Some Dynamic Hilbert-Type Inequalities on Time Scales

Abstract

Throughout this article, we will demonstrate some new generalizations of dynamic Hilbert type inequalities, which are used in various problems involving symmetry. We develop a number of those symmetric inequalities to a general time scale. From these inequalities, as particular cases, we formulate some integral and discrete inequalities that have been demonstrated in the literature and also extend some of the dynamic inequalities that have been achieved in time scales.

1. Introduction

In recent years, Hilbert’s double-series inequality and its integral version [1] (pp. 253–254) has been granted significant attention by many scholars (see, for example, in [2,3,4,5,6,7,8,9,10,11,12]). In particular, B. G. Pachpatte [13] defined a new inequalities close to that of Hilbert as follows. Let $a_s:N_p=\{0,$ $1,$ $2,$ $\dots ,$ $p\}\subset \mathbb{N}\to \mathbb{R}$ and $b_{\vartheta }:N_q=\{0,$ $1,$ $2,$ $\dots ,$ $q\}\subset \mathbb{N}\to \mathbb{R}$ for $p,$ $q\in \mathbb{N}$ and $a\left(0\right)=b\left(0\right)=0.$ Then

$$\begin{array}{ccc}\hfill \sum _{s=1}^p\sum _{\vartheta =1}^q\frac{\left|a_s\right|\left|b_{\vartheta }\right|}{s+\vartheta }& \le & C(p,q){\left(\sum _{s=1}^p(p-s+1){\left|\nabla a_s\right|}^2\right)}^{\frac{1}{2}}\hfill \\ & & \times {\left(\sum _{\vartheta =1}^q(q-\vartheta +1){\left|\nabla b_{\vartheta }\right|}^2\right)}^{\frac{1}{2}},\hfill \end{array}$$

(1)

where $\nabla a_s=a_s-a_{s-1},$ $\nabla b_{\vartheta }=b_{\vartheta }-b_{\vartheta -1}$ and

$$C(p,q)=\frac{1}{2}\sqrt{pq}.$$

An integral version of (1) is established in the next consequence. Let $f\left(s\right)$ and $g\left(\vartheta \right)$ be real-valued continuous functions defined on $I_x=[0,$ $\infty )\subset \mathbb{R}$ and $I_y=[0,$ $\infty )\subset \mathbb{R}$ for $x,$ $y\in I_0=(0,$ $\infty )\subset \mathbb{R}$, respectively, and $f\left(0\right)=g\left(0\right)=0.$ Then

$$\begin{array}{ccc}\hfill {\int }_0^x{\int }_0^y\frac{\left|f\left(s\right)\right|\left|g\left(\vartheta \right)\right|}{s+\vartheta }dsd\vartheta & \le & C^{*}(x,y){\left({\int }_0^x(x-s){\left|f^{\prime }\left(s\right)\right|}^{2}ds\right)}^{\frac{1}{2}}\hfill \\ & & \times {\left({\int }_0^y(y-\vartheta ){\left|g^{\prime }\left(\vartheta \right)\right|}^{2}d\vartheta \right)}^{\frac{1}{2}},\hfill \end{array}$$

(2)

where $u^{\prime }\left(t\right)$ denote the usual derivative of function $u\left(t\right)$ and

$$C^{*}(x,y)=\frac{1}{2}\sqrt{xy}.$$

In [14], Pachpatte gave some generalizations of (1) and (2) as follows. Let $\lambda ,$ $\mu >1$ be constants such that $1/\lambda +1/\mu =1.$ If $a_s:N_p=\{0,$ $1,$ $2,$ $\dots ,$ $p\}\subset \mathbb{N}\to \mathbb{R}$ and $b_{\vartheta }:N_q=\{0,$ $1,$ $2,$ $\dots ,$ $q\}\subset \mathbb{N}\to \mathbb{R}$ for $p,$ $q\in \mathbb{N}$ and $a\left(0\right)=b\left(0\right)=0,$ then

$$\begin{array}{ccc}\hfill \sum _{s=1}^p\sum _{\vartheta =1}^q\frac{\left|a_s\right|\left|b_{\vartheta }\right|}{\mu s^{\lambda -1}+\lambda {\vartheta }^{\mu -1}}& \le & C^{**}(\lambda ,\mu ){\left(\sum _{s=1}^p(p-s+1){\left|\nabla a_s\right|}^{\lambda }\right)}^{\frac{1}{\lambda }}\hfill \\ & & \times {\left(\sum _{\vartheta =1}^q(q-\vartheta +1){\left|\nabla b_{\vartheta }\right|}^{\mu }\right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(3)

where $\nabla a_s=a_s-a_{s-1},$ $\nabla b_{\vartheta }=b_{\vartheta }-b_{\vartheta -1}$ and

$$C^{**}(\lambda ,\mu )=\frac{1}{\lambda \mu }{\left(p\right)}^{\frac{\lambda -1}{\lambda }}{\left(q\right)}^{\frac{\mu -1}{\mu }}.$$

An integral version of (3) is established in the next consequence. Let $\lambda ,$ $\mu >1$ be constants such that $1/\lambda +1/\mu =1.$ If $f\left(s\right)$ and $g\left(\vartheta \right)$ are real-valued continuous functions defined on $I_x=[0,$ $\infty )\subset \mathbb{R}$ and $I_y=[0,$ $\infty )\subset \mathbb{R}$ for $x,$ $y\in I_0=(0,$ $\infty )\subset \mathbb{R}$, respectively, and $f\left(0\right)=g\left(0\right)=0,$ then

$$\begin{array}{ccc}\hfill {\int }_0^x{\int }_0^y\frac{\left|f\left(s\right)\right|\left|g\left(\vartheta \right)\right|}{\mu s^{\lambda -1}+\lambda {\vartheta }^{\mu -1}}dsd\vartheta & \le & D(\lambda ,\mu ){\left({\int }_0^x(x-s){\left|f^{\prime }\left(s\right)\right|}^{\lambda }ds\right)}^{\frac{1}{\lambda }}\hfill \\ & & \times {\left({\int }_0^y(y-\vartheta ){\left|g^{\prime }\left(\vartheta \right)\right|}^{\mu }d\vartheta \right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(4)

where

$$D(\lambda ,\mu )=\frac{1}{\lambda \mu }{\left(x\right)}^{\frac{\lambda -1}{\lambda }}{\left(y\right)}^{\frac{\mu -1}{\mu }}.$$

In [15], the authors gave another generalization of (3) and (4) as follows. Let $\lambda ,$ $\mu >1$ be constants such that $1/\lambda +1/\mu =1.$ If $a_s:N_p=\{0,$ $1,$ $2,$ $\dots ,$ $p\}\subset \mathbb{N}\to \mathbb{R}$ and $b_{\vartheta }:N_q=\{0,$ $1,$ $2,$ $\dots ,$ $q\}\subset \mathbb{N}\to \mathbb{R}$ for $p,$ $q\in \mathbb{N}$ and $a\left(0\right)=b\left(0\right)=0,$ then

$$\begin{array}{ccc}\hfill \sum _{s=1}^p\sum _{\vartheta =1}^q\frac{\left|a_s\right|\left|b_t\right|}{\mu s^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}+\lambda {\vartheta }^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}& \le & D^{*}(\lambda ,\mu ){\left(\sum _{s=1}^p(p-s+1){\left|\nabla a_s\right|}^{\lambda }\right)}^{\frac{1}{\lambda }}\hfill \\ & & \times {\left(\sum _{\vartheta =1}^q(q-\vartheta +1){\left|\nabla b_{\vartheta }\right|}^{\mu }\right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(5)

where $\nabla a_s=a_s-a_{s-1},$ $\nabla b_{\vartheta }=b_{\vartheta }-b_{\vartheta -1}$ and

$$D^{*}(\lambda ,\mu )=\frac{1}{\lambda +\mu }{\left(p\right)}^{\frac{\lambda -1}{\lambda }}{\left(q\right)}^{\frac{\mu -1}{\mu }}.$$

An integral analogue of (5) is established in the next consequence. Let $\lambda ,$ $\mu >1$ be constants such that $1/\lambda +1/\mu =1$ and $f\left(s\right)\in {\mathrm{C}}^1\left[\right[0,$ $x),$ ${\mathbb{R}}^{+}],$ $g\left(\vartheta \right)\in {\mathrm{C}}^1\left[\right[0,$ $y),$ ${\mathbb{R}}^{+}]$ with $f\left(0\right)=g\left(0\right)=0.$ Then

$$\begin{array}{ccc}\hfill {\int }_0^x{\int }_0^y\frac{\left|f\left(s\right)\right|\left|g\left(\vartheta \right)\right|}{\mu s^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}+\lambda {\vartheta }^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}dsd\vartheta & \le & D^{**}(\lambda ,\mu ){\left({\int }_0^x(x-s){\left|f^{\prime }\left(s\right)\right|}^{\lambda }ds\right)}^{\frac{1}{\lambda }}\hfill \\ & & \times {\left({\int }_0^y(y-\vartheta ){\left|g^{\prime }\left(\vartheta \right)\right|}^{\mu }d\vartheta \right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(6)

where

$$D^{**}(\lambda ,\mu )=\frac{1}{\lambda +\mu }{\left(x\right)}^{\frac{\lambda -1}{\lambda }}{\left(y\right)}^{\frac{\mu -1}{\mu }}.$$

On the other hand, Hilger [16] suggested the theory of time scales to unify discrete and continuous analysis, based on which some authors have studied the Hilbert-kind inequalities on time scales (see in [17,18,19,20,21]). In the following, the time scale $\mathbb{T}$ is a non-empty closed subset of $\mathbb{R}$ and defines the time scale interval $[k,$ ${l]}_{\mathbb{T}}$ by

$${[k,l]}_{\mathbb{T}}=[k,l]\cap \mathbb{T}.$$

Let ${\mathrm{C}}_{rd}$ denotes the set of right-dense continuous (rd-continuous), ${\mathrm{CC}}_{rd}$ denotes the set of functions $g({\vartheta }_1,$ ${\vartheta }_2)$ on ${\mathbb{T}}_1\times {\mathbb{T}}_2$ where g is rd-continuous in ${\vartheta }_1$ and ${\vartheta }_2$ and ${\mathrm{CC}}_{rd}^1$ denotes the set of all functions ${\mathrm{CC}}_{rd}$ for which both the ${\Delta }_1$ partial derivative and ${\Delta }_2$ partial derivative exists and are in ${\mathrm{CC}}_{rd}$. For details on calculating the time scales see in [22,23].

The following useful relationships are often used between the time scale calculus $\mathbb{T}$ and the difference calculus $\mathbb{R}$ and the difference calculus $\mathbb{Z}$. Please mind that

(i)

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

$$\sigma \left(\vartheta \right)=\vartheta ,f^{\Delta }\left(\vartheta \right)=f^{\prime }\left(\vartheta \right),{\int }_k^l{f}^{\Delta }\left(\vartheta \right)\Delta \vartheta ={\int }_k^{l}f\left(\vartheta \right)d\vartheta .$$

(7)

(ii)

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

$$\sigma \left(\vartheta \right)=\vartheta +1,f^{\Delta }\left(\vartheta \right)=\Delta f\left(\vartheta \right),{\int }_k^l{f}^{\Delta }\left(\vartheta \right)\Delta \vartheta =\sum _{\vartheta =k}^{l-1}f\left(\vartheta \right).$$

(8)

Within the following, we display some basic lemmas and some algebraic inequalities that play a key role in inaugurating the major findings of this paper.

Lemma 1

(Hölder’s inequality in one dimension [24]).Let $k,$ $l\in \mathbb{T}$ and $\zeta ,$ $\chi \in {\mathrm{C}}_{rd}(\mathbb{T},$ $\mathbb{R}).$ Then

$${\int }_k^l\left|\zeta \left(\vartheta \right)\chi \left(\vartheta \right)\right|\Delta \vartheta \le {\left({\int }_k^l{\left|\zeta \left(\vartheta \right)\right|}^{\lambda }\Delta \vartheta \right)}^{\frac{1}{\lambda }}{\left({\int }_k^l{\left|\chi \left(\vartheta \right)\right|}^{\mu }\Delta \vartheta \right)}^{\frac{1}{\mu }},$$

(9)

where $\lambda ,$ $\mu >1$ and $1/\lambda +1/\mu =1.$

Lemma 2

(Hölder’s inequality in two dimensions [24]).Let $k,$ $l\in \mathbb{T}$ and $\zeta ,$ $\chi \in {\mathrm{CC}}_{rd}^1\left(\right[k,$ ${l]}_{\mathbb{T}}\times [k,$ ${l]}_{\mathbb{T}},$ $\mathbb{R})$. Then,

$${\int }_k^l{\int }_k^l\left|\zeta (s,\vartheta )\chi (s,\vartheta )\right|\Delta s\Delta \vartheta \le {\left({\int }_k^l{\int }_k^l{\left|\zeta (s,\vartheta )\right|}^{\lambda }\Delta s\Delta \vartheta \right)}^{\frac{1}{\lambda }}{\left({\int }_k^l{\int }_k^l{\left|\chi (s,\vartheta )\right|}^{\mu }\Delta s\Delta \vartheta \right)}^{\frac{1}{\mu }},$$

(10)

where $\lambda >1$ and $\mu =1/(\lambda -1).$

Lemma 3

(Jensen’s inequality in one dimension [24]).Let $k,$ $l\in \mathbb{T}$ and $m,$ $n\in \mathbb{R}.$ Assume that $\zeta \in {\mathrm{C}}_{rd}\left(\right[k,$ ${l]}_{\mathbb{T}},$ $(m,$ $n\left)\right)$ and $\chi \in {\mathrm{C}}_{rd}\left(\right[a,$ ${b]}_{\mathbb{T}},$ $\mathbb{R})$ are non-negative with ${\int }_k^l\left|\zeta \left(\vartheta \right)\right|\Delta \vartheta >0.$ If $\Theta \in \mathrm{C}\left(\right(m,$ $n),$ $\mathbb{R})$ ba a convex function, then

$$\Theta \left(\frac{{\int }_k^l\left|\zeta \left(\vartheta \right)\right|\chi \left(\vartheta \right)\Delta \vartheta }{{\int }_k^l\left|\zeta \left(\vartheta \right)\right|\Delta \vartheta }\right)\le \frac{{\int }_k^l\left|\zeta \left(\vartheta \right)\right|\Theta \left(\chi \left(\vartheta \right)\right)\Delta \vartheta }{{\int }_k^l\left|\zeta \left(\vartheta \right)\right|\Delta \vartheta }.$$

(11)

Lemma 4

(Jensen’s inequality in two dimensions [25] Theorem 3.1).Let $s,$ $\vartheta \in \mathrm{R}$ and $-\infty \le p<q\le \infty$. If $\zeta \in {\mathrm{CC}}_{rd}^1(\mathrm{R},$ $(p,$ $q\left)\right)$ and $\Theta :(p,$ $q)\to \mathbb{R}$ be a convex function, then

$$\Theta \left(\frac{{\int }_k^l{\int }_k^l\zeta (s,\vartheta ){\Delta }_{1}s{\Delta }_2\vartheta }{{\int }_k^l{\int }_k^l{\Delta }_{1}s{\Delta }_2\vartheta }\right)\le \frac{{\int }_k^l{\int }_k^l\Theta \left(\zeta (s,\vartheta )\right){\Delta }_{1}s{\Delta }_2\vartheta }{{\int }_k^l{\int }_k^l{\Delta }_{1}s{\Delta }_2\vartheta },$$

(12)

where $\mathrm{R}$ is a rectangle in ${\mathbb{T}}_1\times {\mathbb{T}}_2$ defined by

$$\mathrm{R}=[k,l)\times [m,n)=\left\{\right(s,\vartheta ):s\in [k,l),\vartheta \in [m,n\left)\right\}.$$

Lemma 5

(Fubini’s theorem [26] Theorem 1.1).Let ${\mathbb{T}}_1$ and ${\mathbb{T}}_2$ be two time scales. Suppose that $\zeta :{\mathbb{T}}_1\times {\mathbb{T}}_2\to \mathbb{R}$ is a Δ-integrable function with respect to both time scales. Define

$$\Omega \left(\vartheta \right)={\int }_{{\mathbb{T}}_1}\zeta (s,\vartheta )\Delta s,\vartheta \in {\mathbb{T}}_2,$$

and

$$\Pi \left(s\right)={\int }_{{\mathbb{T}}_2}\zeta (s,\vartheta )\Delta \vartheta ,s\in {\mathbb{T}}_1.$$

Then, Ω is Δ-integrable on ${\mathbb{T}}_2$ and Π is Δ-integrable on ${\mathbb{T}}_1$ and

$${\int }_{{\mathbb{T}}_1}\Delta s{\int }_{{\mathbb{T}}_2}\zeta (s,\vartheta )\Delta \vartheta ={\int }_{{\mathbb{T}}_2}\Delta \vartheta {\int }_{{\mathbb{T}}_1}\zeta (s,\vartheta )\Delta s.$$

(13)

Lemma 6

(Young’s inequality [27]).Let $r>0,$ ${\mu }_q>0$ and ${\sum }_{q=1}^p{\mu }_q={\Omega }_p.$ Then

$${\left\{\prod _{q=1}^p{s}_q^{{\mu }_q}\right\}}^{\frac{1}{{\Omega }_p}}\le {\left\{\frac{1}{{\Omega }_p}\sum _{q=1}^p{\mu }_q{s}_q^r\right\}}^{\frac{1}{r}}.$$

(14)

The symmetry index is the most important parameter when evaluating functional asymmetries in athletes of different disciplines. Hence, the first objective of this paper is to establish a new inequality symmetry to Hilbert’s type inequality. Our findings provide new estimates on time-scale for this form of inequality. During that paper, we must assume that all functions found in the theorems statements are non-negative, right-dense continuous (rd-continuous) and that the integrals considered exist.

2. Main Results

In this section, we state and prove our main results. Namely, we set a time scale model for inequalities (5) and (6). To prove our next theorems, we will assume that $\lambda ,$ $\mu$ be any two real numbers such that $\lambda ,$ $\mu >1$ with $1/\lambda +1/\mu =1.$

2.1. The One Dimension Version

Theorem 1.

Let $s,$ ϑ and $t_0\in \mathbb{T},$ $f\left(s\right)\in {\mathrm{C}}_{rd}^1\left(\right[t_0,$ ${x)}_{\mathbb{T}},$ ${\mathbb{R}}^{+}),$ $g\left(\vartheta \right)\in {\mathrm{C}}_{rd}^1\left(\right[t_0,$ ${y)}_{\mathbb{T}},$ ${\mathbb{R}}^{+})$ and $f\left(t_0\right)=g\left(t_0\right)=0.$ Then, for $s\in [t_0,$ ${x)}_{\mathbb{T}}$ and $\vartheta \in [t_0,$ ${y)}_{\mathbb{T}},$ we have

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\frac{\left|f\left(s\right)\right|\left|g\left(\vartheta \right)\right|}{\mu {(s-t_0)}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}+\lambda {(\vartheta -t_0)}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}\Delta s\Delta \vartheta \hfill \\ \le & E(\lambda ,\mu ){\left({\int }_{t_0}^x(\sigma \left(x\right)-s){\left|f^{\Delta }\left(s\right)\right|}^{\lambda }\Delta s\right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^y(\sigma \left(y\right)-\vartheta ){\left|g^{\Delta }\left(\vartheta \right)\right|}^{\mu }\Delta \vartheta \right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(15)

where

$$E(\lambda ,\mu )=\frac{1}{\lambda +\mu }{(x-t_0)}^{\frac{\lambda -1}{\lambda }}{(y-t_0)}^{\frac{\mu -1}{\mu }},$$

(16)

for $x,$ $y\in I_0=[t_0,$ $\infty )\cap \mathbb{T}.$

Proof. 

From the hypotheses, we have the following two identities hold,

$$\left|f\left(s\right)\right|={\int }_{t_0}^s\left|f^{\Delta }\left(\tau \right)\right|\Delta \tau ,$$

(17)

$$\left|g\left(\vartheta \right)\right|={\int }_{t_0}^{\vartheta }\left|g^{\Delta }\left(\xi \right)\right|\Delta \xi ,$$

(18)

for $s\in [t_0,$ ${x)}_{\mathbb{T}},$ $t\in [t_0,$ ${y)}_{\mathbb{T}}.$ Further, by using Hölder’s integral inequality (9), we have

$$\left|f\left(s\right)\right|\le {(s-t_0)}^{\frac{\lambda -1}{\lambda }}{\left({\int }_{t_0}^s{\left|f^{\Delta }\left(\tau \right)\right|}^{\lambda }\Delta \tau \right)}^{\frac{1}{\lambda }},$$

(19)

$$\left|g\left(\vartheta \right)\right|\le {(\vartheta -t_0)}^{\frac{\mu -1}{\mu }}{\left({\int }_{t_0}^{\vartheta }{\left|g^{\Delta }\left(\xi \right)\right|}^{\mu }\Delta \xi \right)}^{\frac{1}{\mu }}.$$

(20)

By multiplying (19) and (20), we get

$$\left|f\left(s\right)\right|\left|g\left(\vartheta \right)\right|\le {(s-t_0)}^{\frac{\lambda -1}{\lambda }}{(\vartheta -t_0)}^{\frac{\mu -1}{\mu }}{\left({\int }_{t_0}^s{\left|f^{\Delta }\left(\tau \right)\right|}^{\lambda }\Delta \tau \right)}^{\frac{1}{\lambda }}{\left({\int }_{t_0}^{\vartheta }{\left|g^{\Delta }\left(\xi \right)\right|}^{\mu }\Delta \xi \right)}^{\frac{1}{\mu }}.$$

(21)

Using the inequality (14), we note

$${\left(s_1^{{\omega }_1}{s}_2^{{\omega }_2}\right)}^{\frac{r}{{\omega }_1+{\omega }_2}}\le \frac{1}{{\omega }_1+{\omega }_2}\left({\omega }_1{s}_1^r+{\omega }_2{s}_2^r\right).$$

(22)

Now, by setting $s_1={(s-t_0)}^{\lambda -1},$ $s_2={(\vartheta -t_0)}^{\mu -1},$ ${\omega }_1=1/\lambda ,$ ${\omega }_2=1/\mu$ and $r={\omega }_1+{\omega }_2$ in (22), we get

$${(s-t_0)}^{\frac{\lambda -1}{\lambda }}{(\vartheta -t_0)}^{\frac{\mu -1}{\mu }}\le \frac{\lambda \mu }{\lambda +\mu }\left(\frac{{(s-t_0)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}}{\lambda }+\frac{{(\vartheta -t_0)}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\mu }\right).$$

(23)

Substituting (23) into (21) yields

$$\begin{array}{ccc}\hfill \left|f\left(s\right)\right|\left|g\left(\vartheta \right)\right|& \le & \frac{\lambda \mu }{\lambda +\mu }\left(\frac{{(s-t_0)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}}{\lambda }+\frac{{(\vartheta -t_0)}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\mu }\right)\hfill \\ & & \times {\left({\int }_{t_0}^s{\left|f^{\Delta }\left(\tau \right)\right|}^{\lambda }\Delta \tau \right)}^{\frac{1}{\lambda }}{\left({\int }_{t_0}^{\vartheta }{\left|g^{\Delta }\left(\xi \right)\right|}^{\mu }\Delta \xi \right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(24)

Dividing both sides of (24) by the last factor $\mu {(s-t_0)}^{\left[(\lambda -1)(\lambda +\mu )\right]/\lambda \mu }+\lambda {(\vartheta -t_0)}^{{}^{\left[(\mu -1)(\lambda +\mu )\right]/\lambda \mu }},$ we obtain

$$\begin{array}{cc}& \frac{\left|f\left(s\right)\right|\left|g\left(\vartheta \right)\right|}{\mu {(s-t_0)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {(\vartheta -t_0)}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\hfill \\ \le & \frac{\lambda \mu }{\lambda +\mu }{\left({\int }_{t_0}^s{\left|f^{\Delta }\left(\tau \right)\right|}^{\lambda }\Delta \tau \right)}^{\frac{1}{\lambda }}{\left({\int }_{t_0}^{\vartheta }{\left|g^{\Delta }\left(\xi \right)\right|}^{\mu }\Delta \xi \right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(25)

Integrating both sides of (25) and using (9), we find that

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\frac{\left|f\left(s\right)\right|\left|g\left(\vartheta \right)\right|}{\mu {(s-t_0)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {(\vartheta -t_0)}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\Delta s\Delta \vartheta \hfill \\ \le & \frac{\lambda \mu }{\lambda +\mu }{(x-t_0)}^{\frac{\lambda -1}{\lambda }}{(y-t_0)}^{\frac{\mu -1}{\mu }}\left({\int }_{t_0}^x{\left({\int }_{t_0}^s{\left|f^{\Delta }\left(\tau \right)\right|}^{\lambda }\Delta \tau \right)}^{\frac{1}{\lambda }}\Delta s\right)\hfill \\ & \times \left({\int }_{t_0}^y{\left({\int }_{t_0}^{\vartheta }{\left|g^{\Delta }\left(\xi \right)\right|}^{\mu }\Delta \xi \right)}^{\frac{1}{\mu }}\Delta \vartheta \right)\hfill \\ =& E(\lambda ,\mu )\left({\int }_{t_0}^x{\left({\int }_{t_0}^s{\left|f^{\Delta }\left(\tau \right)\right|}^{\lambda }\Delta \tau \right)}^{\frac{1}{\lambda }}\Delta s\right)\left({\int }_{t_0}^y{\left({\int }_{t_0}^{\vartheta }{\left|g^{\Delta }\left(\xi \right)\right|}^{\mu }\Delta \xi \right)}^{\frac{1}{\mu }}\Delta \vartheta \right).\hfill \end{array}$$

(26)

Applying Fubini’s theorem on (26) and by taking advantage of the fact that $\sigma \left(\delta \right)\ge \delta ,$ we conclude that

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\frac{\left|f\left(s\right)\right|\left|g\left(\vartheta \right)\right|}{\mu {(s-t_0)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {(\vartheta -t_0)}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\Delta s\Delta \vartheta \hfill \\ \le & E(\lambda ,\mu ){\left({\int }_{t_0}^x(\sigma \left(x\right)-s){\left|f^{\Delta }\left(s\right)\right|}^{\lambda }\Delta s\right)}^{\frac{1}{\lambda }}{\left({\int }_{t_0}^y(\sigma \left(y\right)-\vartheta ){\left|g^{\Delta }\left(\vartheta \right)\right|}^{\mu }\Delta \vartheta \right)}^{\frac{1}{\mu }},\hfill \end{array}$$

which is equivalent to (15). □

Remark 1.

By setting $1/\lambda +1/\mu =1$ in (22), we obtain

$$\left(s_1^{{\omega }_1}{s}_2^{{\omega }_2}\right)\le \frac{1}{{\omega }_1+{\omega }_2}\left({\omega }_1{s}_1^{{\omega }_1+{\omega }_2}+{\omega }_2{s}_2^{{\omega }_1+{\omega }_2}\right).$$

(27)

Therefore, by applying (27) on the right-hand side of (15) in Theorem 1, we get

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\frac{\left|f\left(s\right)\right|\left|g\left(\vartheta \right)\right|}{\mu {(s-t_0)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {(\vartheta -t_0)}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\Delta s\Delta \vartheta \hfill \\ \le & \frac{\lambda \mu }{{\left(\lambda +\mu \right)}^2}{(x-t_0)}^{\frac{\lambda -1}{\lambda }}{(y-t_0)}^{\frac{\mu -1}{\mu }}\left\{\frac{1}{\lambda }{\left({\int }_{t_0}^x(\sigma \left(x\right)-s){\left|f^{\Delta }\left(s\right)\right|}^{\lambda }\Delta s\right)}^{\frac{\lambda +\mu }{\lambda \mu }}\right.\hfill \\ & \left.+\frac{1}{\mu }{\left({\int }_{t_0}^y(\sigma \left(y\right)-\vartheta ){\left|g^{\Delta }\left(\vartheta \right)\right|}^{\mu }\Delta \vartheta \right)}^{\frac{\lambda +\mu }{\lambda \mu }}\right\}.\hfill \end{array}$$

(28)

Remark 2.

Clearly, for $\mathbb{T}=\mathbb{Z}$ and $t_0=0,$ inequality (28) in Remark 1 reduces to

$$\begin{array}{cc}& \sum _{s=1}^p\sum _{\vartheta =1}^q\frac{\left|a_s\right|\left|b_{\vartheta }\right|}{\mu s^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}+\lambda {\vartheta }^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\hfill \\ \le & \frac{\lambda \mu }{{\left(\lambda +\mu \right)}^2}{\left(p\right)}^{\frac{\lambda -1}{\lambda }}{\left(q\right)}^{\frac{\mu -1}{\mu }}\left\{\frac{1}{\lambda }{\left(\sum _{s=1}^p(p-s+1){\left|\Delta a_s\right|}^{\lambda }\right)}^{\frac{\lambda +\mu }{\lambda \mu }}\right.\hfill \\ & \left.+\frac{1}{\mu }{\left(\sum _{\vartheta =1}^q(q-\vartheta +1){\left|\Delta b_{\vartheta }\right|}^{\mu }\right)}^{\frac{\lambda +\mu }{\lambda \mu }}\right\}.\hfill \end{array}$$

(29)

where $\Delta a_s=a_{s+1}-a_s,$ $\Delta b_{\vartheta }=b_{\vartheta +1}-b_{\vartheta }.$ It is merely a similar variant of the consequence disparity in [15] Remark 1, attributed to Young and Byung.

Remark 3.

For $\mathbb{T}=\mathbb{R}$ and $t_0=0,$ Remark 1 coincides with Remark 2 in [15].

Remark 4.

Inequality (15) is nothing more than a close version of the following inequality established in [28] Theorem 6,

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\frac{F\left(s\right)G\left(\vartheta \right)}{\mu {(s-t_0)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {(\vartheta -t_0)}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\Delta s\Delta \vartheta \hfill \\ \le & E^{*}(\lambda ,\mu ){\left({\int }_{t_0}^x(\sigma \left(x\right)-s){\left[f\left(s\right)\right]}^{\lambda }\Delta s\right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^y(\sigma \left(y\right)-\vartheta ){\left[g\left(\vartheta \right)\right]}^{\mu }\Delta \vartheta \right)}^{\frac{1}{\mu }},\hfill \end{array}$$

where $F\left(s\right)={\int }_{t_0}^{s}f\left(\xi \right)\Delta \xi ,$ $G\left(\vartheta \right)={\int }_{t_0}^{t}g\left(\xi \right)\Delta \xi$ and

$$E^{*}(\lambda ,\mu )=\frac{1}{\lambda +\mu }{(s-t_0)}^{\frac{\lambda -1}{\lambda }}{(\vartheta -t_0)}^{\frac{\mu -1}{\mu }}.$$

Remark 5.

Letting $1/\lambda +1/\mu =1$ in (15), then we get

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\frac{\left|f\left(s\right)\right|\left|g\left(\vartheta \right)\right|}{\mu {(s-t_0)}^{{}^{\lambda -1}}+\lambda {(\vartheta -t_0)}^{{}^{\mu -1}}}\Delta s\Delta \vartheta \hfill \\ \le & E^{**}(\lambda ,\mu ){\left({\int }_{t_0}^x(\sigma \left(x\right)-s){\left|f^{\Delta }\left(s\right)\right|}^{\lambda }\Delta s\right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^y(\sigma \left(y\right)-\vartheta ){\left|g^{\Delta }\left(\vartheta \right)\right|}^{\mu }\Delta \vartheta \right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(30)

where

$$E^{**}(\lambda ,\mu )=\frac{1}{\lambda \mu }{(x-t_0)}^{\frac{\lambda -1}{\lambda }}{(y-t_0)}^{\frac{\mu -1}{\mu }}.$$

It is merely a similar variant of the consequence disparity in [29] Corollary 3.3, attributed to Saker et al.

Remark 6.

Inequality (30) is exactly the time scale form of inequalities (1) and (3) in Theorems (1) and (2), respectively, due to B. G. Pachpatte [14].

Remark 7.

As a particular state of Theorem 1 if $\mathbb{T}=\mathbb{Z}$ and $t_0=0,$ then we have relations (8) and inequality (15) reduce to

$$\begin{array}{cc}& \sum _{s=1}^p\sum _{\vartheta =1}^q\frac{\left|a_s\right|\left|b_{\vartheta }\right|}{\mu s^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}+\lambda {\vartheta }^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\hfill \\ \le & E_0(\lambda ,\mu ){\left(\sum _{s=1}^p(p-s+1){\left|\Delta a_s\right|}^{\lambda }\right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left(\sum _{\vartheta =1}^q(q-\vartheta +1){\left|\Delta b_{\vartheta }\right|}^{\mu }\right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(31)

where $\Delta a_s=a_{s+1}-a_s,$ $\Delta b_{\vartheta }=b_{\vartheta +1}-b_{\vartheta }$ and

$$E_0(\lambda ,\mu )=\frac{1}{\lambda +\mu }{\left(p\right)}^{\frac{\lambda -1}{\lambda }}{\left(q\right)}^{\frac{\mu -1}{\mu }}.$$

This is just a similar version of (5) that premised in the Introduction.

Remark 8.

As a given state of Theorem 1 if $\mathbb{T}=\mathbb{R}$ and $t_0=0,$ inequality (15) reduce to (6).

In what follows, we give a further generalization of (15) obtained in Theorem 1. Until giving our results, we assume that there are two functions $\Phi$ and $\Psi$ which are real-valued, nonnegative, convex, and submultiplicetive functions defined on $\left[0,\infty \right).$ A function $\chi$ is a submultiplicetive if $\chi \left(s\vartheta \right)\le \chi \left(s\right)\chi \left(\vartheta \right)$ for $s,$ $\vartheta \ge 0$.

Theorem 2.

Let $s,$ ϑ and $t_0\in \mathbb{T},$ $f\left(s\right)\in {\mathrm{C}}_{rd}^1\left(\right[t_0,$ ${x)}_{\mathbb{T}},$ ${\mathbb{R}}^{+}),$ $g\left(\vartheta \right)\in {\mathrm{C}}_{rd}^1\left(\right[t_0,$ ${y)}_{\mathbb{T}},$ ${\mathbb{R}}^{+})$ and $f\left(t_0\right)=g\left(t_0\right)=0.$ Suppose $h\left(\tau \right)>0$ on $[t_0,$ ${x)}_{\mathbb{T}}$ and $l\left(\xi \right)>0$ on $[t_0,$ ${y)}_{\mathbb{T}}$ and assume that

$$H\left(s\right)={\int }_{t_0}^s\left|h\left(\tau \right)\right|\Delta \tau \mathit{and}L\left(\vartheta \right)={\int }_{t_0}^{\vartheta }\left|l\left(\xi \right)\right|\Delta \xi .$$

(32)

Then, for $s\in [t_0,$ ${x)}_{\mathbb{T}}$ and $\vartheta \in [t_0,$ ${y)}_{\mathbb{T}},$ we get

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\frac{\Phi \left(\left|f\left(s\right)\right|\right)\Psi \left(\left|g\left(\vartheta \right)\right|\right)}{\mu {(s-t_0)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {(\vartheta -t_0)}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\Delta s\Delta \vartheta \hfill \\ \le & G(\lambda ,\mu ){\left({\int }_{t_0}^x(\sigma \left(x\right)-s){\left(\left|h\left(s\right)\right|\Phi \left(\left|\frac{f^{\Delta }\left(s\right)}{h\left(s\right)}\right|\right)\right)}^{\lambda }\Delta s\right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^y(\sigma \left(y\right)-\vartheta ){\left(\left|l\left(\vartheta \right)\right|\Psi \left(\left|\frac{g^{\Delta }\left(\vartheta \right)}{l\left(\vartheta \right)}\right|\right)\right)}^{\mu }\Delta \vartheta \right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(33)

where

$$G(\lambda ,\mu )=\frac{1}{\lambda +\mu }{\left({\int }_{t_0}^x{\left(\frac{\Phi \left(H\right(s\left)\right)}{H\left(s\right)}\right)}^{\frac{\lambda }{\lambda -1}}\Delta s\right)}^{\frac{\lambda -1}{\lambda }}{\left({\int }_{t_0}^y{\left(\frac{\Psi \left(L\right(\vartheta \left)\right)}{L\left(\vartheta \right)}\right)}^{\frac{\mu }{\mu -1}}\Delta \vartheta \right)}^{\frac{\mu -1}{\mu }},$$

(34)

for $x,$ $y\in I_0=[t_0,$ $\infty )\cap \mathbb{T}.$

Proof. 

Using the two identities (17) and (18) in the proof of Theorem 1 and the properties of $\Phi$ and utilize (11), we obtain

$$\begin{array}{ccc}\hfill \Phi \left(\left|f\left(s\right)\right|\right)& =& \Phi \left(\frac{H\left(s\right){\int }_{t_0}^s\left|h\left(\tau \right)\right|\left|\frac{f^{\Delta }\left(\tau \right)}{h\left(\tau \right)}\right|\Delta \tau }{{\int }_{t_0}^s\left|h\left(\tau \right)\right|\Delta \tau }\right)\hfill \\ & \le & \Phi \left(H\left(s\right)\right)\Phi \left(\frac{{\int }_{t_0}^s\left|h\left(\tau \right)\right|\left|\frac{f^{\Delta }\left(\tau \right)}{h\left(\tau \right)}\right|\Delta \tau }{{\int }_{t_0}^s\left|h\left(\tau \right)\right|\Delta \tau }\right)\hfill \\ & \le & \frac{\Phi \left(H\right(s\left)\right)}{H\left(s\right)}{\int }_{t_0}^s\left|h\left(\tau \right)\right|\Phi \left(\left|\frac{f^{\Delta }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\Delta \tau .\hfill \end{array}$$

(35)

Further, by Hölder’s integral inequality (9), we see that

$$\begin{array}{ccc}\hfill \Phi \left(\left|f\left(s\right)\right|\right)& \le & \frac{\Phi \left(H\right(s\left)\right)}{H\left(s\right)}{\left({\int }_{t_0}^s{\left(1\right)}^{\frac{\lambda }{\lambda -1}}\Delta \tau \right)}^{\frac{\lambda -1}{\lambda }}{\left({\int }_{t_0}^s{\left(\left|h\left(\tau \right)\right|\Phi \left(\left|\frac{f^{\Delta }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right)}^{\lambda }\Delta \tau \right)}^{\frac{1}{\lambda }}\hfill \\ & \le & \frac{\Phi \left(H\right(s\left)\right)}{H\left(s\right)}{\left(s-t_0\right)}^{\frac{\lambda -1}{\lambda }}{\left({\int }_{t_0}^s{\left(\left|h\left(\tau \right)\right|\Phi \left(\left|\frac{f^{\Delta }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right)}^{\lambda }\Delta \tau \right)}^{\frac{1}{\lambda }}.\hfill \end{array}$$

(36)

Likewise, we get

$$\Psi \left(\left|g\left(\vartheta \right)\right|\right)\le \frac{\Psi \left(L\right(\vartheta \left)\right)}{L\left(\vartheta \right)}{\left(\vartheta -t_0\right)}^{\frac{\mu -1}{\mu }}{\left({\int }_{t_0}^t{\left(\left|l\left(\xi \right)\right|\Psi \left(\left|\frac{g^{\Delta }\left(\xi \right)}{l\left(\xi \right)}\right|\right)\right)}^{\mu }\Delta \xi \right)}^{\frac{1}{\mu }}.$$

(37)

By multiplying (36) and (37), we get

$$\begin{array}{ccc}\hfill \Phi \left(\left|f\left(s\right)\right|\right)\Psi \left(\left|g\left(\vartheta \right)\right|\right)& \le & {\left(s-t_0\right)}^{\frac{\lambda -1}{\lambda }}{\left(\vartheta -t_0\right)}^{\frac{\mu -1}{\mu }}\hfill \\ & & \times \frac{\Phi \left(H\right(s\left)\right)}{H\left(s\right)}{\left({\int }_{t_0}^s{\left(\left|h\left(\tau \right)\right|\Phi \left(\left|\frac{f^{\Delta }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right)}^{\lambda }\Delta \tau \right)}^{\frac{1}{\lambda }}\hfill \\ & & \times \frac{\Psi \left(L\right(\vartheta \left)\right)}{L\left(\vartheta \right)}{\left({\int }_{t_0}^{\vartheta }{\left(\left|l\left(\xi \right)\right|\Psi \left(\left|\frac{g^{\Delta }\left(\xi \right)}{l\left(\xi \right)}\right|\right)\right)}^{\mu }\Delta \xi \right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(38)

Applying (22) on the term ${(s-t_0)}^{(\lambda -1)/\lambda }\times {(\vartheta -t_0)}^{(\mu -1)/\mu },$ gives

$$\begin{array}{ccc}\hfill \Phi \left(\left|f\left(s\right)\right|\right)\Psi \left(\left|g\left(\vartheta \right)\right|\right)& \le & \frac{\lambda \mu }{\lambda +\mu }\left(\frac{{(s-t_0)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}}{\lambda }+\frac{{(\vartheta -t_0)}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\mu }\right)\hfill \\ & & \times \frac{\Phi \left(H\right(s\left)\right)}{H\left(s\right)}{\left({\int }_{t_0}^s{\left(\left|h\left(\tau \right)\right|\Phi \left(\left|\frac{f^{\Delta }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right)}^{\lambda }\Delta \tau \right)}^{\frac{1}{\lambda }}\hfill \\ & & \times \frac{\Psi \left(L\right(\vartheta \left)\right)}{L\left(\vartheta \right)}{\left({\int }_{t_0}^{\vartheta }{\left(\left|l\left(\xi \right)\right|\Psi \left(\left|\frac{g^{\Delta }\left(\xi \right)}{l\left(\xi \right)}\right|\right)\right)}^{\mu }\Delta \xi \right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(39)

From (39), we observe that

$$\begin{array}{cc}& \frac{\Phi \left(\left|f\left(s\right)\right|\right)\Psi \left(\left|g\left(\vartheta \right)\right|\right)}{\mu {(s-t_0)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {(\vartheta -t_0)}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\hfill \\ \le & \frac{1}{\lambda +\mu }\left(\frac{\Phi \left(H\right(s\left)\right)}{H\left(s\right)}{\left({\int }_{t_0}^s{\left(\left|h\left(\tau \right)\right|\Phi \left(\left|\frac{f^{\Delta }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right)}^{\lambda }\Delta \tau \right)}^{\frac{1}{\lambda }}\right)\hfill \\ & \times \left(\frac{\Psi \left(L\right(\vartheta \left)\right)}{L\left(\vartheta \right)}{\left({\int }_{t_0}^t{\left(\left|l\left(\xi \right)\right|\Psi \left(\left|\frac{g^{\Delta }\left(\xi \right)}{l\left(\xi \right)}\right|\right)\right)}^{\mu }\Delta \xi \right)}^{\frac{1}{\mu }}\right).\hfill \end{array}$$

(40)

Integrating both sides of (40) and using (9), we find that

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\frac{\Phi \left(\left|f\left(s\right)\right|\right)\Psi \left(\left|g\left(\vartheta \right)\right|\right)}{\mu {(s-t_0)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {(\vartheta -t_0)}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\Delta s\Delta \vartheta \hfill \\ \le & \frac{1}{\lambda +\mu }{\left({\int }_{t_0}^x{\left(\frac{\Phi \left(H\right(s\left)\right)}{H\left(s\right)}\right)}^{\frac{\lambda }{\lambda -1}}\Delta s\right)}^{\frac{\lambda -1}{\lambda }}{\left({\int }_{t_0}^x{\int }_{t_0}^s{\left(\left|h\left(\tau \right)\right|\Phi \left(\left|\frac{f^{\Delta }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right)}^{\lambda }\Delta \tau \Delta s\right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^y{\left(\frac{\Psi \left(L\right(\vartheta \left)\right)}{L\left(\vartheta \right)}\right)}^{\frac{\mu }{\mu -1}}\Delta \vartheta \right)}^{\frac{\mu -1}{\mu }}{\left({\int }_{t_0}^y{\int }_{t_0}^{\vartheta }{\left(\left|l\left(\xi \right)\right|\Psi \left(\left|\frac{g^{\Delta }\left(\xi \right)}{l\left(\xi \right)}\right|\right)\right)}^{\mu }\Delta \xi \Delta \vartheta \right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(41)

Applying Fubini’s theorem on (41) and using $\sigma \left(\delta \right)\ge \delta ,$ we obtain

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\frac{\Phi \left(\left|f\left(s\right)\right|\right)\Psi \left(\left|g\left(\vartheta \right)\right|\right)}{\mu {(s-t_0)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {(\vartheta -t_0)}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\Delta s\Delta \vartheta \hfill \\ \le & \frac{1}{\lambda +\mu }{\left({\int }_{t_0}^x{\left(\frac{\Phi \left(H\right(s\left)\right)}{H\left(s\right)}\right)}^{\frac{\lambda }{\lambda -1}}\Delta s\right)}^{\frac{\lambda -1}{\lambda }}{\left({\int }_{t_0}^y{\left(\frac{\Psi \left(L\right(\vartheta \left)\right)}{L\left(\vartheta \right)}\right)}^{\frac{\mu }{\mu -1}}\Delta \vartheta \right)}^{\frac{\mu -1}{\mu }}\hfill \\ & \times {\left({\int }_{t_0}^x(\sigma \left(x\right)-s){\left(\left|h\left(s\right)\right|\Phi \left(\left|\frac{f^{\Delta }\left(s\right)}{h\left(s\right)}\right|\right)\right)}^{\lambda }\Delta s\right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^y(\sigma \left(y\right)-\vartheta ){\left(\left|l\left(\vartheta \right)\right|\Psi \left(\left|\frac{g^{\Delta }\left(\vartheta \right)}{l\left(\vartheta \right)}\right|\right)\right)}^{\mu }\Delta \vartheta \right)}^{\frac{1}{\mu }}\hfill \\ =& G(\lambda ,\mu ){\left({\int }_{t_0}^x(\sigma \left(x\right)-s){\left(\left|h\left(s\right)\right|\Phi \left(\left|\frac{f^{\Delta }\left(s\right)}{h\left(s\right)}\right|\right)\right)}^{\lambda }\Delta s\right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^y(\sigma \left(y\right)-\vartheta ){\left(\left|l\left(\vartheta \right)\right|\Psi \left(\left|\frac{g^{\Delta }\left(\vartheta \right)}{l\left(\vartheta \right)}\right|\right)\right)}^{\mu }\Delta \vartheta \right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

which is equivalent to (33). □

Remark 9.

By applying (27) on the right-hand side of (33) in Theorem 1, then

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\frac{\Phi \left(\left|f\left(s\right)\right|\right)\Psi \left(\left|g\left(\vartheta \right)\right|\right)}{\mu {(s-t_0)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {(\vartheta -t_0)}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\Delta s\Delta \vartheta \hfill \\ \le & G(\lambda ,\mu )\left\{\frac{1}{\lambda }{\left({\int }_{t_0}^x(\sigma \left(x\right)-s){\left(\left|h\left(s\right)\right|\Phi \left(\left|\frac{f^{\Delta }\left(s\right)}{h\left(s\right)}\right|\right)\right)}^{\lambda }\Delta s\right)}^{\frac{\lambda +\mu }{\lambda \mu }}\right.\hfill \\ & \left.+\frac{1}{\mu }{\left({\int }_{t_0}^y(\sigma \left(y\right)-\vartheta ){\left(\left|l\left(\vartheta \right)\right|\Psi \left(\left|\frac{g^{\Delta }\left(\vartheta \right)}{l\left(\vartheta \right)}\right|\right)\right)}^{\mu }\Delta \vartheta \right)}^{\frac{\lambda +\mu }{\lambda \mu }}\right\}.\hfill \end{array}$$

(42)

Remark 10.

Inequality (33) is actually a related version of the following inequality in [28] Theorem 9,

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\frac{\Phi \left(F\right(s\left)\right)\Psi \left(G\right(\vartheta \left)\right)}{\mu {(s-t_0)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {(\vartheta -t_0)}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\Delta s\Delta \vartheta \hfill \\ \le & G^{*}(\lambda ,\mu ){\left({\int }_{t_0}^x(\sigma \left(x\right)-s){\left(h\left(s\right)\Phi \left(\frac{f\left(s\right)}{h\left(s\right)}\right)\right)}^{\lambda }\Delta s\right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^y(\sigma \left(y\right)-\vartheta ){\left(l\left(\vartheta \right)\Psi \left(\frac{g\left(\vartheta \right)}{l\left(\vartheta \right)}\right)\right)}^{\mu }\Delta \vartheta \right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(43)

where

$$G^{*}(\lambda ,\mu )=\frac{1}{\lambda +\mu }{\left({\int }_{t_0}^x{\left(\frac{\Phi \left(H\right(s\left)\right)}{H\left(s\right)}\right)}^{\frac{\lambda }{\lambda -1}}\Delta s\right)}^{\frac{\lambda -1}{\lambda }}{\left({\int }_{t_0}^y{\left(\frac{\Psi \left(L\right(\vartheta \left)\right)}{L\left(\vartheta \right)}\right)}^{\frac{\mu }{\mu -1}}\Delta \vartheta \right)}^{\frac{\mu -1}{\mu }},$$

$F\left(s\right)={\int }_{t_0}^{s}f\left(\xi \right)\Delta \xi ,$$G\left(\vartheta \right)={\int }_{t_0}^{t}g\left(\xi \right)\Delta \xi ,$$H\left(s\right)={\int }_{t_0}^{s}h\left(\xi \right)\Delta \xi$ and $L\left(\vartheta \right)={\int }_{t_0}^{\vartheta }l\left(\xi \right)\Delta \xi .$

Remark 11.

Letting $1/\lambda +1/\mu =1$ in (33), then we get

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\frac{\Phi \left(\left|f\left(s\right)\right|\right)\Psi \left(\left|g\left(\vartheta \right)\right|\right)}{\mu {(s-t_0)}^{{}^{\lambda -1}}+\lambda {(\vartheta -t_0)}^{{}^{\mu -1}}}\Delta s\Delta \vartheta \hfill \\ \le & G^{**}(\lambda ,\mu ){\left({\int }_{t_0}^x(\sigma \left(x\right)-s){\left(\left|h\left(s\right)\right|\Phi \left(\left|\frac{f^{\Delta }\left(s\right)}{h\left(s\right)}\right|\right)\right)}^{\lambda }\Delta s\right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^y(\sigma \left(y\right)-\vartheta ){\left(\left|l\left(\vartheta \right)\right|\Psi \left(\left|\frac{g^{\Delta }\left(\vartheta \right)}{l\left(\vartheta \right)}\right|\right)\right)}^{\mu }\Delta \vartheta \right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(44)

where

$$G^{**}(\lambda ,\mu )=\frac{1}{\lambda \mu }{\left({\int }_{t_0}^x{\left(\frac{\Phi \left(H\right(s\left)\right)}{H\left(s\right)}\right)}^{\frac{\lambda }{\lambda -1}}\Delta s\right)}^{\frac{\lambda -1}{\lambda }}{\left({\int }_{t_0}^y{\left(\frac{\Psi \left(L\right(\vartheta \left)\right)}{L\left(\vartheta \right)}\right)}^{\frac{\mu }{\mu -1}}\Delta \vartheta \right)}^{\frac{\mu -1}{\mu }}.$$

It is merely a similar variant of the consequence disparity in [29] Theorem 3.2, attributed to Saker et al.

Remark 12.

As a particular state of Theorem 2 if $\mathbb{T}=\mathbb{Z}$ and $t_0=0,$ then we have relations (8) and inequality (33) reduce to

$$\begin{array}{cc}& \sum _{s=1}^p\sum _{\vartheta =1}^q\frac{\Phi \left(\left|a_s\right|\right)\Psi \left(\left|b_{\vartheta }\right|\right)}{\mu s^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}+\lambda t^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\hfill \\ \le & G_0(\lambda ,\mu ){\left(\sum _{s=1}^p(p-s+1){\left(\left|a_s\right|\Phi \left(\left|\frac{\Delta a_s}{a_s}\right|\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left(\sum _{\vartheta =1}^q(q-\vartheta +1){\left(\left|b_{\vartheta }\right|\Phi \left(\left|\frac{\Delta b_{\vartheta }}{b_{\vartheta }}\right|\right)\right)}^{\mu }\right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(45)

where $\Delta a_s=a_{s+1}-a_s,$ $\Delta b_{\vartheta }=b_{\vartheta +1}-b_{\vartheta }$ and

$$G_0(\lambda ,\mu )=\frac{1}{\lambda +\mu }{\left(\sum _{s=1}^p{\left(\frac{\Phi \left(H_s\right)}{H_s}\right)}^{\frac{\lambda }{\lambda -1}}\right)}^{\frac{\lambda -1}{\lambda }}{\left(\sum _{\vartheta =1}^q{\left(\frac{\Psi \left(L_{\vartheta }\right)}{L_{\vartheta }}\right)}^{\frac{\mu }{\mu -1}}\right)}^{\frac{\mu -1}{\mu }}.$$

For $\lambda =\mu =2,$ inequality (45) is just a similar version of Pachpatte’s result [13] Theorem 1.

Remark 13.

As a particular case of Theorem 2, if $\mathbb{T}=\mathbb{R}$ and $t_0=0,$ then we have relations (7) and inequality (33) reduce to

$$\begin{array}{cc}& {\int }_0^x{\int }_0^y\frac{\Phi \left(\left|f\left(s\right)\right|\right)\Psi \left(\left|g\left(\vartheta \right)\right|\right)}{\mu {\left(s\right)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left(\vartheta \right)}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}dsd\vartheta \hfill \\ \le & G_0^{*}(\lambda ,\mu ){\left({\int }_0^x(x-s){\left(\left|h\left(s\right)\right|\Phi \left(\left|\frac{f^{\prime }\left(s\right)}{h\left(s\right)}\right|\right)\right)}^{\lambda }ds\right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_0^y(y-\vartheta ){\left(\left|l\left(\vartheta \right)\right|\Psi \left(\left|\frac{g^{\prime }\left(\vartheta \right)}{l\left(\vartheta \right)}\right|\right)\right)}^{\mu }d\vartheta \right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(46)

where

$$G_0^{*}(\lambda ,\mu )=\frac{1}{\lambda +\mu }{\left({\int }_0^x{\left(\frac{\Phi \left(H\right(s\left)\right)}{H\left(s\right)}\right)}^{\frac{\lambda }{\lambda -1}}ds\right)}^{\frac{\lambda -1}{\lambda }}{\left({\int }_0^y{\left(\frac{\Psi \left(L\right(\vartheta \left)\right)}{L\left(\vartheta \right)}\right)}^{\frac{\mu }{\mu -1}}d\vartheta \right)}^{\frac{\mu -1}{\mu }}.$$

For $\lambda =\mu =2,$ this is Pachpatte’s result [13] Theorem 2.

2.2. The Two Dimension Version

In the next theorems, we define the two independent variable versions of the inequalities given in Theorems 1 and 2. Throughout this paragraph, we are always assuming that ${\mathbb{T}}_1$ and ${\mathbb{T}}_2$ are two defined time scales with (i) $t_0,$ $s,$ $t,$ $x,$ $z\in {\mathbb{T}}_1;$ (ii) $t_0,$ $\vartheta ,$ $r,$ $y,$ $w\in {\mathbb{T}}_2.$ We denote the partial delta derivatives of $u(s,$ $\vartheta )$ with respect to $s,$ $\vartheta$ and $s\vartheta$ by

$$u^{{\Delta }_1}(s,\vartheta )=\frac{\partial u(s,\vartheta )}{{\Delta }_{1}s},u^{{\Delta }_2}(s,\vartheta )=\frac{\partial u(s,\vartheta )}{{\Delta }_2\vartheta }\mathrm{and}{u}^{{\Delta }_1{\Delta }_2}(s,\vartheta )=\frac{{\partial }^{2}u(s,\vartheta )}{{\Delta }_{2}t{\Delta }_1\vartheta }=u^{{\Delta }_2{\Delta }_1}(s,\vartheta ),$$

respectively.

Theorem 3.

Let $f(s,$ $\vartheta )\in {\mathrm{CC}}_{rd}^1\left(\right[t_0,$ ${x)}_{{\mathbb{T}}_1}\times [t_0,$ ${y)}_{{\mathbb{T}}_2},$ ${\mathbb{R}}^{+}),$ $g(t,$ $r)\in {\mathrm{CC}}_{rd}^1\left(\right[t_0,$ ${z)}_{{\mathbb{T}}_1}\times [t_0,$ ${w)}_{{\mathbb{T}}_2},$ ${\mathbb{R}}^{+})$ with $f(s,$ $t_0)=g(t,$ $t_0)=0$ and $f(t_0,$ $\vartheta )=g(t_0,$ $r)=0.$ Then for $(s,$ $\vartheta )\in [t_0,$ ${x)}_{{\mathbb{T}}_1}\times [t_0,$ ${y)}_{{\mathbb{T}}_2}$ and $(t,$ $r)\in [t_0,$ ${z)}_{{\mathbb{T}}_1}\times [t_0,$ ${w)}_{{\mathbb{T}}_2},$ one gets

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\left({\int }_{t_0}^z{\int }_{t_0}^w\frac{\left|f(s,\vartheta )\right|\left|g(t,r)\right|}{\mu {\left[(s-t_0)(\vartheta -t_0)\right]}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left[(t-t_0)(r-t_0)\right]}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\Delta }_{1}t{\Delta }_{2}r\right){\Delta }_{1}s{\Delta }_2\vartheta \hfill \\ \le & R(\lambda ,\mu ){\left({\int }_{t_0}^x{\int }_{t_0}^y(\sigma \left(x\right)-s)(\sigma \left(y\right)-\vartheta ){\left|\frac{{\partial }^{2}f(s,\vartheta )}{{\Delta }_{1}s{\Delta }_2\vartheta }\right|}^{\lambda }{\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^z{\int }_{t_0}^w(\sigma \left(w\right)-t)(\sigma \left(z\right)-r){\left|\frac{{\partial }^{2}g(t,r)}{{\Delta }_{1}t{\Delta }_{2}r}\right|}^{\mu }{\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(47)

where

$$R(\lambda ,\mu )=\frac{1}{\lambda +\mu }{\left[(x-t_0)(y-t_0)\right]}^{\frac{\lambda -1}{\lambda }}{\left[(w-t_0)(z-t_0)\right]}^{\frac{\mu -1}{\mu }},$$

for $x,$ $z\in I_0=[t_0,$ $\infty )\cap {\mathbb{T}}_1$ and $y,$ $w\in I_0^{*}=[t_0,$ $\infty )\cap {\mathbb{T}}_2.$

Proof. 

From the hypotheses, we have the following two identities hold,

$$\left|f(s,\vartheta )\right|={\int }_{t_0}^s{\int }_{t_0}^{\vartheta }\left|\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }\right|{\Delta }_1\xi {\Delta }_2\eta ,$$

(48)

$$\left|g(t,r)\right|={\int }_{t_0}^t{\int }_{t_0}^r\left|\frac{{\partial }^{2}g(\sigma ,\tau )}{{\Delta }_1\sigma {\Delta }_2\tau }\right|{\Delta }_1\sigma {\Delta }_2\tau .$$

(49)

Further, by using Hölder’s integral inequality (10), we find that

$$\left|f(s,\vartheta )\right|\le {\left[(s-t_0)(\vartheta -t_0)\right]}^{\frac{\lambda -1}{\lambda }}{\left({\int }_{t_0}^s{\int }_{t_0}^{\vartheta }{\left|\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }\right|}^{\lambda }{\Delta }_1\xi {\Delta }_2\eta \right)}^{\frac{1}{\lambda }},$$

(50)

and

$$\left|g(t,r)\right|\le {\left[(t-t_0)(r-t_0)\right]}^{\frac{\mu -1}{\mu }}{\left({\int }_{t_0}^t{\int }_{t_0}^r{\left|\frac{{\partial }^{2}g(\sigma ,\tau )}{{\Delta }_1\sigma {\Delta }_2\tau }\right|}^{\mu }{\Delta }_1\sigma {\Delta }_2\tau \right)}^{\frac{1}{\mu }}.$$

(51)

By multiplying (50) and (51), we get

$$\begin{array}{ccc}\hfill \left|f(s,\vartheta )\right|\left|g(t,r)\right|& \le & {\left[(s-t_0)(\vartheta -t_0)\right]}^{\frac{\lambda -1}{\lambda }}{\left[(t-t_0)(r-t_0)\right]}^{\frac{\mu -1}{\mu }}\hfill \\ & & \times {\left({\int }_{t_0}^s{\int }_{t_0}^{\vartheta }{\left|\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }\right|}^{\lambda }{\Delta }_1\xi {\Delta }_2\eta \right)}^{\frac{1}{\lambda }}\hfill \\ & & \times {\left({\int }_{t_0}^t{\int }_{t_0}^r{\left|\frac{{\partial }^{2}g(\sigma ,\tau )}{{\Delta }_1\sigma {\Delta }_2\tau }\right|}^{\mu }{\Delta }_1\sigma {\Delta }_2\tau \right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(52)

Applying (22) on the term ${\left[(s-t_0)(\vartheta -t_0)\right]}^{(\lambda -1)/\lambda }$ and the term ${\left[(t-t_0)(r-t_0)\right]}^{(\mu -1)/\mu },$ gives

$$\begin{array}{ccc}\hfill \left|f(s,\vartheta )\right|\left|g(t,r)\right|& \le & \frac{\lambda \mu }{\lambda +\mu }\left(\frac{{\left[(s-t_0)(\vartheta -t_0)\right]}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}}{\lambda }+\frac{{\left[(t-t_0)(r-t_0)\right]}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}{\mu }\right)\hfill \\ & & \times {\left({\int }_{t_0}^s{\int }_{t_0}^{\vartheta }{\left|\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }\right|}^{\lambda }{\Delta }_1\xi {\Delta }_2\eta \right)}^{\frac{1}{\lambda }}\hfill \\ & & \times {\left({\int }_{t_0}^t{\int }_{t_0}^r{\left|\frac{{\partial }^{2}g(\sigma ,\tau )}{{\Delta }_1\sigma {\Delta }_2\tau }\right|}^{\mu }{\Delta }_1\sigma {\Delta }_2\tau \right)}^{\frac{1}{\mu }}\hfill \\ & =& \frac{\lambda \mu }{\lambda +\mu }\left(\frac{\mu {\left[(s-t_0)(\vartheta -t_0)\right]}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left[(t-t_0)(r-t_0)\right]}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}{\lambda \mu }\right)\hfill \\ & & \times {\left({\int }_{t_0}^s{\int }_{t_0}^{\vartheta }{\left|\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }\right|}^{\lambda }{\Delta }_1\xi {\Delta }_2\eta \right)}^{\frac{1}{\lambda }}\hfill \\ & & \times {\left({\int }_{t_0}^t{\int }_{t_0}^r{\left|\frac{{\partial }^{2}g(\sigma ,\tau )}{{\Delta }_1\sigma {\Delta }_2\tau }\right|}^{\mu }{\Delta }_1\sigma {\Delta }_2\tau \right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(53)

Dividing both sides of (53) by $\mu {\left[(s-t_0)(\vartheta -t_0)\right]}^{\left[(\lambda -1)(\lambda +\mu )\right]/\lambda \mu }+\lambda {\left[(t-t_0)(r-t_0)\right]}^{{}^{\left[(\mu -1)(\lambda +\mu )\right]/\lambda \mu }},$ we obtain

$$\begin{array}{cc}& \frac{\left|f(s,\vartheta )\right|\left|g(t,r)\right|}{\mu {\left[(s-t_0)(\vartheta -t_0)\right]}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left[(t-t_0)(r-t_0)\right]}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\hfill \\ \le & \frac{1}{\lambda +\mu }{\left({\int }_{t_0}^s{\int }_{t_0}^t{\left|\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }\right|}^{\lambda }{\Delta }_1\xi {\Delta }_2\eta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^t{\int }_{t_0}^r{\left|\frac{{\partial }^{2}g(\sigma ,\tau )}{{\Delta }_1\sigma {\Delta }_2\tau }\right|}^{\mu }{\Delta }_1\sigma {\Delta }_2\tau \right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(54)

Integrating both sides of (54) and using (10), we see that

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\left({\int }_{t_0}^z{\int }_{t_0}^w\frac{\left|f(s,\vartheta )\right|\left|g(t,r)\right|}{\mu {\left[(s-t_0)(\vartheta -t_0)\right]}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left[(t-t_0)(r-t_0)\right]}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\Delta }_{1}t{\Delta }_{2}r\right){\Delta }_{1}s{\Delta }_2\vartheta \hfill \\ \le & \frac{1}{\lambda +\mu }{\left[(x-t_0)(y-t_0)\right]}^{\frac{\lambda -1}{\lambda }}{\left[(z-t_0)(w-t_0)\right]}^{\frac{\mu -1}{\mu }}\hfill \\ & \times {\left({\int }_{t_0}^x{\int }_{t_0}^y\left({\int }_{t_0}^s{\int }_{t_0}^{\vartheta }{\left|\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }\right|}^{\lambda }{\Delta }_1\xi {\Delta }_2\eta \right){\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^z{\int }_{t_0}^w\left({\int }_{t_0}^t{\int }_{t_0}^r{\left|\frac{{\partial }^{2}g(\sigma ,\tau )}{{\Delta }_1\sigma {\Delta }_2\tau }\right|}^{\mu }{\Delta }_1\sigma {\Delta }_2\tau \right){\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{1}{\mu }}\hfill \\ =& R(\lambda ,\mu ){\left({\int }_{t_0}^x{\int }_{t_0}^y\left({\int }_{t_0}^s{\int }_{t_0}^{\vartheta }{\left|\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }\right|}^{\lambda }{\Delta }_1\xi {\Delta }_2\eta \right){\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^z{\int }_{t_0}^w\left({\int }_{t_0}^t{\int }_{t_0}^r{\left|\frac{{\partial }^{2}g(\sigma ,\tau )}{{\Delta }_1\sigma {\Delta }_2\tau }\right|}^{\mu }{\Delta }_1\sigma {\Delta }_2\tau \right){\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(55)

Applying Fubini’s theorem on (55), we conclude that

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\left({\int }_{t_0}^z{\int }_{t_0}^w\frac{\left|f(s,\vartheta )\right|\left|g(t,r)\right|}{\mu {\left[(s-t_0)(\vartheta -t_0)\right]}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left[(t-t_0)(r-t_0)\right]}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\Delta }_{1}t{\Delta }_{2}r\right){\Delta }_{1}s{\Delta }_2\vartheta \hfill \\ \le & R(\lambda ,\mu ){\left({\int }_{t_0}^x{\int }_{t_0}^y(x-s)(y-\vartheta ){\left|\frac{{\partial }^{2}f(s,\vartheta )}{{\Delta }_{1}s{\Delta }_2\vartheta }\right|}^{\lambda }{\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^z{\int }_{t_0}^w(z-t)(w-r){\left|\frac{{\partial }^{2}g(t,r)}{{\Delta }_{1}t{\Delta }_{2}r}\right|}^{\mu }{\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{1}{\mu }},\hfill \end{array}$$

by using the fact that $\sigma \left(\delta \right)\ge \delta ,$ one gets

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\left({\int }_{t_0}^z{\int }_{t_0}^w\frac{\left|f(s,\vartheta )\right|\left|g(t,r)\right|}{\mu {\left[(s-t_0)(\vartheta -t_0)\right]}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left[(t-t_0)(r-t_0)\right]}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\Delta }_{1}t{\Delta }_{2}r\right){\Delta }_{1}s{\Delta }_2\vartheta \hfill \\ \le & R(\lambda ,\mu ){\left({\int }_{t_0}^x{\int }_{t_0}^y(\sigma \left(x\right)-s)(\sigma \left(y\right)-t){\left|\frac{{\partial }^{2}f(s,\vartheta )}{{\Delta }_{1}s{\Delta }_2\vartheta }\right|}^{\lambda }{\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^z{\int }_{t_0}^w(\sigma \left(w\right)-t)(\sigma \left(z\right)-r){\left|\frac{{\partial }^{2}g(t,r)}{{\Delta }_{1}t{\Delta }_{2}r}\right|}^{\mu }{\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{1}{\mu }},\hfill \end{array}$$

which proves (47). This completes the proof. □

Remark 14.

Applying (27) on the right-hand side of (47) in Theorem 3 gives

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\left({\int }_{t_0}^z{\int }_{t_0}^w\frac{\left|f(s,\vartheta )\right|\left|g(t,r)\right|}{\mu {\left[(s-t_0)(\vartheta -t_0)\right]}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left[(t-t_0)(r-t_0)\right]}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\Delta }_{1}t{\Delta }_{2}r\right){\Delta }_{1}s{\Delta }_2\vartheta \hfill \\ \le & R(\lambda ,\mu )\left\{\frac{1}{\lambda }{\left({\int }_{t_0}^x{\int }_{t_0}^y(\sigma \left(x\right)-s)(\sigma \left(y\right)-\vartheta ){\left|\frac{{\partial }^{2}f(s,\vartheta )}{{\Delta }_{1}s{\Delta }_2\vartheta }\right|}^{\lambda }{\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{\lambda +\mu }{\lambda \mu }}\right.\hfill \\ & \left.+\frac{1}{\mu }{\left({\int }_{t_0}^z{\int }_{t_0}^w(\sigma \left(z\right)-t)(\sigma \left(w\right)-r){\left|\frac{{\partial }^{2}g(t,r)}{{\Delta }_{1}t{\Delta }_{2}r}\right|}^{\mu }{\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{\lambda +\mu }{\lambda \mu }}\right\}.\hfill \end{array}$$

(56)

Remark 15.

Letting $1/\lambda +1/\mu =1$ in (47), then we get

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\left({\int }_{t_0}^z{\int }_{t_0}^w\frac{\left|f(s,\vartheta )\right|\left|g(t,r)\right|}{\mu {\left[(s-t_0)(\vartheta -t_0)\right]}^{{}^{(\lambda -1)}}+\lambda {\left[(t-t_0)(r-t_0)\right]}^{{}^{(\mu -1)}}}{\Delta }_{1}t{\Delta }_{2}r\right){\Delta }_{1}s{\Delta }_2\vartheta \hfill \\ \le & R^{*}(\lambda ,\mu ){\left({\int }_{t_0}^x{\int }_{t_0}^y(\sigma \left(x\right)-s)(\sigma \left(y\right)-\vartheta ){\left|\frac{{\partial }^{2}f(s,\vartheta )}{{\Delta }_{1}s{\Delta }_2\vartheta }\right|}^{\lambda }{\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^z{\int }_{t_0}^w(\sigma \left(w\right)-t)(\sigma \left(z\right)-r){\left|\frac{{\partial }^{2}g(t,r)}{{\Delta }_{1}t{\Delta }_{2}r}\right|}^{\mu }{\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(57)

where

$$R^{*}(\lambda ,\mu )=\frac{1}{\lambda \mu }{\left[(x-t_0)(y-t_0)\right]}^{\frac{\lambda -1}{\lambda }}{\left[(w-t_0)(z-t_0)\right]}^{\frac{\mu -1}{\mu }},$$

which is exactly the time scale version of inequalities (8) and (10) in Theorems (3) and (4), respectively, due to B. G. Pachpatte [14].

Remark 16.

As a particular state of Theorem 3 if ${\mathbb{T}}_1={\mathbb{T}}_2=\mathbb{Z}$ and $t_0=0,$ then we have relations (8) and inequality (47) reduce to

$$\begin{array}{cc}& \sum _{s=1}^x\sum _{\vartheta =1}^y\left(\sum _{t=1}^z\sum _{r=1}^w\frac{\left|a_{s,\vartheta }\right|\left|b_{t,r}\right|}{\mu {\left(s\vartheta \right)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left(tr\right)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}}\right)\hfill \\ \le & R_0(\lambda ,\mu ){\left(\sum _{s=1}^x\sum _{\vartheta =1}^y(x-s+1)(y-\vartheta +1){\left|{\Delta }_2{\Delta }_1\left(a_{s,\vartheta }\right)\right|}^{\lambda }\right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left(\sum _{t=1}^z\sum _{r=1}^w(z-t+1)(w-r+1){\left|{\Delta }_2{\Delta }_1\left(b_{t,r}\right)\right|}^{\mu }\right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(58)

where the operators ${\Delta }_1\left(a_{s,\vartheta }\right)=a_{s+1,\vartheta }-a_{s,\vartheta },$ ${\Delta }_1\left(b_{t,r}\right)=b_{t+1,r}-b_{t,r},$ ${\Delta }_2{\Delta }_1\left(a_{s,\vartheta }\right)={\Delta }_2\left({\Delta }_1\left(a_{s,\vartheta }\right)\right)={\Delta }_1\left({\Delta }_2\left(a_{s,\vartheta }\right)\right),$ ${\Delta }_2{\Delta }_1\left(b_{t,r}\right)={\Delta }_2\left({\Delta }_1\left(b_{t,r}\right)\right)={\Delta }_1\left({\Delta }_2\left(b_{t,r}\right)\right)$ and

$$R_0(\lambda ,\mu )=\frac{1}{\lambda +\mu }{\left(xy\right)}^{\frac{\lambda -1}{\lambda }}{\left(zw\right)}^{\frac{\mu -1}{\mu }}.$$

It is merely a similar variant of the consequence disparity in [15] Theorem 2.3.

Remark 17.

As a particular state of Theorem 3, if ${\mathbb{T}}_1={\mathbb{T}}_2=\mathbb{R}$ and $t_0=0,$ we have relations (7) and inequality (47) reduce to

$$\begin{array}{cc}& {\int }_0^x{\int }_0^y\left({\int }_0^z{\int }_0^w\frac{\left|f(s,\vartheta )\right|\left|g(t,r)\right|}{\mu {\left(s\vartheta \right)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left(tr\right)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}}dtdr\right)dsd\vartheta \hfill \\ \le & R_0^{*}(\lambda ,\mu ){\left({\int }_0^x{\int }_0^y(x-s)(y-\vartheta ){\left|D_2{D}_{1}f(s,\vartheta )\right|}^{\lambda }dsd\vartheta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_0^z{\int }_0^w(z-t)(w-r){\left|D_2{D}_{1}g(t,r)\right|}^{\mu }dtdr\right)}^{\frac{1}{\mu }},\hfill \end{array}$$

where

$$R_0^{*}(\lambda ,\mu )=\frac{1}{{\lambda }_{+}\mu }{\left(xy\right)}^{\frac{\lambda -1}{\lambda }}{\left(zw\right)}^{\frac{\mu -1}{\mu }},$$

and $D_{1}u(s,$ $\vartheta )=\left(\partial /\partial s\right)u(s,$ $\vartheta ),$ $D_{2}u(s,$ $\vartheta )=\left(\partial /\partial \vartheta \right)u(s,$ $\vartheta ),$ $D_2{D}_{1}u(s,$ $\vartheta )=D_2{D}_{1}u(s,$ $\vartheta )=\left({\partial }^2/\partial s\partial \vartheta \right)u(s,$ $\vartheta )$, which is the same result inequality due to Young and Byung in [15] Theorem 2.4.

Theorem 4.

Let $f(s,$ $\vartheta )$ and $g(t,$ $r)$ with $f(s,$ $t_0)=g(t,$ $t_0)=0$ and $f(t_0,$ $\vartheta )=g(t_0,$ $r)=0,$ be as in Theorem 3 and $h(\xi ,$ $\eta )>0,$ $l(\sigma ,$ $\tau )>0$. Furthermore, assume that

$$H(s,\vartheta )={\int }_{t_0}^s{\int }_{t_0}^{\vartheta }\left|h(\xi ,\eta )\right|{\Delta }_1\xi {\Delta }_2\eta \mathit{and}L(t,r)={\int }_{t_0}^t{\int }_{t_0}^r\left|l(\sigma ,\tau )\right|{\Delta }_1\sigma {\Delta }_2\tau .$$

(59)

Then for $(s,$ $v)\in [t_0,$ ${x)}_{{\mathbb{T}}_1}\times [t_0,$ ${y)}_{{\mathbb{T}}_2}$ and $(t,$ $r)\in [t_0,$ ${z)}_{{\mathbb{T}}_1}\times [t_0,$ ${w)}_{{\mathbb{T}}_2},$ one gets

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\left({\int }_{t_0}^z{\int }_{t_0}^w\frac{\Phi \left(\left|f(s,\vartheta )\right|\right)\Psi \left(\left|g(t,r)\right|\right)}{\mu {\left[(s-t_0)(\vartheta -t_0)\right]}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left[(t-t_0)(r-t_0)\right]}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\Delta }_{1}t{\Delta }_{2}r\right){\Delta }_{1}s{\Delta }_2\vartheta \hfill \\ \le & S(\lambda ,\mu ){\left({\int }_{t_0}^x{\int }_{t_0}^y(\sigma \left(x\right)-s)(\sigma \left(y\right)-\vartheta ){\left(\left|h(s,\vartheta )\right|\Phi \left(\left|\frac{\frac{{\partial }^{2}f(s,\vartheta )}{{\Delta }_{1}s{\Delta }_2\vartheta }}{h(s,\vartheta )}\right|\right)\right)}^{\lambda }{\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^z{\int }_{t_0}^w(\sigma \left(z\right)-t)(\sigma \left(w\right)-r){\left(\left|l(t,r)\right|\Psi \left(\left|\frac{\frac{{\partial }^{2}g(t,r)}{{\Delta }_{1}t{\Delta }_{2}r}}{l(t,r)}\right|\right)\right)}^{\mu }{\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(60)

where

$$S(\lambda ,\mu )=\frac{1}{\lambda +\mu }{\left({\int }_{t_0}^x{\int }_{t_0}^y{\left(\frac{\Phi \left(H\right(s,\vartheta \left)\right)}{H(s,t)}\right)}^{\frac{\lambda }{\lambda -1}}{\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{\lambda -1}{\lambda }}{\left({\int }_{t_0}^z{\int }_{t_0}^w{\left(\frac{\Psi \left(L\right(t,r\left)\right)}{L(t,r)}\right)}^{\frac{\mu }{\mu -1}}{\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{\mu -1}{\mu }}.$$

Proof. 

Using the two identities (48) and (49) in the proof of Theorem 3 and the properties of the function $\Phi$ and utilize (12), we obtain

$$\begin{array}{ccc}\hfill \Phi \left(\left|f(s,\vartheta )\right|\right)& =& \Phi \left(\frac{H(s,\vartheta ){\int }_{t_0}^s{\int }_{t_0}^{\vartheta }\left|h(\xi ,\eta )\right|\left(\left|\frac{\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }}{h(\xi ,\eta )}\right|\right){\Delta }_1\xi {\Delta }_2\eta }{{\int }_{t_0}^s{\int }_{t_0}^{\vartheta }\left|h(\xi ,\eta )\right|{\Delta }_1\xi {\Delta }_2\eta }\right)\hfill \\ & \le & \Phi \left(H(s,\vartheta )\right)\Phi \left(\frac{{\int }_{t_0}^s{\int }_{t_0}^{\vartheta }\left|h(\xi ,\eta )\right|\left(\left|\frac{\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }}{h(\xi ,\eta )}\right|\right){\Delta }_1\xi {\Delta }_2\eta }{{\int }_{t_0}^s{\int }_{t_0}^{\vartheta }\left|h(\xi ,\eta )\right|{\Delta }_1\xi {\Delta }_2\eta }\right)\hfill \\ & \le & \frac{\Phi \left(H\right(s,\vartheta \left)\right)}{H(s,\vartheta )}{\int }_{t_0}^s{\int }_{t_0}^{\vartheta }\left|h(\xi ,\eta )\right|\Phi \left(\left|\frac{\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }}{h(\xi ,\eta )}\right|\right){\Delta }_1\xi {\Delta }_2\eta .\hfill \end{array}$$

(61)

Applying (10) with indices $\lambda$ and $\lambda /(\lambda -1)$ on the right-hand side of (61), we have

$$\begin{array}{ccc}\hfill \Phi \left(\left|f(s,\vartheta )\right|\right)& \le & {\left[(s-t_0)(\vartheta -t_0)\right]}^{\frac{\lambda -1}{\lambda }}\frac{\Phi \left(H\right(s,\vartheta \left)\right)}{H(s,\vartheta )}\hfill \\ & & \times {\left({\int }_{t_0}^s{\int }_{t_0}^{\vartheta }{\left(\left|h(\xi ,\eta )\right|\Phi \left(\left|\frac{\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }}{h(\xi ,\eta )}\right|\right)\right)}^{\lambda }{\Delta }_1\xi {\Delta }_2\eta \right)}^{\frac{1}{\lambda }}.\hfill \end{array}$$

(62)

Analogously,

$$\begin{array}{ccc}\hfill \Psi \left(\left|g(t,r)\right|\right)& \le & {\left[(t-t_0)(r-t_0)\right]}^{\frac{\mu -1}{\mu }}\frac{\Psi \left(L\right(t,r\left)\right)}{L(t,r)}\hfill \\ & & \times {\left({\int }_{t_0}^t{\int }_{t_0}^r{\left(\left|l(\sigma ,\tau )\right|\Psi \left(\left|\frac{\frac{{\partial }^{2}g(\sigma ,\tau )}{{\Delta }_1\sigma {\Delta }_2\tau }}{l(\sigma ,\tau )}\right|\right)\right)}^{\mu }{\Delta }_1\sigma {\Delta }_2\tau \right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(63)

Thus, form (62) and (63), it can be acquired that

$$\begin{array}{cc}& \Phi \left(\left|f(s,\vartheta )\right|\right)\Psi \left(\left|g(t,r)\right|\right)\hfill \\ \le & {\left[(s-t_0)(\vartheta -t_0)\right]}^{\frac{\lambda -1}{\lambda }}{\left[(t-t_0)(r-t_0)\right]}^{\frac{\mu -1}{\mu }}\hfill \\ & \times \frac{\Phi \left(H\right(s,\vartheta \left)\right)}{H(s,\vartheta )}{\left({\int }_{t_0}^s{\int }_{t_0}^{\vartheta }{\left(\left|h(\xi ,\eta )\right|\Phi \left(\left|\frac{\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }}{h(\xi ,\eta )}\right|\right)\right)}^{\lambda }{\Delta }_1\xi {\Delta }_2\eta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times \frac{\Psi \left(L\right(t,r\left)\right)}{L(t,r)}{\left({\int }_{t_0}^t{\int }_{t_0}^r{\left(\left|l(\sigma ,\tau )\right|\Psi \left(\left|\frac{\frac{{\partial }^{2}g(\sigma ,\tau )}{{\Delta }_1\sigma {\Delta }_2\tau }}{l(\sigma ,\tau )}\right|\right)\right)}^{\mu }{\Delta }_1\sigma {\Delta }_2\tau \right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(64)

Applying (22) on the term ${\left[(s-t_0)(\vartheta -t_0)\right]}^{(\lambda -1)/\lambda }$ and the term ${\left[(t-t_0)(r-t_0)\right]}^{(\mu -1)/\mu }$ gives

$$\begin{array}{cc}& \Phi \left(\left|f(s,\vartheta )\right|\right)\Psi \left(\left|g(t,r)\right|\right)\hfill \\ \le & \frac{\lambda \mu }{\lambda +\mu }\left(\frac{{\left[(s-t_0)(\vartheta -t_0)\right]}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}}{\lambda }+\frac{{\left[(t-t_0)(r-t_0)\right]}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}{\mu }\right)\hfill \\ & \times \frac{\Phi \left(H\right(s,\vartheta \left)\right)}{H(s,\vartheta )}{\left({\int }_{t_0}^s{\int }_{t_0}^{\vartheta }{\left(\left|h(\xi ,\eta )\right|\Phi \left(\left|\frac{\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }}{h(\xi ,\eta )}\right|\right)\right)}^{\lambda }{\Delta }_1\xi {\Delta }_2\eta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times \frac{\Psi \left(L\right(t,r\left)\right)}{L(t,r)}{\left({\int }_{t_0}^t{\int }_{t_0}^r{\left(\left|l(\sigma ,\tau )\right|\Psi \left(\left|\frac{\frac{{\partial }^{2}g(\sigma ,\tau )}{{\Delta }_1\sigma {\Delta }_2\tau }}{l(\sigma ,\tau )}\right|\right)\right)}^{\mu }{\Delta }_1\sigma {\Delta }_2\tau \right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(65)

From (65), we observe that

$$\begin{array}{cc}& \frac{\Phi \left(\left|f(s,\vartheta )\right|\right)\Psi \left(\left|g(t,r)\right|\right)}{\mu {\left[(s-t_0)(\vartheta -t_0)\right]}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left[(t-t_0)(r-t_0)\right]}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\hfill \\ \le & \frac{1}{\lambda +\mu }\frac{\Phi \left(H\right(s,\vartheta \left)\right)}{H(s,\vartheta )}{\left({\int }_{t_0}^s{\int }_{t_0}^{\vartheta }{\left(\left|h(\xi ,\eta )\right|\Phi \left(\left|\frac{\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }}{h(\xi ,\eta )}\right|\right)\right)}^{\lambda }{\Delta }_1\xi {\Delta }_2\eta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times \frac{\Psi \left(L\right(t,r\left)\right)}{L(t,r)}{\left({\int }_{t_0}^t{\int }_{t_0}^r{\left(\left|l(\sigma ,\tau )\right|\Psi \left(\left|\frac{\frac{{\partial }^{2}g(\sigma ,\tau )}{{\Delta }_1\sigma {\Delta }_2\tau }}{l(\sigma ,\tau )}\right|\right)\right)}^{\mu }{\Delta }_1\sigma {\Delta }_2\tau \right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(66)

Integrating both sides of (66) and using (10) again with respect to $\lambda ,$ $\lambda /(\lambda -1)$ and $\mu ,$ $\mu /(\mu -1),$ respectively, we may write

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\left({\int }_{t_0}^z{\int }_{t_0}^w\frac{\Phi \left(\left|f(s,\vartheta )\right|\right)\Psi \left(\left|g(t,r)\right|\right)}{\mu {\left[(s-t_0)(\vartheta -t_0)\right]}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left[(t-t_0)(r-t_0)\right]}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\Delta }_{1}t{\Delta }_{2}r\right){\Delta }_{1}s{\Delta }_2\vartheta \hfill \\ \le & \frac{1}{\lambda +\mu }{\left({\int }_{t_0}^x{\int }_{t_0}^y{\left(\frac{\Phi \left(H\right(s,\vartheta \left)\right)}{H(s,\vartheta )}\right)}^{\frac{\lambda }{\lambda -1}}{\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{\lambda -1}{\lambda }}{\left({\int }_{t_0}^z{\int }_{t_0}^w{\left(\frac{\Psi \left(L\right(t,r\left)\right)}{L(t,r)}\right)}^{\frac{\mu }{\mu -1}}{\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{\mu -1}{\mu }}\hfill \\ & \times {\left({\int }_{t_0}^x{\int }_{t_0}^y\left({\int }_{t_0}^s{\int }_{t_0}^{\vartheta }{\left(\left|h(\xi ,\eta )\right|\Phi \left(\left|\frac{\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }}{h(\xi ,\eta )}\right|\right)\right)}^{\lambda }{\Delta }_1\xi {\Delta }_2\eta \right){\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^z{\int }_{t_0}^w\left({\int }_{t_0}^t{\int }_{t_0}^r{\left(\left|l(\sigma ,\tau )\right|\Psi \left(\left|\frac{\frac{{\partial }^{2}g(\sigma ,\tau )}{{\Delta }_1\sigma {\Delta }_2\tau }}{l(\sigma ,\tau )}\right|\right)\right)}^{\mu }{\Delta }_1\xi {\Delta }_2\eta \right){\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{1}{\mu }}\hfill \\ =& S(\lambda ,\mu ){\left({\int }_{t_0}^x{\int }_{t_0}^y\left({\int }_{t_0}^s{\int }_{t_0}^{\vartheta }{\left(\left|h(\xi ,\eta )\right|\Phi \left(\left|\frac{\frac{{\partial }^{2}f(\xi ,\eta )}{{\Delta }_1\xi {\Delta }_2\eta }}{h(\xi ,\eta )}\right|\right)\right)}^{\lambda }{\Delta }_1\xi {\Delta }_2\eta \right){\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^z{\int }_{t_0}^w\left({\int }_{t_0}^t{\int }_{t_0}^r{\left(\left|l(\sigma ,\tau )\right|\Psi \left(\left|\frac{\frac{{\partial }^{2}g(\sigma ,\tau )}{{\Delta }_1\sigma {\Delta }_2\tau }}{l(\sigma ,\tau )}\right|\right)\right)}^{\mu }{\Delta }_1\sigma {\Delta }_2\tau \right){\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{1}{\mu }}.\hfill \end{array}$$

(67)

Applying Fubini’s theorem on (67) and using $\sigma \left(\delta \right)\ge \delta ,$ we get

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\left({\int }_{t_0}^z{\int }_{t_0}^w\frac{\Phi \left(\left|f(s,\vartheta )\right|\right)\Psi \left(\left|g(t,r)\right|\right)}{\mu {\left[(s-t_0)(\vartheta -t_0)\right]}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left[(t-t_0)(r-t_0)\right]}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\Delta }_{1}t{\Delta }_{2}r\right){\Delta }_{1}s{\Delta }_2\vartheta \hfill \\ \le & S(\lambda ,\mu ){\left({\int }_{t_0}^x{\int }_{t_0}^y(x-s)(y-\vartheta ){\left(\left|h(s,\vartheta )\right|\Phi \left(\left|\frac{\frac{{\partial }^{2}f(s,\vartheta )}{{\Delta }_{1}s{\Delta }_2\vartheta }}{h(s,t)}\right|\right)\right)}^{\lambda }{\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^z{\int }_{t_0}^w(z-t)(w-r){\left(\left|\left(l\right(t,r)\right|\Psi \left(\left|\frac{\frac{{\partial }^{2}g(t,r)}{{\Delta }_{1}t{\Delta }_{2}r}}{l(t,r)}\right|\right)\right)}^{\mu }{\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{1}{\mu }}\hfill \\ \le & S(\lambda ,\mu ){\left({\int }_{t_0}^x{\int }_{t_0}^y(\sigma \left(x\right)-s)(\sigma \left(y\right)-\vartheta ){\left(\left|h(s,\vartheta )\right|\Phi \left(\left|\frac{\frac{{\partial }^{2}f(s,\vartheta )}{{\Delta }_{1}s{\Delta }_2\vartheta }}{h(s,\vartheta )}\right|\right)\right)}^{\lambda }{\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^z{\int }_{t_0}^w(\sigma \left(z\right)-t)(\sigma \left(w\right)-r){\left(\left|\left(l\right(t,r)\right|\Psi \left(\left|\frac{\frac{{\partial }^{2}g(t,r)}{{\Delta }_{1}t{\Delta }_{2}r}}{l(t,r)}\right|\right)\right)}^{\mu }{\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{1}{\mu }},\hfill \end{array}$$

which is (60). This completes the proof. □

Remark 18.

By applying (27) on the right-hand side of (60) in Theorem 4, then

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\left({\int }_{t_0}^z{\int }_{t_0}^w\frac{\Phi \left(\left|f(s,\vartheta )\right|\right)\Psi \left(\left|g(t,r)\right|\right)}{\mu {\left[(s-t_0)(t-t_0)\right]}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left[(t-t_0)(r-t_0)\right]}^{{}^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\Delta }_{1}t{\Delta }_{2}r\right){\Delta }_{1}s{\Delta }_{2}t\hfill \\ \le & S(\lambda ,\mu )\left\{\frac{1}{\lambda }{\left({\int }_{t_0}^x{\int }_{t_0}^y(\sigma \left(x\right)-s)(\sigma \left(y\right)-\vartheta ){\left(\left|h(s,\vartheta )\right|\Phi \left(\left|\frac{\frac{{\partial }^{2}f(s,\vartheta )}{{\Delta }_{1}s{\Delta }_{2}t}}{h(s,\vartheta )}\right|\right)\right)}^{\lambda }{\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{\lambda +\mu }{\lambda \mu }}\right.\hfill \\ & \left.+\frac{1}{\mu }{\left({\int }_{t_0}^z{\int }_{t_0}^w(\sigma \left(z\right)-t)(\sigma \left(w\right)-r){\left(\left|\left(l\right(t,r)\right|\Psi \left(\left|\frac{\frac{{\partial }^{2}g(t,r)}{{\Delta }_{1}t{\Delta }_{2}r}}{l(t,r)}\right|\right)\right)}^{\mu }{\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{\lambda +\mu }{\lambda \mu }}\right\}.\hfill \end{array}$$

Remark 19.

Letting $1/\lambda +1/\mu =1$ in (60), then we get

$$\begin{array}{cc}& {\int }_{t_0}^x{\int }_{t_0}^y\left({\int }_{t_0}^z{\int }_{t_0}^w\frac{\Phi \left(\left|f(s,\vartheta )\right|\right)\Psi \left(\left|g(t,r)\right|\right)}{\mu {\left[(s-t_0)(\vartheta -t_0)\right]}^{{}^{(\lambda -1)}}+\lambda {\left[(t-t_0)(r-t_0)\right]}^{{}^{(\mu -1)}}}{\Delta }_{1}t{\Delta }_{2}r\right){\Delta }_{1}s{\Delta }_2\vartheta \hfill \\ \le & S^{*}(\lambda ,\mu ){\left({\int }_{t_0}^x{\int }_{t_0}^y(\sigma \left(x\right)-s)(\sigma \left(y\right)-\vartheta ){\left(\left|h(s,\vartheta )\right|\Phi \left(\left|\frac{\frac{{\partial }^{2}f(s,\vartheta )}{{\Delta }_{1}s{\Delta }_2\vartheta }}{h(s,\vartheta )}\right|\right)\right)}^{\lambda }{\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_{t_0}^z{\int }_{t_0}^w(\sigma \left(z\right)-t)(\sigma \left(w\right)-r){\left(\left|\left(l\right(t,r)\right|\Psi \left(\left|\frac{\frac{{\partial }^{2}g(t,r)}{{\Delta }_{1}t{\Delta }_{2}r}}{l(t,r)}\right|\right)\right)}^{\mu }{\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{1}{\mu }},\hfill \end{array}$$

where

$$S^{*}(\lambda ,\mu )=\frac{1}{\lambda \mu }{\left({\int }_{t_0}^x{\int }_{t_0}^y{\left(\frac{\Phi \left(H\right(s,\vartheta \left)\right)}{H(s,\vartheta )}\right)}^{\frac{\lambda }{\lambda -1}}{\Delta }_{1}s{\Delta }_2\vartheta \right)}^{\frac{\lambda -1}{\lambda }}{\left({\int }_{t_0}^z{\int }_{t_0}^w{\left(\frac{\Psi \left(L\right(t,r\left)\right)}{L(t,r)}\right)}^{\frac{\mu }{\mu -1}}{\Delta }_{1}t{\Delta }_{2}r\right)}^{\frac{\mu -1}{\mu }}.$$

Remark 20.

As a particular state of Theorem 4 if ${\mathbb{T}}_1={\mathbb{T}}_2=\mathbb{Z}$ and $t_0=0,$ then we have relation (8) and inequality (60) reduce to

$$\begin{array}{cc}& \sum _{s=1}^x\sum _{\vartheta =1}^y\left(\sum _{t=1}^z\sum _{r=1}^w\frac{\Phi \left(\left|a_{s,\vartheta }\right|\right)\Psi \left(\left|b_{t,r}\right|\right)}{\mu {\left(s\vartheta \right)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left(tr\right)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}}\right)\hfill \\ \le & S_0(\lambda ,\mu ){\left(\sum _{s=1}^x\sum _{\vartheta =1}^y(x-s+1)(y-t+1){\left(\left|h(s,\vartheta )\right|\Phi \left(\left|\frac{{\Delta }_2{\Delta }_1\left(a_{s,\vartheta }\right)}{h(s,\vartheta )}\right|\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left(\sum _{t=1}^z\sum _{r=1}^w(z-t+1)(w-r+1){\left(\left|\left(l\right(t,r)\right|\Psi \left(\left|\frac{{\Delta }_2{\Delta }_1{b}_{t,r}}{l(t,r)}\right|\right)\right)}^{\mu }\right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(68)

where the operators ${\Delta }_1\left(a_{s,\vartheta }\right)=a_{s+1,\vartheta }-a_{s,\vartheta },$ ${\Delta }_1\left(b_{t,r}\right)=b_{t+1,r}-b_{t,r},$ ${\Delta }_2{\Delta }_1\left(a_{s,\vartheta }\right)={\Delta }_2\left({\Delta }_1\left(a_{s,\vartheta }\right)\right)={\Delta }_1\left({\Delta }_2\left(a_{s,\vartheta }\right)\right),$ ${\Delta }_2{\Delta }_1{b}_{t,r}={\Delta }_2\left({\Delta }_1\left(b_{t,r}\right)\right)={\Delta }_1\left({\Delta }_2\left(b_{t,r}\right)\right)$ and

$$S_0(\lambda ,\mu )=\frac{1}{\lambda +\mu }{\left(\sum _{s=1}^x\sum _{\vartheta =1}^y{\left(\frac{\Phi \left(H_{s,\vartheta }\right)}{H_{s,\vartheta }}\right)}^{\frac{\lambda }{\lambda -1}}\right)}^{\frac{\lambda -1}{\lambda }}{\left(\sum _{t=1}^z\sum _{r=1}^w{\left(\frac{\Psi \left(L_{t,r}\right)}{L_{t,r}}\right)}^{\frac{\mu }{\mu -1}}\right)}^{\frac{\mu -1}{\mu }}.$$

Remark 21.

As a particular state of Theorem 4 if ${\mathbb{T}}_1={\mathbb{T}}_2=\mathbb{R}$ and $t_0=0,$ we have relations (7) and inequality (60) reduce to

$$\begin{array}{cc}& {\int }_0^x{\int }_0^y\left({\int }_0^z{\int }_0^w\frac{\Phi \left(\left|f(s,\vartheta )\right|\right)\Psi \left(\left|g(t,r)\right|\right)}{\mu {\left(s\vartheta \right)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda {\left(tr\right)}^{{}^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}}dtdr\right)dsd\vartheta \hfill \\ \le & S_0^{*}(\lambda ,\mu ){\left({\int }_0^x{\int }_0^y(x-s)(y-\vartheta ){\left(\left|h(s,\vartheta )\right|\Phi \left(\left|\frac{D_2{D}_{1}f(s,\vartheta )}{h(s,\vartheta )}\right|\right)\right)}^{\lambda }dsd\vartheta \right)}^{\frac{1}{\lambda }}\hfill \\ & \times {\left({\int }_0^z{\int }_0^w(z-t)(w-r){\left(\left|\left(l\right(t,r)\right|\Psi \left(\left|\frac{D_2{D}_{1}g(t,r)}{l(t,r)}\right|\right)\right)}^{\mu }dtdr\right)}^{\frac{1}{\mu }},\hfill \end{array}$$

(69)

where

$$S_0^{*}(\lambda ,\mu )=\frac{1}{\lambda +\mu }{\left({\int }_0^x{\int }_0^y{\left(\frac{\Phi \left(H\right(s,\vartheta \left)\right)}{H(s,\vartheta )}\right)}^{\frac{\lambda }{\lambda -1}}dsd\vartheta \right)}^{\frac{\lambda -1}{\lambda }}{\left({\int }_0^z{\int }_0^w{\left(\frac{\Psi \left(L\right(t,r\left)\right)}{L(t,r)}\right)}^{\frac{\mu }{\mu -1}}dtdr\right)}^{\frac{\mu -1}{\mu }},$$

and $D_{1}u(s,$ $\vartheta )=\left(\partial /\partial s\right)u(s,$ $\vartheta ),$ $D_{2}u(s,$ $\vartheta )=\left(\partial /\partial t\right)u(s,$ $\vartheta ),$ $D_2{D}_{1}u(s,$ $\vartheta )=D_2{D}_{1}u(s,$ $\vartheta )=\left({\partial }^2/\partial s\partial \vartheta \right)u(s,$ $\vartheta )$.

3. Conclusions

In the context of this article, we presented generalizations of symmetric Hilbert-type inequalities on time scales. Our consequences are considered in rather general forms and contain several special integral and discrete of symmetric inequalities. The technique is based on the applications of well-known inequalities and new tools from time scale calculus, which is used in various problems involving symmetry for Hilbert-type inequalities. For future work, we can present such inequalities by using Riemann–Liouville type fractional integrals and fractional derivatives on time scales, which has many applications of symmetric and asymmetric Hilbert-type inequalities. It will also be very interesting to present such inequalities on quantum calculus.