Hardy–Leindler, Yang and Hwang Inequalities for Functions of Several Variables via Time Scale Calculus

Abstract

In this paper, Hardy–Leindler, Hardy–Yang and Hwang type inequalities are extended on time scales calculus. These extensions are depending upon use of symmetric multiple delta integrals. The target is achieved by utilizing some inequalities in literature along with mathematical induction principle and Fubini’s theorem on time scales. The obtained inequalities are discussed in discrete, continuous and quantum calculus in search of applications. Particular cases of proved results include Hardy, Copson, Hardy–Littlewood, Levinson and Bennett-type inequalities for symmetric sums.

1. Introduction

In 1920, Hardy proved following discrete inequality [1].

$$\sum _{\kappa =1}^{\infty }{(\frac{1}{\kappa }\sum _{\iota =1}^{\kappa }a\left(\iota \right))}^p\le {(\frac{p}{p-1})}^p\sum _{\kappa =1}^{\infty }{a}^p\left(\kappa \right),p>1,$$

(1)

where ${\left\{a\left(\kappa \right)\right\}}_{\kappa =1}^{\infty }$ is the sequence of nonnegative real numbers. The constant in (1) is best possible. This inequality asserts that ${\sum }_{\kappa =1}^{\infty }{\left(\frac{1}{\kappa }{\sum }_{\iota =1}^{\kappa }a\left(\iota \right)\right)}^p<\infty$ whenever ${\sum }_{\kappa =1}^{\infty }{a}^p\left(\kappa \right)<\infty .$

In 1928, Copson [2] gave extension to Hardy’s discrete inequality (1) and established the following:

If $r\ge c>1,a\left(\kappa \right)>0$ and $\zeta \left(\kappa \right)>0,$ then for $\kappa \ge 1,$

$$\begin{array}{c}\hfill \sum _{\kappa =1}^{\infty }\left(\frac{\zeta \left(\kappa \right)}{{(\Omega (\kappa \left)\right)}^c}\right){\left(\sum _{\iota =1}^{\kappa }\zeta \left(\iota \right)a\left(\iota \right)\right)}^r\le {\left(\frac{r}{c-1}\right)}^r\sum _{\kappa =1}^{\infty }\zeta \left(\kappa \right){(\Omega \left(\kappa \right))}^{r-c}{a}^r\left(\kappa \right),\end{array}$$

(2)

where $\Omega \left(\kappa \right)={\sum }_{\iota =1}^{\kappa }\zeta \left(\iota \right)$ and for $r>1\ge c>0,$

$$\begin{array}{c}\hfill \sum _{\kappa =1}^{\infty }\left(\frac{\zeta \left(\kappa \right)}{{(\Omega (\kappa \left)\right)}^c}\right){\left(\sum _{\iota =\kappa }^{\infty }\zeta \left(\iota \right)a\left(\iota \right)\right)}^r\le {\left(\frac{r}{1-c}\right)}^r\sum _{\kappa =1}^{\infty }\zeta \left(\kappa \right){(\Omega \left(\kappa \right))}^{r-c}{a}^r\left(\kappa \right).\end{array}$$

(3)

In 1970 Leindler [3] proposed the following inequality analogous to that of Copson’s inequality (3), by changing the parameters on weight function. He proved that, if ${\sum }_{\iota =\kappa }^{\infty }\zeta \left(\iota \right)<\infty ,p>1$ and $0\le c<1$, then

$$\begin{array}{c}\hfill \sum _{\kappa =1}^{\infty }\frac{\zeta \left(\kappa \right)}{{\left({\Omega }^{*}\left(\kappa \right)\right)}^c}{\left(\sum _{\iota =1}^{\kappa }\zeta \left(\iota \right)a\left(\iota \right)\right)}^p\le \left(\frac{p}{1-c}\right)\sum _{\kappa =1}^{\infty }\zeta \left(\kappa \right){\left({\Omega }^{*}\left(\kappa \right)\right)}^{p-c}{a}^p\left(\kappa \right),\end{array}$$

(4)

where ${\Omega }^{*}\left(\kappa \right)={\sum }_{\iota =\kappa }^{\infty }\zeta \left(\iota \right).$

In 1987 Bennett [4] proved that if ${\sum }_{\iota =\kappa }^{\infty }\zeta \left(\iota \right)<\infty$ and $1<c\le p$, then

$$\begin{array}{c}\hfill \sum _{\kappa =1}^{\infty }\frac{\zeta \left(\kappa \right)}{{\left({\Omega }^{*}\left(\kappa \right)\right)}^c}{\left(\sum _{\iota =\kappa }^{\infty }\zeta \left(\iota \right)a\left(\iota \right)\right)}^p\le \left(\frac{p}{c-1}\right)\sum _{\kappa =1}^{\infty }\zeta \left(\kappa \right){\left({\Omega }^{*}\left(\kappa \right)\right)}^{p-c}{a}^p\left(\kappa \right).\end{array}$$

(5)

Some Hardy-type inequalities for classical integrals are the following:

In [5], Hardy established the integral version of his inequality (1) by using the calculus of variations stated as; if $g\left(\varsigma \right)$ is a positive integrable function over any finite interval $(0,\varsigma )$, and $g^p$ is an integrable function over $(0,\infty )$ for $p>1$, then

$$\begin{array}{c}\hfill \underset{0}{\overset{\infty }{\int }}{\left(\frac{1}{\varsigma }\underset{0}{\overset{\varsigma }{\int }}g\left(s\right)ds\right)}^{p}d\varsigma \le {\left(\frac{p}{p-1}\right)}^p\underset{0}{\overset{\infty }{\int }}{g}^p\left(\varsigma \right)d\varsigma .\end{array}$$

(6)

The constant in (6) is best possible. In 1928, Hardy [6] generalized the inequality (6) and proved that if $g\left(\varsigma \right)>0$ is a positive integrable function over any finite interval $(0,\varsigma )$, $g^p\left(\varsigma \right)$ is an integrable function over $(0,\infty ),$ then for $\gamma ,p>1,$ we obtain

$$\begin{array}{c}\hfill \underset{0}{\overset{\infty }{\int }}\frac{1}{{\varsigma }^{\gamma }}{\left(\underset{0}{\overset{\varsigma }{\int }}g\left(s\right)ds\right)}^{p}d\varsigma \le {\left(\frac{p}{\gamma -1}\right)}^p\underset{0}{\overset{\infty }{\int }}\frac{1}{{\varsigma }^{\gamma -p}}{g}^p\left(\varsigma \right)d\varsigma ,\end{array}$$

(7)

and for $p>1$, $0<\gamma \le 1,$

$$\begin{array}{c}\hfill \underset{0}{\overset{\infty }{\int }}\frac{1}{{\varsigma }^{\gamma }}{\left(\underset{\varsigma }{\overset{\infty }{\int }}g\left(s\right)ds\right)}^{p}d\varsigma \le {\left(\frac{p}{1-\gamma }\right)}^p\underset{0}{\overset{\infty }{\int }}\frac{1}{{\varsigma }^{\gamma -p}}{g}^p\left(\varsigma \right)d\varsigma .\end{array}$$

(8)

In 1964, Levinsen [7] extended the Hardy continuous inequality (6) by applying Jensen’s inequality. Precisely, he proved that, if $\varphi \left(u\right)$ is a real-valued positive convex function for $u>0,$ $p>1,$ $g\left(\varsigma \right)>0$, $\lambda \left(\varsigma \right)>0$ for $\varsigma >0$, and there exists a constant $K>0$ such that

$$\begin{array}{c}\hfill p-1+\frac{{\lambda }^{\prime }\left(\varsigma \right)\Lambda \left(\varsigma \right)}{{\lambda }^2\left(\varsigma \right)}\ge \frac{p}{K},\end{array}$$

for all $\varsigma >0$, then

$$\begin{array}{c}\hfill \underset{0}{\overset{\infty }{\int }}\varphi \left(\frac{1}{\Lambda \left(\varsigma \right)}\underset{0}{\overset{\varsigma }{\int }}\lambda \left(s\right)g\left(s\right)ds\right)d\varsigma \le K^p\underset{0}{\overset{p}{\int }}\varphi \left(g\right(\varsigma \left)\right)d\varsigma ,\end{array}$$

(9)

where $\Lambda \left(\varsigma \right)=\underset{0}{\overset{\varsigma }{\int }}\lambda \left(s\right)ds.$

In 1976 Copson [8] proved the integral forms of his inequalities (2) and (3) which can be considered as generalizations of the inequalities (6) and (8). In particular, Copson proved that if $p\le 1$ and $\gamma >1$, then

$$\begin{array}{c}\hfill \underset{0}{\overset{\infty }{\int }}\frac{\lambda \left(\varsigma \right)}{{\Lambda }^{\gamma }\left(\varsigma \right)}{\Phi }^p\left(\varsigma \right)d\varsigma \le {\left(\frac{p}{\gamma -1}\right)}^p\underset{0}{\overset{\infty }{\int }}\frac{\lambda \left(\varsigma \right)}{{\Lambda }^{\gamma -p}\left(\varsigma \right)}{g}^p\left(\varsigma \right)d\varsigma ,\end{array}$$

(10)

where $\Lambda \left(\varsigma \right)=\underset{0}{\overset{\varsigma }{\int }}\lambda \left(s\right)ds$ and $\Phi \left(\varsigma \right)=\underset{0}{\overset{\varsigma }{\int }}\lambda \left(s\right)g\left(s\right)ds$, and if $p\le 1$, $0\le \gamma <1$, then

$$\begin{array}{c}\hfill \underset{0}{\overset{\infty }{\int }}\frac{\lambda \left(\varsigma \right)}{{\Lambda }^{\gamma }\left(\varsigma \right)}{\overline{\Phi }}^p\left(\varsigma \right)d\varsigma \le {\left(\frac{p}{1-\gamma }\right)}^p\underset{0}{\overset{\infty }{\int }}\frac{\lambda \left(\varsigma \right)}{{\Lambda }^{\gamma -p}\left(\varsigma \right)}{g}^p\left(\varsigma \right)d\varsigma ,\end{array}$$

(11)

where $\overline{\Phi }\left(\varsigma \right)=\underset{\varsigma }{\overset{\infty }{\int }}\lambda \left(s\right)g\left(s\right)ds$.

In 1999 Yang and Hwang [9] generalized the inequality (9) due to Levinson and proved that, if $p>1$, $\lambda \left(\varsigma \right)$, $q\left(\varsigma \right)$, $g\left(\varsigma \right)$ are non-negative functions and there exists a constant $K>0$ such that

$$\begin{array}{c}\hfill p-1+\frac{q^{\prime }\left(\varsigma \right)\Lambda \left(\varsigma \right)}{q^2\left(\varsigma \right)\lambda \left(\varsigma \right)}\ge \frac{p}{K},\end{array}$$

for all $\varsigma >0,$ then

$$\begin{array}{c}\hfill \underset{0}{\overset{\infty }{\int }}\lambda \left(\varsigma \right){\left(\frac{\Phi \left(\varsigma \right)}{\Lambda \left(\varsigma \right)}\right)}^{p}d\varsigma \le K^p\underset{0}{\overset{\infty }{\int }}\lambda \left(\varsigma \right)g^p\left(\varsigma \right)d\varsigma ,\end{array}$$

(12)

where $\Phi \left(\varsigma \right)=\underset{0}{\overset{\varsigma }{\int }}\lambda \left(s\right)q\left(s\right)g\left(s\right)ds$ and $\Lambda \left(\varsigma \right)=\underset{0}{\overset{\varsigma }{\int }}\lambda \left(s\right)g\left(s\right)ds.$

S. Hilger have introduced the theory of time scales (unification as well extension of discrete and continuous calculus) in [10]. Some developments in this direction can be found in [11,12]. Different authors have adopted different techniques to study dynamic inequalities on time scales. S. H. Saker with different co-authors verified many variants of Hardy inequalities on time scales in [1,5,13,14,15,16,17]. Dynamic Hardy-type inequalities with non-conjugate parameters are studied in [18] and dynamic inequalities in quotients with general kernels and measures are proved in [19]. In [20,21], the delta fractional integrals are used instead of classical delta integrals to generalize Hardy-type inequalities for convex and superquadratic functions, respectively. Next, we have included few dynamical Hardy inequalities.

In [22], P. Rehak proved the time scale version of (6) and found its applications to half-linear dynamic equations. His inequality is the following one: if $p>1$, g is a nonnegative rd-continuous function and the delta integral $\underset{a}{\overset{\infty }{\int }}{g}^p\left(\varsigma \right)\Delta \varsigma$ exists as a finite number, then

$$\begin{array}{c}\hfill \underset{a}{\overset{\infty }{\int }}{\left(\frac{1}{\sigma \left(\varsigma \right)-a}\underset{a}{\overset{\varsigma }{\int }}g\left(s\right)\Delta s\right)}^p\Delta \varsigma \le {\left(\frac{p}{p-1}\right)}^p\underset{a}{\overset{\infty }{\int }}{g}^p\left(\varsigma \right)\Delta \varsigma .\end{array}$$

(13)

They also found that if, in addition, $\frac{\mu \left(\varsigma \right)}{\varsigma }\to 0$ as $t\to \infty$, then the constant in (13) is the best possible. In [16] the authors proved the time scale version of (7) which is given by

$$\begin{array}{c}\hfill \underset{a}{\overset{\infty }{\int }}\frac{1}{{\varsigma }^{\gamma }}{\left(\underset{a}{\overset{\varsigma }{\int }}g\left(s\right)\Delta s\right)}^p\Delta \varsigma \le {\left(\frac{p{K}^{\gamma }}{\gamma -1}\right)}^p\underset{a}{\overset{\infty }{\int }}\frac{1}{{\varsigma }^{\gamma -p}}{g}^p\left(\varsigma \right)\Delta \varsigma .\end{array}$$

(14)

where $p>1,\gamma >1$ and there exists a constant $K>0,$ with $\frac{t}{\sigma \left(t\right)}>\frac{1}{K}$, for $t\in \mathbb{T}$. They also proved that if $p>1$, and $\gamma <1$, then

$$\begin{array}{c}\hfill \underset{a}{\overset{\infty }{\int }}\frac{1}{{\sigma }^{\gamma }\left(\varsigma \right)}{\left(\underset{\varsigma }{\overset{\infty }{\int }}g\left(s\right)\Delta s\right)}^p\Delta \varsigma \le {\left(\frac{p}{1-\gamma }\right)}^p\underset{a}{\overset{\infty }{\int }}\frac{1}{{\sigma }^{\gamma -p}\left(\varsigma \right)}{g}^p\left(\varsigma \right)\Delta \varsigma .\end{array}$$

(15)

Hardy–Leindler-type inequalities in the settings of time scale calculus ([15], Theorems 2.9 and 2.10) are the following: for any time scale $\mathbb{T}$, let $a\in$${[0,\infty )}_{\mathbb{T}},$$0\le c<1,$ $r>1$, and $\zeta$ and g be non-negative functions. If

$$\begin{array}{c}\hfill \begin{array}{cccc}\Lambda \left(\varsigma \right)=\underset{\varsigma }{\overset{\infty }{\int }}\zeta \left(s\right)\Delta s<\infty ,& and& \overline{\Psi }\left(\varsigma \right):=\underset{a}{\overset{\varsigma }{\int }}\zeta \left(s\right)g\left(s\right)\Delta s<\infty & \begin{array}{cc}for& \begin{array}{cc}any& \varsigma \in {\left[a,\infty \right)}_{\mathbb{T}},\end{array}\end{array}\end{array}\end{array}$$

then

$$\begin{array}{c}\hfill {\int }_a^{\infty }\frac{\zeta \left(\varsigma \right)}{{(\Lambda \left(\varsigma \right))}^c}{\left({\overline{\Psi }}^{\sigma }\left(\varsigma \right)\right)}^r\Delta \varsigma \le \frac{r}{1-c}{\int }_a^{\infty }\frac{{\left({\overline{\Psi }}^{\sigma }\left(\varsigma \right)\right)}^{r-1}}{{(\Lambda \left(\varsigma \right))}^{1-c}}\zeta \left(\varsigma \right)g\left(\varsigma \right)\Delta \varsigma ,\end{array}$$

and

$$\begin{array}{c}\hfill {\int }_a^{\infty }\frac{\zeta \left(\varsigma \right)}{{(\Lambda \left(\varsigma \right))}^c}{\left({\overline{\Psi }}^{\sigma }\left(\varsigma \right)\right)}^r\Delta \varsigma \le {(\frac{r}{1-c})}^r{\int }_a^{\infty }\frac{\zeta \left(\varsigma \right)}{{(\Lambda \left(\varsigma \right))}^{c-r}}{g}^r\left(\varsigma \right)\Delta \varsigma .\end{array}$$

(16)

For any time scale $\mathbb{T}$, let $a\in {[0,\infty )}_{\mathbb{T}}$, $r\ge c>1,$ and $\zeta$ and g be non-negative. If

$$\begin{array}{c}\hfill \begin{array}{cccc}\Lambda \left(\varsigma \right)=\underset{\varsigma }{\overset{\infty }{\int }}\zeta \left(s\right)\Delta s<\infty ,& and& \Psi \left(\varsigma \right):=\underset{\varsigma }{\overset{\infty }{\int }}\zeta \left(s\right)g\left(s\right)\Delta s<\infty ,& \begin{array}{cc}for& \begin{array}{cc}any& \varsigma \in {\left[a,\infty \right)}_{\mathbb{T}},\end{array}\end{array}\end{array}\end{array}$$

then

$$\begin{array}{c}\hfill {\int }_a^{\infty }\frac{\zeta \left(\varsigma \right)}{{(\Lambda \left(\varsigma \right))}^c}{(\Psi \left(\varsigma \right))}^r\Delta \varsigma \le \frac{r}{c-1}{\int }_a^{\infty }\frac{{(\Psi \left(\varsigma \right))}^{r-1}}{{(\Lambda \left(\varsigma \right))}^{c-1}}\zeta \left(\varsigma \right)g\left(\varsigma \right)\Delta \varsigma ,\end{array}$$

and

$$\begin{array}{c}\hfill {\int }_a^{\infty }\frac{\zeta \left(\varsigma \right)}{{(\Lambda \left(\varsigma \right))}^c}{(\Psi \left(\varsigma \right))}^r\Delta \varsigma \le {(\frac{r}{c-1})}^r{\int }_a^{\infty }\frac{\zeta \left(\varsigma \right)}{{(\Lambda \left(\varsigma \right))}^{c-r}}{g}^r\left(\varsigma \right)\Delta \varsigma .\end{array}$$

(17)

However, the next results consist of Hardy–Yang- and Hwang-type inequalities ([17], Theorem 2.1, Theorem 2.2), respectively. Denote a time scale by $\mathbb{T}$ with $a\in [0,\infty )$. Let $1<\gamma \le r$ and $q\left(v\right)$ be an growing function on $[a,\infty )$. Additionally adopt that there occurs a fixed $k>0$ such that,

$$\gamma -1+\frac{q^{\Delta }\left(v\right){\Lambda }^{\sigma }\left(v\right){\Phi }^r\left(v\right)}{\lambda \left(v\right){\left(q^{\sigma }\left(v\right)\right)}^{{}^2}{\left({\Phi }^{\sigma }\left(v\right)\right)}^r}\ge \frac{r}{k},$$

then

$$\underset{a}{\overset{\infty }{\int }}\frac{\lambda \left(v\right)}{{\left({\Lambda }^{\sigma }\left(v\right)\right)}^{\gamma }}{\left({\Phi }^{\sigma }\left(v\right)\right)}^r\Delta v\le k^r\underset{a}{\overset{\infty }{\int }}\frac{{\left({\Lambda }^{\sigma }\left(v\right)\right)}^{\gamma \left(r-1\right)}}{\Lambda {\left(v\right)}^{\left(\gamma -1\right)r}}\lambda \left(v\right)g^r\left(v\right)\Delta v,$$

(18)

where

$$\begin{array}{c}\hfill \Phi \left(v\right):={\int }_a^v\lambda \left(s\right)q\left(s\right)f\left(s\right)\Delta s<\infty ,\end{array}$$

and

$$\begin{array}{c}\hfill \Lambda \left(v\right):={\int }_a^v\lambda \left(s\right)q^{\sigma }\left(s\right)\Delta s<\infty .\end{array}$$

For any time scale $\mathbb{T}$, let $a\in {[0,\infty )}_{\mathbb{T}}$, $p>1$, $0\le \gamma \le 1$ and $\lambda \left(t\right)$, $q\left(t\right)$ be increasing functions on ${[a,\infty )}_{\mathbb{T}}$. Furthermore, for a constant $k>0$. Consider

$$1-\gamma +\frac{q^{\Delta }\left(t\right){\Lambda }^{\sigma }\left(t\right)}{\lambda \left(t\right)q^2\left(t\right)}\ge \frac{p}{k},t\in {[a,\infty )}_{\mathbb{T}},$$

then

$${\int }_a^{\infty }\frac{\lambda \left(t\right)}{{\left({\Lambda }^{\sigma }\left(t\right)\right)}^{\gamma }}{\left(\overline{\Phi }\left(t\right)\right)}^p\Delta t\le k^p{\int }_a^{\infty }{\left({\Lambda }^{\sigma }\left(t\right)\right)}^{(p-\gamma )}\lambda \left(t\right)g^p\left(t\right)\Delta t,$$

(19)

where

$$\begin{array}{c}\hfill \overline{\Phi }\left(t\right):={\int }_t^{\infty }\lambda \left(s\right)q\left(s\right)g\left(s\right)\Delta s,\end{array}$$

and

$$\begin{array}{c}\hfill \Lambda \left(t\right):={\int }_a^t\lambda \left(s\right)q\left(s\right)\Delta s.\end{array}$$

Dynamic calculus on time scales for functions depending upon more than one variable along with many applications is compiled in [11]. Study of Hardy-type inequalities on time scales via convexity in several variables can be found in [23]. Some refinements of results in [23] including multidimensional dynamic inequalities with general kernels and measures are given in [24]. Further some results concerning with multivariate Hardy type inequalities can be seen in [25,26,27,28].

Motivated by this trend of developing inequalities for function of several variables, Hardy–Leindler-, Hardy–Yang- and Hwang-type inequalities are extended for multiple delta integrals in this paper. The organization of this paper is described as follows: Section 2 deals the precepts related to basic essentials of time scales calculus. Section 3 consists of the main results of the paper, in which Hardy–Leindler-type inequalities for function of several variables are proved and special cases are studied for symmetric sums and multiple classical integrals by assigning particular values to time scales and by making restrictions to the functions involved in proved inequalities. Hardy–Yang- and Hwang-type inequalities for functions of more than one variable along with applications are also part of Section 3. Finally concluding remarks are given in Section 5.

2. Preliminaries

A time scale is made up of closed sets of real numbers [12]. Therefore, for sets of real numbers, counting numbers are the models of time scales. However, for the sets of rational numbers, complex numbers and open interval are not time scales. The operator $\sigma :\mathbb{T}\to \mathbb{T}$, for $s\in \mathbb{T}$ is called forward jump operator if $\sigma \left(s\right):=inf\left\{a\in \mathbb{T};a>s\right\}$ and the operator $\rho :\mathbb{T}\to \mathbb{T}$, for $s\in \mathbb{T}$, is called backward jump operator if $\rho \left(s\right):=sup\left\{a\in \mathbb{T};a<s\right\}$. If the point $s\in \mathbb{T}$ satisfies $\sigma \left(s\right)>s$, then it is right scattered and if $\rho \left(s\right)<s$, then it is left scattered. If a point is left scattered and right scattered simultaneously, then it is called an isolated point. Similarly, if $s<sup\mathbb{T}$ and $\sigma \left(s\right)=s$, then it is right dense, and it is left dense if $s>inf\mathbb{T}$ and $\rho \left(s\right)=s$. The points which are left dense as well as right dense are called dense points. Let $\mu ,\nu :\mathbb{T}\to \mathbb{R}$ be defined by $\mu \left(\kappa \right)=\sigma \left(\kappa \right)-\kappa$ and $\nu \left(\kappa \right)=\kappa -\rho \left(\kappa \right)$ for $\forall \kappa \in \mathbb{T}$ are forward and backward graininess functions, respectively. Suppose a function $g:\mathbb{T}\to \mathbb{R}$ satisfies:

• g is continuous at right dense points on $\mathbb{T}$;
• the left hand limits exist and are finite at left dense points on $\mathbb{T}$;
  then, g is right-dense continuous (rd-continuous) on $\mathbb{T}$.

The set of all $rd$ continuous functions is denoted by $C_{rd}\left(\mathbb{T}\right)$.

Define

$${\mathbb{T}}^k=\left\{\begin{array}{c}\mathbb{T}\setminus (\rho (\sup \mathbb{T}),sup\mathbb{T}],\mathrm{if}\sup \mathbb{T}<\infty ,\\ \mathbb{T},\mathrm{otherwise}.\end{array}\right.$$

Assume $\omega :\mathbb{T}\to \mathbb{R}$ is a function and let $\varsigma \in {\mathbb{T}}^k.$ Then we define ${\omega }^{\Delta }\left(\varsigma \right)$ to be the number (provided it exists) with the property that for given $ϵ>0,$ there is a neighborhood P of $\varsigma$ (i.e., $P=(\varsigma -\delta ,\varsigma +\delta )\cap \mathbb{T}$ for some $\delta >0$) such that

$$|[\omega \left(\sigma \left(\varsigma \right)\right)-\omega \left(r\right)]-{\omega }^{\Delta }\left(\varsigma \right)[\sigma \left(\varsigma \right)-r]|\le ϵ|\sigma \left(\varsigma \right)-r|,$$

holds for all $r\in P.$ We call $\omega$ is delta differentiable at $\varsigma$ or ${\omega }^{\Delta }\left(\varsigma \right)$ is the delta (or Hilger) derivative of $\omega$ at $\varsigma$.

A function $W:\mathbb{T}\to \mathbb{R}$ is called an antiderivative of $w:\mathbb{T}\to \mathbb{R}$ provided

$$W^{\Delta }\left(\varsigma \right)=w\left(\varsigma \right)$$

holds for all $\varsigma \in {\mathbb{T}}^k.$

For $W^{\Delta }\left(\varsigma \right)=w\left(\varsigma \right),$$\varsigma \in {\mathbb{T}}^k$, the delta integral of w is stated as:

$${\int }_l^{\varsigma }w\left(s\right)\Delta s=W\left(\varsigma \right)-W\left(l\right),\mathrm{for}l\in \mathbb{T}.$$

Moreover, for $w\in {\mathbb{C}}_{rd}({\mathbb{T}}^k,\mathbb{R}),$ the Cauchy integral

$$W\left(\varsigma \right):={\int }_{{\varsigma }_0}^{\varsigma }w\left(s\right)\Delta s,$$

exists for ${\varsigma }_0\in {\mathbb{T}}^k$ and satisfies $W^{\Delta }\left(\varsigma \right)=w\left(\varsigma \right)$.

An indefinite integral is defined as:

$${\int }_t^{\infty }w\left(\varsigma \right)\Delta \varsigma =\underset{p\to \infty }{lim}{\int }_t^{p}w\left(\varsigma \right)\Delta \varsigma \mathrm{for}t\in {\mathbb{T}}^k.$$

Fubini’s Theorem [29]:

Let there exist two time scales measure spaces $(B,P,{\sigma }_{\Delta })$ and $(A,P,{\kappa }_{\Delta })$ which have finite dimensions. If $\eta :B\times A\to \mathbb{R}$ is ${\sigma }_{\Delta }\times {\kappa }_{\Delta }$ integrable function and if we define the function $\varphi \left(p\right)={\int }_B\eta (l,p)\Delta l$ for almost every $p\in A$ and $\tau \left(l\right)={\int }_A\eta (l,p)\Delta p$ for almost every $l\in B$, then $\Phi$ is ${\kappa }_{\Delta }$ integrable on $A,\tau$ is ${\sigma }_{\Delta }$ integrable on B, and

$$\begin{array}{c}\hfill {\int }_B\Delta l{\int }_A\eta (l,p)\Delta p={\int }_A\Delta p{\int }_B\eta (l,p)\Delta l.\end{array}$$

(20)

3. Hardy–Leindler-Type Inequalities for Multiple Delta Integrals

In the sequel, following notations are used:

${\mathit{\varsigma }}_{\mathbf{n}}=({\varsigma }_1,\dots ,{\varsigma }_n),$${\mathbf{s}}_{\mathbf{n}}=(s_1,\dots ,s_n)$ and ${\left(0,\infty \right)}_{{\mathbb{T}}_{\iota }}=\left(0,\infty \right)\cap {\mathbb{T}}_{\iota }.$

3.1. Hardy–Leindler-Type Inequality $r>1,0\le c_{\iota }<1$

Theorem 1.

Let ${\mathbb{T}}_{\iota }$ be a time scale, ${\zeta }_{\iota }:{\mathbb{T}}_{\iota }\to {\mathbb{R}}^{+}$ is such that ${\Lambda }_{\iota }\left({\varsigma }_{\iota }\right):={\int }_{{\varsigma }_{\iota }}^{\infty }{\zeta }_{\iota }\left(s_{\iota }\right)\Delta s_{\iota },$ exists with ${\Lambda }_{\iota }(\infty )=0$ for $\iota =1,2,\dots ,n$ and $g:{\mathbb{T}}_1\times {\mathbb{T}}_2\cdots \times {\mathbb{T}}_n\to {\mathbb{R}}^{+}$ is such that

$$\overline{\Psi }\left({\varsigma }_{\mathbf{n}}\right):={\int }_{a_1}^{{\varsigma }_1}\cdots {\int }_{a_n}^{{\varsigma }_n}\prod _{\iota =1}^n{\zeta }_{\iota }\left(s_{\iota }\right)g\left({\varsigma }_{\mathbf{n}}\right)\Delta {\varsigma }_n\cdots \Delta {\varsigma }_1,$$

(21)

exists. Then for $a_{\iota }\in {[0,\infty )}_{{\mathbb{T}}_{\iota }},$ we have

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{\iota =1}^n\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\Lambda }_{\iota }^{c_{\iota }}\left({\varsigma }_{\iota }\right)}\right]{\left({\overline{\Psi }}^{{\sigma }_1\cdots {\sigma }_n}\left({\mathit{\varsigma }}_{\mathbf{n}}\right)\right)}^r\Delta {\varsigma }_n\cdots \Delta {\varsigma }_1\hfill \\ \hfill \le \prod _{\iota =1}^n\left[{\left(\frac{r}{1-c_{\iota }}\right)}^r\right]{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{\iota =1}^n\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\Lambda }_{\iota }^{c_{\iota }-r}\left({\varsigma }_{\iota }\right)}\right]g^r\left({\mathit{\varsigma }}_{\mathbf{n}}\right)\Delta {\varsigma }_n\cdots \Delta {\varsigma }_1.\end{array}$$

(22)

Proof. 

To prove the result we use the principle of mathematical induction. For $n=1,$ the statement is true by (16). Assume that (22) holds for $1\le n\le v$. To prove the result for $n=v+1$, the left-hand side of (22) can be written as

$${\int }_{a_1}^{\infty }\cdots {\int }_{a_v}^{\infty }\prod _{\iota =1}^v\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\Lambda }_{\iota }^{c_{\iota }}\left({\varsigma }_{\iota }\right)}\right]\times \left\{{\int }_{a_{v+1}}^{\infty }\frac{{\zeta }_{v+1}\left({\varsigma }_{v+1}\right)}{{\Lambda }_{v+1}^{c_{v+1}}\left({\varsigma }_{v+1}\right)}{\left({\overline{\Psi }}^{{\sigma }_1\cdots {\sigma }_{v+1}}\left({\mathit{\varsigma }}_{\mathbf{v}+\mathbf{1}}\right)\right)}^r\Delta {\varsigma }_{v+1}\right\}\Delta {\varsigma }_v\cdots \Delta {\varsigma }_1.$$

(23)

Denote

$$\begin{array}{c}\hfill I_{v+1}={\int }_{a_{v+1}}^{\infty }\frac{{\zeta }_{v+1}\left({\varsigma }_{v+1}\right)}{{\Lambda }_{v+1}^{c_{v+1}}\left({\varsigma }_{v+1}\right)}{\left({\overline{\Psi }}^{{\sigma }_1\cdots {\sigma }_{v+1}}\left({\mathit{\varsigma }}_{\mathbf{v}+\mathbf{1}}\right)\right)}^r\Delta {\varsigma }_{v+1}.\end{array}$$

(24)

Use (16) in (24) with respect for fix $\left({\mathit{\varsigma }}_{\mathbf{v}}\right)\in {\mathbb{T}}_1\times \cdots \times {\mathbb{T}}_v$ to get,

$$\begin{array}{c}\hfill I_{v+1}\le {\left(\frac{r}{1-c_{v+1}}\right)}^r{\int }_{a_{v+1}}^{\infty }\frac{{\zeta }_{v+1}\left({\varsigma }_{v+1}\right){\left({\overline{\Psi }}^{{\sigma }_1\cdots {\sigma }_v}({\mathit{\varsigma }}_{\mathbf{v}},{\varsigma }_{v+1})\right)}^r}{{\left[{\Lambda }_{v+1}\left({\varsigma }_{v+1}\right)\right]}^{c_{v+1}-r}}\Delta {\varsigma }_{v+1}.\end{array}$$

(25)

Use (25) in (23) and use (20) v-times on the right-hand side of the resultant inequality to obtain

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_v}^{\infty }\left(\prod _{\iota =1}^v\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\Lambda }_{\iota }^{c_{\iota }}\left({\varsigma }_{\iota }\right)}\right)\left\{{\int }_{a_{v+1}}^{\infty }\frac{{\zeta }_{v+1}\left({\varsigma }_{v+1}\right){\left({\overline{\Psi }}^{{\sigma }_1\cdots {\sigma }_{v+1}}({\mathit{\varsigma }}_{\mathbf{v}},{\varsigma }_{v+1})\right)}^r}{{\left({\Lambda }_{v+1}\left({\varsigma }_{v+1}\right)\right)}^{c_{v+1}}}\Delta {\varsigma }_{v+1}\right\}\Delta {\varsigma }_v\cdots \Delta {\varsigma }_1\hfill \\ \le {\left(\frac{r}{1-c_{v+1}}\right)}^r{\int }_{a_{v+1}}^{\infty }\frac{{\zeta }_{v+1}\left({\varsigma }_{v+1}\right)}{{\Lambda }_{v+1}^{c_{v+1}-r}\left({\varsigma }_{v+1}\right)}\\ \hfill \times \left\{{\int }_{a_1}^{\infty }\cdots {\int }_{a_v}^{\infty }\prod _{\iota =1}^v\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\Lambda }_{\iota }^{c_{\iota }}\left({\varsigma }_{\iota }\right)}{\left({\overline{\Psi }}^{{\sigma }_1\dots {\sigma }_v}({\mathit{\varsigma }}_{\mathbf{v}},{\varsigma }_{v+1})\right)}^r\Delta {\varsigma }_v\cdots \Delta {\varsigma }_1\right\}\Delta {\varsigma }_{v+1}.\end{array}$$

Use induction hypothesis for fixed ${\varsigma }_{v+1}\in {\mathbb{T}}_{v+1}$ on ${\overline{\Psi }}^{{\sigma }_1\cdots {\sigma }_v}({\mathit{\varsigma }}_{\mathbf{v}},{\varsigma }_{v+1})$ instead of ${\overline{\Psi }}_v^{{\sigma }_1\cdots {\sigma }_v}\left({\mathit{\varsigma }}_{\mathbf{v}}\right)$ to obtain

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_{v+1}}^{\infty }\prod _{\iota =1}^{v+1}\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\left({\Lambda }_{\iota }\left({\varsigma }_{\iota }\right)\right)}^{c_{\iota }}}\right]{\left({\overline{\Psi }}^{{\sigma }_1\cdots {\sigma }_{v+1}}\left({\mathit{\varsigma }}_{\mathbf{v}+\mathbf{1}}\right)\right)}^r\Delta {\varsigma }_v\cdots \Delta {\varsigma }_1\hfill \\ \hfill \le \prod _{\iota =1}^{v+1}\left[{\left(\frac{r}{1-c_{\iota }}\right)}^r\right]{\int }_{a_1}^{\infty }\cdots {\int }_{a_{v+1}}^{\infty }\prod _{\iota =1}^{v+1}\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\left({\Lambda }_{\iota }\left({\varsigma }_{\iota }\right)\right)}^{c_{\iota }-r}}\right]g^r\left({\varsigma }_{\mathbf{v}+\mathbf{1}}\right)\Delta {\varsigma }_{v+1}\cdots \Delta {\varsigma }_1.\end{array}$$

Hence by the principle of mathematical induction inequality (22) is true for all natural numbers n. □

Example 1.

Choose ${\mathbb{T}}_{\iota }=\mathbb{R}$ for all $\iota =1,\dots ,n,$ in Theorem 1. In this case (22) takes the form,

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{\iota =1}^n\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\left({\Lambda }_{\iota }\left({\varsigma }_{\iota }\right)\right)}^{c_{\iota }}}\right](\overline{\Psi }{\left({\varsigma }_{\mathbf{n}}\right))}^{r}d{\varsigma }_n\cdots d{\varsigma }_1\hfill \\ \hfill \le \prod _{\iota =1}^n\left[{\left(\frac{r}{1-c_{\iota }}\right)}^r\right]{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{\iota =1}^n\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\left({\Lambda }_{\iota }\left({\varsigma }_{\iota }\right)\right)}^{c_{\iota }-r}}\right]g^r({\varsigma }_1,\dots ,{\varsigma }_n)d{\varsigma }_n\cdots d{\varsigma }_1.\end{array}$$

The following result is extension of (4).

Example 2.

Choose ${\mathbb{T}}_{\iota }=h_{\iota }\mathbb{N},$ $h_{\iota }>0,$ $a_{\iota }=h_{\iota }$, $\forall \iota \in \{1,\dots ,n\}$, in Theorem 1. In this case (22) takes the form,

$$\begin{array}{c}\sum _{p_1=1}^{\infty }\cdots \sum _{p_n=1}^{\infty }\prod _{\iota =1}^n\left(\frac{{\zeta }_{\iota }\left(h_i{p}_{\iota }\right)}{{\left({\Lambda }_{\iota }\left(h_i{p}_{\iota }\right)\right)}^{c_{\iota }}}{h}_{\iota }\right){\overline{\Psi }}^r(h_1(p_1+1),\dots ,h_n(p_n+1))\hfill \\ \hfill \le \prod _{\iota =1}^n{\left(\frac{r}{1-c_{\iota }}\right)}^r\sum _{p_1=1}^{\infty }\cdots \sum _{p_n=1}^{\infty }\prod _{\iota =1}^n\left(\frac{{\zeta }_{\iota }\left(h_i{p}_{\iota }\right)}{{\left({\Lambda }_{\iota }\left(h_i{p}_{\iota }\right)\right)}^{c_{\iota }-r}}{h}_{\iota }\right)g^r(h_1{p}_1,\cdots ,h_n{p}_n),\end{array}$$

where $\overline{\Psi }(h_1{p}_1,\dots ,h_n{p}_n)={\sum }_{{\alpha }_1=1}^{p_1}\cdots {\sum }_{{\alpha }_n=1}^{p_n}{\prod }_{\iota =1}^n{h}_{\iota }{\zeta }_{\iota }\left(h_{\iota }{\alpha }_{\iota }\right)g(h_1{\alpha }_1,\dots ,h_n{\alpha }_n).$

Example 3.

Choose ${\mathbb{T}}_{\iota }=q^{\mathbb{N}}=\left\{q_{\iota }^{{\alpha }_{\iota }}:{\alpha }_{\iota }\in \mathbb{N},q_{\iota }>1\right\}$ and $a_{\iota }=q_{\iota }$, $\forall \iota \in \{1,\dots ,n\}$ in Theorem 1. In this case (22) takes the form,

$$\begin{array}{c}\sum _{{\alpha }_1=1}^{\infty }\cdots \sum _{{\alpha }_n=1}^{\infty }\prod _{\iota =1}^n\left(\frac{q_{\iota }^{{\alpha }_{\iota }}{\zeta }_{\iota }\left(q_{\iota }^{{\alpha }_{\iota }}\right)}{\left({\Lambda }_{\iota }{\left(q_{\iota }^{{\alpha }_{\iota }}\right)}^{c_{\iota }}\right)}\right){\overline{\Psi }}^r(q_1^{{\alpha }_1+1},\dots ,q_n^{{\alpha }_n+1})\hfill \\ \hfill \le \prod _{\iota =1}^n{\left(\frac{r}{1-c_{\iota }}\right)}^r\sum _{{\alpha }_1=1}^{\infty }\cdots \sum _{{\alpha }_n=1}^{\infty }\prod _{\iota =1}^n\left(\frac{q_{\iota }^{{\alpha }_{\iota }}{\zeta }_{\iota }\left(q_{\iota }^{{\alpha }_{\iota }}\right)}{\left({\Lambda }_{\iota }{\left(q_{\iota }^{{\alpha }_{\iota }}\right)}^{c_{\iota }-p}\right)}\right)g^r(q_1^{{\alpha }_1},\dots ,q_n^{{\alpha }_n}),\end{array}$$

where $\overline{\Psi }(q_1^{{\alpha }_1+1},\dots ,q_n^{{\alpha }_n+1})={\sum }_{{\beta }_1=1}^{{\alpha }_1}\cdots {\sum }_{{\beta }_n=1}^{{\alpha }_n}{\prod }_{\iota =1}^n\left({\zeta }_{\iota }\left(q_{\iota }^{{\beta }_{\iota }}\right)q_{\iota }^{{\beta }_{\iota }}(q_{\iota }-1)\right)g(q_1^{{\beta }_1},\dots ,q_n^{{\beta }_n}).$

3.2. Hardy–Leindler-Type Inequality for $r\ge c_{\iota }>1$

Theorem 2.

Let ${\mathbb{T}}_{\iota }$ be a time scale, ${\zeta }_{\iota }:{\mathbb{T}}_{\iota }\to {\mathbb{R}}^{+}$ is such that ${\Lambda }_{\iota }\left({\varsigma }_{\iota }\right):={\int }_{{\varsigma }_{\iota }}^{\infty }{\zeta }_{\iota }\left(s_{\iota }\right)\Delta s_{\iota }$ exists for $\iota =1,2,\dots ,n$ and $g:{\mathbb{T}}_1\times {\mathbb{T}}_2\times \cdots \times {\mathbb{T}}_n\to {\mathbb{R}}^{+}$ is such that

$$\Psi \left({\varsigma }_{\mathbf{n}}\right):={\int }_{{\varsigma }_1}^{\infty }\cdots {\int }_{{\varsigma }_n}^{\infty }\prod _{\iota =1}^n{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)g\left({\varsigma }_{\mathbf{n}}\right)\Delta {\varsigma }_n\cdots \Delta {\varsigma }_1,$$

exists. Then for $a_{\iota }\in {[0,\infty )}_{{\mathbb{T}}_{\iota }}$, we have

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{\iota =1}^n\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\left({\Lambda }_{\iota }\left({\varsigma }_{\iota }\right)\right)}^{c_{\iota }}}\right]{(\Psi \left({\varsigma }_{\mathbf{n}}\right))}^r\Delta {\varsigma }_n\cdots \Delta {\varsigma }_1\hfill \\ \hfill \le \prod _{\iota =1}^n\left[{\left(\frac{r}{c_{\iota }-1}\right)}^r\right]{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{\iota =1}^n\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\left({\Lambda }_{\iota }\left({\varsigma }_{\iota }\right)\right)}^{c_{\iota }-r}}\right]g^r\left({\varsigma }_{\mathbf{n}}\right)\Delta {\varsigma }_n\cdots \Delta {\varsigma }_1.\end{array}$$

(26)

Proof. 

To prove the result, we use the principle of mathematical induction. For $n=1,$ the statement is true by (17). Assume that (26) holds for $1\le n\le v$. To prove the result for $n=v+1$, the left hand side of (26) can be written as

$$\begin{array}{c}\hfill {\int }_{a_1}^{\infty }\cdots {\int }_{a_v}^{\infty }\prod _{\iota =1}^v\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\left({\Lambda }_{\iota }\left({\varsigma }_{\iota }\right)\right)}^{c_{\iota }}}\right]\left\{{\int }_{a_{v+1}}^{\infty }\frac{{\zeta }_{v+1}\left({\varsigma }_{v+1}\right)}{\left({\Lambda }_{v+1}^{c_{v+1}}\left({\varsigma }_{v+1}\right)\right)}(\Psi \left({\varsigma }_{\mathbf{v}+\mathbf{1}}\right)){}^r\Delta {\varsigma }_{v+1}\right\}\Delta {\varsigma }_v\cdots \Delta {\varsigma }_1.\end{array}$$

(27)

Denote

$$\begin{array}{c}\hfill I_{v+1}={\int }_{a_{v+1}}^{\infty }\frac{{\zeta }_{v+1}\left({\varsigma }_{v+1}\right)}{{\Lambda }_{v+1}^{c_{v+1}}\left({\varsigma }_{v+1}\right)}{(\Psi \left({\varsigma }_{\mathbf{v}+\mathbf{1}}\right))}^r\Delta {\varsigma }_{v+1}.\end{array}$$

(28)

Apply (17) on (28) with respect to ${\varsigma }_{v+1}\in {\mathbb{T}}_{v+1}$ for fix $\left({\varsigma }_{\mathbf{v}}\right)\in {\mathbb{T}}_1\times \cdots \times {\mathbb{T}}_v$ to obtain,

$$\begin{array}{c}\hfill I_{v+1}\le {\left(\frac{r}{c_{v+1}-1}\right)}^r{\int }_{a_{v+1}}^{\infty }\frac{{\zeta }_{v+1}\left({\varsigma }_{v+1}\right){[\Psi ({\mathit{\varsigma }}_{\mathbf{v}},{\varsigma }_{v+1})]}^r}{{\left[{\Lambda }_{v+1}\left({\varsigma }_{v+1}\right)\right]}^{c_{v+1}-r}}\Delta {\varsigma }_{v+1},\end{array}$$

(29)

where

$$\Psi ({\varsigma }_1,\dots ,{\varsigma }_v)={\int }_{{\varsigma }_1}^{\infty }\cdots {\int }_{{\varsigma }_v}^{\infty }\prod _{\iota =1}^v{\zeta }_{\iota }\left({\eta }_{\iota }\right)g\left({\eta }_{\mathbf{v}}\right)\Delta {\eta }_v\cdots \Delta {\eta }_1.$$

Substitute (29) in (27) and use (20) v-times on the right hand side of the resultant inequality to obtain

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_v}^{\infty }\prod _{\iota =1}^v\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\left({\Lambda }_{\iota }\left({\varsigma }_{\iota }\right)\right)}^{c_{\iota }}}\right]\left\{{\int }_{a_{v+1}}^{\infty }\frac{{\zeta }_{v+1}\left({\varsigma }_{v+1}\right)}{\left({\Lambda }_{v+1}^{c_{v+1}}\left({\varsigma }_{v+1}\right)\right)}{(\Psi \left({\varsigma }_{\mathbf{v}+\mathbf{1}}\right))}^r\Delta {\varsigma }_{v+1}\right\}\Delta {\varsigma }_v\cdots \Delta {\varsigma }_1\hfill \\ \le {\left(\frac{r}{c_{v+1}-1}\right)}^r{\int }_{a_{v+1}}^{\infty }\frac{{\zeta }_{v+1}\left({\varsigma }_{v+1}\right)}{{\left[{\Lambda }_{v+1}\left({\varsigma }_{v+1}\right)\right]}^{c_{v+1}-r}}\\ \hfill \times \left\{{\int }_{a_1}^{\infty }\cdots {\int }_{a_v}^{\infty }\prod _{\iota =1}^v\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\left({\Lambda }_{\iota }\left({\varsigma }_{\iota }\right)\right)}^{c_{\iota }}}\right]{[\Psi ({\varsigma }_{\mathbf{v}},{\varsigma }_{v+1})]}^r\Delta {\varsigma }_v\cdots \Delta {\varsigma }_1\right\}\Delta {\varsigma }_{v+1}.\end{array}$$

Use induction hypothesis for fixed ${\varsigma }_{v+1}\in {\mathbb{T}}_{v+1}$ on $\Psi ({\varsigma }_{\mathbf{v}},{\varsigma }_{v+1})$ instead of $\Psi \left({\varsigma }_{\mathbf{v}}\right)$ to obtain

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_v}^{\infty }{\int }_{a_{v+1}}^{\infty }\prod _{\iota =1}^{v+1}\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\Lambda }_{\iota }^{c_{\iota }}\left({\varsigma }_{\iota }\right)}\right]{(\Psi \left({\varsigma }_{\mathbf{v}+\mathbf{1}}\right))}^r\Delta {\varsigma }_{v+1}\cdots \Delta {\varsigma }_1\hfill \\ \hfill \le \prod _{\iota =1}^{v+1}\left[{\left(\frac{r}{c_{\iota }-1}\right)}^r\right]{\int }_{a_1}^{\infty }\cdots {\int }_{a_{v+1}}^{\infty }\prod _{\iota =1}^{v+1}\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\left({\Lambda }_{\iota }\left({\varsigma }_{\iota }\right)\right)}^{c_{\iota }-r}}\right]g^r\left({\varsigma }_{\mathbf{v}+\mathbf{1}}\right)\Delta {\varsigma }_{v+1}\cdots \Delta {\varsigma }_1.\end{array}$$

Hence, by the principle of mathematical induction, (26) is true for all natural numbers n. □

Example 4.

Choose ${\mathbb{T}}_{\iota }={\mathbb{R}}_{+}$ for all $\iota =1,\dots ,n$ in Theorem 2. In this case (26) takes the form,

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{\iota =1}^n\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\left({\Lambda }_{\iota }\left({\varsigma }_{\iota }\right)\right)}^{c_{\iota }}}\right]{(\Psi \left({\mathit{\varsigma }}_{\mathbf{n}}\right))}^{r}d{\varsigma }_n\cdots d{\varsigma }_1\hfill \\ \hfill \le \prod _{\iota =1}^n\left[{\left(\frac{r}{c_{\iota }-1}\right)}^r\right]{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{\iota =1}^n\left[\frac{{\zeta }_{\iota }\left({\varsigma }_{\iota }\right)}{{\left({\Lambda }_{\iota }\left({\varsigma }_{\iota }\right)\right)}^{c_{\iota }-r}}\right]g^r\left({\mathit{\varsigma }}_{\mathbf{n}}\right)d{\varsigma }_n\cdots d{\varsigma }_1.\end{array}$$

The following result is generalization of (5).

Example 5.

Take ${\mathbb{T}}_{\iota }=h_{\iota }\mathbb{N}$ for $h_{\iota }>0$ in Theorem 2 and $a_{\iota }=h_{\iota }$, $\forall \iota \in \{1,\dots ,n\}$. In this case (26) takes the form,

$$\begin{array}{c}\sum _{p_1=1}^{\infty }\dots \sum _{p_n=1}^{\infty }\prod _{\iota =1}^n\left(\frac{{\zeta }_{\iota }\left(h_i{p}_{\iota }\right)}{{\left({\Lambda }_{\iota }\left(h_i{p}_{\iota }\right)\right)}^{c_{\iota }}}{h}_{\iota }\right){\Psi }^r(h_1\left(p_1\right),\dots ,h_n\left(p_n\right))\hfill \\ \hfill \le \prod _{\iota =1}^n{\left(\frac{r}{c_{\iota }-1}\right)}^r\sum _{p_1=1}^{\infty }\dots \sum _{p_n=1}^{\infty }\prod _{\iota =1}^n\left(\frac{{\zeta }_{\iota }\left(h_i{p}_{\iota }\right)}{{\left({\Lambda }_{\iota }\left(h_i{p}_{\iota }\right)\right)}^{c_{\iota }-r}}{h}_{\iota }\right)g^r(h_1{p}_1,\dots ,h_n{p}_n),\end{array}$$

where $\Psi (h_1\left(p_1\right),\dots ,h_n\left(p_n\right))={\sum }_{{\alpha }_1=p_1}^{\infty }\cdots {\sum }_{{\alpha }_n=p_n}^{\infty }{\prod }_{\iota =1}^n{h}_{\iota }{\zeta }_{\iota }\left(h_{\iota }{\alpha }_{\iota }\right)g(h_1{\alpha }_1,\dots ,h_n{\alpha }_n).$

Example 6.

Let ${\mathbb{T}}_{\iota }=q_{\iota }^{\mathbb{N}}=\left\{q_{\iota }^{{\alpha }_{\iota }}:{\alpha }_{\iota }\in \mathbb{N},q_{\iota }>1\right\}$ and $a_{\iota }=q_{\iota }$, $\forall \iota \in \{1,\dots ,n\}$ in Theorem 2. In this case (26) takes the form,

$$\begin{array}{c}\sum _{{\alpha }_1=1}^{\infty }\cdots \sum _{{\alpha }_n=1}^{\infty }\prod _{\iota =1}^n\left(\frac{q_{\iota }^{{\alpha }_{\iota }}{\zeta }_{\iota }\left(q_{\iota }^{{\alpha }_{\iota }}\right)}{\left({\Lambda }_{\iota }{\left(q_{\iota }^{{\alpha }_{\iota }}\right)}^{c_{\iota }}\right)}\right){\Psi }^r(q_1^{{\alpha }_1},\dots ,q_n^{{\alpha }_n})\hfill \\ \hfill \le \prod _{\iota =1}^n{\left(\frac{r}{c_{\iota }-1}\right)}^r\sum _{{\alpha }_1=1}^{\infty }\cdots \sum _{{\alpha }_n=1}^{\infty }\prod _{\iota =1}^n\left(\frac{q_{\iota }^{{\alpha }_{\iota }}{\zeta }_{\iota }\left(q_{\iota }^{{\alpha }_{\iota }}\right)}{\left({\Lambda }_{\iota }{\left(q_{\iota }^{{\alpha }_{\iota }}\right)}^{c_{\iota }-p}\right)}\right)g^r(q_1^{{\alpha }_1},\dots ,q_n^{{\alpha }_n}),\end{array}$$

where $\Psi (q_1^{{\alpha }_1},\dots ,q_n^{{\alpha }_n})={\sum }_{{\beta }_1={\alpha }_1}^{\infty }\cdots {\sum }_{{\beta }_n={\alpha }_n}^{\infty }{\prod }_{\iota =1}^n{\zeta }_{\iota }\left(q_{\iota }^{{\beta }_{\iota }}\right)q_{\iota }^{{\beta }_{\iota }}(q_{\iota }-1)g(q_1^{{\beta }_1},\dots ,q_n^{{\beta }_n}).$

4. Hardy–Yang- and Hwang-Type Inequalities on Time Scales for Functions of Several Variables

Theorem 3.

Let ${\mathbb{T}}_i$ denotes a time scale, $a_i\in {[0,\infty )}_{{\mathbb{T}}_i}$, $q_i\left(t_i\right)$ and ${\lambda }_i\left(t_i\right)$ are increasing functions on ${[a_i,\infty )}_{{\mathbb{T}}_i}$ for $i\in \{1,\dots ,n\}$. Furthermore assume that there exists constant $k_i>0$ such that

$${\gamma }_i-1+\prod _{i=1}^n\frac{q_i^{{\Delta }_i}\left({\varsigma }_i\right){\Lambda }_i^{{\sigma }_i}\left({\varsigma }_i\right){\Phi }^p\left({\varsigma }_{\mathbf{n}}\right)}{{\lambda }_i\left({\varsigma }_i\right){\left(q_i^{{\sigma }_i}\left({\varsigma }_i\right)\right)}^2{\left({\Phi }^{{\sigma }_1\cdots {\sigma }_n}\left({\varsigma }_{\mathbf{n}}\right)\right)}^p}⩾\frac{p}{k_i}.$$

Then for $1<{\gamma }_i\le p$, we have,

$$\begin{array}{c}\hfill {\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{i=1}^n\frac{{\lambda }_i\left({\varsigma }_i\right)}{{\left({\Lambda }_i^{{\sigma }_i}{t}_i\right)}^{{\gamma }_i}}{\left({\Phi }^{{\sigma }_1\cdots {\sigma }_n}\left({\varsigma }_{\mathbf{n}}\right)\right)}^p\Delta {\varsigma }_1\cdots \Delta {\varsigma }_n\\ \hfill \le k^{np}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{i=1}^n\frac{{\left({\Lambda }_i^{{\sigma }_i}\left({\varsigma }_i\right)\right)}^{{\gamma }_i(p-1)}}{{\left({\Lambda }_i\left({\varsigma }_i\right)\right)}^{({\gamma }_i-1)p}}{\lambda }_i\left({\varsigma }_i\right)g^p\left({\varsigma }_{\mathbf{n}}\right)\Delta {\varsigma }_1\cdots \Delta {\varsigma }_n,\end{array}$$

(30)

where,

$$\begin{array}{c}\hfill \Phi ({\varsigma }_1,\dots ,{\varsigma }_n)={\int }_{a_1}^{{\varsigma }_1}\cdots {\int }_{a_n}^{{\varsigma }_n}\prod _{i=1}^n{\lambda }_i\left(s_i\right)q_i\left(s_i\right)g\left(s_{\mathbf{n}}\right)\Delta s_1\cdots \Delta s_n,\end{array}$$

and

$$\begin{array}{c}\hfill {\Lambda }_i\left({\varsigma }_i\right):={\int }_{a_1}^{{\varsigma }_1}\cdots {\int }_{a_n}^{{\varsigma }_n}\prod _{i=1}^n{\lambda }_i\left(s_i\right)q_i^{{\sigma }_i}\left(s_i\right)\Delta s_1\cdots \Delta s_n.\end{array}$$

Proof. 

The result is proved by using the principal of mathematical induction. For $n=1$, statement is true by (18). Suppose for $n=v$, (30) holds. To prove for $n=v+1$, take L.H.S of (30) in the following form:

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_v}^{\infty }\prod _{i=1}^v\frac{{\lambda }_i\left({\varsigma }_i\right)}{{\left({\Lambda }_i^{{\sigma }_i}\left(t_i\right)\right)}^{{\gamma }_i}}\hfill \\ \hfill \times \left({\int }_{a_{v+1}}^{\infty }\frac{{\lambda }_{v+1}\left({\varsigma }_{v+1}\right)}{{\left({\Lambda }_{v+1}^{{\sigma }_{v+1}}\left({\varsigma }_{v+1}\right)\right)}^{{\gamma }_{v+1}}}{\left({\Phi }^{{\sigma }_1\cdots {\sigma }_{v+1}}({\varsigma }_{\mathbf{v}},{\varsigma }_{v+1})\right)}^p\Delta {\varsigma }_{v+1}\right)\Delta {\varsigma }_1\cdots \Delta {\varsigma }_v.\end{array}$$

(31)

Denote

$$\begin{array}{c}\hfill I_{v+1}={\int }_{a_{v+1}}^{\infty }\frac{{\lambda }_{v+1}\left({\varsigma }_{v+1}\right)}{{\Lambda }_{v+1}^{{\sigma }_{v+1}}\left(t_{v+1}\right){)}^{{\gamma }_{v+1}}}{\left({\Phi }^{{\sigma }_1\cdots {\sigma }_{v+1}}({\varsigma }_{\mathbf{v}},{\varsigma }_{v+1})\right)}^p\Delta {\varsigma }_{v+1}.\end{array}$$

(32)

Use (18) in (32) with respect to ${\varsigma }_{v+1}\in {\mathbb{T}}_{v+1}$ for fix ${\varsigma }_{\mathbf{v}}\in {\mathbb{T}}_1\times \dots \times {\mathbb{T}}_v$ to obtain

$$I_{v+1}\le k^p{\int }_{a_{v+1}}^{\infty }\frac{{\left({\Lambda }_{v+1}^{{\sigma }_{v+1}}\left({\varsigma }_{v+1}\right)\right)}^{{\gamma }_{v+1}(p-1)}}{{\Lambda }_{v+1}^{({\gamma }_{v+1}-1)p}\left({\varsigma }_{v+1}\right)}{\lambda }_{v+1}\left({\varsigma }_{v+1}\right){\left({\Phi }^{{\sigma }_1\cdots {\sigma }_v}({\varsigma }_{\mathbf{v}},{\varsigma }_{v+1})\right)}^p\Delta {\varsigma }_{v+1}.$$

(33)

Substitute (33) in (31) and use Fubini’s Theorem v-time on the right hand side of the resultant inequality to obtain

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_v}^{\infty }\prod _{i=1}^v\frac{{\lambda }_i\left({\varsigma }_i\right)}{{\left({\Lambda }_i^{{\sigma }_i}\left(t_i\right)\right)}^{{\gamma }_i}}{I}_{v+1}\Delta {\varsigma }_1\cdots \Delta {\varsigma }_v\hfill \\ \le k^p{\int }_{a_{v+1}}^{\infty }\frac{({\Lambda }_{v+1}^{{\sigma }_{v+1}}{\left({\varsigma }_{v+1}\right)}^{{\gamma }_{v+1}(p-1)}}{{\Lambda }_{v+1}^{({\gamma }_{v+1}-1)p}\left({\varsigma }_{v+1}\right)}{\lambda }_{v+1}\left({\varsigma }_{v+1}\right)\\ \hfill \times \left({\int }_{a_1}^{\infty }\cdots {\int }_{a_v}^{\infty }\prod _{i=1}^v\frac{{\lambda }_i\left({\varsigma }_i\right)}{{\left({\Lambda }_i^{{\sigma }_i}\left({\varsigma }_i\right)\right)}^{{\gamma }_i}}{\left({\Phi }^{{\sigma }_1\cdots {\sigma }_v}({\varsigma }_{\mathbf{v}},{\varsigma }_{v+1})\right)}^p\Delta {\varsigma }_1\cdots \Delta {\varsigma }_v\right)\Delta {\varsigma }_{v+1}.\end{array}$$

By using induction hypothesis for ${\Phi }^{{\sigma }_1\cdots {\sigma }_v}({\varsigma }_{\mathbf{v}},{\varsigma }_{v+1})$ instead for ${\Phi }^{{\sigma }_1\cdots {\sigma }_v}\left({\varsigma }_{\mathbf{v}}\right)$ with fix ${\varsigma }_{v+1}\in {\mathbb{T}}_{v+1}$, we obtain

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_v}^{\infty }\prod _{i=1}^{v+1}\frac{{\lambda }_i\left({\varsigma }_i\right)}{{\left({\Lambda }_i^{{\sigma }_i}{\varsigma }_i\right)}^{{\gamma }_i}}{\Phi }^p\left({\varsigma }_{\mathbf{v}+\mathbf{1}}\right)\Delta {\varsigma }_1\cdots \Delta {\varsigma }_{v+1}\hfill \\ \hfill ⩽k^{(v+1)p}{\int }_{a_1}^{\infty }\cdots {\int }_{a_v}^{\infty }\prod _{i=1}^{v+1}{\left({\Lambda }_i^{{\sigma }_i}\left({\varsigma }_i\right)\right)}^{p-{\gamma }_i}{\lambda }_i\left({\varsigma }_i\right)g^p\left({\varsigma }_{\mathbf{v}+\mathbf{1}}\right)\Delta {\varsigma }_1\cdots \Delta {\varsigma }_{v+1}.\end{array}$$

Hence, by induction principal statement is true for all positive integers n. □

Remark 1.

As a special case of Theorem 3 when $q_i\left({\varsigma }_i\right)=1$, then (30) becomes the following Copson type inequality

$$\begin{array}{c}\hfill {\int }_{a_1}^{\infty }\cdots {\int }_{a_2}^{\infty }\prod _{i=1}^n\frac{{\lambda }_i\left({\varsigma }_i\right)}{{\left({\Lambda }_i^{{\sigma }_i}\left(t_i\right)\right)}^{{\gamma }_i}}{\left(\Phi \left({\varsigma }_{\mathbf{n}}\right)\right)}^p\Delta {\varsigma }_1\cdots \Delta {\varsigma }_n\\ \hfill \le \frac{p}{{\gamma }_i-1}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{i=1}^n({\Lambda }_i^{-{\gamma }_i+1}\left({\varsigma }_i\right)){\lambda }_i\left({\varsigma }_i\right)g^p\left({\varsigma }_{\mathbf{n}}\right)\Delta {\varsigma }_1\cdots \Delta {\varsigma }_n.\end{array}$$

(34)

For $n=1,$ (34) is given in [15].

Remark 2.

As a special case of Theorem 3 when $q_i\left({\varsigma }_i\right)={\lambda }_i\left({\varsigma }_i\right)=1$, then (30) becomes

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{i=1}^n\frac{1}{{({\sigma }_i-a_i)}^{{\gamma }_i}}{\left({\int }_{a_1}^{{\sigma }_1\left({\varsigma }_1\right)}\cdots {\int }_{a_n}^{{\sigma }_n\left({\varsigma }_n\right)}g\left(s_{\mathbf{n}}\right)\Delta s_1\cdots \Delta s_n\right)}^p\Delta {\varsigma }_1\cdots \Delta {\varsigma }_n\hfill \\ \hfill \le \prod _{i=1}^n{\left(\frac{p}{{\gamma }_i-1}\right)}^{np}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{i=n}^n\frac{{({\sigma }_i\left({\varsigma }_i\right)-a_i)}^{{\gamma }_i(p-1)}}{{({\varsigma }_i-a_i)}^{({\gamma }_i-1)p}}{g}^p\left({\varsigma }_{\mathbf{n}}\right)\Delta {\varsigma }_1\cdots \Delta {\varsigma }_n.\end{array}$$

(35)

For $n=1,$ (35) is given in [30].

Remark 3.

Assume that $q_i\left({\varsigma }_i\right)=1$, ${\gamma }_i=p$, if their exist constants ${\gamma }_i>1$, with

$$\begin{array}{c}\hfill \prod _{i=1}^n\frac{{\Lambda }_i\left({\varsigma }_i\right)}{{\Lambda }_i^{{\sigma }_i}\left({\varsigma }_i\right)}\ge \prod _{i=1}^n\frac{1}{{\gamma }_i},for{\varsigma }_i\in {[a_i,\infty )}_{{\mathbb{T}}_i},\end{array}$$

then (30) becomes

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{i=1}^n{\lambda }_i\left({\varsigma }_i\right){\left(\frac{\Phi \left({\varsigma }_{\mathbf{n}}\right)}{{\Lambda }_i^{{\sigma }_i}\left({\varsigma }_i\right)}\right)}^p\Delta {\varsigma }_1\cdots \Delta {\varsigma }_n\hfill \\ \hfill \le \prod _{i=1}^n{\left(\frac{p{\gamma }_i^{p-1}}{p-1}\right)}^{np}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }{\lambda }_i\left({\varsigma }_i\right)g^p\left({\varsigma }_{\mathbf{n}}\right)\Delta {\varsigma }_1\cdots \Delta {\varsigma }_n.\end{array}$$

(36)

Remark 4.

As a special case of (36) when ${\lambda }_i\left({\varsigma }_i\right)=1$, one obtains

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }{\left(\prod _{i=1}^n\frac{1}{({\sigma }_i\left({\varsigma }_i\right)-a_i)}{\int }_{a_1}^{{\sigma }_1\left({\varsigma }_1\right)},\dots ,{\int }_{a_n}^{{\sigma }_n\left({\varsigma }_n\right)}\prod _{i=1}^{n}g(s_1,\dots ,s_n)\Delta s_1\cdots \Delta s_n\right)}^p\Delta {\varsigma }_1\cdots \Delta {\varsigma }_n\hfill \\ \hfill \le \prod _{i=n}^n{\left(\frac{p{\gamma }_i^{p-1}}{p-1}\right)}^{np}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{i=1}^n{g}^p({\varsigma }_1,\dots ,{\varsigma }_n)\Delta {\varsigma }_1\cdots \Delta {\varsigma }_n.\end{array}$$

Example 7.

In addition to the assumptions of Theorem 3, choose ${\mathbb{T}}_i=l_i^{\mathbb{N}}$ for all $i\in \{1,\dots ,n\}$ and $p>1$ then (30) takes the form

$$\begin{array}{c}\sum _{{\alpha }_1=1}^{\infty }\cdots \sum _{{\alpha }_n=1}^{\infty }\prod _{i=1}^n\frac{{\lambda }_i\left({l_i}^{{\alpha }_i}\right)}{{\left({\Lambda }_i\left({l_i}^{{\alpha }_i+1}\right)\right)}^{{\gamma }_i}}{\left(\Phi ({l_1}^{{\alpha }_1+1},\dots ,{l_n}^{{\alpha }_n+1})\right)}^p(l_i-1){l_i}^{{\alpha }_i}\hfill \\ \hfill \le k^{np}\sum _{{\alpha }_1=1}^{\infty }\cdots \sum _{{\alpha }_n=1}^{\infty }\prod _{i=1}^n\frac{{\left({\Lambda }_i\left({l_i}^{{\alpha }_i+1}\right)\right)}^{{\gamma }_i(p-1)}}{{\left({\Lambda }_i\left({l_i}^{{\alpha }_i}\right)\right)}^{({\gamma }_i-1)p}}{\lambda }_i\left({l_i}^{{\alpha }_i}\right)g^p({l_1}^{{\alpha }_1},\dots ,{l_n}^{{\alpha }_n})(l_i-1){l_i}^{{\alpha }_i}.\end{array}$$

where,

$$\Phi ({l_1}^{{\alpha }_1},\dots ,{l_n}^{{\alpha }_n})=\sum _{\alpha =1}^{m_1}\cdots \sum _{{\alpha }_n=1}^{m_n}\prod _{i=1}^n{\lambda }_i\left({l_i}^{m_i}\right)q_i\left({l_i}^{m_i}\right)g({l_1}^{{\alpha }_1},\dots ,{l_n}^{{\alpha }_n})(l_i-1){l_i}^{{\alpha }_i}.$$

When $\mathbb{T}=\mathbb{R}$, we have the following results:

Corollary 1.

Let $1<{\gamma }_i\le p$ and $q_i\left({\varsigma }_i\right)$ be increasing function on $[a_i,\infty ]$. Furthermore assume that there exist constants $k_i>0$, such that

$$\prod _{i=1}^n\left({\gamma }_i-1+\frac{q_i^{\prime }\left({\varsigma }_i\right){\Lambda }_i\left({\varsigma }_i\right)}{{\lambda }_i\left({\varsigma }_i\right)q_i^2\left(t_i\right)}\right)\ge \prod _{i=1}^n\frac{p}{k_i},for{t}_i\ge a_i.$$

Then

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{i=1}^n\frac{{\lambda }_i\left({\varsigma }_i\right)}{{\Lambda }_i^{{\gamma }_i}\left({\varsigma }_i\right)}{\Phi }^p\left({\varsigma }_{\mathbf{n}}\right)d{\varsigma }_1\cdots d{\varsigma }_n\hfill \\ \hfill \le \prod _{i=1}^n{k}^{np}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{i=1}^n{\left({\Lambda }_i^{{\sigma }_i}\left({\varsigma }_i\right)\right)}^{(p-{\gamma }_i)}{\lambda }_i\left({\varsigma }_i\right)g^p\left({\varsigma }_{\mathbf{n}}\right)d{\varsigma }_1\cdots d{\varsigma }_n,\end{array}$$

(37)

where

$$\begin{array}{c}\hfill \Phi \left({\varsigma }_{\mathbf{n}}\right)={\int }_{a_1}^{{\varsigma }_1}\cdots {\int }_{a_n}^{{\varsigma }_n}\prod _{i=1}^n{\lambda }_i\left(s_i\right)q_i\left(s_i\right)g\left(s_{\mathbf{n}}\right)d{\varsigma }_1\cdots d{\varsigma }_n,\end{array}$$

and

$$\begin{array}{c}\hfill {\Lambda }_i\left({\varsigma }_i\right):={\int }_{a_1}^{{\varsigma }_1}\cdots {\int }_{a_n}^{{\varsigma }_n}\prod _{i=1}^n{\lambda }_i\left(s_i\right)q_i\left(s_i\right)d{s}_1\cdots d{s}_n.\end{array}$$

For single valued function it coincides with (12).

Remark 5.

When ${\gamma }_i=p>1$ and $a_i=0$ and ${\lambda }_i\left({\varsigma }_i\right)=1$, then (37) reduces to the following Levinson-type inequality

$$\begin{array}{c}{\int }_0^{\infty }\cdots {\int }_0^{\infty }\Phi \left({\int }_0^{{\varsigma }_1}\cdots {\int }_0^{{\varsigma }_n}\prod _{i=1}^{n}g(s_1,\dots ,s_n)d{s}_1\cdots d{s}_n\right)d{\varsigma }_1\cdots d{\varsigma }_n\hfill \\ \hfill \le k^{np}{\int }_0^{\infty }\cdots {\int }_0^{\infty }\Phi \left(g({\varsigma }_1,\dots ,{\varsigma }_n)\right)d{\varsigma }_1\cdots d{\varsigma }_n.\end{array}$$

(38)

(38) is special form of (9) for $n=1.$

Remark 6.

When $q_i\left({\varsigma }_i\right)=1$, then (37) reduces to the following Copson integral inequality for multiple integrals

$$\begin{array}{c}{\int }_0^b\cdots {\int }_0^b\prod _{i=1}^n\frac{{\lambda }_i\left({\varsigma }_i\right)}{{\Lambda }_i^{{\gamma }_i}\left({\varsigma }_i\right)}{\Phi }_n^p\left({\varsigma }_{\mathbf{n}}\right)d{\varsigma }_1\cdots d{t}_n\hfill \\ \hfill \le \prod _{i=1}^n\left(\frac{p}{({\gamma }_i-1)}\right){\int }_0^b\cdots {\int }_0^b\prod _{i=1}^n\frac{{\lambda }_i\left({\varsigma }_i\right)}{{\Lambda }_i^{{\gamma }_i-p}\left({\varsigma }_i\right)}{g}^p\left({\varsigma }_{\mathbf{n}}\right)d{\varsigma }_1\cdots d{\varsigma }_n.\end{array}$$

which is extension of (10).

Remark 7.

When ${\lambda }_i\left({\varsigma }_i\right)=q_i\left({\varsigma }_i\right)=1$ and $a_i=0$, then (37) becomes the following Hardy inequality

$$\begin{array}{c}{\int }_0^{\infty }\cdots {\int }_0^{\infty }\prod _{i=1}^n\frac{1}{{\left({\varsigma }_i\right)}^{{\gamma }_i}}{\left({\int }_0^{{\varsigma }_1}\cdots {\int }_0^{{\varsigma }_n}g(s_1,\dots ,s_n)d{s}_1\cdots d{s}_n\right)}^{p}d{\varsigma }_1\cdots d{\varsigma }_n\hfill \\ \hfill \le \prod _{i=1}^n{\left(\frac{p}{({\gamma }_i-1)}\right)}^p{\int }_0^{\infty }\cdots {\int }_0^{\infty }\prod _{i=1}^n\frac{1}{{\left({\varsigma }_i\right)}^{{\gamma }_i-p}}{g}^p({\varsigma }_1,\dots ,{\varsigma }_n)d{\varsigma }_1\cdots d{\varsigma }_n,{\gamma }_i>1.\end{array}$$

which is (7) for $n=1.$

Remark 8.

When ${\gamma }_i=p$ and ${\lambda }_i\left({\varsigma }_i\right)=q_i\left({\varsigma }_i\right)=1$, we obtain extension to the classical Hardy type inequality (6).

Theorem 4.

Let ${\mathbb{T}}_i$ denotes a time scale. Assume $a_i\in {[0,\infty )}_{{\mathbb{T}}_i}$ and $q_i\left({\varsigma }_i\right)$, ${\lambda }_i\left(t_i\right)$ are increasing functions on ${[a_i,\infty )}_{{\mathbb{T}}_i}$ for $i\in \{1,\dots ,n\}$. Furthermore assume that there exist constants $k_i>0$ such that

$${1-\gamma }_i-\frac{q_i^{{\Delta }_i}\left({\varsigma }_i\right){{\Lambda }_i}^{{\sigma }_i}\left({\varsigma }_i\right)}{{\lambda }_i\left({\varsigma }_i\right){q_i}^2\left({\varsigma }_i\right)}\ge \frac{p}{k_i},\forall i$$

then for $1<{\gamma }_i\le p$,

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{i=1}^n\frac{{\lambda }_i\left({\varsigma }_i\right)}{{\left({\Lambda }_i^{{\sigma }_i}\left({\varsigma }_i\right)\right)}^{{\gamma }_i}}{\left(\overline{\Phi }\left({\varsigma }_{\mathbf{n}}\right)\right)}^p\Delta {\varsigma }_1\cdots \Delta {\varsigma }_n\hfill \\ \hfill \le k^{np}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{i=1}^n{\left({\Lambda }_i^{{\sigma }_i}\left({\varsigma }_i\right)\right)}^{p-{\gamma }_i}{\lambda }_i\left({\varsigma }_i\right)g^p\left({\varsigma }_{\mathbf{n}}\right)\Delta {\varsigma }_1\cdots \Delta {\varsigma }_n,\end{array}$$

(39)

where

$$\begin{array}{c}\hfill \overline{\Phi }\left({\varsigma }_{\mathbf{n}}\right):={\int }_{{\varsigma }_1}^{\infty }\cdots {\int }_{{\varsigma }_n}^{\infty }{\lambda }_i\left(s_i\right)q_i\left(s_i\right)g\left(s_{\mathbf{n}}\right)\Delta s_1\cdots \Delta s_n,\end{array}$$

and

$$\begin{array}{c}\hfill {\Lambda }_i\left({\varsigma }_i\right):={\int }_{a_i}^{{\varsigma }_i}{\lambda }_i\left(s_i\right)q_i\left(s_i\right)\Delta s_i.\end{array}$$

Proof. 

The result is proved by using the principal of mathematical induction. For $n=1,$ statement is true by (19). Suppose for $n=v$, (39) holds. To prove for $n=v+1,$ take L.H.S of (39) in the following form:

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_v}^{\infty }\prod _{i=1}^v\frac{{\lambda }_i\left({\varsigma }_i\right)}{{\left({{\Lambda }_i}^{{\sigma }_i}\left({\varsigma }_i\right)\right)}^{{\gamma }_i}}\hfill \\ \hfill \times \left({\int }_{a_{v+1}}^{\infty }{\lambda }_{v+1}\left({\varsigma }_{v+1}\right){\left({\Lambda }_{v+1}^{{\sigma }_{v+1}}\left({\varsigma }_{v+1}\right)\right)}^{{\gamma }_{v+1}}{\left(\overline{\Phi }\left({\varsigma }_{\mathbf{v}+\mathbf{1}}\right)\right)}^p\Delta {\varsigma }_{v+1}\right)\Delta {\varsigma }_1\cdots \Delta {\varsigma }_v.\end{array}$$

(40)

Denote

$$\begin{array}{c}\hfill I_{v+1}={\int }_{a_{v+1}}^{\infty }{\lambda }_{v+1}\left({\varsigma }_{v+1}\right){\left({\Lambda }_{v+1}^{{\sigma }_{v+1}}\left({\varsigma }_{v+1}\right)\right)}^{{\gamma }_{v+1}}{\left(\overline{\Phi }\left({\varsigma }_{\mathbf{v}+\mathbf{1}}\right)\right)}^p\Delta {\varsigma }_{v+1}.\end{array}$$

(41)

Use (19) in (41) with respect for fix ${\varsigma }_v\in {\mathbb{T}}_1\times \dots \times {\mathbb{T}}_v$ to obtain

$$\begin{array}{c}\hfill I_{v+1}\le k^p{\int }_{a_{v+1}}^{\infty }{\left({\Lambda }_{v+1}^{{\sigma }_{v+1}}\left({\varsigma }_{v+1}\right)\right)}^{p-{\gamma }_{v+1}}{\lambda }_{v+1}\left({\varsigma }_{v+1}\right){\left(\overline{\Phi }(t_{\mathbf{v}},t_{v+1})\right)}^p\Delta {\varsigma }_{v+1}.\end{array}$$

(42)

Use (42) in (40) and use (20) v-times on the right hand side of the resultant inequality to obtain

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_v}^{\infty }\prod _{i=1}^v\frac{{\lambda }_i\left({\varsigma }_i\right)}{{\left({{\Lambda }_i}^{{\sigma }_i}\left(t_i\right)\right)}^{{\gamma }_i}}{I}_{v+1}\Delta {\varsigma }_1\cdots \Delta {\varsigma }_v\le k^p{\int }_{a_{v+1}}^{\infty }{\left({\Lambda }_{v+1}^{{\sigma }_{v+1}}\left({\varsigma }_{v+1}\right)\right)}^{p-{\gamma }_{v+1}}{\lambda }_{v+1}\left({\varsigma }_{v+1}\right)\hfill \\ \hfill \times \left({\int }_{a_1}^{\infty }\cdots {\int }_{a_v}^{\infty }\prod _{i=1}^v\frac{{\lambda }_i\left({\varsigma }_i\right)}{{\left({\Lambda }_i^{{\sigma }_i}\left({\varsigma }_i\right)\right)}^{{\gamma }_i}}{\left(\overline{\Phi }({\varsigma }_v,{\varsigma }_{v+1})\right)}^p\Delta {\varsigma }_v...\Delta {\varsigma }_v\right)\Delta {\varsigma }_{v+1}.\end{array}$$

By using induction hypothesis for $\overline{\Phi }({\varsigma }_{\mathbf{v}},{\varsigma }_{v+1})$ instead of $\overline{\Phi }\left({\varsigma }_{\mathbf{v}}\right)$ with fix $t_{v+1}\in {\mathbb{T}}_{v+1}$, one obtain

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_{v+1}}^{\infty }\prod _{i=1}^{v+1}\frac{{\lambda }_i\left({\varsigma }_i\right)}{{\left({\Lambda }_i^{{\sigma }_i}\left({\varsigma }_i\right)\right)}^{{\gamma }_i}}{\left(\overline{\Phi }\left({\varsigma }_{\mathbf{v}+\mathbf{1}}\right)\right)}^p\Delta {\varsigma }_1\cdots \Delta {\varsigma }_{v+1}\hfill \\ \hfill \le k^{(v+1)p}{\int }_{a_1}^{\infty }\cdots {\int }_{a_{v+1}}^{\infty }\prod _{i=1}^{v+1}{\left({\Lambda }_i^{{\sigma }_i}\left({\varsigma }_i\right)\right)}^{p-{\gamma }_i}{\lambda }_i\left({\varsigma }_i\right)g^p\left({\varsigma }_{\mathbf{v}+\mathbf{1}}\right)\Delta {\varsigma }_1\cdots \Delta {\varsigma }_{v+1}.\end{array}$$

Hence, by induction principal, the statement is true for all positive integers n. □

Remark 9.

Let $q_i\left({\varsigma }_i\right)={\lambda }_i,$ ${\lambda }_i=1$ in Theorem 4 we have the time scale version of Hardy Littlewood inequality

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{i=1}^n\frac{1}{{({\sigma }_i-a_i)}^{{\gamma }_i}}{\left({\int }_{{\varsigma }_1}^{\infty }\cdots {\int }_{{\varsigma }_n}^{\infty }g(s_1,\dots ,s_n)\Delta s_1\cdots \Delta s_n\right)}^p\Delta {\varsigma }_1\cdots \Delta {\varsigma }_n\hfill \\ \hfill \le k^{np}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\frac{1}{{({\sigma }_i-a_i)}^{{\gamma }_i-p}}{g}^p({\varsigma }_1,\dots ,{\varsigma }_n)\Delta {\varsigma }_1\cdots \Delta {\varsigma }_n.\end{array}$$

Note that when $\mathbb{T}=\mathbb{R},$ we have the following result:

Example 8.

If $p>1$, $0<{\gamma }_i\le 1$ and there exist constants $k_i>0$, such that

$$1-{\gamma }_i-\prod _{i=1}^n\frac{\acute{q_i}\left({\varsigma }_i\right){\Lambda }_i\left({\varsigma }_i\right)}{{\lambda }_i\left({\varsigma }_i\right){q_i}^2\left({\varsigma }_i\right)}\ge \prod _{i=1}^n\frac{p}{k_i},{\varsigma }_i\in {[a_i,\infty )}_{{\mathbb{T}}_i},$$

then

$$\begin{array}{c}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{i=1}^n\frac{{\lambda }_i\left({\varsigma }_i\right)}{{\left({\Lambda }_i\left({\varsigma }_i\right)\right)}^{{\gamma }_i}}{\left(\overline{\Phi }({\varsigma }_1.,\dots ,{\varsigma }_n)\right)}^{p}d{\varsigma }_1\cdots d{\varsigma }_n\hfill \\ \hfill \le k^{np}{\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }\prod _{i=1}^n{\left({\Lambda }_i\left({\varsigma }_i\right)\right)}^{(p-{\gamma }_i)}{g}^p({\varsigma }_1,\dots ,{\varsigma }_n)d{\varsigma }_1\cdots d{\varsigma }_n,\end{array}$$

where

$$\begin{array}{c}\hfill \overline{\Phi }({\varsigma }_1,\dots ,{\varsigma }_n)={\int }_{a_1}^{{\varsigma }_1}\cdots {\int }_{a_n}^{{\varsigma }_n}\prod _{i=1}^n{\lambda }_i\left(s_i\right)q_i\left(s_i\right)g(s_1,\dots ,s_n)d{s}_1\cdots d{s}_n,\end{array}$$

and

$$\begin{array}{c}\hfill {\Lambda }_i\left({\varsigma }_i\right):={\int }_{a_1}^{{\varsigma }_1}\cdots {\int }_{a_n}^{{\varsigma }_n}\prod _{i=1}^n{\lambda }_i\left(s_i\right)q_i\left(s_i\right)g(s_1,\dots ,s_n)d{s}_1\cdots d{s}_n.\end{array}$$

5. Conclusions

Some multidimensional dynamic inequalities of Hardy–Leindler-, Hardy–Yang- and Hwang-type on time scales are derived in present paper. To demonstrate the validity of the results, we have deduced several Hardy-type inequalities given in the literature from newly obtained results, which include some dynamic Hardy inequalities for functions of one variable from [15,16,17,22,30]. By choosing ${\mathbb{T}}_{\iota }=\mathbb{R}\forall \iota \in \{1,\dots ,n\}$ for all in obtained results, we succeed to obtain multivariate generalized forms of some inequalities from [1,2,3,4], whereas substitution of ${\mathbb{T}}_{\iota }=\mathbb{N}\forall \iota \in \{1,\dots ,n\}$ produces some multidimensional discrete Hardy-type inequalities from [5,6,7,8,9]. Moreover, main results are also discussed by fixing ${\mathbb{T}}_{\iota }=q^{\mathbb{N}}\forall \iota \in \{1,\dots ,n\},q>1,$ which are helpful to find all Hardy-, Copson-, Levinson-, and Bennett-type inequalities in quantum calculus as well. It is also worth mentioning here that the results of the paper can be reconstructed with the help of nabla calculus. Future plan may carry some multidimensional Hardy type inequalities involving convex combination of delta and nabla integrals based upon these results.