the property (5) of Theorem 4 and using the assumption ${\lambda }_j\left({\epsilon }_j\right)=0$, we deduce that

$$\begin{array}{ccc}\hfill {\int }_{{\xi }_j}^{{\epsilon }_j}\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|\Delta z_j& \ge & \left|{\int }_{{\xi }_j}^{{\epsilon }_j}{\lambda }_j^{\Delta }\left(z_j\right)\Delta z_j\right|\hfill \\ & =& \left|{\lambda }_j\left({\epsilon }_j\right)-{\lambda }_j\left({\xi }_j\right)\right|=\left|{\lambda }_j\left({\xi }_j\right)\right|\hfill \end{array}$$

and then

$$\prod _{j=1}^s\left|{\lambda }_j\left({\xi }_j\right)\right|\le \prod _{j=1}^s{\int }_{{\xi }_j}^{{\epsilon }_j}\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|\Delta z_j.$$

(20)

Applying (13) on ${\int }_{{\xi }_j}^{{\epsilon }_j}\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|\Delta z_j$ with $l_j,r_j>1,$ $\Im \left(z_j\right)=\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|$ and ${\rm Y}\left(z_j\right)=1,$ we have

$$\begin{array}{ccc}\hfill {\int }_{{\xi }_j}^{{\epsilon }_j}\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|\Delta z_j& \le & {\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\right)}^{\frac{1}{r_j}}{\left({\int }_{{\xi }_j}^{{\epsilon }_j}\Delta z_j\right)}^{\frac{1}{l_j}}\hfill \\ & =& {\left({\epsilon }_j-{\xi }_j\right)}^{\frac{1}{l_j}}{\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\right)}^{\frac{1}{r_j}}\hfill \end{array}$$

and then

$$\begin{array}{ccc}\hfill \prod _{j=1}^s{\int }_{{\xi }_j}^{{\epsilon }_j}\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|\Delta z_j& \le & \prod _{j=1}^s{\left({\epsilon }_j-{\xi }_j\right)}^{\frac{1}{l_j}}{\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\right)}^{\frac{1}{r_j}}\hfill \\ & =& \prod _{j=1}^s{\left({\epsilon }_j-{\xi }_j\right)}^{\frac{1}{l_j}}\prod _{j=1}^s{\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\right)}^{\frac{1}{r_j}}.\hfill \end{array}$$

(21)

By substituting (21) into (20) and applying (16) on ${\prod }_{j=1}^s{\left({\epsilon }_j-{\xi }_j\right)}^{(1/l_j)}$ with $w_j={\epsilon }_j-{\xi }_j,$ we acquire

$$\begin{array}{ccc}\hfill \prod _{j=1}^s\left|{\lambda }_j\left({\xi }_j\right)\right|& \le & \prod _{j=1}^s{\left({\epsilon }_j-{\xi }_j\right)}^{\frac{1}{l_j}}\prod _{j=1}^s{\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\right)}^{\frac{1}{r_j}}\hfill \\ & \le & \frac{{\left({\sum }_{j=1}^s\frac{{\epsilon }_j-{\xi }_j}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}{{\left(s-{\sum }_{j=1}^s\frac{1}{r_j}\right)}^{s-{\sum }_{j=1}^s\frac{1}{r_j}}}\prod _{j=1}^s{\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\right)}^{\frac{1}{r_j}}.\hfill \end{array}$$

(22)

Dividing (22) on ${\left({\sum }_{j=1}^s\frac{{\epsilon }_j-{\xi }_j}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}$ and integrating over ${\xi }_j$ from $a_j$ to ${\epsilon }_j,$ $j=1,2,...,s,$ we conclude that

$$\begin{array}{cc}\hfill & {\int }_{a_s}^{{\epsilon }_s}...{\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{j=1}^s\left|{\lambda }_j\left({\xi }_j\right)\right|}{{\left({\sum }_{j=1}^s\frac{{\epsilon }_j-{\xi }_j}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}\Delta {\xi }_1...\Delta {\xi }_s\hfill \\ \hfill & \le {\left(s-\sum _{j=1}^s\frac{1}{r_j}\right)}^{{\sum }_{j=1}^s\frac{1}{r_j}-s}{\int }_{a_s}^{{\epsilon }_s}...{\int }_{a_1}^{{\epsilon }_1}\prod _{j=1}^s{\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\right)}^{\frac{1}{r_j}}\Delta {\xi }_1...\Delta {\xi }_s\hfill \\ \hfill & ={\left(s-\sum _{j=1}^s\frac{1}{r_j}\right)}^{{\sum }_{j=1}^s\frac{1}{r_j}-s}\prod _{j=1}^s{\int }_{a_j}^{{\epsilon }_j}{\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\right)}^{\frac{1}{r_j}}\Delta {\xi }_j.\hfill \end{array}$$

(23)

Again, using (13) on ${\int }_{a_j}^{{\epsilon }_j}{\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\right)}^{\frac{1}{r_j}}\Delta {\xi }_j$ with $l_j,r_j>1,$ $\Im \left({\xi }_j\right)={\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\right)}^{\frac{1}{r_j}}$ and ${\rm Y}\left({\xi }_j\right)=1,$ we get

$$\begin{array}{ccc}\hfill {\int }_{a_j}^{{\epsilon }_j}{\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\right)}^{\frac{1}{r_j}}\Delta {\xi }_j& \le & {\left({\int }_{a_j}^{{\epsilon }_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\Delta {\xi }_j\right)}^{\frac{1}{r_j}}{\left({\int }_{a_j}^{{\epsilon }_j}\Delta {\xi }_j\right)}^{\frac{1}{l_j}}\hfill \\ & =& {\left({\epsilon }_j-a_j\right)}^{\frac{1}{l_j}}{\left({\int }_{a_j}^{{\epsilon }_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\Delta {\xi }_j\right)}^{\frac{1}{r_j}},\hfill \end{array}$$

and then

$$\begin{array}{ccc}& & \prod _{j=1}^s{\int }_{a_j}^{{\epsilon }_j}{\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\right)}^{\frac{1}{r_j}}\Delta {\xi }_j\hfill \\ & \le & \prod _{j=1}^s{\left({\epsilon }_j-a_j\right)}^{\frac{1}{l_j}}{\left({\int }_{a_j}^{{\epsilon }_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\Delta {\xi }_j\right)}^{\frac{1}{r_j}}\hfill \\ & =& \prod _{j=1}^s{\left({\epsilon }_j-a_j\right)}^{\frac{1}{l_j}}\prod _{j=1}^s{\left({\int }_{a_j}^{{\epsilon }_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\Delta {\xi }_j\right)}^{\frac{1}{r_j}}.\hfill \end{array}$$

(24)

Substituting (24) into (23), we obtain

$$\begin{array}{cc}\hfill & {\int }_{a_s}^{{\epsilon }_s}...{\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{j=1}^s\left|{\lambda }_j\left({\xi }_j\right)\right|}{{\left({\sum }_{j=1}^s\frac{{\epsilon }_j-{\xi }_j}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}\Delta {\xi }_1...\Delta {\xi }_s\hfill \\ \hfill & \le {\left(s-\sum _{j=1}^s\frac{1}{r_j}\right)}^{{\sum }_{j=1}^s\frac{1}{r_j}-s}\prod _{j=1}^s{\left({\epsilon }_j-a_j\right)}^{\frac{1}{l_j}}\prod _{j=1}^s{\left({\int }_{a_j}^{{\epsilon }_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\Delta {\xi }_j\right)}^{\frac{1}{r_j}}.\hfill \end{array}$$

(25)

Now, using (12) on ${\int }_{a_j}^{{\epsilon }_j}\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\right)\Delta {\xi }_j$ with $\omega \left({\xi }_j\right)={\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j$ and ${\kappa }^{\Delta }\left({\xi }_j\right)=1,$ we find that

$$\begin{array}{ccc}& & {\int }_{a_j}^{{\epsilon }_j}\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\right)\Delta {\xi }_j\hfill \\ & =& {\left.\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left(z_j\right)\right|}^{r_j}\Delta z_j\right)\kappa \left({\xi }_j\right)\right|}_{a_j}^{{\epsilon }_j}+{\int }_{a_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left({\xi }_j\right)\right|}^{r_j}{\kappa }^{\sigma }\left({\xi }_j\right)\Delta {\xi }_j\hfill \\ & =& {\int }_{a_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left({\xi }_j\right)\right|}^{r_j}\left(\sigma \left({\xi }_j\right)-a_j\right)\Delta {\xi }_j,\hfill \end{array}$$

(26)

where $\kappa \left({\xi }_j\right)={\xi }_j-a_j.$ Combining (26) with (25), we get

$$\begin{array}{ccc}& & {\int }_{a_s}^{{\epsilon }_s}...{\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{j=1}^s\left|{\lambda }_j\left({\xi }_j\right)\right|}{{\left({\sum }_{j=1}^s\frac{{\epsilon }_j-{\xi }_j}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}\Delta {\xi }_1...\Delta {\xi }_s\hfill \\ & \le & {\left(s-\sum _{j=1}^s\frac{1}{r_j}\right)}^{{\sum }_{j=1}^s\frac{1}{r_j}-s}\prod _{j=1}^s{\left({\epsilon }_j-a_j\right)}^{\frac{1}{l_j}}\hfill \\ & & \times \prod _{j=1}^s{\left({\int }_{a_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left({\xi }_j\right)\right|}^{r_j}\left(\sigma \left({\xi }_j\right)-a_j\right)\Delta {\xi }_j\right)}^{\frac{1}{r_j}}\hfill \\ & =& A\prod _{j=1}^s{\left({\int }_{a_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\Delta }\left({\xi }_j\right)\right|}^{r_j}\left(\sigma \left({\xi }_j\right)-a_j\right)\Delta {\xi }_j\right)}^{\frac{1}{r_j}}.\hfill \end{array}$$

Hence, (22) is proved. □

Corollary 1. 

Let $\mathbb{T}=\mathbb{Z}$ in Theorem 6, $a_j,{\epsilon }_j\in \mathbb{N},$ $l_j,r_j>1$ such that $1/r_j+1/l_j=1$ and ${\lambda }_j$ be real sequences with ${\lambda }_j\left({\epsilon }_j\right)=0;$ $j=1,2,...,s.$ Then $\sigma \left({\xi }_j\right)={\xi }_j+1$ and

$$\sum _{{\xi }_1=a_1}^{{\epsilon }_1-1}\sum _{{\xi }_2=a_2}^{{\epsilon }_2-1}...\sum _{{\xi }_s=a_s}^{{\epsilon }_s-1}\frac{{\prod }_{j=1}^s\left|{\lambda }_j\left({\xi }_j\right)\right|}{{\left({\sum }_{j=1}^s\frac{{\epsilon }_j-{\xi }_j}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}\le A\prod _{j=1}^s{\left(\sum _{{\xi }_j=a_j}^{{\epsilon }_j-1}\left({\xi }_j-a_j+1\right){\left|\Delta {\lambda }_j\left({\xi }_j\right)\right|}^{r_j}\right)}^{\frac{1}{r_j}}.$$

Here, Δ is the forward difference operator and A is specified as in (19).

Corollary 2. 

Let $\mathbb{T}=\mathbb{R}$in Theorem 6, $a_j,{\epsilon }_j\in \mathbb{R},$ $l_j,r_j>1$ such that $1/r_j+1/l_j=1$ and ${\lambda }_j\in \mathrm{C}(\left[a_j,{\epsilon }_j\right],\mathbb{R})$ with ${\lambda }_j\left({\epsilon }_j\right)=0;$ $j=1,2,...,s.$ Then $\sigma \left({\xi }_j\right)={\xi }_j$ and

$${\int }_{a_s}^{{\epsilon }_s}...{\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{j=1}^s\left|{\lambda }_j\left({\xi }_j\right)\right|}{{\left({\sum }_{j=1}^s\frac{{\epsilon }_j-{\xi }_j}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}d{\xi }_1...d{\xi }_s\le A\prod _{j=1}^s{\left({\int }_{a_j}^{{\epsilon }_j}{\left|{\lambda }_j^{\prime }\left({\xi }_j\right)\right|}^{r_j}\left({\xi }_j-a_j\right)d{\xi }_j\right)}^{\frac{1}{r_j}},$$

where A is given by (19).

Corollary 3. 

Let $\mathbb{T}=q^{\stackrel{-}{\mathbb{Z}}}$ for $q>1,$ $l_j,r_j>1$ such that $1/r_j+1/l_j=1$ and ${\lambda }_j$ be real sequences with ${\lambda }_j\left({\epsilon }_j\right)=0;$ $j=1,2,...,s.$ Then $\sigma \left({\xi }_j\right)=q{\xi }_j$ and

$$\begin{array}{ccc}& & \sum _{{\xi }_s={log}_q{a}_s}^{{log}_q{\epsilon }_s-1}...\sum _{{\xi }_1={log}_q{a}_1}^{{log}_q{\epsilon }_1-1}\frac{{(q-1)}^n{\prod }_{j=1}^s{\xi }_j\left|{\lambda }_j\left({\xi }_j\right)\right|}{{\left({\sum }_{j=1}^s\frac{{\epsilon }_j-{\xi }_j}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}\hfill \\ & \le & A\prod _{j=1}^s{\left(\sum _{{\xi }_j={log}_q{a}_j}^{{log}_q{\epsilon }_j-1}(q-1)\left(q{\xi }_j-a_j\right){\xi }_j{\left|{\Delta }_q{\lambda }_j\left({\xi }_j\right)\right|}^{r_j}\right)}^{\frac{1}{r_j}},\hfill \end{array}$$

where A is given by (19) and

$${\Delta }_q{\lambda }_j\left({\xi }_j\right)=\frac{{\lambda }_j\left(q{\xi }_j\right)-{\lambda }_j\left({\xi }_j\right)}{\left(q-1\right){\xi }_j}\forall {\xi }_j\in \mathbb{T}\setminus \left\{0\right\}.$$

In the following, we generalize the last theorem for two variables.

Theorem 7. 

Let $a_j,{\epsilon }_j,{ϵ}_j\in \mathbb{T},$ $l_j,r_j>1$ such that $1/l_j+1/r_j=1,$ ${\lambda }_j\in C{\mathrm{C}}_{rd}^{\prime }({\left[a_j,{\epsilon }_j\right]}_{\mathbb{T}}\times {\left[a_j,{ϵ}_j\right]}_{\mathbb{T}},$ $\mathbb{R})$ with ${\lambda }_j({\tau }_j,{\epsilon }_j)={\lambda }_j({ϵ}_j,{\xi }_j)=0$ for ${\xi }_j\in {\left[a_j,{\epsilon }_j\right]}_{\mathbb{T}}$ and ${\tau }_j\in {\left[a_j,{ϵ}_j\right]}_{\mathbb{T}};$ $j=1,2,...,s.$ Then

$$\begin{array}{cc}\hfill & {\int }_{a_s}^{{ϵ}_s}{\int }_{a_1}^{{ϵ}_1}...{\int }_{a_s}^{{\epsilon }_s}{\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{j=1}^s\left|{\lambda }_j({\tau }_j,{\xi }_j)\right|}{{\left({\sum }_{j=1}^s\frac{\left({ϵ}_j-{\tau }_j\right)({\epsilon }_j-{\xi }_j)}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}{\Delta }_2{\xi }_1...{\Delta }_2{\xi }_s{\Delta }_1{\tau }_1...{\Delta }_1{\tau }_s\hfill \\ \hfill & \le B\prod _{j=1}^s{\left({\int }_{a_j}^{{ϵ}_j}{\int }_{a_j}^{{\epsilon }_j}\left(\sigma \left({\tau }_j\right)-a_j\right)\left(\sigma \left({\xi }_j\right)-a_j\right){\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\xi }_j)\right|}^{r_j}{\Delta }_2{\xi }_j{\Delta }_1{\tau }_j\right)}^{\frac{1}{r_j}},\hfill \end{array}$$

(27)

where

$$B={\left(s-\sum _{j=1}^s\frac{1}{r_j}\right)}^{{\sum }_{j=1}^s\frac{1}{r_j}-s}\prod _{j=1}^s{\left({ϵ}_j-a_j\right)}^{\frac{1}{l_j}}{\left({\epsilon }_j-a_j\right)}^{\frac{1}{l_j}}.$$

(28)

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

Proof. 

Applying the property (5) of Theorem 4, Fubini’s theorem and using the hypothesis ${\lambda }_j({\tau }_j,{\epsilon }_j)={\lambda }_j({ϵ}_j,{\xi }_j)=0$, we get

$$\begin{array}{ccc}\hfill {\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(t_j,{\vartheta }_j)\right|{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j& \ge & \left|{\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j){\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right|\hfill \\ & =& \left|{\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left[{\lambda }_j^{{\Delta }_2}(z_j,{\vartheta }_j)\right]}^{{\Delta }_1}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right|\hfill \\ & =& \left|{\int }_{{\xi }_j}^{{\epsilon }_j}\left({\int }_{{\tau }_j}^{{ϵ}_j}{\left[{\lambda }_j^{{\Delta }_2}(z_j,{\vartheta }_j)\right]}^{{\Delta }_1}{\Delta }_1{z}_j\right){\Delta }_2{\vartheta }_j\right|\hfill \\ & =& \left|{\int }_{{\xi }_j}^{{\epsilon }_j}\left({\lambda }_j^{{\Delta }_2}({ϵ}_j,{\vartheta }_j)-{\lambda }_j^{{\Delta }_2}({\tau }_j,{\vartheta }_j)\right){\Delta }_2{\vartheta }_j\right|\hfill \\ & =& \left|{\lambda }_j({ϵ}_j,{\epsilon }_j)-{\lambda }_j({ϵ}_j,{\xi }_j)+{\lambda }_j({\tau }_j,{\xi }_j)-{\lambda }_j({\tau }_j,{\epsilon }_j)\right|\hfill \\ & =& \left|{\lambda }_j({\tau }_j,{\xi }_j)\right|,\hfill \end{array}$$

and then

$$\prod _{j=1}^s\left|{\lambda }_j({\tau }_j,{\xi }_j)\right|\le \prod _{j=1}^s{\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j.$$

(29)

Applying (14) on ${\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j$ with $l_j,r_j>1,$ $f(z_j,{\vartheta }_j)=1$ and $g(z_j,{\vartheta }_j)=\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|,$ we see that

$$\begin{array}{ccc}& & {\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\hfill \\ & \le & {\left({\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right)}^{\frac{1}{r_j}}{\left({\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right)}^{\frac{1}{l_j}}\hfill \\ & =& {\left({ϵ}_j-{\tau }_j\right)}^{\frac{1}{l_j}}{\left({\epsilon }_j-{\xi }_j\right)}^{\frac{1}{l_j}}{\left({\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right)}^{\frac{1}{r_j}},\hfill \end{array}$$

and then

$$\begin{array}{ccc}& & \prod _{j=1}^s{\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\hfill \\ & \le & \prod _{j=1}^s{\left({ϵ}_j-{\tau }_j\right)}^{\frac{1}{l_j}}{\left({\epsilon }_j-{\xi }_j\right)}^{\frac{1}{l_j}}{\left({\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right)}^{\frac{1}{r_j}}\hfill \\ & =& \prod _{j=1}^s{\left({ϵ}_j-{\tau }_j\right)}^{\frac{1}{l_j}}{\left({\epsilon }_j-{\xi }_j\right)}^{\frac{1}{l_j}}\prod _{j=1}^s{\left({\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right)}^{\frac{1}{r_j}}.\hfill \end{array}$$

(30)

Substituting (30) into (29) and applying (16) on $w_j=\left({ϵ}_j-{\tau }_j\right)\left({\epsilon }_j-{\xi }_j\right),$ we obtain

$$\begin{array}{ccc}\hfill \prod _{j=1}^s\left|{\lambda }_j({\tau }_j,{\xi }_j)\right|& \le & \prod _{j=1}^s{\left({ϵ}_j-{\tau }_j\right)}^{\frac{1}{l_j}}{\left({\epsilon }_j-{\xi }_j\right)}^{\frac{1}{l_j}}\prod _{j=1}^s{\left({\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right)}^{\frac{1}{r_j}}\hfill \\ & \le & \frac{{\left({\sum }_{j=1}^s\frac{\left({ϵ}_j-{\tau }_j\right)({\epsilon }_j-{\xi }_j)}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}{{\left(s-{\sum }_{j=1}^s\frac{1}{r_j}\right)}^{s-{\sum }_{j=1}^s\frac{1}{r_j}}}\prod _{j=1}^s{\left({\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right)}^{\frac{1}{r_j}}.\hfill \end{array}$$

(31)

Dividing (31) on ${\left({\sum }_{j=1}^s\frac{\left({ϵ}_j-{\tau }_j\right)({\epsilon }_j-{\xi }_j)}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}$, integrating over ${\xi }_j$ from $a_j$ to ${\epsilon }_j$ and over ${\tau }_j$ from $a_j$ to ${ϵ}_j$for $j=1,2,...,s,$ we conclude that

$$\begin{array}{cc}\hfill & {\int }_{a_s}^{{ϵ}_s}{\int }_{a_1}^{{ϵ}_1}...{\int }_{a_s}^{{\epsilon }_s}{\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{j=1}^s\left|{\lambda }_j({\tau }_j,{\xi }_j)\right|}{{\left({\sum }_{j=1}^s\frac{\left({ϵ}_j-{\tau }_j\right)({\epsilon }_j-{\xi }_j)}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}{\Delta }_2{\xi }_1...{\Delta }_2{\xi }_s{\Delta }_1{\tau }_1...{\Delta }_1{\tau }_s\hfill \\ \hfill & \le {\left(s-\sum _{j=1}^s\frac{1}{r_j}\right)}^{{\sum }_{j=1}^s\frac{1}{r_j}-s}\hfill \\ \hfill & \times {\int }_{a_s}^{{ϵ}_s}{\int }_{a_1}^{{ϵ}_1}...{\int }_{a_s}^{{\epsilon }_s}{\int }_{a_1}^{{\epsilon }_1}\prod _{j=1}^s{\left({\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right)}^{\frac{1}{r_j}}{\Delta }_2{\xi }_1...{\Delta }_2{\xi }_s{\Delta }_1{\tau }_1...{\Delta }_1{\tau }_s\hfill \\ \hfill & ={\left(s-\sum _{j=1}^s\frac{1}{r_j}\right)}^{{\sum }_{j=1}^s\frac{1}{r_j}-s}\hfill \\ \hfill & \times \prod _{j=1}^s{\int }_{a_j}^{{ϵ}_j}{\int }_{a_j}^{{\epsilon }_j}{\left({\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right)}^{\frac{1}{r_j}}{\Delta }_2{\xi }_j{\Delta }_1{\tau }_j.\hfill \end{array}$$

(32)

Again, using (14) on ${\int }_{a_j}^{{ϵ}_j}{\int }_{a_j}^{{\epsilon }_j}{\left({\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right)}^{\frac{1}{r_j}}{\Delta }_2{\xi }_j{\Delta }_1{\tau }_j$ with exponents $r_j,l_j>1$ and $f({\xi }_j,{\tau }_j)=1,$

$$g({\xi }_j,{\tau }_j)={\left({\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right)}^{\frac{1}{r_j}},$$

we observe that

$$\begin{array}{cc}\hfill & {\int }_{a_j}^{{ϵ}_j}{\int }_{a_j}^{{\epsilon }_j}{\left({\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right)}^{\frac{1}{r_j}}{\Delta }_2{\xi }_j{\Delta }_1{\tau }_j\hfill \\ \hfill & \le {\left({\int }_{a_j}^{{ϵ}_j}{\int }_{a_j}^{{\epsilon }_j}{\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j{\Delta }_2{\xi }_j{\Delta }_1{\tau }_j\right)}^{\frac{1}{r_j}}{\left({\int }_{a_j}^{{ϵ}_j}{\int }_{a_j}^{{\epsilon }_j}{\Delta }_2{\xi }_j{\Delta }_1{\tau }_j\right)}^{\frac{1}{l_j}}\hfill \\ \hfill & ={\left({ϵ}_j-a_j\right)}^{\frac{1}{l_j}}{\left({\epsilon }_j-a_j\right)}^{\frac{1}{l_j}}{\left({\int }_{a_j}^{{ϵ}_j}{\int }_{a_j}^{{\epsilon }_j}{\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j{\Delta }_2{\xi }_j{\Delta }_1{\tau }_j\right)}^{\frac{1}{r_j}},\hfill \end{array}$$

and then

$$\begin{array}{cc}\hfill & \prod _{j=1}^s{\int }_{a_j}^{{ϵ}_j}{\int }_{a_j}^{{\epsilon }_j}{\left({\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right)}^{\frac{1}{r_j}}{\Delta }_2{\xi }_j{\Delta }_1{\tau }_j\hfill \\ \hfill & \le \prod _{j=1}^s{\left({ϵ}_j-a_j\right)}^{\frac{1}{l_j}}{\left({\epsilon }_j-a_j\right)}^{\frac{1}{l_j}}{\left({\int }_{a_j}^{{ϵ}_j}{\int }_{a_j}^{{\epsilon }_j}{\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j{\Delta }_2{\xi }_j{\Delta }_1{\tau }_j\right)}^{\frac{1}{r_j}}\hfill \\ \hfill & =\prod _{j=1}^s{\left({ϵ}_j-a_j\right)}^{\frac{1}{l_j}}{\left({\epsilon }_j-a_j\right)}^{\frac{1}{l_j}}\hfill \\ \hfill & \times \prod _{j=1}^s{\left({\int }_{a_j}^{{ϵ}_j}{\int }_{a_j}^{{\epsilon }_j}{\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j{\Delta }_2{\xi }_j{\Delta }_1{\tau }_j\right)}^{\frac{1}{r_j}}.\hfill \end{array}$$

(33)

Substituting (33) into (32) and applying the Fubini theorem, we see that

$$\begin{array}{cc}\hfill & {\int }_{a_s}^{{ϵ}_s}{\int }_{a_1}^{{ϵ}_1}...{\int }_{a_s}^{{\epsilon }_s}{\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{j=1}^s\left|{\lambda }_j({\tau }_j,{\xi }_j)\right|}{{\left({\sum }_{j=1}^s\frac{\left({ϵ}_j-{\tau }_j\right)({\epsilon }_j-{\xi }_j)}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}{\Delta }_2{\xi }_1...{\Delta }_2{\xi }_s{\Delta }_1{\tau }_1...{\Delta }_1{\tau }_s\hfill \\ \hfill & \le {\left(s-\sum _{j=1}^s\frac{1}{r_j}\right)}^{{\sum }_{j=1}^s\frac{1}{r_j}-s}\prod _{j=1}^s{\left({ϵ}_j-a_j\right)}^{\frac{1}{l_j}}{\left({\epsilon }_j-a_j\right)}^{\frac{1}{l_j}}\hfill \\ \hfill & \times \prod _{j=1}^s{\left({\int }_{a_j}^{{ϵ}_j}{\int }_{a_j}^{{\epsilon }_j}{\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j{\Delta }_2{\xi }_j{\Delta }_1{\tau }_j\right)}^{\frac{1}{r_j}}\hfill \\ \hfill & ={\left(s-\sum _{j=1}^s\frac{1}{r_j}\right)}^{{\sum }_{j=1}^s\frac{1}{r_j}-s}\prod _{j=1}^s{\left({ϵ}_j-a_j\right)}^{\frac{1}{l_j}}{\left({\epsilon }_j-a_j\right)}^{\frac{1}{l_j}}\hfill \\ \hfill & \times \prod _{j=1}^s{\left({\int }_{a_j}^{{\epsilon }_j}\left({\int }_{a_j}^{{ϵ}_j}\left[{\int }_{{\tau }_j}^{{ϵ}_j}{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j{\Delta }_1{z}_j\right]{\Delta }_1{\tau }_j\right){\Delta }_2{\xi }_j\right)}^{\frac{1}{r_j}}.\hfill \end{array}$$

(34)

Now, by applying (12) on ${\int }_{a_j}^{{ϵ}_j}\left({\int }_{{\tau }_j}^{{ϵ}_j}\left[{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j\right]{\Delta }_1{z}_j\right){\Delta }_1{\tau }_j$ with

$$\omega \left({\tau }_j\right)={\int }_{{\tau }_j}^{{ϵ}_j}\left[{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j\right]{\Delta }_1{z}_j\mathrm{and}{\kappa }^{\Delta }\left({\tau }_j\right)=1,$$

we find that

$$\begin{array}{ccc}& & {\int }_{a_j}^{{ϵ}_j}\left({\int }_{{\tau }_j}^{{ϵ}_j}\left[{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j\right]{\Delta }_1{z}_j\right){\Delta }_1{\tau }_j\hfill \\ & =& {\left.\kappa \left({\tau }_j\right){\int }_{{\tau }_j}^{{ϵ}_j}\left[{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j\right]{\Delta }_1{z}_j\right|}_{a_j}^{{ϵ}_j}\hfill \\ & & +{\int }_{a_j}^{{ϵ}_j}{\kappa }^{\sigma }\left({\tau }_j\right)\left[{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j\right]{\Delta }_1{\tau }_j\hfill \\ & =& {\int }_{a_j}^{{ϵ}_j}\left[\sigma \left({\tau }_j\right)-a_j\right]\left[{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j\right]{\Delta }_1{\tau }_j.\hfill \end{array}$$

(35)

where $\kappa \left({\tau }_j\right)={\tau }_j-a_j.$ By integrating (34) over ${\xi }_j$ from $a_j$ to ${\epsilon }_j$ and using the Fubini theorem, we have

$$\begin{array}{ccc}& & {\int }_{a_j}^{{\epsilon }_j}{\int }_{a_j}^{{ϵ}_j}\left({\int }_{{\tau }_j}^{{ϵ}_j}\left[{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j\right]{\Delta }_1{z}_j\right){\Delta }_1{\tau }_j{\Delta }_2{\xi }_j\hfill \\ & =& {\int }_{a_j}^{{\epsilon }_j}{\int }_{a_j}^{{ϵ}_j}\left[\sigma \left({\tau }_j\right)-a_j\right]\left[{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j\right]{\Delta }_1{\tau }_j{\Delta }_2{\xi }_j\hfill \\ & =& {\int }_{a_j}^{{ϵ}_j}{\int }_{a_j}^{{\epsilon }_j}\left[\sigma \left({\tau }_j\right)-a_j\right]\left[{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j\right]{\Delta }_2{\xi }_j{\Delta }_1{\tau }_j\hfill \\ & =& {\int }_{a_j}^{{ϵ}_j}\left[\sigma \left({\tau }_j\right)-a_j\right]\left({\int }_{a_j}^{{\epsilon }_j}\left[{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j\right]{\Delta }_2{\xi }_j\right){\Delta }_1{\tau }_j.\hfill \end{array}$$

(36)

Again, using (12) on the term ${\int }_{a_j}^{{\epsilon }_j}\left[{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j\right]{\Delta }_2{\xi }_j$ with

$$\omega \left({\tau }_j\right)={\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j\mathrm{and}{\kappa }^{\Delta }\left({\xi }_j\right)=1,$$

we see that

$$\begin{array}{ccc}& & {\int }_{a_j}^{{\epsilon }_j}\left[{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j\right]{\Delta }_2{\xi }_j\hfill \\ & =& {\left.\kappa \left({\xi }_j\right)\left({\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j\right)\right|}_{a_j}^{{\epsilon }_j}\hfill \\ & & +{\int }_{a_j}^{{\epsilon }_j}{\kappa }^{\sigma }\left({\xi }_j\right){\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\xi }_j)\right|}^{r_j}{\Delta }_2{\xi }_j\hfill \\ & =& {\int }_{a_j}^{{\epsilon }_j}\left(\sigma \left({\xi }_j\right)-a_j\right){\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\xi }_j)\right|}^{r_j}{\Delta }_2{\xi }_j.\hfill \end{array}$$

(37)

where $\kappa \left({\xi }_j\right)={\xi }_j-a_j.$ Substituting (37) into (36) and applying Fubini’s theorem, we obtain

$$\begin{array}{ccc}& & {\int }_{a_j}^{{\epsilon }_j}{\int }_{a_j}^{{ϵ}_j}\left({\int }_{{\tau }_j}^{{ϵ}_j}\left[{\int }_{{\xi }_j}^{{\epsilon }_j}{\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}(z_j,{\vartheta }_j)\right|}^{r_j}{\Delta }_2{\vartheta }_j\right]{\Delta }_1{z}_j\right){\Delta }_1{\tau }_j{\Delta }_2{\xi }_j\hfill \\ & =& {\int }_{a_j}^{{ϵ}_j}\left[\sigma \left({\tau }_j\right)-a_j\right]\left({\int }_{a_j}^{{\epsilon }_j}\left(\sigma \left({\xi }_j\right)-a_j\right){\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\xi }_j)\right|}^{r_j}{\Delta }_2{\xi }_j\right){\Delta }_1{\tau }_j\hfill \\ & =& {\int }_{a_j}^{{ϵ}_j}{\int }_{a_j}^{{\epsilon }_j}\left[\sigma \left({\tau }_j\right)-a_j\right]\left(\sigma \left({\xi }_j\right)-a_j\right){\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\xi }_j)\right|}^{r_j}{\Delta }_2{\xi }_j{\Delta }_1{\tau }_j\hfill \\ & =& {\int }_{a_j}^{{\epsilon }_j}{\int }_{a_j}^{{ϵ}_j}\left[\sigma \left({\tau }_j\right)-a_j\right]\left(\sigma \left({\xi }_j\right)-a_j\right){\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\xi }_j)\right|}^{r_j}{\Delta }_1{\tau }_j{\Delta }_2{\xi }_j.\hfill \end{array}$$

(38)

Substituting (38) into (34), we get

$$\begin{array}{cc}\hfill & {\int }_{a_s}^{{ϵ}_s}{\int }_{a_1}^{{ϵ}_1}...{\int }_{a_s}^{{\epsilon }_s}{\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{j=1}^s\left|{\lambda }_j({\tau }_j,{\xi }_j)\right|}{{\left({\sum }_{j=1}^s\frac{\left({ϵ}_j-{\tau }_j\right)({\epsilon }_j-{\xi }_j)}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}{\Delta }_2{\xi }_1...{\Delta }_2{\xi }_s{\Delta }_1{\tau }_1...{\Delta }_1{\tau }_s\hfill \\ \hfill & \le {\left(s-\sum _{j=1}^s\frac{1}{r_j}\right)}^{{\sum }_{j=1}^s\frac{1}{r_j}-s}\prod _{j=1}^s{\left({ϵ}_j-a_j\right)}^{\frac{1}{l_j}}{\left({\epsilon }_j-a_j\right)}^{\frac{1}{l_j}}\hfill \\ \hfill & \times \prod _{j=1}^s{\left({\int }_{a_j}^{{\epsilon }_j}{\int }_{a_j}^{{ϵ}_j}\left(\sigma \left({\tau }_j\right)-a_j\right)\left(\sigma \left({\xi }_j\right)-a_j\right){\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\xi }_j)\right|}^{r_j}{\Delta }_2{\xi }_j{\Delta }_1{\tau }_j\right)}^{\frac{1}{r_j}}\hfill \\ \hfill & =B\prod _{j=1}^s{\left({\int }_{a_j}^{{\epsilon }_j}{\int }_{a_j}^{{ϵ}_j}\left(\sigma \left({\tau }_j\right)-a_j\right)\left(\sigma \left({\xi }_j\right)-a_j\right){\left|{\lambda }_j^{{\Delta }_2{\Delta }_1}({\tau }_j,{\xi }_j)\right|}^{r_j}{\Delta }_2{\xi }_j{\Delta }_1{\tau }_j\right)}^{\frac{1}{r_j}}.\hfill \end{array}$$

Hence, (27) is proved. □

Corollary 4. 

Let $\mathbb{T}=\mathbb{Z}$ in Theorem 7, $a_j,{\epsilon }_j,{ϵ}_j\in \mathbb{Z},$ $r_j,l_j>1$ such that $1/r_j+1/l_j=1$ and ${\lambda }_j$be real sequenceswith ${\lambda }_j({\tau }_j,{\epsilon }_j)={\lambda }_j({ϵ}_j,{\xi }_j)=0$ for ${\xi }_j\in \left[a_j,{\epsilon }_j\right]$ and ${\tau }_j\in \left[a_j,{ϵ}_j\right],$ where $j=1,2,...,s.$ Then $\sigma \left({\tau }_j\right)={\tau }_j+1,$ $\sigma \left({\xi }_j\right)={\xi }_j+1$ and

$$\begin{array}{cc}\hfill & \sum _{{\tau }_s=a_s}^{{ϵ}_s-1}\sum _{{\tau }_1=a_1}^{{ϵ}_1-1}...\sum _{{\xi }_s=a_s}^{{\epsilon }_s-1}\sum _{{\xi }_1=a_1}^{{\epsilon }_1-1}\frac{{\prod }_{j=1}^s\left|{\lambda }_j({\tau }_j,{\xi }_j)\right|}{{\left({\sum }_{j=1}^s\frac{\left({ϵ}_j-{\tau }_j\right)({\epsilon }_j-{\xi }_j)}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}\hfill \\ \hfill & \le B\prod _{j=1}^s{\left(\sum _{{\xi }_j=a_j}^{{\epsilon }_j-1}\sum _{{\tau }_j=a_j}^{{ϵ}_j-1}\left({\tau }_j-a_j+1\right)\left({\xi }_j-a_j+1\right){\left|{\Delta }_2{\Delta }_1{\lambda }_j({\tau }_j,{\xi }_j)\right|}^{r_j}\right)}^{\frac{1}{r_j}},\hfill \end{array}$$

where B is given by (28).

Corollary 5. 

Let $\mathbb{T}=\mathbb{R}$ in Theorem 7, $a_j,{\epsilon }_j,{ϵ}_j\in \mathbb{R},$ $r_j,l_j>1$ such that $1/r_j+1/l_j=1$ and ${\lambda }_j\in C{\mathrm{C}}^{\prime }(\left[a_j,{\epsilon }_j\right]\times \left[a_j,{ϵ}_j\right],\mathbb{R})$ with ${\lambda }_j({\tau }_j,{\epsilon }_j)={\lambda }_j({ϵ}_j,{\xi }_j)=0$ for ${\xi }_j\in {\left[a_j,{\epsilon }_j\right]}_{\mathbb{T}}$ and ${\tau }_j\in {\left[a_j,{ϵ}_j\right]}_{\mathbb{T}},$ where $j=1,2,...,s.$ Then $\sigma \left({\tau }_j\right)={\tau }_j,$ $\sigma \left({\xi }_j\right)={\xi }_j$ and

$$\begin{array}{cc}\hfill & {\int }_{a_s}^{{ϵ}_s}{\int }_{a_1}^{{ϵ}_1}...{\int }_{a_s}^{{\epsilon }_s}{\int }_{a_1}^{{\epsilon }_1}\frac{{\prod }_{j=1}^s\left|{\lambda }_j({\tau }_j,{\xi }_j)\right|}{{\left({\sum }_{j=1}^s\frac{\left({ϵ}_j-{\tau }_j\right)({\epsilon }_j-{\xi }_j)}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}d{\xi }_1...d{\xi }_{s}d{\tau }_1...d{\tau }_s\hfill \\ \hfill & \le B\prod _{j=1}^s{\left({\int }_{a_j}^{{\epsilon }_j}{\int }_{a_j}^{{ϵ}_j}\left({\tau }_j-a_j\right)\left({\xi }_j-a_j\right){\left|\frac{{\partial }^2{\lambda }_j({\tau }_j,{\xi }_j)}{\partial {\xi }_j\partial {\tau }_j}\right|}^{r_j}d{\xi }_{j}d{\tau }_j\right)}^{\frac{1}{r_j}},\hfill \end{array}$$

where B is given by (28).

Corollary 6. 

Let $\mathbb{T}=q^{\stackrel{-}{\mathbb{Z}}}$ for $q>1,$ $a_j,{\epsilon }_j,{ϵ}_j\in \mathbb{T},$ $r_j,l_j>1$ such that $1/r_j+1/l_j=1$ and ${\lambda }_j$are real sequences with ${\lambda }_j({\tau }_j,{\epsilon }_j)={\lambda }_j({ϵ}_j,{\xi }_j)=0$ for ${\xi }_j\in \left[a_j,{\epsilon }_j\right]$ and ${\tau }_j\in \left[a_j,{ϵ}_j\right],$ where $j=1,2,...,s.$ Then $\sigma \left({\tau }_j\right)=q{\tau }_j,$ $\sigma \left({\xi }_j\right)=q{\xi }_j$ and

$$\begin{array}{cc}\hfill & \sum _{{\tau }_s={log}_q{a}_s}^{{log}_q{ϵ}_s-1}\sum _{{\tau }_1={log}_q{a}_1}^{{log}_q{ϵ}_1-1}...\sum _{{\xi }_s={log}_q{a}_s}^{{log}_q{\epsilon }_{s-1}}\sum _{{\xi }_1={log}_q{a}_1}^{{log}_q{\epsilon }_{1-1}}\frac{{(q-1)}^{2n}{\prod }_{j=1}^s{\tau }_j{\xi }_j\left|{\lambda }_j({\tau }_j,{\xi }_j)\right|}{{\left({\sum }_{j=1}^s\frac{\left({ϵ}_j-{\tau }_j\right)({\epsilon }_j-{\xi }_j)}{l_j}\right)}^{{\sum }_{j=1}^s\frac{1}{l_j}}}\hfill \\ \hfill & \le B\prod _{j=1}^s{\left(\sum _{{\tau }_j={log}_q{a}_j}^{{log}_q{ϵ}_j-1}\sum _{{\xi }_j={log}_q{a}_j}^{{log}_q{\epsilon }_{j-1}}\left(q{\tau }_j-a_j\right)\left(q{\xi }_j-a_j\right){(q-1)}^2{\tau }_j{\xi }_j{\left|{\Delta }_q^2{\Delta }_q^1{\lambda }_j({\tau }_j,{\xi }_j)\right|}^{r_j}\right)}^{\frac{1}{r_j}}.\hfill \end{array}$$

Here B is given by (28) and the ${\Delta }_q^1-$derivative of $\lambda (\tau ,\xi )$ is the ${\Delta }_q-$derivative with respect to the first variable τ and the ${\Delta }_q^2-$derivative of $\lambda (\tau ,\xi )$ is the ${\Delta }_q-$derivative with respect to the second variable $\xi .$

4. Conclusions

In this study, a generalization of the Hilbert-type inequalities within the framework of time scales in delta calculus. We should note that we used different conditions from some previous results, thus, various refinements of the classic Hilbert-type inequalities are obtained. Throughout the work, it is shown that some known results from the literature are obtained as particular cases of ours. For future work, we will be able to present such inequalities by employing nabla calculus and diamond-$\alpha$ calculus for $\alpha \in (0,1)$. Additionally, it will be very fascinating to present similar inequalities on time scales using Riemann–Liouville type fractional integrals.