Dynamic Hilbert-Type Inequalities with Fenchel-Legendre Transform

Abstract

Our work is based on the multiple inequalities illustrated in 2020 by Hamiaz and Abuelela. With the help of a Fenchel-Legendre transform, which is used in various problems involving symmetry, we generalize a number of those inequalities to a general time scale. Besides that, in order to get new results as special cases, we will extend our results to continuous and discrete calculus.

1. Introduction

In 2020, Hamiaz and Abuelela [1] have studied the following discrete inequalities:

Theorem 1.

Suppose $q,$ $p⩾1,$ $\alpha ⩾\beta ⩾\frac{1}{2}$ and ${\left(b_m\right)}_m\ge 0$ ${\left(a_n\right)}_n\ge 0$ are sequences of real numbers. Define $A_n={\sum }_{s=1}^n{a}_s,$ $B_m={\sum }_{t=1}^m{b}_t.$ Then

$$\begin{array}{c}\hfill \sum _{n=1}^k\sum _{m=1}^r{\displaystyle \frac{A_n^{2p}{B}_m^{2q}}{h\left(n\right)+h^{*}\left(m\right)}}⩽C_1^{*}(p,q)\left(\sum _{n=1}^k(k-n+1){\left(a_n{A}_n^{p-1}\right)}^2\right)\\ \hfill \times \left(\sum _{m=1}^r(r-m+1){\left(b_m{B}_m^{q-1}\right)}^2\right)\end{array}$$

and

$$\begin{array}{cc}\hfill \sum _{n=1}^k\sum _{m=1}^r{\displaystyle \frac{A_n^p{B}_m^q}{{\left({|h\left(n\right)|}^{\frac{1}{2\beta }}+{|h^{*}\left(m\right)|}^{\frac{1}{2\beta }}\right)}^{\alpha }}}& ⩽\sum _{n=1}^k\sum _{m=1}^r{\displaystyle \frac{A_n^p{B}_m^q}{\sqrt{h\left(n\right)+h^{*}\left(m\right)}}}\hfill \\ & ⩽C_2^{*}(p,q,k,r){\left(\sum _{n=1}^k(k-n+1){\left(a_n{A}_n^{p-1}\right)}^2\right)}^{\frac{1}{2}}\hfill \\ & \times {\left(\sum _{m=1}^r(r-m+1){\left(b_m{B}_m^{q-1}\right)}^2\right)}^{\frac{1}{2}}\hfill \end{array}$$

unless $\left(a_n\right)$ or $\left(b_m\right)$ is null, where

$$C_1^{*}(p,q)={\left(pq\right)}^2\mathit{and}{C}_2^{*}(p,q,r,k)=pq\sqrt{kr}.$$

Hilger [2] suggested time scales theory to unify discrete and continuous analysis. More Hilbert-type inequalities and other types can be seen in [1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36], see also [37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53]. For more details on time scales calculus see [54].

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

(i) 

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

$$\sigma \left(t\right)=t,\mu \left(t\right)=0,f^{\Delta }\left(t\right)=f^{\prime }\left(t\right),{\int }_a^{b}f\left(t\right)\Delta t={\int }_a^{b}f\left(t\right)dt.$$

(1)

(ii) 

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

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

(2)

Next is Hölder’s and Jensen’s inequality:

Lemma 1

([19]). Let $a,b\in \mathbb{T}$ and f, $g\in C_{rd}({[a,b]}_{\mathbb{T}},[0,\infty ))$. If p, $q>1$ with $\frac{1}{p}+\frac{1}{q}=1$, then

$${\int }_a^{b}f\left(t\right)g\left(t\right)\Delta t\le {\left[{\int }_a^b{f}^p\left(t\right)\Delta t\right]}^{\frac{1}{p}}{\left[{\int }_a^b{g}^q\left(t\right)\Delta t\right]}^{\frac{1}{q}}.$$

Lemma 2

( [19]). Let a, $b\in \mathbb{T}$ and $\breve{c}$, $\breve{d}\in \mathbb{R}$. Assume that $g\in C_{rd}\left({[a,b]}_{\mathbb{T}},[\breve{c},\breve{d}]\right)$ and $r\in C_{rd}\left({[a,b]}_{\mathbb{T}},\mathbb{R}\right)$ are nonnegative with ${\int }_a^{b}r\left(t\right)\Delta t>0$. If $\breve{\mathrm{\Phi }}\in C_{rd}\left((\breve{c},\breve{d}),\mathbb{R}\right)$ be a convex function, then

$$\breve{\mathrm{\Phi }}\left({\displaystyle \frac{{\int }_a^{b}g\left(t\right)r\left(t\right)\Delta t}{{\int }_a^{b}r\left(t\right)\Delta t}}\right)⩽{\displaystyle \frac{{\int }_a^{b}r\left(t\right)\breve{\mathrm{\Phi }}\left(g\left(t\right)\right)\Delta t}{{\int }_a^{b}r\left(t\right)\Delta t}}.$$

Now, we present the Fenchel-Legendre transform and refer, for example, to [11,12,13], for more details.

Definition 1.

Assuming $h:{\mathbb{R}}^n⟶\mathbb{R}\cup \{+\infty \}$ is a function: $h\ne +\infty$ i.e., $Dom\left(h\right)=\{x\in {\mathbb{R}}^n,|h\left(x\right)<\infty \}\ne \varnothing$. Then the Fenchel-Legendre transform is defined as:

$$h^{*}:{\mathbb{R}}^n⟶\mathbb{R}\cup \{+\infty \},y⟶h^{*}\left(y\right)=sup\{<y,x>-h\left(x\right),x\in Dom\left(h\right)\}$$

(3)

where $<.,.>$ is the scalar product on ${\mathbb{R}}^n$. The mapping $h⟶h^{*}$ is often be called the conjugate operation.

The domain of $h^{*}$ is the set of slopes of all the affine functions minorizing the function h over ${\mathbb{R}}^n$. An equivalent formula for (3) is introduced as follows:

Corollary 1.

Assuming $h:{\mathbb{R}}^n⟶\mathbb{R}$ is differentiable, strictly convex and 1-coercive function. Then

$$h^{*}\left(y\right)=<y,{(\nabla h)}^{-1}\left(y\right)>-h\left({(\nabla h)}^{-1}\left(y\right)\right),$$

(4)

∀$y\in Dom\left(h^{*}\right)$, where $<.,.>$ denotes the scalar product on ${\mathbb{R}}^n$.

Lemma 3

( [13]). Let h be a function and $h^{*}$ its Fenchel-Legendre transform. Then

$$<x,y>⩽h\left(x\right)+h^{*}\left(y\right),$$

(5)

for all $x\in Dom\left(h\right)$, and $y\in Dom\left(h^{*}\right).$

In addition, we will use the following definition and lemma as we will see in the proof of our results:

Definition 2.

The function $\breve{\mathrm{\Phi }}$ is said to be a submultiplicative on $[0,\infty )$ if

$$\breve{\mathrm{\Phi }}\left(xy\right)⩽\breve{\mathrm{\Phi }}\left(x\right)\breve{\mathrm{\Phi }}\left(y\right),\mathrm{for}\mathrm{all}x,y⩾0.$$

(6)

Lemma 4

([20]). Assuming $\mathbb{T}$ is a time scale with $x,a\in \mathbb{T}$ such that $x⩾a.$ If $f⩾0$ and $\tilde{\alpha }⩾1,$ then

$${\left({\int }_a^{\sigma \left(x\right)}f\left(\check{\tau }\right)\Delta \check{\tau }\right)}^{\tilde{\alpha }}⩽\tilde{\alpha }{\int }_a^{\sigma \left(x\right)}f\left(\eta \right){\left({\int }_a^{\sigma \left(\eta \right)}f\left(\check{\tau }\right)\Delta \check{\tau }\right)}^{\tilde{\alpha }-1}\Delta \eta .$$

(7)

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

Lemma 5

(Fubini’s Theorem, see [55]). Assume that $(X,{\sum }_1,{\mu }_{\Delta })$ and $(Y,{\sum }_2,{\nu }_{\Delta })$ are two finite-dimensional time scales measure spaces. Moreover, suppose that $f:X\times Y\to \mathbb{R}$ is a delta integrable function and define the functions

$${\widehat{\pi }}_1\left(y\right)={\int }_{X}f(x,y)d{\mu }_{\Delta }\left(x\right),y\in Y,$$

and

$${\widehat{\pi }}_2\left(x\right)={\int }_{X}f(x,y)d{\nu }_{\Delta }\left(y\right),x\in X.$$

Then ${\widehat{\pi }}_1$ is delta integrable on Y and ${\widehat{\pi }}_2$ is delta integrable on X and

$${\int }_{X}d{\mu }_{\Delta }\left(x\right){\int }_{Y}f(x,y)d{\nu }_{\Delta }\left(y\right)={\int }_{Y}d{\nu }_{\Delta }\left(y\right){\int }_{X}f(x,y)d{\mu }_{\Delta }\left(x\right).$$

In this manuscript, by using Fubini’s theorem and the Fenchel-Legendre transform, which is used in various problems involving symmetry, we extend the discrete results proved in [1] on time scales. We start from the inequalities treated in the Theorem 1. Our results can be applied to give more general forms of some previously proved inequalities through substituting h and $h^{*}$ by suitable functions as we will see in the following two sections.

The following section contains our main results.

2. Main Results

We start by establishing the following useful inequality:

Lemma 6.

Assume x and $y\in \mathbb{R}$ such that $x+y⩾1,$ then for $\gamma >0,$ and $\alpha ⩾\beta ⩾\frac{1}{2}$, we get

$${(x+y)}^{\frac{1}{\gamma }}⩽{\left({|x|}^{\frac{1}{2\beta }}+{|y|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{\gamma }}.$$

(8)

Proof. 

For $x+y⩾1$ and $\frac{\alpha }{\beta }⩾1,$ we have

$$\begin{array}{c}\hfill {(x+y)}^{\frac{1}{2}}⩽{\left[{(x+y)}^{\frac{1}{2}}\right]}^{\frac{\alpha }{\beta }}={\left[{(x+y)}^{\frac{1}{2\beta }}\right]}^{\alpha }⩽{\left[(|x|+{|y|)}^{\frac{1}{2\beta }}\right]}^{\alpha }.\end{array}$$

(9)

From ${\left(|x|+|y|\right)}^{\frac{1}{n}}⩽{|x|}^{\frac{1}{n}}+{|y|}^{\frac{1}{n}}$, for all $n⩾1$. Thus, from (9), and since $2\beta ⩾1,$ we obtain:

$$\begin{array}{c}\hfill {(x+y)}^{\frac{1}{2}}⩽{\left[(|x|+{|y|)}^{\frac{1}{2\beta }}\right]}^{\alpha }⩽{\left[{|x|}^{\frac{1}{2\beta }}+{|y|}^{\frac{1}{2\beta }}\right]}^{\alpha }.\end{array}$$

(10)

Now, since $\gamma >0$, by taking the power $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\gamma $}\right.$ for both sides of (10), we get:

$$\begin{array}{c}\hfill {(x+y)}^{\frac{1}{\gamma }}⩽{\left({|x|}^{\frac{1}{2\beta }}+{|y|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{\gamma }}.\end{array}$$

This proves our claim. ☐

In the next theorems, we will let $p>1$, $q>1$ and $\frac{1}{q}+\frac{1}{p}=1.$

Theorem 2.

Let $\mathbb{T}$ be a time scale with $L⩾1,$ $K⩾1$ and $s,$ $t,$ $t_0,$ $x,$ y $\in \mathbb{T}.$ Assume $a\left(\check{\tau }\right)\ge 0$ and $b\left(\check{\tau }\right)\ge 0$ are right-dense continuous functions on the time scales intervals ${[t_0,x]}_{\mathbb{T}}$ and ${[t_0,y]}_{\mathbb{T}}$ respectively and define

$$A\left(s\right):={\int }_{t_0}^{s}a\left(\check{\tau }\right)\Delta \check{\tau },andB\left(t\right):={\int }_{t_0}^{t}b\left(\check{\tau }\right)\Delta \check{\tau },$$

then for $\sigma \left(s\right)\in {[t_0,x]}_{\mathbb{T}}$ and $\sigma \left(t\right)\in {[t_0,y]}_{\mathbb{T}},$ we have that

$$\begin{array}{c}{\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{A^{qK}\left(\sigma \left(s\right)\right)B^{qL}\left(\sigma \left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2q\alpha }{p}}}}\Delta s\Delta t\hfill \\ ⩽C_1(L,K,q)\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(a\left(s\right)A^{K-1}(\sigma \left(s\right)\right)}^q\Delta s\right)\hfill \\ \times \left({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){\left(b\left(t\right)B^{L-1}(\sigma \left(t\right)\right)}^q\Delta t\right)\hfill \end{array}$$

(11)

and

$$\begin{array}{c}{\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)}{{\left({|h(\sigma \left(s\right)-t_0)|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ ⩽C_2(L,K,p){\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(A^{K-1}\left(\sigma \left(s\right)\right)a\left(s\right)\right)}^q\Delta s\right)}^{\frac{1}{q}}\hfill \\ \times ({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){{\left(B^{L-1}\left(\sigma \left(t\right)\right)b\left(t\right)\right)}^q\Delta t)}^{\frac{1}{q}}\hfill \end{array}$$

(12)

where $C_1(L,K,q)={\left(KL\right)}^q\mathit{and}{C}_2(L,K,p)=KL{(x-t_0)}^{\frac{1}{p}}{(y-t_0)}^{\frac{1}{p}}.$

Proof. 

By using the inequality (7), we obtain

$$A^K\left(\sigma \left(s\right)\right)⩽K{\int }_{t_0}^{\sigma \left(s\right)}a\left(\eta \right)A^{K-1}(\sigma \left(\eta \right)\Delta \eta ,$$

(13)

$$B^L\left(\sigma \left(t\right)\right)⩽L{\int }_{t_0}^{\sigma \left(t\right)}b\left(\eta \right)B^{L-1}\left(\sigma \left(\eta \right)\right)\Delta \eta .$$

(14)

We use Lemma 1. Then from (13), we get

$$A^K\left(\sigma \left(s\right)\right)⩽K{(\sigma \left(s\right)-t_0)}^{\frac{1}{p}}{\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(a\left(\eta \right)A^{K-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)}^{\frac{1}{q}}.$$

(15)

We use Lemma 1. Then from (14), we also have

$$B^L\left(\sigma \left(t\right)\right)⩽L{(\sigma \left(t\right)-t_0)}^{\frac{1}{p}}{\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(b\left(\eta \right)B^{L-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)}^{\frac{1}{q}}.$$

(16)

From (15) and (16), we get

$$\begin{array}{c}\hfill A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)⩽KL{(\sigma \left(s\right)-t_0)}^{\frac{1}{p}}{(\sigma \left(t\right)-t_0)}^{\frac{1}{p}}\\ \hfill \times {\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(a\left(\eta \right)A^{K-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)}^{\frac{1}{q}}\\ \hfill \times {\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(b\left(\eta \right)B^{L-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)}^{\frac{1}{q}}.\end{array}$$

(17)

From inequality (17), we have

$$\begin{array}{c}\hfill A^{qK}\left(\sigma \left(s\right)\right)B^{qL}\left(\sigma \left(t\right)\right)⩽{\left(KL\right)}^q{(\sigma \left(s\right)-t_0)}^{\frac{q}{p}}{(\sigma \left(t\right)-t_0)}^{\frac{q}{p}}\\ \hfill \times \left({\int }_{t_0}^{\sigma \left(s\right)}{\left(a\left(\eta \right)A^{K-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)\\ \hfill \times \left({\int }_{t_0}^{\sigma \left(t\right)}{\left(b\left(\eta \right)B^{L-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right).\end{array}$$

(18)

Using Lemma 3 in (17) and (18) gives

$$\begin{array}{c}\hfill A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)⩽KL{\left(h(\sigma \left(s\right)-t_0)+h^{*}(\sigma \left(t\right)-t_0)\right)}^{\frac{1}{p}}\\ \hfill \times {\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(a\left(\eta \right)A^{K-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)}^{\frac{1}{q}}\\ \hfill \times {\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(b\left(\eta \right)B^{L-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)}^{\frac{1}{q}},\end{array}$$

(19)

$$\begin{array}{c}\hfill A^{qK}\left(\sigma \left(s\right)\right)B^{qL}\left(\sigma \left(t\right)\right)⩽{\left(KL\right)}^q{\left(h(\sigma \left(s\right)-t_0)+h^{*}(\sigma \left(t\right)-t_0)\right)}^{\frac{q}{p}}\\ \hfill \times \left({\int }_{t_0}^{\sigma \left(s\right)}{\left(a\left(\eta \right)A^{K-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)\\ \hfill \times \left({\int }_{t_0}^{\sigma \left(t\right)}{\left(b\left(\eta \right)B^{L-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right).\end{array}$$

(20)

Using Lemma 6 in (19) and (20) gives

$$\begin{array}{c}\hfill A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)⩽KL{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}\\ \hfill \times {\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(a\left(\eta \right)A^{K-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)}^{\frac{1}{q}}\\ \hfill \times {\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(b\left(\eta \right)B^{L-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)}^{\frac{1}{q}},\end{array}$$

(21)

$$\begin{array}{c}\hfill A^{qK}\left(\sigma \left(s\right)\right)B^{qL}\left(\sigma \left(t\right)\right)⩽{\left(KL\right)}^q{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2q\alpha }{p}}\\ \hfill \times \left({\int }_{t_0}^{\sigma \left(s\right)}{\left(a\left(\eta \right)A^{K-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)\\ \hfill \times \left({\int }_{t_0}^{\sigma \left(t\right)}{\left(b\left(\eta \right)B^{L-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right).\end{array}$$

(22)

Dividing both sides of (21) and (22) by ${\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}$ and ${\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2q\alpha }{p}}$ respectively, we get that

$$\begin{array}{c}\hfill \frac{A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}⩽KL{\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(a\left(\eta \right)A^{K-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)}^{\frac{1}{q}}\\ \hfill \times {\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(b\left(\eta \right)B^{L-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)}^{\frac{1}{q}},\end{array}$$

(23)

$$\begin{array}{c}\hfill \frac{A^{qK}\left(\sigma \left(s\right)\right)B^{qL}\left(\sigma \left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2q\alpha }{p}}}⩽{\left(KL\right)}^q\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(a\left(\eta \right)A^{K-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)\\ \hfill \times \left({\int }_{t_0}^{\sigma \left(t\right)}{\left(b\left(\eta \right)B^{L-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right).\end{array}$$

(24)

From (23) by using Lemma 1 we obtain

$$\begin{array}{c}{\int }_{t_0}^x{\int }_{t_0}^y\frac{A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}\Delta s\Delta t\hfill \\ ⩽KL{(x-t_0)}^{\frac{1}{p}}{(y-t_0)}^{\frac{1}{p}}{\int }_{t_0}^x\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(a\left(\eta \right)A^{K-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)\Delta s{)}^{\frac{1}{q}}\hfill \\ \times {\int }_{t_0}^y\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(b\left(\eta \right)B^{L-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)\Delta t{)}^{\frac{1}{q}}.\hfill \end{array}$$

(25)

From (24), we get

$$\begin{array}{c}\hfill {\int }_{t_0}^x{\int }_{t_0}^y\frac{A^{qK}\left(\sigma \left(s\right)\right)B^{qL}\left(\sigma \left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2q\alpha }{p}}}\Delta s\Delta t\\ \hfill ⩽{\left(KL\right)}^q{\int }_{t_0}^x\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(a\left(\eta \right)A^{K-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)\Delta s)\\ \hfill \times {\int }_{t_0}^y\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(b\left(\eta \right)B^{L-1}(\sigma \left(\eta \right)\right)}^q\Delta \eta \right)\Delta t).\end{array}$$

(26)

Applying Fubini’s Theorem on (25) and (26) gives

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y\frac{A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}\Delta s\Delta t\hfill \\ \hfill & ⩽KL{(x-t_0)}^{\frac{1}{p}}{(y-t_0)}^{\frac{1}{p}}{\left({\int }_{t_0}^x(x-\sigma \left(s\right)){\left(a\left(s\right)A^{K-1}(\sigma \left(s\right)\right)}^q\Delta s\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times {\left({\int }_{t_0}^y(y-\sigma \left(t\right)){\left(b\left(t\right)B^{L-1}(\sigma \left(t\right)\right)}^q\Delta t\right)}^{\frac{1}{q}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y\frac{A^{qK}\left(\sigma \left(s\right)\right)B^{qL}\left(\sigma \left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2q\alpha }{p}}}\Delta s\Delta t\hfill \\ \hfill & ⩽{\left(KL\right)}^q\left({\int }_{t_0}^x(x-\sigma \left(s\right)){\left(a\left(s\right)A^{K-1}(\sigma \left(s\right)\right)}^q\right)\Delta t)\hfill \\ \hfill & \times \left({\int }_{t_0}^y(y-\sigma \left(t\right)){\left(b\left(t\right)B^{L-1}(\sigma \left(t\right)\right)}^q\Delta t\right).\hfill \end{array}$$

Using the facts $\sigma \left(x\right)⩾x$, $\sigma \left(y\right)⩾y$ yields

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y\frac{A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}\Delta s\Delta t\hfill \\ \hfill & ⩽C_2(L,K,p){\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(a\left(s\right)A^{K-1}(\sigma \left(s\right)\right)}^q\Delta s\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times {\left({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){\left(b\left(t\right)B^{L-1}(\sigma \left(t\right)\right)}^q\Delta t\right)}^{\frac{1}{q}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y\frac{A^{qK}\left(\sigma \left(s\right)\right)B^{qL}\left(\sigma \left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2q\alpha }{p}}}\Delta s\Delta t\hfill \\ \hfill & ⩽C_1(L,K,q)\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(a\left(s\right)A^{K-1}(\sigma \left(s\right)\right)}^2\Delta s\right)\hfill \\ \hfill & \times \left({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){\left(b\left(t\right)B^{L-1}(\sigma \left(t\right)\right)}^q\Delta t\right).\hfill \end{array}$$

This completes the proof. ☐

Theorem 3.

Let $a\left(\check{\tau }\right),$ $b\left(\eta \right),$ $A\left(s\right)$ and $B\left(t\right)$ be defined as in Theorem 2, thus

$$\begin{array}{c}{\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{A^q\left(\sigma \left(s\right)\right)B^q\left(\sigma \left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2q\alpha }{p}}}}\Delta s\Delta t\hfill \\ ⩽\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right))a^q\left(s\right)\Delta s\right)\left({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right))b^q\left(t\right)\Delta t\right)\hfill \end{array}$$

and

$$\begin{array}{c}{\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{A\left(\sigma \right(s\left)\right)B\left(\sigma \right(t\left)\right)}{{\left({|h(\sigma \left(s\right)-t_0)|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ ⩽{(x-t_0)}^{\frac{1}{p}}{(y-t_0)}^{\frac{1}{p}}{\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right))a^q\left(s\right)\Delta s\right)}^{\frac{1}{q}}{\left({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right))b^q\left(t\right)\Delta t\right)}^{\frac{1}{q}}.\hfill \end{array}$$

Proof. 

Put $K=L=1$ in (11) and (12). This completes the proof. ☐

In Theorem 2, if we choose $\mathbb{T}=\mathbb{R}$, then we have relation (1) and the next results:

Corollary 2.

If $a\left(s\right)\ge 0$, $b\left(t\right)\ge 0$. Define $A\left(s\right):={\int }_0^{s}a\left(\eta \right)d\eta$ and $B\left(t\right):={\int }_0^{t}b\left(\eta \right)d\eta$, then

$$\begin{array}{cc}\hfill & {\int }_0^x{\int }_0^y{\displaystyle \frac{A^{qK}\left(s\right)B^{qL}\left(t\right)}{{\left({|h\left(s\right)|}^{\frac{1}{2\beta }}+{|h^{*}\left(t\right)|}^{\frac{1}{2\beta }}\right)}^{\frac{2q\alpha }{p}}}}dsdt\hfill \\ \hfill & ⩽C_1(L,K,q)\left({\int }_0^x(x-s){\left(a\left(s\right)A^{K-1}\left(s\right)\right)}^{q}ds\right)\hfill \\ \hfill & \times \left({\int }_0^y(y-t){\left(b\left(t\right)B^{L-1}\left(t\right)\right)}^{q}dt\right).\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\int }_0^x{\int }_0^y{\displaystyle \frac{A^K\left(s\right)B^L\left(t\right)}{{\left({|h(s)|}^{\frac{1}{2\beta }}+{|h^{*}\left(t\right)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}dsdt\hfill \\ \hfill & ⩽C_3(L,K,p){\left({\int }_0^x(x-s){\left(A^{K-1}\left(s\right)a\left(s\right)\right)}^{q}ds\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times ({\int }_0^y(y-t){{\left(B^{L-1}\left(t\right)b\left(t\right)\right)}^{q}dt)}^{\frac{1}{q}}\hfill \end{array}$$

where

$$C_3(L,K,p)=KL{\left(xy\right)}^{\frac{1}{p}}.$$

In Theorem 2, if we chose $\mathbb{T}=\mathbb{Z}$, then we get (2), and the next result:

Corollary 3.

If $a\left(n\right)\ge$ and $b\left(m\right)\ge 0$. Define

$$A\left(n\right)=\sum _{s=0}^{n}a\left(s\right),B\left(m\right)=\sum _{k=0}^{m}b\left(k\right).$$

Then

$$\begin{array}{c}\hfill \sum _{n=1}^N\sum _{m=1}^M{\displaystyle \frac{A^{qL}\left(n\right)B^{qK}\left(m\right)}{{\left({|h(n+1)|}^{\frac{1}{2\beta }}+{|h^{*}(m+1)|}^{\frac{1}{2\beta }}\right)}^{\frac{2q\alpha }{p}}}}⩽C_1(K,L,q)\left(\sum _{n=1}^N(N+1-(n+1)){\left(a\left(n\right)A^{L-1}\left(n\right)\right)}^q\right)\\ \hfill \times \left(\sum _{m=1}^M(M+1-(m+1)){\left(b\left(m\right)B^{L-1}\left(m\right)\right)}^q\right)\end{array}$$

and

$$\begin{array}{c}\hfill \sum _{n=1}^N\sum _{m=1}^M{\displaystyle \frac{A^L\left(n\right)B^K\left(m\right)}{{\left({|h(n+1)|}^{\frac{1}{2\beta }}+{|h^{*}(m+1)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}⩽C_4(K,L,p){\left(\sum _{n=1}^N(N+1-(n+1)){\left(a\left(n\right)A^{L-1}\left(n\right)\right)}^q\right)}^{\frac{1}{q}}\\ \hfill \times {\left(\sum _{m=1}^M(M+1-(m+1)){\left(b\left(m\right)B^{L-1}\left(m\right)\right)}^q\right)}^{\frac{1}{q}}\end{array}$$

where

$$C_4(K,L,p)=KL{\left(NM\right)}^{\frac{1}{p}}.$$

Remark 1.

Taking $p=q=2$ in Corollary 3 gives the result due to Hamiaz and Abuelela ([1], Theorem 3).

Corollary 4.

With the hypotheses of Theorem 2 we have:

$$\begin{array}{c}{\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{A^{qK}\left(\sigma \left(s\right)\right)B^{qL}\left(\sigma \left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2q\alpha }{p}}}}\Delta s\Delta t\\ ⩽C_1(L,K,q)\{h\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(a\left(s\right)A^{K-1}(\sigma \left(s\right)\right)}^q\Delta s\right)\\ +h^{*}\left({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){\left(b\left(t\right)B^{L-1}(\sigma \left(t\right)\right)}^q\Delta t\right)\}\end{array}$$

and

$$\begin{array}{c}{\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)}{{\left({|h(\sigma \left(s\right)-t_0)|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\\ ⩽C_2(L,K,p)\{h\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(A^{K-1}\left(\sigma \left(s\right)\right)a\left(s\right)\right)}^q\Delta s\right)\\ +h^{*}{\left({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){\left(B^{L-1}\left(\sigma \left(t\right)\right)b\left(t\right)\right)}^q\Delta t)\right\}}^{\frac{1}{q}}.\end{array}$$

Proof. 

Using the Fenchel-Young inequality (5) in (11) and (12). This proves the claim. ☐

Theorem 4.

Assuming the time scale $\mathbb{T}$ with $s,$ $t,$ $t_0,$ $x,$ $y\in \mathbb{T},$ $A\left(s\right)$ and $B\left(t\right)$ are defined as in Theorem 2. Suppose $f\left(\check{\tau }\right)\ge 0$ and $g\left(\eta \right)\ge 0$ are right-dense continuous functions on ${[t_0,x]}_{\mathbb{T}}$ and ${[t_0,y]}_{\mathbb{T}}$ respectively. Suppose that $\breve{\mathrm{\Phi }}\ge 0$ and $\breve{\mathrm{\Psi }}\ge 0$ are convex, and submultiplicative functions on $[0,\infty ).$ Furthermore assume that

$$F\left(s\right):={\int }_{t_0}^{s}f\left(\check{\tau }\right)\Delta \check{\tau },andG\left(t\right):={\int }_{t_0}^{t}g\left(\eta \right)\Delta \eta ,$$

(27)

then for $\sigma \left(s\right)\in {[t_0,x]}_{\mathbb{T}}$ and $\sigma \left(t\right)\in {[t_0,y]}_{\mathbb{T}},$ we have that

$$\begin{array}{c}{\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)}{{\left({|h(\sigma \left(s\right)-t_0)|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\\ ⩽M_1\left(p\right){\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(f\left(s\right)\breve{\mathrm{\Phi }}\left({\displaystyle \frac{a\left(s\right)}{f\left(s\right)}}\right)\right)}^q\Delta s\right)}^{\frac{1}{q}}\\ \times ({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){{\left(g\left(t\right)\breve{\mathrm{\Psi }}\left(\frac{b\left(t\right)}{g\left(t\right)}\right)\right)}^q\Delta t)}^{\frac{1}{q}}\end{array}$$

(28)

where

$$M_1\left(p\right)={\left\{{\int }_{t_0}^x{\left({\displaystyle \frac{\breve{\mathrm{\Phi }}\left(F^{\sigma }\left(s\right)\right)}{F^{\sigma }\left(s\right)}}\right)}^p\Delta s\right\}}^{\frac{1}{p}}{\left\{{\int }_{t_0}^y{\left({\displaystyle \frac{\breve{\mathrm{\Psi }}\left(G^{\sigma }\left(t\right)\right)}{G^{\sigma }\left(t\right)}}\right)}^p\Delta t\right\}}^{\frac{1}{p}}.$$

Proof. 

From the properties of $\breve{\mathrm{\Phi }}$ and using (2), we obtain

$$\begin{array}{cc}\hfill \breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)& =\breve{\mathrm{\Phi }}\left({\displaystyle \frac{F\left(\sigma \left(s\right)\right){\int }_{t_0}^{\sigma \left(s\right)}f\left(\check{\tau }\right)\frac{a\left(\check{\tau }\right)}{f\left(\check{\tau }\right)}\Delta \check{\tau }}{{\int }_{t_0}^{\sigma \left(s\right)}f\left(\check{\tau }\right)\Delta \check{\tau }}}\right)\hfill \\ \hfill & ⩽\breve{\mathrm{\Phi }}(F\left(\sigma \left(s\right)\right)\breve{\mathrm{\Phi }}\left({\displaystyle \frac{{\int }_{t_0}^{\sigma \left(s\right)}f\left(\check{\tau }\right)\frac{a\left(\check{\tau }\right)}{f\left(\check{\tau }\right)}\Delta \check{\tau }}{{\int }_{t_0}^{\sigma \left(s\right)}f\left(\check{\tau }\right)\Delta \check{\tau }}}\right)\hfill \\ \hfill & ⩽{\displaystyle \frac{\breve{\mathrm{\Phi }}(F\left(\sigma \left(s\right)\right)}{F\left(\sigma \right(s\left)\right)}}{\int }_{t_0}^{\sigma \left(s\right)}f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left({\displaystyle \frac{a\left(\check{\tau }\right)}{f\left(\check{\tau }\right)}}\right)\Delta \check{\tau }.\hfill \end{array}$$

(29)

Using (1) in (29), we see that

$$\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)⩽{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(F^{\sigma }\left(s\right)\right)}{F^{\sigma }\left(s\right)}}{(\sigma \left(s\right)-t_0)}^{\frac{1}{p}}{\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(\check{\tau }\right)}{f\left(\check{\tau }\right)}\right]\right)}^q\Delta \check{\tau }\right)}^{\frac{1}{q}}.$$

(30)

In addition, from the convexity and submultiplicative property of $\breve{\mathrm{\Psi }}$, we get by using (2) and (1):

$$\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)⩽{\displaystyle \frac{\breve{\mathrm{\Psi }}\left(G^{\sigma }\left(t\right)\right)}{G^{\sigma }\left(t\right)}}{(\sigma \left(t\right)-t_0)}^{\frac{1}{p}}{\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^q\Delta \eta \right)}^{\frac{1}{q}}.$$

(31)

From (30) and (31), we have

$$\begin{array}{cc}\hfill \breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)⩽& {(\sigma \left(s\right)-t_0)}^{\frac{1}{p}}{(\sigma \left(t\right)-t_0)}^{\frac{1}{p}}({\displaystyle \frac{\breve{\mathrm{\Phi }}\left(F^{\sigma }\left(s\right)\right)}{F^{\sigma }\left(s\right)}}{\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(\check{\tau }\right)}{f\left(\check{\tau }\right)}\right]\right)}^q\Delta \check{\tau }\right)}^{\frac{1}{q}}\hfill \\ & \times ({\displaystyle \frac{\breve{\mathrm{\Psi }}\left(G^{\sigma }\left(t\right)\right)}{G^{\sigma }\left(t\right)}}{\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^q\Delta \eta \right)}^{\frac{1}{q}}\hfill \end{array}$$

(32)

Using (5) on ${(\sigma \left(s\right)-t_0)}^{\frac{1}{p}}{(\sigma \left(t\right)-t_0)}^{\frac{1}{p}}$ gives:

$$\begin{array}{cc}\hfill \breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)⩽& {\left(h(\sigma \left(s\right)-t_0)+h^{*}(\sigma \left(t\right)-t_0)\right)}^{\frac{1}{p}}({\displaystyle \frac{\breve{\mathrm{\Phi }}\left(F^{\sigma }\left(s\right)\right)}{F^{\sigma }\left(s\right)}}{\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(\check{\tau }\right)}{f\left(\check{\tau }\right)}\right]\right)}^q\Delta \check{\tau }\right)}^{\frac{1}{q}}\hfill \\ & \times ({\displaystyle \frac{\breve{\mathrm{\Psi }}\left(G^{\sigma }\left(t\right)\right)}{G^{\sigma }\left(t\right)}}{\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^q\Delta \eta \right)}^{\frac{1}{q}}.\hfill \end{array}$$

(33)

Applying Lemma 6 on the right hand side of (33), we see that

$$\begin{array}{cc}\hfill & \breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)⩽{\left(|h(\sigma \left(s\right)-t_0{|}^{\frac{1}{2\beta }}+|h^{*}(\sigma \left(t\right)-t_0){|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}\hfill \\ \hfill & \times ({\displaystyle \frac{\breve{\mathrm{\Phi }}\left(F^{\sigma }\left(s\right)\right)}{F^{\sigma }\left(s\right)}}{\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(\check{\tau }\right)}{f\left(\check{\tau }\right)}\right]\right)}^q\Delta \check{\tau }\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times ({\displaystyle \frac{\breve{\mathrm{\Psi }}\left(G^{\sigma }\left(t\right)\right)}{G^{\sigma }\left(t\right)}}{\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^q\Delta \eta \right)}^{\frac{1}{q}}.\hfill \end{array}$$

(34)

From (34), we have

$$\begin{array}{c}{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0{|}^{\frac{1}{2\beta }}+|h^{*}(\sigma \left(t\right)-t_0){|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}⩽({\displaystyle \frac{\breve{\mathrm{\Phi }}\left(F^{\sigma }\left(s\right)\right)}{F^{\sigma }\left(s\right)}}{\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(\check{\tau }\right)}{f\left(\check{\tau }\right)}\right]\right)}^q\Delta \check{\tau }\right)}^{\frac{1}{q}}\hfill \\ \times ({\displaystyle \frac{\breve{\mathrm{\Psi }}\left(G^{\sigma }\left(t\right)\right)}{G^{\sigma }\left(t\right)}}{\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^q\Delta \eta \right)}^{\frac{1}{q}}.\hfill \end{array}$$

(35)

From (35), we obtain

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0{|}^{\frac{1}{2\beta }}+|h^{*}(\sigma \left(t\right)-t_0){|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ \hfill & ⩽{\int }_{t_0}^x({\displaystyle \frac{\breve{\mathrm{\Phi }}\left(F^{\sigma }\left(s\right)\right)}{F^{\sigma }\left(s\right)}}{\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(\check{\tau }\right)}{f\left(\check{\tau }\right)}\right]\right)}^q\Delta \check{\tau }\right)}^{\frac{1}{q}}\Delta s\hfill \\ \hfill & \times {\int }_{t_0}^y({\displaystyle \frac{\breve{\mathrm{\Psi }}\left(G^{\sigma }\left(t\right)\right)}{G^{\sigma }\left(t\right)}}{\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^q\Delta \eta \right)}^{\frac{1}{q}}\Delta t.\hfill \end{array}$$

(36)

From (36), by using (1), we have

$$\begin{array}{c}\hfill {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0{|}^{\frac{1}{2\beta }}+|h^{*}(\sigma \left(t\right)-t_0){|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\\ \hfill ⩽{\left\{{\int }_{t_0}^x{\left({\displaystyle \frac{\breve{\mathrm{\Phi }}\left(F^{\sigma }\left(s\right)\right)}{F^{\sigma }\left(s\right)}}\right)}^p\Delta s\right\}}^{\frac{1}{p}}{\left({\int }_{t_0}^x{\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(\check{\tau }\right)}{f\left(\check{\tau }\right)}\right]\right)}^q\Delta \check{\tau }\Delta s\right)}^{\frac{1}{q}}\\ \hfill \times {\left\{{\int }_{t_0}^y{\left({\displaystyle \frac{\breve{\mathrm{\Psi }}\left(G^{\sigma }\left(t\right)\right)}{G^{\sigma }\left(t\right)}}\right)}^p\Delta t\right\}}^{\frac{1}{p}}{\left({\int }_{t_0}^y{\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^q\Delta \eta \Delta t\right)}^{\frac{1}{q}}.\end{array}$$

(37)

From (37), by using (5), we obtain

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0{|}^{\frac{1}{2\beta }}+|h^{*}(\sigma \left(t\right)-t_0){|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ \hfill & ⩽M_1\left(p\right){\left({\int }_{t_0}^x(x-\sigma \left(s\right)){\left(f\left(s\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(s\right)}{f\left(s\right)}\right]\right)}^q\Delta s\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times {\left({\int }_{t_0}^y(y-\sigma \left(t\right)){\left(g\left(t\right)\breve{\mathrm{\Psi }}\left[\frac{b\left(t\right)}{g\left(t\right)}\right]\right)}^q\Delta t\right)}^{\frac{1}{q}}.\hfill \end{array}$$

By using the facts $\sigma \left(x\right)⩾x$ and $\sigma \left(y\right)⩾y,$ we obtain

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)}{{\left(|h(\sigma \left(s\right)-t_0{|}^{\frac{1}{2\beta }}+|h^{*}(\sigma \left(t\right)-t_0){|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ \hfill & ⩽M_1\left(p\right){\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(f\left(s\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(s\right)}{f\left(s\right)}\right]\right)}^q\Delta s\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times {\left({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){\left(g\left(t\right)\breve{\mathrm{\Psi }}\left[\frac{b\left(t\right)}{g\left(t\right)}\right]\right)}^q\Delta t\right)}^{\frac{1}{q}}\hfill \end{array}$$

where

$$M_1\left(p\right)={\left\{{\int }_{t_0}^x{\left({\displaystyle \frac{\breve{\mathrm{\Phi }}\left(F^{\sigma }\left(s\right)\right)}{F^{\sigma }\left(s\right)}}\right)}^p\Delta s\right\}}^{\frac{1}{p}}{\left\{{\int }_{t_0}^y{\left({\displaystyle \frac{\breve{\mathrm{\Psi }}\left(G^{\sigma }\left(t\right)\right)}{G^{\sigma }\left(t\right)}}\right)}^p\Delta t\right\}}^{\frac{1}{p}}.$$

This completes the proof. ☐

In Theorem 4, taking $\mathbb{T}=\mathbb{R}$, we have (1) and the result:

Corollary 5.

Assume that $a\left(s\right)\ge 0,$ $b\left(t\right)\ge 0,$ $f\left(\check{\tau }\right)\ge 0$ and $g\left(\eta \right)\ge 0$, we define

$$A\left(s\right):={\int }_0^{s}a\left(\eta \right)d\eta ,B\left(t\right):={\int }_0^{t}b\left(\eta \right)d\eta ,F\left(s\right):={\int }_0^{s}f\left(\check{\tau }\right)d\check{\tau },andG\left(t\right):={\int }_0^{t}g\left(\eta \right)d\eta .$$

Then

$$\begin{array}{cc}\hfill {\int }_0^x{\int }_0^y{\displaystyle \frac{\breve{\mathrm{\Phi }}(A\left(s\right)\breve{\mathrm{\Psi }}\left(B\left(t\right)\right)}{{\left({|h(s)|}^{\frac{1}{2\beta }}+{|h^{*}\left(t\right)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}dsdt& ⩽M_2\left(p\right){\left({\int }_0^x(x-s){\left(f\left(s\right)\breve{\mathrm{\Phi }}\left({\displaystyle \frac{a\left(s\right)}{f\left(s\right)}}\right)\right)}^{q}ds\right)}^{\frac{1}{q}}\hfill \\ & \times ({\int }_0^y(y-t){{\left(g\left(t\right)\breve{\mathrm{\Psi }}\left(\frac{b\left(t\right)}{g\left(t\right)}\right)\right)}^{q}dt)}^{\frac{1}{q}}\hfill \end{array}$$

where

$$M_2\left(p\right)={\left\{{\int }_0^x{\left({\displaystyle \frac{\breve{\mathrm{\Phi }}\left(F\left(s\right)\right)}{F\left(s\right)}}\right)}^{p}ds\right\}}^{\frac{1}{p}}{\left\{{\int }_0^y{\left({\displaystyle \frac{\breve{\mathrm{\Psi }}\left(G\left(t\right)\right)}{G\left(t\right)}}\right)}^{p}dt\right\}}^{\frac{1}{p}}.$$

In Theorem 4, taking $\mathbb{T}=\mathbb{Z}$, gives (2) and the result:

Corollary 6.

Assume that $a\left(n\right)\ge 0$, $b\left(m\right)\ge 0$, $f\left(n\right)\ge 0$, $g\left(m\right)\ge 0$ are sequences of real numbers. Define

$$A\left(n\right)=\sum _{s=0}^{n}a\left(s\right),B\left(m\right)=\sum _{k=0}^{m}b\left(k\right),F\left(n\right)=\sum _{s=0}^{n}f\left(s\right)\mathit{and}G\left(m\right)=\sum _{k=0}^{m}g\left(k\right).$$

Then

$$\begin{array}{c}\hfill \sum _{n=1}^N\sum _{m=1}^M{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A\left(n\right)\right)\breve{\mathrm{\Psi }}\left(B\left(m\right)\right)}{{\left({|h(n+1)|}^{\frac{1}{2\beta }}+{|h^{*}(m+1)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}⩽M_3\left(p\right){\left\{\sum _{n=1}^N(N+1-(n+1)){\left(f\left(n\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(n\right)}{f\left(n\right)}\right]\right)}^q\right\}}^{\frac{1}{q}}\\ \hfill \times {\left\{\sum _{m=1}^M(M+1-(m+1)){\left(g\left(m\right)\breve{\mathrm{\Psi }}\left[\frac{b\left(m\right)}{g\left(m\right)}\right]\right)}^q\right\}}^{\frac{1}{q}}\end{array}$$

where

$$M_3\left(p\right)={\left\{\sum _{n=1}^N{\left(\frac{\breve{\mathrm{\Phi }}(F\left(n\right)}{F\left(n\right)}\right)}^p\right\}}^{\frac{1}{p}}{\left\{\sum _{m=1}^M{\left(\frac{\breve{\mathrm{\Psi }}(G\left(m\right)}{G\left(m\right)}\right)}^p\right\}}^{\frac{1}{p}}$$

Remark 2.

In Corollary 6, if $p=q=2$ we get the result due to Hamiaz and Abuelela ([1], Theorem 5).

Corollary 7.

Under the hypotheses of Theorem 4 the following inequality hold:

$$\begin{array}{c}{\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)}{{\left({|h(\sigma \left(s\right)-t_0)|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ ⩽M_1\left(p\right)[h\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(f\left(s\right)\breve{\mathrm{\Phi }}\left({\displaystyle \frac{a\left(s\right)}{f\left(s\right)}}\right)\right)}^q\Delta s\right)\hfill \\ +h^{*}{\left({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){\left(g\left(t\right)\breve{\mathrm{\Psi }}\left(\frac{b\left(t\right)}{g\left(t\right)}\right)\right)}^q\Delta t)\right]}^{\frac{1}{q}}.\hfill \end{array}$$

Proof. 

Using (5) in (28). This proves our claim. ☐

Lemma 7.

With hypotheses of Theorem 4, we get:

$$\begin{array}{c}{\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}{\left(A^{\sigma }\left(s\right)\right)}^2\breve{\mathrm{\Psi }}{\left(B^{\sigma }\left(t\right)\right)}^2}{\left(h(\sigma \left(s\right)-t_0)+h^{*}(\sigma \left(t\right)-t_0)\right)}}\Delta s\Delta t\hfill \\ ⩽M_4{\left\{{\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(f\left(s\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(s\right)}{f\left(s\right)}\right]\right)}^4\Delta s\right\}}^{\frac{1}{2}}{\left\{{\int }_{t_0}^y(\sigma \left(t\right)-\sigma \left(t\right)){\left(g\left(t\right)\breve{\mathrm{\Psi }}\left[\frac{b\left(t\right)}{g\left(t\right)}\right]\right)}^4\Delta t\right\}}^{\frac{1}{2}}\hfill \end{array}$$

(38)

where

$$M_4={\left\{{\int }_{t_0}^x\left({\displaystyle \frac{\breve{\mathrm{\Phi }}{\left(F^{\sigma }\left(s\right)\right)}^4}{{\left(F^{\sigma }\left(s\right)\right)}^4}}\right)(\sigma \left(s\right)-t_0)\Delta s\right\}}^{\frac{1}{2}}{\left\{{\int }_{t_0}^y\left({\displaystyle \frac{\breve{\mathrm{\Psi }}{\left(G^{\sigma }\left(t\right)\right)}^4}{{\left(G^{\sigma }\left(t\right)\right)}^4}}\right)(\sigma \left(t\right)-t_0)\Delta t\right\}}^{\frac{1}{2}}.$$

(39)

Proof. 

From (30) and (31) and by using Fenchel-Young inequality with $p=q=2$ we have

$$\begin{array}{c}\breve{\mathrm{\Phi }}{\left(A^{\sigma }\left(s\right)\right)}^2\breve{\mathrm{\Psi }}{\left(B^{\sigma }\left(t\right)\right)}^2\hfill \\ ⩽\left(h(\sigma \left(s\right)-t_0)+h^{*}(\sigma \left(t\right)-t_0)\right)({\displaystyle \frac{\breve{\mathrm{\Phi }}{\left(F^{\sigma }\left(s\right)\right)}^2}{{\left(F^{\sigma }\left(s\right)\right)}^2}}\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(\check{\tau }\right)}{f\left(\check{\tau }\right)}\right]\right)}^2\Delta \check{\tau }\right)\hfill \\ \times ({\displaystyle \frac{\breve{\mathrm{\Psi }}{\left(G^{\sigma }\left(t\right)\right)}^2}{{\left(G^{\sigma }\left(t\right)\right)}^2}}\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^2\Delta \eta \right).\hfill \end{array}$$

(40)

From (40), by using (1) with $p=q=2$, we obtain

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}{\left(A^{\sigma }\left(s\right)\right)}^2\breve{\mathrm{\Psi }}{\left(B^{\sigma }\left(t\right)\right)}^2}{\left(h(\sigma \left(s\right)-t_0)+h^{*}(\sigma \left(t\right)-t_0)\right)}}\Delta s\Delta t\hfill \\ \hfill & ⩽{\int }_{t_0}^x({\displaystyle \frac{\breve{\mathrm{\Phi }}{\left(F^{\sigma }\left(s\right)\right)}^2}{{\left(F^{\sigma }\left(s\right)\right)}^2}}\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(\check{\tau }\right)}{f\left(\check{\tau }\right)}\right]\right)}^2\Delta \check{\tau }\right)\Delta s\hfill \\ \hfill & \times {\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Psi }}{\left(G^{\sigma }\left(t\right)\right)}^2}{{\left(G^{\sigma }\left(t\right)\right)}^2}}\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^2\Delta \eta \right)\Delta t\hfill \\ \hfill & ⩽{\int }_{t_0}^x\left({\displaystyle \frac{\breve{\mathrm{\Phi }}{\left(F^{\sigma }\left(s\right)\right)}^2}{{\left(F^{\sigma }\left(s\right)\right)}^2}}\right){(\sigma \left(s\right)-t_0)}^{\frac{1}{2}}{\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(\check{\tau }\right)}{f\left(\check{\tau }\right)}\right]\right)}^4\Delta \check{\tau }\right)}^{\frac{1}{2}})\Delta s\hfill \\ \hfill & \times {\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Psi }}{\left(G^{\sigma }\left(t\right)\right)}^2}{{\left(G^{\sigma }\left(t\right)\right)}^2}}{(\sigma \left(t\right)-t_0)}^{\frac{1}{2}}{\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^4\Delta \eta \right)}^{\frac{1}{2}}\Delta t\hfill \\ \hfill & ⩽{\left\{{\int }_{t_0}^x\left({\displaystyle \frac{\breve{\mathrm{\Phi }}{\left(F^{\sigma }\left(s\right)\right)}^4}{{\left(F^{\sigma }\left(s\right)\right)}^4}}\right)(\sigma \left(s\right)-t_0)\Delta s\right\}}^{\frac{1}{2}}{\left\{{\int }_{t_0}^x\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(\check{\tau }\right)}{f\left(\check{\tau }\right)}\right]\right)}^4\Delta \check{\tau }\right)\Delta s\right\}}^{\frac{1}{2}}\hfill \\ \hfill & \times {\left\{{\int }_{t_0}^y\left({\displaystyle \frac{\breve{\mathrm{\Psi }}{\left(G^{\sigma }\left(t\right)\right)}^4}{{\left(G^{\sigma }\left(t\right)\right)}^4}}\right)(\sigma \left(t\right)-t_0)\Delta t\right\}}^{\frac{1}{2}}{\left\{{\int }_{t_0}^y\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^4\Delta \eta \right)\Delta t\right\}}^{\frac{1}{2}}.\hfill \end{array}$$

(41)

Applying (5) on (41), we obtain

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}{\left(A^{\sigma }\left(s\right)\right)}^2\breve{\mathrm{\Psi }}{\left(B^{\sigma }\left(t\right)\right)}^2}{\left(h(\sigma \left(s\right)-t_0)+h^{*}(\sigma \left(t\right)-t_0)\right)}}\Delta s\Delta t\hfill \\ \hfill & ⩽{\left\{{\int }_{t_0}^x\left({\displaystyle \frac{\breve{\mathrm{\Phi }}{\left(F^{\sigma }\left(s\right)\right)}^4}{{\left(F^{\sigma }\left(s\right)\right)}^4}}\right)(\sigma \left(s\right)-t_0)\Delta s\right\}}^{\frac{1}{2}}{\left\{{\int }_{t_0}^x(x-\sigma \left(s\right)){\left(f\left(s\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(s\right)}{f\left(s\right)}\right]\right)}^4\Delta s\right\}}^{\frac{1}{2}}\hfill \\ \hfill & \times {\left\{{\int }_{t_0}^y\left({\displaystyle \frac{\breve{\mathrm{\Psi }}{\left(G^{\sigma }\left(t\right)\right)}^4}{{\left(G^{\sigma }\left(t\right)\right)}^4}}\right)(\sigma \left(t\right)-t_0)\Delta t\right\}}^{\frac{1}{2}}{\left\{{\int }_{t_0}^y(t-\sigma \left(t\right)){\left(g\left(t\right)\breve{\mathrm{\Psi }}\left[\frac{b\left(t\right)}{g\left(t\right)}\right]\right)}^4\Delta t\right\}}^{\frac{1}{2}}\hfill \\ \hfill & =M_4{\left\{{\int }_{t_0}^x(x-\sigma \left(s\right)){\left(f\left(s\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(s\right)}{f\left(s\right)}\right]\right)}^4\Delta s\right\}}^{\frac{1}{2}}{\left\{{\int }_{t_0}^y(t-\sigma \left(t\right)){\left(g\left(t\right)\breve{\mathrm{\Psi }}\left[\frac{b\left(t\right)}{g\left(t\right)}\right]\right)}^4\Delta t\right\}}^{\frac{1}{2}}.\hfill \end{array}$$

Since $\sigma \left(x\right)⩾x$ and $\sigma \left(y\right)⩾y$, from the last inequality above, we have

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}{\left(A^{\sigma }\left(s\right)\right)}^2\breve{\mathrm{\Psi }}{\left(B^{\sigma }\left(t\right)\right)}^2}{\left(h(\sigma \left(s\right)-t_0)+h^{*}(\sigma \left(t\right)-t_0)\right)}}\Delta s\Delta t\hfill \\ \hfill & ⩽M_4{\left\{{\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(f\left(s\right)\breve{\mathrm{\Phi }}\left[\frac{a\left(s\right)}{f\left(s\right)}\right]\right)}^4\Delta s\right\}}^{\frac{1}{2}}{\left\{{\int }_{t_0}^y(\sigma \left(t\right)-\sigma \left(t\right)){\left(g\left(t\right)\breve{\mathrm{\Psi }}\left[\frac{b\left(t\right)}{g\left(t\right)}\right]\right)}^4\Delta t\right\}}^{\frac{1}{2}}\hfill \end{array}$$

where $M_4$ defined as in (39). This proves our claim. ☐

Theorem 5.

Assume the time scale $\mathbb{T}$ with $t,$ $s,$ $x_0,$ $t_0,$ $y\in \mathbb{T}.$ Suppose that $b\left(\check{\tau }\right)\ge 0$ and $a\left(\check{\tau }\right)\ge 0$ are right-dense continuous functions on ${[t_0,y]}_{\mathbb{T}}$ and ${[t_0,x]}_{\mathbb{T}}$. Let $G,$ $F,$ $g,$ $f,$ $\breve{\mathrm{\Psi }}$ and $\breve{\mathrm{\Phi }}$ be as assumed in Theorem 4. Furthermore assume that

$$A\left(s\right):=\frac{1}{F\left(s\right)}{\int }_{t_0}^{s}a\left(\check{\tau }\right)f\left(\check{\tau }\right)\Delta \check{\tau },andB\left(t\right):=\frac{1}{G\left(t\right)}{\int }_{t_0}^{t}b\left(\eta \right)g\left(\eta \right)\Delta \eta ,$$

(42)

then for $\sigma \left(s\right)\in {[t_0,x]}_{\mathbb{T}}$ and $\sigma \left(t\right)\in {[t_0,y]}_{\mathbb{T}},$ we have that

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)F^{\sigma }\left(s\right)G^{\sigma }\left(t\right)}{{\left({|h(\sigma \left(s\right)-t_0)|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ \hfill & ⩽M_5\left(p\right){\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(f\left(s\right)\breve{\mathrm{\Phi }}\left(a\left(s\right)\right)\right)}^q\Delta s\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times ({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){{\left(g\left(t\right)\breve{\mathrm{\Psi }}\left(b\left(t\right)\right)\right)}^q\Delta t)}^{\frac{1}{q}}\hfill \end{array}$$

(43)

where

$$M_5\left(p\right)={(x-t_0)}^{\frac{1}{p}}{(y-t_0)}^{\frac{1}{p}}.$$

(44)

Proof. 

From (42), we see that

$$\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)=\breve{\mathrm{\Phi }}\left(\frac{1}{F^{\sigma }\left(s\right)}{\int }_{t_0}^{\sigma \left(s\right)}f\left(\check{\tau }\right)a\left(\check{\tau }\right)\Delta \check{\tau }\right).$$

(45)

Applying (1) on (45), we obtain

$$\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)⩽\frac{{(\sigma \left(s\right)-t_0)}^{\frac{1}{p}}}{F^{\sigma }\left(s\right)}{\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[a\left(\check{\tau }\right)\right]\right)}^q\Delta \check{\tau }\right)}^{\frac{1}{q}}.$$

(46)

From (46), we get

$$\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)F^{\sigma }\left(s\right)⩽{(\sigma \left(s\right)-t_0)}^{\frac{1}{p}}{\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[a\left(\check{\tau }\right)\right]\right)}^q\Delta \check{\tau }\right)}^{\frac{1}{q}}.$$

(47)

Similarly, we obtain

$$\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)G^{\sigma }\left(t\right)⩽{(\sigma \left(t\right)-t_0)}^{\frac{1}{p}}{\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[b\left(\eta \right)\right]\right)}^q\Delta \eta \right)}^{\frac{1}{q}}.$$

(48)

From (47) and (48), we observe that

$$\begin{array}{c}\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)G^{\sigma }\left(t\right)F^{\sigma }\left(s\right)⩽{(\sigma \left(s\right)-t_0)}^{\frac{1}{p}}{(\sigma \left(t\right)-t_0)}^{\frac{1}{p}}\\ \times {\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[a\left(\check{\tau }\right)\right]\right)}^q\Delta \check{\tau }\right)}^{\frac{1}{q}}{\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[b\left(\eta \right)\right]\right)}^q\Delta \eta \right)}^{\frac{1}{q}}.\end{array}$$

(49)

Applying the Lemma 3 on the term ${(\sigma \left(s\right)-t_0)}^{\frac{1}{p}}{(\sigma \left(t\right)-t_0)}^{\frac{1}{p}},$ gives:

$$\begin{array}{cc}\hfill \breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)G^{\sigma }\left(t\right)F^{\sigma }\left(s\right)& ⩽{\left(h(\sigma \left(s\right)-t_0)+h^{*}(\sigma \left(t\right)-t_0)\right)}^{\frac{1}{p}}{\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[a\left(\check{\tau }\right)\right]\right)}^q\Delta \check{\tau }\right)}^{\frac{1}{q}}\hfill \\ & \times {\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[b\left(\eta \right)\right]\right)}^q\Delta \eta \right)}^{\frac{1}{q}}.\hfill \end{array}$$

(50)

From 6 and (50), we obtain

$$\begin{array}{c}\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)G^{\sigma }\left(t\right)F^{\sigma }\left(s\right)⩽{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}\\ \times {\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[a\left(\check{\tau }\right)\right]\right)}^q\Delta \check{\tau }\right)}^{\frac{1}{q}}{\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[b\left(\eta \right)\right]\right)}^q\Delta \eta \right)}^{\frac{1}{q}}.\end{array}$$

(51)

Dividing both sides of (51) by ${\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}},$ we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)G^{\sigma }\left(t\right)F^{\sigma }\left(s\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}& ⩽{\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[a\left(\check{\tau }\right)\right]\right)}^q\Delta \check{\tau }\right)}^{\frac{1}{q}}\hfill \\ & \times {\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[b\left(\eta \right)\right]\right)}^q\Delta \eta \right)}^{\frac{1}{q}}.\hfill \end{array}$$

(52)

Taking delta-integral for (52), yields:

$$\begin{array}{c}{\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)G^{\sigma }\left(t\right)F^{\sigma }\left(s\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ ⩽\left({\int }_{t_0}^x{\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[a\left(\check{\tau }\right)\right]\right)}^q\Delta \check{\tau }\right)}^{\frac{1}{q}}\Delta s\right)\left({\int }_{t_0}^y{\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[b\left(\eta \right)\right]\right)}^q\Delta \eta \right)}^{\frac{1}{q}}\Delta t\right).\hfill \end{array}$$

(53)

Using (1) in (53), yield:

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)G^{\sigma }\left(t\right)F^{\sigma }\left(s\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ \hfill & ⩽{(x-t_0)}^{\frac{1}{p}}{(y-t_0)}^{\frac{1}{p}}{\left({\int }_{t_0}^x\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[a\left(\check{\tau }\right)\right]\right)}^q\Delta \check{\tau }\right)\Delta s\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times {\left({\int }_{t_0}^y\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[b\left(\eta \right)\right]\right)}^q\Delta \eta \right)\Delta t\right)}^{\frac{1}{q}}\hfill \\ \hfill & =M_5\left(p\right){\left({\int }_{t_0}^x\left({\int }_{t_0}^{\sigma \left(s\right)}{\left(f\left(\check{\tau }\right)\breve{\mathrm{\Phi }}\left[a\left(\check{\tau }\right)\right]\right)}^q\Delta \check{\tau }\right)\Delta s\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times {\left({\int }_{t_0}^y\left({\int }_{t_0}^{\sigma \left(t\right)}{\left(g\left(\eta \right)\breve{\mathrm{\Psi }}\left[b\left(\eta \right)\right]\right)}^q\Delta \eta \right)\Delta t\right)}^{\frac{1}{q}},\hfill \end{array}$$

(54)

where $M_5$ defined as in (44). From (5) and (54), we get:

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)G^{\sigma }\left(t\right)F^{\sigma }\left(s\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ \hfill & =M_5\left(p\right){\left({\int }_{t_0}^x(x-\sigma \left(s\right)){\left(f\left(s\right)\breve{\mathrm{\Phi }}\left[a\left(s\right)\right]\right)}^q\Delta s\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times {\left({\int }_{t_0}^y(y-\sigma \left(t\right)){\left(g\left(t\right)\breve{\mathrm{\Psi }}\left[b\left(t\right)\right]\right)}^q\Delta t\right)}^{\frac{1}{q}}.\hfill \end{array}$$

By using the fact $\sigma \left(x\right)⩾x$ and $\sigma \left(y\right)⩾y,$ we obtain

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)G^{\sigma }\left(t\right)F^{\sigma }\left(s\right)}{{\left(|h(\sigma \left(s\right)-t_0){|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ \hfill & =M_5\left(p\right){\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(f\left(s\right)\breve{\mathrm{\Phi }}\left[a\left(s\right)\right]\right)}^q\Delta s\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times {\left({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){\left(g\left(t\right)\breve{\mathrm{\Psi }}\left[b\left(t\right)\right]\right)}^q\Delta t\right)}^{\frac{1}{q}}.\hfill \end{array}$$

This completes the proof. ☐

Taking $\mathbb{T}=\mathbb{R}$ in Theorem 5 with relation (1), we have:

Corollary 8.

Assume $g\left(t\right)\ge 0$, $b\left(t\right)\ge 0,$ $f\left(s\right)\ge 0$, $a\left(s\right)\ge 0$. Define

$$A\left(s\right):=\frac{1}{F\left(s\right)}{\int }_0^{s}f\left(\check{\tau }\right)a\left(\check{\tau }\right)d\check{\tau }\mathit{and}B\left(t\right):=\frac{1}{G\left(t\right)}{\int }_0^{t}g\left(\check{\tau }\right)b\left(\check{\tau }\right)d\check{\tau },$$

$$F\left(s\right):={\int }_0^{s}f\left(\check{\tau }\right)d\check{\tau }\mathit{and}G\left(t\right):={\int }_0^{t}g\left(\check{\tau }\right)d\check{\tau }.$$

Then

$$\begin{array}{cc}\hfill {\int }_0^x{\int }_0^y{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B\left(t\right)\right)F\left(s\right)G\left(t\right)}{{\left({|h(s)|}^{\frac{1}{2\beta }}+{|h^{*}\left(t\right)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}dsdt& ⩽M_6\left(p\right){\left({\int }_0^x(x-s){\left(f\left(s\right)\breve{\mathrm{\Phi }}\left(a\left(s\right)\right)\right)}^{q}ds\right)}^{\frac{1}{q}}.\hfill \\ & \times ({\int }_0^y(y-t){{\left(g\left(t\right)\breve{\mathrm{\Psi }}\left(b\left(t\right)\right)\right)}^{q}dt)}^{\frac{1}{q}}\hfill \end{array}$$

where

$$M_6\left(p\right)={\left(x\right)}^{\frac{1}{p}}{\left(y\right)}^{\frac{1}{p}}$$

Taking $\mathbb{T}=\mathbb{Z}$ in Theorem 5 with relation (2), gives:

Corollary 9.

Assume $g\left(n\right)\ge 0$, $b\left(n\right)\ge 0,$ $f\left(n\right)\ge 0$, $a\left(n\right)\ge 0$. Define

$$A\left(n\right):=\frac{1}{F\left(n\right)}\sum _{s=0}^{n}f\left(s\right)a\left(s\right)\mathit{and}B\left(m\right):=\frac{1}{G\left(m\right)}\sum _{k=0}^{m}g\left(k\right)b\left(k\right).$$

$$F\left(n\right):=\sum _{s=0}^{n}f\left(s\right)\mathit{and}G\left(m\right):=\sum _{k=0}^{m}g\left(k\right).$$

Then

$$\begin{array}{cc}\hfill \sum _{n=1}^N\sum _{m=1}^M{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A\left(n\right)\right)\breve{\mathrm{\Psi }}\left(B\left(m\right)\right)F\left(n\right)G\left(m\right)}{{\left({|h(n+1)|}^{\frac{1}{2\beta }}+{|h^{*}(m+1)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}& ⩽M_7\left(p\right){\left(\sum _{n=1}^N(N+1-(n+1)){\left(f\left(n\right)\breve{\mathrm{\Phi }}\left(a\left(n\right)\right)\right)}^q\right)}^{\frac{1}{q}}.\hfill \\ & \times (\sum _{m=1}^M(M+1-(m+1)){{\left(g\left(m\right)\breve{\mathrm{\Psi }}\left(b\left(m\right)\right)\right)}^q)}^{\frac{1}{q}}\hfill \end{array}$$

where

$$M_7\left(p\right)={\left(NM\right)}^{\frac{1}{p}}.$$

Remark 3.

In Corollary 9, if $p=q=2$ we get the result due to Hamiaz and Abuelela ([1], Theorem 7).

Corollary 10.

With the hypotheses of Theorem 5, we get:

$$\begin{array}{c}{\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{\breve{\mathrm{\Phi }}\left(A^{\sigma }\left(s\right)\right)\breve{\mathrm{\Psi }}\left(B^{\sigma }\left(t\right)\right)F^{\sigma }\left(s\right)G^{\sigma }\left(t\right)}{{\left({|h(\sigma \left(s\right)-t_0)|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\\ ⩽M_5\left(p\right)\{h\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(f\left(s\right)\breve{\mathrm{\Phi }}\left(a\left(s\right)\right)\right)}^q\Delta s\right)\\ +h^{*}{\left({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){\left(g\left(t\right)\breve{\mathrm{\Psi }}\left(b\left(t\right)\right)\right)}^q\Delta t)\right\}}^{\frac{1}{q}}.\end{array}$$

where $M_5$ defined as in (44).

Proof. 

We apply the Fenchel-Young inequality (5) in (43). This completes the proof. ☐

3. Some Applications

We can apply our inequalities to obtain different formulas of Hilbert-type inequalities by suggesting $h^{*}\left(y\right)$ and $h\left(x\right)$ by some functions:

In (12), as a special case, if we take $h\left(x\right)={\displaystyle \frac{x^2}{2}}$, we have $h^{*}\left(x\right)={\displaystyle \frac{x^2}{2}}$ see [12], we get

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)}{{\left({|h(\sigma \left(s\right)-t_0)|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ \hfill & ={\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)}{{\left({(\sigma \left(s\right)-t_0))}^{\frac{1}{\beta }}+{(\sigma \left(t\right)-t_0)}^{\frac{1}{\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ \hfill & ⩽{\left(\frac{1}{2}\right)}^{\frac{\alpha }{p\beta }}{C}_2(L,K,p){\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(A^{K-1}\left(\sigma \left(s\right)\right)a\left(s\right)\right)}^q\Delta s\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times {\left({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){\left(B^{L-1}\left(\sigma \left(t\right)\right)b\left(t\right)\right)}^q\Delta t\right)}^{\frac{1}{q}},\hfill \end{array}$$

(55)

where $C_2(L,K,p)$ defined as in Theorem 2. Consequently, for $\alpha =\beta =1,$ inequality (55) produces

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)}{{\left((\sigma \left(s\right)-t_0))+(\sigma \left(t\right)-t_0)\right)}^{\frac{2}{p}}}}\Delta s\Delta t\hfill \\ \hfill & ⩽{\left(\frac{1}{2}\right)}^{\frac{1}{p}}{C}_2(L,K,p){\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(A^{K-1}\left(\sigma \left(s\right)\right)a\left(s\right)\right)}^q\Delta s\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times ({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){{\left(B^{L-1}\left(\sigma \left(t\right)\right)b\left(t\right)\right)}^q\Delta t)}^{\frac{1}{q}}.\hfill \end{array}$$

(56)

On the other hand if we take $h\left(n\right)=\frac{n^r}{r},r>1,$ then $h^{*}\left(m\right)=\frac{m^k}{k}$ where $\frac{1}{r}+\frac{1}{k}=1$ and $n,m{\mathbb{R}}_{+},$ then (12) gives

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)}{{\left({|h(\sigma \left(s\right)-t_0)|}^{\frac{1}{2\beta }}+{|h^{*}(\sigma \left(t\right)-t_0)|}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ \hfill & ={\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)}{{\left({\left(k{(\sigma \left(s\right)-t_0)}^r\right)}^{\frac{1}{2\beta }}+{\left(r{(\sigma \left(t\right)-t_0)}^k\right)}^{\frac{1}{2\beta }}\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ \hfill & ⩽{\left(\frac{1}{rk}\right)}^{\frac{\alpha }{p\beta }}{C}_2(L,K,p){\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(A^{K-1}\left(\sigma \left(s\right)\right)a\left(s\right)\right)}^q\Delta s\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times ({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){{\left(B^{L-1}\left(\sigma \left(t\right)\right)b\left(t\right)\right)}^q\Delta t)}^{\frac{1}{q}}.\hfill \end{array}$$

(57)

Clearly, when $\beta =\frac{1}{2\alpha },$ the inequality (57) becomes

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)}{{\left({\left(k{(\sigma \left(s\right)-t_0)}^r\right)}^{\alpha }+{\left(r{(\sigma \left(t\right)-t_0)}^k\right)}^{\alpha }\right)}^{\frac{2\alpha }{p}}}}\Delta s\Delta t\hfill \\ \hfill & ⩽{\left(\frac{1}{rk}\right)}^{\frac{2{\alpha }^2}{p}}{C}_2(L,K,p){\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(A^{K-1}\left(\sigma \left(s\right)\right)a\left(s\right)\right)}^q\Delta s\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times ({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){{\left(B^{L-1}\left(\sigma \left(t\right)\right)b\left(t\right)\right)}^q\Delta t)}^{\frac{1}{q}}.\hfill \end{array}$$

(58)

If $\beta =\alpha =1$. From (57), we get

$$\begin{array}{cc}\hfill & {\int }_{t_0}^x{\int }_{t_0}^y{\displaystyle \frac{A^K\left(\sigma \left(s\right)\right)B^L\left(\sigma \left(t\right)\right)}{{\left({\left(k{(\sigma \left(s\right)-t_0)}^r\right)}^{\frac{1}{2}}+{\left(r{(\sigma \left(t\right)-t_0)}^k\right)}^{\frac{1}{2}}\right)}^{\frac{2}{p}}}}\Delta s\Delta t\hfill \\ \hfill & ⩽{\left(\frac{1}{rk}\right)}^{\frac{1}{p}}{C}_2(L,K,p){\left({\int }_{t_0}^x(\sigma \left(x\right)-\sigma \left(s\right)){\left(A^{K-1}\left(\sigma \left(s\right)\right)a\left(s\right)\right)}^q\Delta s\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times ({\int }_{t_0}^y(\sigma \left(y\right)-\sigma \left(t\right)){{\left(B^{L-1}\left(\sigma \left(t\right)\right)b\left(t\right)\right)}^q\Delta t)}^{\frac{1}{q}}.\hfill \end{array}$$

4. Conclusions

In this paper, with the help of a Fenchel-Legendre transform, which is used in various problems involving symmetry, we generalized a number of Hilbert-type inequalities to a general time scale. Besides that, in order to obtain some new inequalities as special cases, we also extended our inequalities to discrete and continuous calculus. In the future, we can generalize these inequalities in a different way by using other mathematical tools.