Some New Generalized Inequalities of Hardy Type Involving Several Functions on Time Scale Nabla Calculus

Abstract

In this article, we establish several new generalized Hardy-type inequalities involving several functions on time-scale nabla calculus. Furthermore, we derive some new multidimensional Hardy-type inequalities on time scales nabla calculus. The main results are proved by applying Minkowski’s inequality, Jensen’s inequality and Arithmetic Mean–Geometric Mean inequality. As a special case of our results, when $\mathbb{T}=\mathbb{R},$ we obtain refinements of some well-known continuous inequalities and when $\mathbb{T}=\mathbb{N}$, the results which are essentially new.

1. Introduction

In [1], Hardy proved that

$$\sum _{l=1}^{\infty }{\left(\frac{1}{l}\sum _{i=1}^{l}a\left(i\right)\right)}^q\le {\left(\frac{q}{q-1}\right)}^q\sum _{l=1}^{\infty }{a}^q\left(l\right),q>1,$$

(1)

where $a\left(l\right)\ge 0$ for $l\ge 1$ and ${\sum }_{l=1}^{\infty }{a}^q\left(l\right)<\infty$.

In [2], Hardy proved the continuous case of (1) in the following form

$${\int }_0^{\infty }{\left(\frac{1}{\theta }{\int }_0^{\theta }f\left(\tau \right)d\tau \right)}^{q}d\theta \le {\left(\frac{q}{q-1}\right)}^q{\int }_0^{\infty }{f}^q\left(\theta \right)d\theta ,q>1,$$

(2)

where $f\ge 0$ and integrable over any finite interval $(0,\theta ),$ $\theta \in (0,\infty ),$$f\in L^q(0,\infty )$ and the constant ${\left(q/\left(q-1\right)\right)}^q$ in (1) and (2) is sharp.

In [3], Kaijser et al. established that if $\Phi$ is a convex function on ${\mathbb{R}}^{+}$, then

$${\int }_0^{\infty }\Phi \left(\frac{1}{\lambda }{\int }_0^{\lambda }\varpi \left(\eta \right)d\eta \right)\frac{d\lambda }{\lambda }\le {\int }_0^{\infty }\Phi \left(\varpi \left(\lambda \right)\right)\frac{d\lambda }{\lambda },$$

(3)

where $\varpi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a locally integrable positive function. In [4], Čižmešija et al. generalized (3) in the following form

$${\int }_0^r\varkappa \left(\lambda \right)\Phi \left(\frac{1}{\lambda }{\int }_0^{\lambda }\varpi \left(\eta \right)d\eta \right)\frac{d\lambda }{\lambda }\le {\int }_0^r\omega \left(\lambda \right)\Phi \left(\varpi \left(\lambda \right)\right)\frac{d\lambda }{\lambda },$$

where $\varkappa :(0,r)\to \mathbb{R},$$0<r\le \infty ,$ is a non-negative function, such that $\lambda \to \varkappa \left(\lambda \right)/{\lambda }^2$ is locally integrable, $\Phi$ is a convex function, $\varpi :(0,r)\to \mathbb{R}$$\forall \lambda \in (0,r)$ and

$$\omega \left(\eta \right)=\eta {\int }_{\eta }^r\frac{\varkappa \left(\lambda \right)}{{\lambda }^2}d\lambda ,\mathrm{for}\eta \in (0,r).$$

In [5], Kaijser et al. explicated that if $\varkappa :(0,r)\to \mathbb{R},$$\varrho :(0,r)\times (0,r)\to \mathbb{R},$ $0<r\le \infty$ are positive functions, such that $0<\Lambda \left(\eta \right)={\int }_0^{\eta }\varrho (\eta ,\vartheta )d\vartheta <\infty ,$$\eta \in (0,r)$, $\Phi$ is a convex function on $I\subseteq \mathbb{R}$, and

$$\omega \left(\lambda \right)=\lambda {\int }_{\lambda }^r\varkappa \left(\eta \right)\frac{\varrho (\eta ,\lambda )}{\Lambda \left(\eta \right)}\frac{d\eta }{\eta }<\infty ,\lambda \in (0,r),$$

then

$${\int }_0^r\varkappa \left(\lambda \right)\Phi \left(A_{\varrho }\varpi \left(\lambda \right)\right)\frac{d\lambda }{\lambda }\le {\int }_0^r\omega \left(\lambda \right)\Phi \left(\varpi \left(\lambda \right)\right)\frac{d\lambda }{\lambda },$$

(4)

where $\varpi :(0,r)\to \mathbb{R}$ is a function with values in $I,$ and

$$A_{\varrho }\varpi \left(\lambda \right)=\frac{1}{\Lambda \left(\lambda \right)}{\int }_0^{\lambda }\varrho (\lambda ,\vartheta )\varpi \left(\vartheta \right)d\vartheta ,\lambda \in (0,r).$$

Additionally, in [5] it is established that if $1<p\le q<\infty ,s\in (1,p)$ and $0<r<\infty ,$$\varrho :{\mathbb{R}}^{+}\times {\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a non-negative kernel, $\varkappa \left(\lambda \right)\ge 0$ and $\omega \left(\lambda \right)\ge 0$ are weighted functions, and

$${\left({\int }_0^r{\left[\Phi \left(A_{\varrho }\varpi \left(\lambda \right)\right)\right]}^q\varkappa \left(\lambda \right)\frac{d\lambda }{\lambda }\right)}^{\frac{1}{q}}\le C{\left[{\int }_0^r{\Phi }^p\left(\varpi \left(\lambda \right)\right)\omega \left(\lambda \right)\frac{d\lambda }{\lambda }\right]}^{\frac{1}{p}},$$

(5)

holds for all $\varpi \left(\lambda \right)\ge 0,$$\lambda \in [0,r]$ and $C>0,$ if

$$A\left(s\right)=\underset{0<\vartheta <r}{sup}{\left[\Omega \left(\vartheta \right)\right]}^{\frac{s-1}{p}}{\left({\int }_{\vartheta }^r{\left(\frac{\varrho (\lambda ,\vartheta )}{\Lambda \left(\lambda \right)}\right)}^q{\left[\Omega \left(\lambda \right)\right]}^{\frac{q\left(p-s\right)}{p}}\varkappa \left(\lambda \right)\frac{d\lambda }{\lambda }\right)}^{\frac{1}{q}}<\infty ,$$

where

$$\Omega \left(\vartheta \right)={\int }_0^{\vartheta }{\left[\omega \left(\eta \right)\right]}^{\frac{-1}{p-1}}{\eta }^{\frac{1}{p-1}}d\eta .$$

In the last few decades, researchers discovered the time-scale calculus which unifies the continuous and discrete calculus. A time scale $\mathbb{T}$ is an arbitrary, non-empty closed subset of the real numbers $\mathbb{R}$. Many authors established some new dynamic inequalities on $\mathbb{T}$; see the books [6,7] and the papers [8,9,10,11,12].

In [13], Özkan et al. demonstrated that if $0\le r<y\le \infty ,$$u\in C_{rd}\left([r,y),\mathbb{R}\right)$ is a non-negative function, such that ${\int }_t^y\frac{u\left(\lambda \right)}{\left(\lambda -r\right)\left(\sigma \left(\lambda \right)-r\right)}\Delta \lambda$ exists as a finite number, $\Phi$ is continuous and convex,$f\in C_{rd}\left([r,y),\mathbb{R}\right)$ and

$$v\left(t\right)=(t-r){\int }_t^y\frac{u\left(\lambda \right)}{\left(\lambda -r\right)\left(\sigma \left(\lambda \right)-r\right)}\Delta \lambda ,t\in [r,y),$$

then

$${\int }_r^{y}u\left(\lambda \right)\Phi \left(\frac{1}{\sigma \left(\lambda \right)-r}{\int }_r^{\sigma \left(\lambda \right)}f\left(t\right)\Delta t\right)\frac{\Delta \lambda }{\lambda -r}\le {\int }_r^{y}v\left(\lambda \right)\Phi \left(f\left(\lambda \right)\right)\frac{\Delta \lambda }{\lambda -r}.$$

(6)

They also proved that if $u\in C_{rd}\left([y,\infty ),\mathbb{R}\right)$ is a non-negative function, and

$$v\left(t\right)=\frac{1}{t}{\int }_b^{t}u\left(\lambda \right)\Delta \lambda ,t\in [y,\infty ),$$

then

$${\int }_r^{y}u\left(\lambda \right)\Phi \left(\frac{1}{\sigma \left(\lambda \right)-r}{\int }_r^{\sigma \left(\lambda \right)}f\left(t\right)\Delta t\right)\frac{\Delta \lambda }{\lambda -r}\le {\int }_r^{y}v\left(\lambda \right)\Phi \left(f\left(\lambda \right)\right)\frac{\Delta \lambda }{\lambda -r},$$

holds for all $f\in C_{rd}\left([y,\infty ),\mathbb{R}\right).$

In [14], the authors proved the time-scale version of (4) as follows. Let $k(\lambda ,\theta )\in C_{rd}\left([r,y)\times [r,y),\mathbb{R}\right),$$u\in C_{rd}\left([r,y),\mathbb{R}\right)$ be non-negative functions, $f\in C_{rd}\left([r,y),\mathbb{R}\right),$ $\Phi$ is a continuous and convex function, and

$$v\left(t\right)=(t-r){\int }_t^y\frac{k(\lambda ,t)}{K\left(\sigma \right(\lambda ),\lambda )}u\left(\lambda \right)\frac{\Delta \lambda }{\lambda -r},t\in [r,y).$$

Then,

$${\int }_r^{y}u\left(\lambda \right)\Phi \left(A_{k}f(\sigma \left(\lambda \right),\lambda )\right)\frac{\Delta \lambda }{\lambda -r}\le {\int }_r^{y}v\left(\lambda \right)\Phi \left(f\left(\lambda \right)\right)\frac{\Delta \lambda }{\lambda -r},$$

(7)

where

$$A_{k}f(t,\beta )=\frac{1}{K(t,\beta )}{\int }_r^{t}k(\beta ,\theta )f\left(\theta \right)\Delta \theta ,K(t,\beta ):={\int }_r^{t}k(\beta ,\theta )\Delta \theta .$$

Our aim in this study is to generalize (4) on time-scale nabla calculus of power $\eta \ge 1$ in the form

$${\int }_r^y{\chi }^{\eta }\left(A_{\varrho }\varpi \left(\zeta \right)\right)\frac{\varkappa \left(\zeta \right)}{\rho \left(\zeta \right)-r}\nabla \zeta \le {\left(\frac{B}{A}\right)}^{\eta }{\left({\int }_r^y\chi \left(\varpi \left(\vartheta \right)\right)\frac{\omega \left(\vartheta \right)}{\rho \left(\vartheta \right)-r}\nabla \vartheta \right)}^{\eta },$$

where $A,B$ are positive constants. We will also establish the last inequality for several functions. Furthermore, we will prove the last inequality in multidimensions on time-scales nabla calculus.

The paper proceeds as follows. In Section 2, we state some properties concerning the time-scales nabla calculus needed in Section 3, where we prove the main results. Our main results when $\mathbb{T}\to \mathbb{R}$, we obtain (4) proved by Kaijser et al. [5] and when $\mathbb{T}\to \mathbb{N}$, we obtain a new discrete inequality.

2. Preliminaries and Basic Lemmas

For a time scale $\mathbb{T}$, we define the backward jump operator as $\rho \left(\gamma \right)=sup\{s\in \mathbb{T}:s<\gamma \}.$ Additionally, we define a mapping $\nu :\mathbb{T}\to {\mathbb{R}}^{+}$ by $\nu \left(\gamma \right)=\gamma -\rho \left(\gamma \right)$, such that if $\varpi$ is nabla differentiable at $\gamma ,$ then $\nu \left(\gamma \right){\varpi }^{\nabla }\left(\gamma \right)=\varpi \left(\gamma \right)-{\varpi }^{\rho }\left(\gamma \right).$ For more details about $\mathbb{T}$ calculus, see ([6,7]).

The nabla derivative of $\varkappa \omega$ and $\varkappa /\omega$ (where $\omega \left(\gamma \right){\omega }^{\rho }\left(\gamma \right)\ne 0$) are given by

$$\begin{array}{cc}\hfill {\left(\varkappa \omega \right)}^{\nabla }\left(\gamma \right)& ={\varkappa }^{\nabla }\left(\gamma \right)\omega \left(\gamma \right)+{\varkappa }^{\rho }\left(\gamma \right){\omega }^{\nabla }\left(\gamma \right)\hfill \\ \hfill & =\varkappa \left(\gamma \right){\omega }^{\nabla }\left(\gamma \right)+{\varkappa }^{\nabla }\left(\gamma \right){\omega }^{\rho }\left(\gamma \right),\hfill \end{array}$$

and

$${\left(\frac{\varkappa }{\omega }\right)}^{\nabla }\left(\gamma \right)=\frac{{\varkappa }^{\nabla }\left(\gamma \right)\omega \left(\gamma \right)-\varkappa \left(\gamma \right){\omega }^{\nabla }\left(\gamma \right)}{\omega \left(\gamma \right){\omega }^{\rho }\left(\gamma \right)}.$$

Definition 1

([6]). A function $F:\mathbb{T}\to \mathbb{R}$ is a nabla antiderivative of $\varpi :\mathbb{T}\to \mathbb{R}$ if $F^{\nabla }\left(t\right)=\varpi \left(t\right)$ holds $\forall t\in \mathbb{T}$. Hence, we have

$${\int }_r^t\varpi \left(\gamma \right)\nabla \gamma =F\left(t\right)-F\left(r\right)\forall t\in \mathbb{T}.$$

Theorem 1 

([6]). 0If $r,y\in \mathbb{T},$$\alpha \in \mathbb{R}$ and $\varpi ,\lambda$ are ld-continuous functions, then

(1)

${\int }_r^y\left[\varpi \left(\gamma \right)+\lambda \left(\gamma \right)\right]\nabla \gamma ={\int }_r^y\varpi \left(\gamma \right)\nabla \gamma +{\int }_r^y\lambda \left(\gamma \right)\nabla \gamma ;$

(2)

${\int }_r^y\alpha \varpi \left(\gamma \right)\nabla \gamma =\alpha {\int }_r^y\varpi \left(\gamma \right)\nabla \gamma ;$

(3)

${\int }_r^r\varpi \left(\gamma \right)\nabla \gamma =0.$

The integration by parts formula on time scales nabla calculus [6] is

$${\int }_r^y\varkappa \left(\gamma \right){\omega }^{\nabla }\left(\gamma \right)\nabla \gamma ={\left[\varkappa \left(\gamma \right)\omega \left(\gamma \right)\right]}_r^y-{\int }_r^y{\varkappa }^{\nabla }\left(\gamma \right){\omega }^{\rho }\left(\gamma \right)\nabla \gamma .$$

(8)

The Arithmetic Mean–Geometric Mean inequality is given by

$${\left[{\lambda }_1\left(\zeta \right){\lambda }_2\left(\zeta \right)...{\lambda }_n\left(\zeta \right)\right]}^{\frac{1}{n}}\le \frac{{\sum }_{k=1}^n{\lambda }_k\left(\zeta \right)}{n}.$$

(9)

where ${\lambda }_1\left(\zeta \right),...,{\lambda }_n\left(\zeta \right),$$n\ge 1$ are non-negative functions.

In 2008, Ferreira et al. [15] proved Minkowski’s inequality on diamond alpha time scales. As a special case of this inequality (when $\alpha =0$), we get Minkowski’s inequality on time-scale nabla calculus as follows.

Lemma 1

([15]). Let $r,y\in \mathbb{T}$ and $f,g$ be non-negative functions. Then,

$${\left({\int }_r^y{\left(f\left(\gamma \right)+g\left(\gamma \right)\right)}^p\nabla \gamma \right)}^{\frac{1}{p}}\le {\left({\int }_r^y{f}^p\left(\gamma \right)\nabla \gamma \right)}^{\frac{1}{p}}+{\left({\int }_r^y{g}^p\left(\gamma \right)\nabla \gamma \right)}^{\frac{1}{p}},$$

(10)

for $p\ge 1.$

Lemma 2

([16]). Let $\varkappa ,\omega$ and ϖ be non-negative functions on $\Omega ,{\rm Y}$ and $\Omega \times {\rm Y},$ respectively. If $\alpha \ge 1,$ then

$$\begin{array}{cc}\hfill & {\left({\int }_{\Omega }\varkappa \left(\lambda \right){\left({\int }_{{\rm Y}}\varpi (\lambda ,\vartheta )\omega \left(\vartheta \right)\nabla \vartheta \right)}^{\alpha }\nabla \lambda \right)}^{\frac{1}{\alpha }}\hfill \\ \hfill & \le {\int }_{{\rm Y}}\omega \left(\vartheta \right){\left({\int }_{\Omega }{\varpi }^{\alpha }(\lambda ,\vartheta )\varkappa \left(\lambda \right)\nabla \lambda \right)}^{\frac{1}{\alpha }}\nabla \vartheta .\hfill \end{array}$$

(11)

Theorem 2

([16]). Let ${\epsilon }_i,{\varsigma }_i\in \mathbb{T},$$i=1,2,..,m,$ $\gamma \ge 1,$$\varkappa :{\mathbb{T}}^m\times {\mathbb{T}}^m\to \mathbb{R}$ and $w,h:{\mathbb{T}}^m\to \mathbb{R}$ be non-negative rd-continuous functions. Then,

$$\begin{array}{cc}\hfill & {\left[{\int }_{{\epsilon }_1}^{{\varsigma }_1}...{\int }_{{\epsilon }_m}^{{\varsigma }_m}w\left(\mathbf{y}\right){\left({\int }_{{\epsilon }_1}^{{\varsigma }_1}...{\int }_{{\epsilon }_m}^{{\varsigma }_m}h\left(\mathbf{z}\right)\varkappa (\mathbf{y},\mathbf{z})\nabla \mathbf{z}\right)}^{\gamma }\nabla \mathbf{y}\right]}^{\frac{1}{\gamma }}\hfill \\ \hfill & \le {\int }_{{\epsilon }_1}^{{\varsigma }_1}...{\int }_{{\epsilon }_m}^{{\varsigma }_m}h\left(\mathbf{z}\right){\left({\int }_{{\epsilon }_1}^{{\varsigma }_1}...{\int }_{{\epsilon }_m}^{{\varsigma }_m}w\left(\mathbf{y}\right){\varkappa }^{\gamma }(\mathbf{y},\mathbf{z})\nabla \mathbf{y}\right)}^{\frac{1}{\gamma }}\nabla \mathbf{z},\hfill \end{array}$$

(12)

where $\nabla \mathbf{y}=\nabla y_1...\nabla y_m,$$\varkappa (\mathbf{y},\mathbf{z})=\varkappa (y_1,...,y_m,z_1,...,z_m),$ $w\left(\mathbf{z}\right)=w\left(z_1,...,z_m\right)$ and $h\left(\mathbf{z}\right)=h\left(z_1,...,z_m\right).$

In [17], Jensen’s inequality is proved for the diamond$-\alpha$ time scale. In the case, $\alpha =0,$ this inequality can be written in nabla time-scale calculus as follows.

Lemma 3

([17]). Let $r,y\in \mathbb{T},$$h\in {\mathrm{C}}_{ld}({[r,y]}_{\mathbb{T}},\mathbb{R}),$$u:{[r,y]}_{\mathbb{T}}\to (c,d),$ $c,d\in \mathbb{R}$ be ld-continuous and Φ be continuous and convex. Then,

$$\Phi \left(\frac{1}{{\int }_r^{y}h\left(\vartheta \right)\nabla \vartheta }{\int }_r^{y}h\left(\gamma \right)u\left(\gamma \right)\nabla \gamma \right)\le \frac{1}{{\int }_r^{y}h\left(\vartheta \right)\nabla \vartheta }{\int }_r^{y}h\left(\gamma \right)\Phi \left(u\left(\gamma \right)\right)\nabla \gamma .$$

(13)

If Φ is a concave function, then (13) will be reversed.

Theorem 3

([17]). Let ${\epsilon }_i,{\varsigma }_i\in \mathbb{T},$$i=1,2,...,m,$ $g:{\mathbb{T}}^m\to (c,d),$$c,d\in \mathbb{R}$ be ld-continuous and Φ be continuous and convex. Then,

$$\begin{array}{cc}\hfill & \Phi \left(\frac{1}{{\int }_{{\epsilon }_1}^{{\varsigma }_1}...{\int }_{{\epsilon }_m}^{{\varsigma }_m}\varrho (\mathbf{y},\mathbf{z})\nabla \mathbf{z}}{\int }_{{\epsilon }_1}^{{\varsigma }_1}...{\int }_{{\epsilon }_m}^{{\varsigma }_m}\varrho (\mathbf{y},\mathbf{z})g\left(\mathbf{z}\right)\nabla \mathbf{z}\right)\hfill \\ \hfill & \le \frac{1}{{\int }_{{\epsilon }_1}^{{\varsigma }_1}...{\int }_{{\epsilon }_m}^{{\varsigma }_m}\varrho (\mathbf{y},\mathbf{z})\nabla \mathbf{z}}{\int }_{{\epsilon }_1}^{{\varsigma }_1}...{\int }_{{\epsilon }_m}^{{\varsigma }_m}\varrho (\mathbf{y},\mathbf{z})\Phi \left(g\left(\mathbf{z}\right)\right)\nabla \mathbf{z},\hfill \end{array}$$

(14)

where $\nabla \mathbf{z}=\nabla z_1...\nabla z_m,$$\varrho (\mathbf{y},\mathbf{z})=\varrho (y_1,...,y_m,z_1,...,z_m)$ and $g\left(\mathbf{z}\right)=g\left(z_1,...,z_m\right).$

3. Main Results

Throughout this section, we will assume that the functions (without mention) are non-negative ld-continuous functions and the integrals in the statements of the theorems are convergent. We define the general Hardy operator $A_{\varrho }$ as follows

$$A_{\varrho }\varpi \left(\lambda \right)=\frac{1}{\Lambda \left(\lambda \right)}{\int }_r^y\varrho (\lambda ,\vartheta )\varpi \left(\vartheta \right)\nabla \vartheta ,\Lambda \left(\lambda \right)={\int }_r^y\varrho (\lambda ,\vartheta )\nabla \vartheta ,$$

where $\lambda >r$ and $\varpi \in {\mathrm{C}}_{ld}({[r,y]}_{\mathbb{T}},{\mathbb{R}}^{+})$ and $\varrho (\lambda ,\vartheta )\in {\mathrm{C}}_{ld}({[r,y]}_{\mathbb{T}}\times {[r,y]}_{\mathbb{T}},{\mathbb{R}}^{+}).$

Now, we state and prove our main results.

Theorem 4.

Let $r,y\in \mathbb{T},$$\eta \ge 1$ and $\varkappa ,\omega$ be weighted functions, such that

$$\omega \left(\vartheta \right)=\left(\rho \left(\vartheta \right)-r\right){\left({\int }_r^y{\left(\frac{\varrho (\lambda ,\vartheta )}{\Lambda \left(\lambda \right)}\right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \right)}^{\frac{1}{\eta }}.$$

(15)

Furthermore, assume that $\chi ,\xi$ defined on $(c,d),$$-\infty <c<d<\infty$ and ξ is a convex function, such that

$$A\xi \left(\lambda \right)\le \chi \left(\lambda \right)\le B\xi \left(\lambda \right),c<\lambda <d,$$

(16)

where $A,B$ are positive constants; then

$${\int }_r^y{\chi }^{\eta }\left(A_{\varrho }\varpi \left(\lambda \right)\right)\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \le {\left(\frac{B}{A}\right)}^{\eta }{\left({\int }_r^y\chi \left(\varpi \left(\vartheta \right)\right)\frac{\omega \left(\vartheta \right)}{\rho \left(\vartheta \right)-r}\nabla \vartheta \right)}^{\eta },$$

(17)

holds for the non-negative function $\varpi .$

Proof. 

Using (16) and applying (13) (where $\xi$ is convex), we have

$$\begin{array}{cc}\hfill & {\int }_r^y{\chi }^{\eta }\left(A_{\varrho }\varpi \left(\lambda \right)\right)\varkappa \left(\lambda \right)\frac{\nabla \lambda }{\rho \left(\lambda \right)-r}\hfill \\ \hfill & ={\int }_r^y{\chi }^{\eta }\left(\frac{1}{\Lambda \left(\lambda \right)}{\int }_r^y\varrho (\lambda ,\vartheta )\varpi \left(\vartheta \right)\nabla \vartheta \right)\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \hfill \\ \hfill & \le B^{\eta }{\int }_r^y{\xi }^{\eta }\left(\frac{1}{\Lambda \left(\lambda \right)}{\int }_r^y\varrho (\lambda ,\vartheta )\varpi \left(\vartheta \right)\nabla \vartheta \right)\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \hfill \\ \hfill & \le B^{\eta }{\int }_r^y\frac{1}{{\Lambda }^{\eta }\left(\lambda \right)}{\left({\int }_r^y\varrho (\lambda ,\vartheta )\xi \left(\varpi \left(\vartheta \right)\right)\nabla \vartheta \right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda .\hfill \end{array}$$

(18)

Applying (11) on the term

$${\int }_r^y\frac{1}{{\Lambda }^{\eta }\left(\lambda \right)}{\left({\int }_r^y\varrho (\lambda ,\vartheta )\xi \left(\varpi \left(\vartheta \right)\right)\nabla \vartheta \right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda ,$$

with $\eta \ge 1,$ we see that

$$\begin{array}{cc}\hfill & {\left({\int }_r^y\frac{1}{{\Lambda }^{\eta }\left(\lambda \right)}{\left({\int }_r^y\varrho (\lambda ,\vartheta )\xi \left(\varpi \left(\vartheta \right)\right)\nabla \vartheta \right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \right)}^{\frac{1}{\eta }}\hfill \\ \hfill & \le {\int }_r^y\xi \left(\varpi \left(\vartheta \right)\right){\left({\int }_r^y{\left(\frac{\varrho (\lambda ,\vartheta )}{\Lambda \left(\lambda \right)}\right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \right)}^{\frac{1}{\eta }}\nabla \vartheta ,\hfill \end{array}$$

then

$$\begin{array}{cc}\hfill & {\int }_r^y\frac{1}{{\Lambda }^{\eta }\left(\lambda \right)}{\left({\int }_r^y\varrho (\lambda ,\vartheta )\xi \left(\varpi \left(\vartheta \right)\right)\nabla \vartheta \right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \hfill \\ \hfill & \le {\left[{\int }_r^y\xi \left(\varpi \left(\vartheta \right)\right){\left({\int }_r^y{\left(\frac{\varrho (\lambda ,\vartheta )}{\Lambda \left(\lambda \right)}\right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \right)}^{\frac{1}{\eta }}\nabla \vartheta \right]}^{\eta }\hfill \\ \hfill & ={\left[{\int }_r^y\xi \left(\varpi \left(\vartheta \right)\right)\frac{1}{\rho \left(\vartheta \right)-r}\left(\rho \left(\vartheta \right)-r\right){\left({\int }_r^y{\left(\frac{\varrho (\lambda ,\vartheta )}{\Lambda \left(\lambda \right)}\right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \right)}^{\frac{1}{\eta }}\nabla \vartheta \right]}^{\eta }.\hfill \end{array}$$

(19)

Substituting (19) into (18), we have from (15) that

$$\begin{array}{cc}\hfill & {\int }_r^y{\chi }^{\eta }\left(A_{\varrho }\varpi \left(\lambda \right)\right)\varkappa \left(\lambda \right)\frac{\nabla \lambda }{\rho \left(\lambda \right)-r}\hfill \\ \hfill & \le B^{\eta }{\left[{\int }_r^y\xi \left(\varpi \left(\vartheta \right)\right)\frac{1}{\rho \left(\vartheta \right)-r}\left(\rho \left(\vartheta \right)-r\right){\left({\int }_r^y{\left(\frac{\varrho (\lambda ,\vartheta )}{\Lambda \left(\lambda \right)}\right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \right)}^{\frac{1}{\eta }}\nabla \vartheta \right]}^{\eta }\hfill \\ \hfill & =B^{\eta }{\left[{\int }_r^y\xi \left(\varpi \left(\vartheta \right)\right)\frac{1}{\rho \left(\vartheta \right)-r}\omega \left(\vartheta \right)\nabla \vartheta \right]}^{\eta },\hfill \end{array}$$

and then, we get from (16) that

$${\int }_r^y{\chi }^{\eta }\left(A_{\varrho }\varpi \left(\lambda \right)\right)\varkappa \left(\lambda \right)\frac{\nabla \lambda }{\rho \left(\lambda \right)-r}\le {\left(\frac{B}{A}\right)}^{\eta }{\left[{\int }_r^y\chi \left(\varpi \left(\vartheta \right)\right)\frac{1}{\rho \left(\vartheta \right)-r}\omega \left(\vartheta \right)\nabla \vartheta \right]}^{\eta },$$

which is (17). □

Corollary 1.

If $A=B$ and $\eta =1,$ then

$${\int }_r^y\chi \left(A_{\varrho }\varpi \left(\lambda \right)\right)\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \le \left({\int }_r^y\chi \left(\varpi \left(\vartheta \right)\right)\frac{\omega \left(\vartheta \right)}{\rho \left(\vartheta \right)-r}\nabla \vartheta \right),$$

where

$$\omega \left(\vartheta \right)=\left(\rho \left(\vartheta \right)-r\right)\left({\int }_r^y\left(\frac{\varrho (\lambda ,\vartheta )}{\Lambda \left(\lambda \right)}\right)\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \right).$$

Remark 1.

If $\mathbb{T}=\mathbb{N},$$r=0,$ then $\rho \left(n\right)=n-1$ and (17) reduces to

$$\begin{array}{cc}\hfill & \sum _{n=1}^N\chi \left(\frac{1}{{\sum }_{m=1}^n\varrho (n,m)}\sum _{m=1}^n\varrho (n,m)\varpi \left(m\right)\right)\frac{\varkappa \left(n\right)}{n-1}\hfill \\ \hfill & \le \left[\sum _{n=1}^N\chi \left(\varpi \left(n\right)\right)\frac{\omega \left(n\right)}{n-1}\right],\mathrm{for}N\in \mathbb{N}.\hfill \end{array}$$

Remark 2.

If $\mathbb{T}=\mathbb{R},$$r=0,$ then $\rho \left(\zeta \right)=\zeta ,$ and we have

$${\int }_r^y\chi \left(A_{\varrho }\varpi \left(\lambda \right)\right)\frac{\varkappa \left(\lambda \right)}{\lambda }d\lambda \le \left({\int }_r^y\chi \left(\varpi \left(\vartheta \right)\right)\frac{\omega \left(\vartheta \right)}{\vartheta }d\vartheta \right),$$

where

$$\omega \left(\vartheta \right)=\vartheta \left({\int }_r^y\left(\frac{\varrho (\lambda ,\vartheta )}{\Lambda \left(\lambda \right)}\right)\frac{\varkappa \left(\lambda \right)}{\lambda }d\lambda \right).$$

Remark 3.

If $\varrho (\lambda ,\vartheta )=\left\{\begin{array}{c}0,\lambda \in [r,\vartheta ),\\ f(\lambda ,\vartheta ),\lambda \in [\vartheta ,y].\end{array}\right.,$ we get the inequality (4) proved by Kaijser et al. [5].

The following theorem is proved for several functions.

Theorem 5.

Let $r,y\in \mathbb{T},$$\eta \ge 1$ and $\varkappa ,\omega$ be as in Theorem 4, such that

$$\omega \left(\vartheta \right)=\left(\rho \left(\vartheta \right)-r\right){\left({\int }_r^y{\left(\frac{\varrho (\lambda ,\vartheta )}{\Lambda \left(\lambda \right)}\right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \right)}^{\frac{1}{\eta }}.$$

(20)

Furthermore, assume that ${\varpi }_k,k=1,2,...,n$ and $\chi ,\xi$ are as in Theorem 4, such that

$$A\xi \left(\lambda \right)\le \chi \left(\lambda \right)\le B\xi \left(\lambda \right),$$

(21)

where $A,B$ are positive constants, then

$$\begin{array}{cc}\hfill & {\int }_r^y{\left[{\Pi }_{k=1}^n\chi \left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)\right]}^{\frac{\eta }{n}}\varkappa \left(\lambda \right)\frac{\nabla \lambda }{\rho \left(\lambda \right)-r}\hfill \\ \hfill & \le {\left(\frac{B}{nA}\right)}^{\eta }{\left(\sum _{k=1}^n\left[{\int }_r^y\chi \left({\varpi }_k\left(\vartheta \right)\right)\frac{1}{\rho \left(\vartheta \right)-r}\omega \left(\vartheta \right)\nabla \vartheta \right]\right)}^{\eta },\hfill \end{array}$$

(22)

holds for $n\ge 1.$

Proof. 

Applying (Arithmetic Mean–Geometric Mean) inequality (9), we see that

$$\begin{array}{cc}\hfill & {\left[{\Pi }_{k=1}^n\chi \left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)\right]}^{\frac{1}{n}}\hfill \\ \hfill & ={\left[\chi \left(A_{\varrho }{\varpi }_1\left(\lambda \right)\right)\chi \left(A_{\varrho }{\varpi }_2\left(\lambda \right)\right)...\chi \left(A_{\varrho }{\varpi }_n\left(\lambda \right)\right)\right]}^{\frac{1}{n}}\hfill \\ \hfill & \le \frac{{\sum }_{k=1}^n\chi \left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)}{n}.\hfill \end{array}$$

Then, we obtain

$$\begin{array}{cc}\hfill & {\int }_r^y{\left[{\Pi }_{k=1}^n\chi \left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)\right]}^{\frac{\eta }{n}}\varkappa \left(\lambda \right)\frac{\nabla \lambda }{\rho \left(\lambda \right)-r}\hfill \\ \hfill & \le {\int }_r^y{\left(\frac{{\sum }_{k=1}^n\chi \left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)}{n}\right)}^{\eta }\varkappa \left(\lambda \right)\frac{\nabla \lambda }{\rho \left(\lambda \right)-r}\hfill \\ \hfill & ={\left(\frac{1}{n}\right)}^{\eta }{\int }_r^y{\left(\sum _{k=1}^n\chi \left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)\right)}^{\eta }\varkappa \left(\lambda \right)\frac{\nabla \lambda }{\rho \left(\lambda \right)-r}.\hfill \end{array}$$

(23)

By applying (10) (where $\eta \ge 1$), we observe that

$$\begin{array}{cc}\hfill & {\left({\int }_r^y{\left(\sum _{k=1}^n\chi \left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)\right)}^{\eta }\varkappa \left(\lambda \right)\frac{\nabla \lambda }{\rho \left(\lambda \right)-r}\right)}^{\frac{1}{\eta }}\hfill \\ \hfill & ={\left({\int }_r^y\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}{\left[\chi \left(A_{\varrho }{\varpi }_1\left(\lambda \right)\right)+...+\chi \left(A_{\varrho }{\varpi }_n\left(\lambda \right)\right)\right]}^{\eta }\nabla \lambda \right)}^{\frac{1}{\eta }}\hfill \\ \hfill & \le {\left({\int }_r^y\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}{\chi }^{\eta }\left(A_{\varrho }{\varpi }_1\left(\lambda \right)\right)\nabla \lambda \right)}^{\frac{1}{\eta }}+...+{\left({\int }_r^y\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}{\chi }^{\eta }\left(A_{\varrho }{\varpi }_n\left(\lambda \right)\right)\nabla \lambda \right)}^{\frac{1}{\eta }}\hfill \\ \hfill & =\sum _{k=1}^n{\left({\int }_r^y\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}{\chi }^{\eta }\left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)\nabla \lambda \right)}^{\frac{1}{\eta }},\hfill \end{array}$$

and then

$$\begin{array}{cc}\hfill & {\int }_r^y{\left(\sum _{k=1}^n\chi \left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)\right)}^{\eta }\varkappa \left(\lambda \right)\frac{\nabla \lambda }{\rho \left(\lambda \right)-r}\hfill \\ \hfill & \le {\left[\sum _{k=1}^n{\left({\int }_r^y\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}{\chi }^{\eta }\left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)\nabla \lambda \right)}^{\frac{1}{\eta }}\right]}^{\eta }.\hfill \end{array}$$

(24)

Substituting (24) into (23), we have that

$$\begin{array}{cc}\hfill & {\int }_r^y{\left[{\Pi }_{k=1}^n\chi \left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)\right]}^{\frac{\eta }{n}}\varkappa \left(\lambda \right)\frac{\nabla \lambda }{\rho \left(\lambda \right)-r}\hfill \\ \hfill & \le {\left(\frac{1}{n}\right)}^{\eta }{\left[\sum _{k=1}^n{\left({\int }_r^y\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}{\chi }^{\eta }\left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)\nabla \lambda \right)}^{\frac{1}{\eta }}\right]}^{\eta }.\hfill \end{array}$$

(25)

Using (21) and applying (13), we get

$$\begin{array}{cc}\hfill & {\int }_r^y{\chi }^{\eta }\left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)\varkappa \left(\lambda \right)\frac{\nabla \lambda }{\rho \left(\lambda \right)-r}\hfill \\ \hfill & ={\int }_r^y{\chi }^{\eta }\left(\frac{1}{\Lambda \left(\lambda \right)}{\int }_r^y\varrho (\lambda ,\vartheta ){\varpi }_k\left(\vartheta \right)\nabla \vartheta \right)\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \hfill \\ \hfill & \le B^{\eta }{\int }_r^y{\xi }^{\eta }\left(\frac{1}{\Lambda \left(\lambda \right)}{\int }_r^y\varrho (\lambda ,\vartheta ){\varpi }_k\left(\vartheta \right)\nabla \vartheta \right)\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \hfill \\ \hfill & \le B^{\eta }{\int }_r^y\frac{1}{{\Lambda }^{\eta }\left(\lambda \right)}{\left({\int }_r^y\varrho (\lambda ,\vartheta )\xi \left({\varpi }_k\left(\vartheta \right)\right)\nabla \vartheta \right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda .\hfill \end{array}$$

(26)

Applying (11) on the term

$${\int }_r^y\frac{1}{{\Lambda }^{\eta }\left(\lambda \right)}{\left({\int }_r^y\varrho (\lambda ,\vartheta )\xi \left({\varpi }_k\left(\vartheta \right)\right)\nabla \vartheta \right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda ,$$

with $\eta \ge 1,$ we see that

$$\begin{array}{cc}\hfill & {\left({\int }_r^y\frac{1}{{\Lambda }^{\eta }\left(\lambda \right)}{\left({\int }_r^y\varrho (\lambda ,\vartheta )\xi \left({\varpi }_k\left(\vartheta \right)\right)\nabla \vartheta \right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \right)}^{\frac{1}{\eta }}\hfill \\ \hfill & \le {\int }_r^y\xi \left({\varpi }_k\left(\vartheta \right)\right){\left({\int }_r^y{\left(\frac{\varrho (\lambda ,\vartheta )}{\Lambda \left(\lambda \right)}\right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \right)}^{\frac{1}{\eta }}\nabla \vartheta ,\hfill \end{array}$$

then

$$\begin{array}{cc}\hfill & {\int }_r^y\frac{1}{{\Lambda }^{\eta }\left(\lambda \right)}{\left({\int }_r^y\varrho (\lambda ,\vartheta )\xi \left({\varpi }_k\left(\vartheta \right)\right)\nabla \vartheta \right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \hfill \\ \hfill & \le {\left[{\int }_r^y\xi \left({\varpi }_k\left(\vartheta \right)\right){\left({\int }_r^y{\left(\frac{\varrho (\lambda ,\vartheta )}{\Lambda \left(\lambda \right)}\right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \right)}^{\frac{1}{\eta }}\nabla \vartheta \right]}^{\eta }\hfill \\ \hfill & =\left[{\int }_r^y\xi \left({\varpi }_k\left(\vartheta \right)\right)\frac{1}{\rho \left(\vartheta \right)-r}\left(\rho \left(\vartheta \right)-r\right)\right.\hfill \\ \hfill & {\left.\times {\left({\int }_r^y{\left(\frac{\varrho (\lambda ,\vartheta )}{\Lambda \left(\lambda \right)}\right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \right)}^{\frac{1}{\eta }}\nabla \vartheta \right]}^{\eta }.\hfill \end{array}$$

(27)

Substituting (27) into (26) and using (20), we get

$$\begin{array}{cc}\hfill & {\int }_r^y{\chi }^{\eta }\left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)\varkappa \left(\lambda \right)\frac{\nabla \lambda }{\rho \left(\lambda \right)-r}\hfill \\ \hfill & \le B^{\eta }\left[{\int }_r^y\xi \left({\varpi }_k\left(\vartheta \right)\right)\frac{1}{\rho \left(\vartheta \right)-r}\left(\rho \left(\vartheta \right)-r\right)\right.\hfill \\ \hfill & \times {\left.{\left({\int }_r^y{\left(\frac{\varrho (\lambda ,\vartheta )}{\Lambda \left(\lambda \right)}\right)}^{\eta }\frac{\varkappa \left(\lambda \right)}{\rho \left(\lambda \right)-r}\nabla \lambda \right)}^{\frac{1}{\eta }}\nabla \vartheta \right]}^{\eta }\hfill \\ \hfill & =B^{\eta }{\left[{\int }_r^y\xi \left({\varpi }_k\left(\vartheta \right)\right)\frac{1}{\rho \left(\vartheta \right)-r}\omega \left(\vartheta \right)\nabla \vartheta \right]}^{\eta }.\hfill \end{array}$$

From (21), we obtain

$$\begin{array}{cc}\hfill & {\int }_r^y{\chi }^{\eta }\left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)\varkappa \left(\lambda \right)\frac{\nabla \lambda }{\rho \left(\lambda \right)-r}\hfill \\ \hfill & \le {\left(\frac{B}{A}\right)}^{\eta }{\left[{\int }_r^y\chi \left({\varpi }_k\left(\vartheta \right)\right)\frac{1}{\rho \left(\vartheta \right)-r}\omega \left(\vartheta \right)\nabla \vartheta \right]}^{\eta }.\hfill \end{array}$$

Substituting the last inequality into (25), we get

$$\begin{array}{cc}\hfill & {\int }_r^y{\left[{\Pi }_{k=1}^n\chi \left(A_{\varrho }{\varpi }_k\left(\lambda \right)\right)\right]}^{\frac{\eta }{n}}\varkappa \left(\lambda \right)\frac{\nabla \lambda }{\rho \left(\lambda \right)-r}\hfill \\ \hfill & \le {\left(\frac{B}{nA}\right)}^{\eta }{\left(\sum _{k=1}^n\left[{\int }_r^y\chi \left({\varpi }_k\left(\vartheta \right)\right)\frac{1}{\rho \left(\vartheta \right)-r}\omega \left(\vartheta \right)\nabla \vartheta \right]\right)}^{\eta },\hfill \end{array}$$

which is (22). □

Remark 4.

If $n=1,$ we get Theorem 4.

Multidimensional Inequalities on Time Scales

In the following section, we define

$$A_{\varrho }\varpi \left(\mathbf{y}\right)=\frac{1}{\Lambda \left(\mathbf{y}\right)}{\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\varrho (\mathbf{y},\mathbf{z})\varpi \left(\mathbf{z}\right)\nabla \mathbf{z},\Lambda \left(\mathbf{y}\right)={\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\varrho (\mathbf{y},\mathbf{z})\nabla \mathbf{z},$$

where $\varrho (\mathbf{y},\mathbf{z})=\varrho (y_1,...,y_m,z_1,...,z_m),$$\nabla \mathbf{y}=\nabla y_1...\nabla y_m$ and $\varpi \left(\mathbf{z}\right)=\varpi (z_1,...,z_m).$

Theorem 6.

Let ${\epsilon }_i,{ϵ}_i\in \mathbb{T},$ $i=1,2,...,m,$$\eta \ge 1$ and $\varkappa ,\omega$ be as in Theorem 4, such that

$$\begin{array}{cc}\hfill \omega \left(\mathbf{z}\right)& =\left(\rho \left(z_1\right)-{\epsilon }_1\right)...\left(\rho \left(z_m\right)-{\epsilon }_m\right)\hfill \\ \hfill & \times {\left({\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}{\left(\frac{\varrho (\mathbf{y},\mathbf{z})}{\Lambda \left(\mathbf{y}\right)}\right)}^{\eta }\frac{\varkappa \left(\mathbf{y}\right)}{(\rho \left(y_1\right)-{\epsilon }_1)...\left(\rho \left(y_m\right)-{\epsilon }_m\right)}\nabla \mathbf{y}\right)}^{\frac{1}{\eta }}.\hfill \end{array}$$

(28)

In addition, assume that $\chi ,\xi$ are as in Theorem 4, such that

$$A\xi \left(\mathbf{y}\right)\le \chi \left(\mathbf{y}\right)\le B\xi \left(\mathbf{y}\right),$$

(29)

where $A,B$ are positive constants, then

$$\begin{array}{cc}\hfill & {\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}{\chi }^{\eta }\left(A_{\varrho }\varpi \left(\mathbf{y}\right)\right)\varkappa \left(\mathbf{y}\right)\frac{\nabla \mathbf{y}}{\left(\rho \left(y_1\right)-{\epsilon }_1\right)...\left(\rho \left(y_m\right)-{\epsilon }_m\right)}\hfill \\ \hfill & \le {\left(\frac{B}{A}\right)}^{\eta }{\left[{\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\chi \left(\varpi \left(\mathbf{z}\right)\right)\frac{1}{\left(\rho \left(z_1\right)-{\epsilon }_1\right)...\left(\rho \left(z_m\right)-{\epsilon }_m\right)}\omega \left(\mathbf{z}\right)\nabla \mathbf{z}\right]}^{\eta },\hfill \end{array}$$

(30)

holds for the non-negative function $\varpi .$

Proof. 

Using (29) and applying (14), we get

$$\begin{array}{cc}\hfill & {\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}{\chi }^{\eta }\left(A_{\varrho }\varpi \left(\mathbf{y}\right)\right)\varkappa \left(\mathbf{y}\right)\frac{\nabla \mathbf{y}}{\left(\rho \left(y_1\right)-{\epsilon }_1\right)...\left(\rho \left(y_m\right)-{\epsilon }_m\right)}\hfill \\ \hfill & ={\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}{\chi }^{\eta }\left(\frac{1}{\Lambda \left(\mathbf{y}\right)}{\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\varrho (\mathbf{y},\mathbf{z})\varpi \left(\mathbf{z}\right)\nabla \mathbf{z}\right)\hfill \\ \hfill & \times \frac{\varkappa \left(\mathbf{y}\right)}{\left(\rho \left(y_1\right)-{\epsilon }_1\right)...\left(\rho \left(y_m\right)-{\epsilon }_m\right)}\nabla \mathbf{y}\hfill \\ \hfill & \le B^{\eta }{\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}{\xi }^{\eta }\left(\frac{1}{\Lambda \left(\mathbf{y}\right)}{\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\varrho (\mathbf{y},\mathbf{z})\varpi \left(\mathbf{z}\right)\nabla \mathbf{z}\right)\hfill \\ \hfill & \times \frac{\varkappa \left(\mathbf{y}\right)}{\left(\rho \left(y_1\right)-{\epsilon }_1\right)...\left(\rho \left(y_m\right)-{\epsilon }_m\right)}\nabla \mathbf{y}\hfill \\ \hfill & \le B^{\eta }{\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\frac{1}{{\Lambda }^{\eta }\left(\mathbf{y}\right)}{\left({\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\varrho (\mathbf{y},\mathbf{z})\xi \left(\varpi \left(\mathbf{z}\right)\right)\nabla \mathbf{z}\right)}^{\eta }\hfill \\ \hfill & \times \frac{\varkappa \left(\mathbf{y}\right)}{\left(\rho \left(y_1\right)-{\epsilon }_1\right)...\left(\rho \left(y_m\right)-{\epsilon }_m\right)}\nabla \mathbf{y}.\hfill \end{array}$$

(31)

Applying (12) on the term

$$\begin{array}{cc}\hfill & {\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\frac{1}{{\Lambda }^{\eta }\left(\mathbf{y}\right)}{\left({\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\varrho (\mathbf{y},\mathbf{z})\xi \left(\varpi \left(\mathbf{z}\right)\right)\nabla \mathbf{z}\right)}^{\eta }\hfill \\ \hfill & \times \frac{\varkappa \left(\mathbf{y}\right)}{\left(\rho \left(y_1\right)-{\epsilon }_1\right)...\left(\rho \left(y_m\right)-{\epsilon }_m\right)}\nabla \mathbf{y},\hfill \end{array}$$

with $\eta \ge 1,$ we see that

$$\begin{array}{cc}\hfill & \left[{\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\frac{1}{{\Lambda }^{\eta }\left(\mathbf{y}\right)}{\left({\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\varrho (\mathbf{y},\mathbf{z})\xi \left(\varpi \left(\mathbf{z}\right)\right)\nabla \mathbf{z}\right)}^{\eta }\right.\hfill \\ \hfill & \times {\left.\frac{\varkappa \left(\mathbf{y}\right)}{\left(\rho \left(y_1\right)-{\epsilon }_1\right)...\left(\rho \left(y_m\right)-{\epsilon }_m\right)}\nabla \mathbf{y}\right]}^{\frac{1}{\eta }}\hfill \\ \hfill & \le {\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\xi \left(\varpi \left(\mathbf{z}\right)\right){\left({\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}{\left(\frac{\varrho (\mathbf{y},\mathbf{z})}{\Lambda \left(\mathbf{y}\right)}\right)}^{\eta }\frac{\varkappa \left(\mathbf{y}\right)}{\left(\rho \left(y_1\right)-{\epsilon }_1\right)...\left(\rho \left(y_m\right)-{\epsilon }_m\right)}\nabla \mathbf{y}\right)}^{\frac{1}{\eta }}\nabla \mathbf{z},\hfill \end{array}$$

then

$$\begin{array}{cc}\hfill & {\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\frac{1}{{\Lambda }^{\eta }\left(\mathbf{y}\right)}{\left({\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\varrho (\mathbf{y},\mathbf{z})\xi \left(\varpi \left(\mathbf{z}\right)\right)\nabla \mathbf{z}\right)}^{\eta }\hfill \\ \hfill & \times \frac{\varkappa \left(\mathbf{y}\right)}{\left(\rho \left(y_1\right)-{\epsilon }_1\right)...\left(\rho \left(y_m\right)-{\epsilon }_m\right)}\nabla \mathbf{y}\hfill \\ \hfill & \le \left[{\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\xi \left(\varpi \left(\mathbf{z}\right)\right)\right.\hfill \\ \hfill & \times {\left.{\left({\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}{\left(\frac{\varrho (\mathbf{y},\mathbf{z})}{\Lambda \left(\mathbf{y}\right)}\right)}^{\eta }\frac{\varkappa \left(\mathbf{y}\right)}{\left(\rho \left(y_1\right)-{\epsilon }_1\right)...\left(\rho \left(y_m\right)-{\epsilon }_m\right)}\nabla \mathbf{y}\right)}^{\frac{1}{\eta }}\nabla \mathbf{z}\right]}^{\eta }.\hfill \end{array}$$

(32)

Substituting (32) into (31), we have from (28) that

$$\begin{array}{cc}\hfill & {\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}{\chi }^{\eta }\left(A_{\varrho }\varpi \left(\mathbf{y}\right)\right)\varkappa \left(\mathbf{y}\right)\frac{\nabla \mathbf{y}}{\left(\rho \left(y_1\right)-{\epsilon }_1\right)...\left(\rho \left(y_m\right)-{\epsilon }_m\right)}\hfill \\ \hfill & \le B^{\eta }\left[{\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\xi \left(\varpi \left(\mathbf{z}\right)\right)\right.\hfill \\ \hfill & \times {\left.{\left({\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}{\left(\frac{\varrho (\mathbf{y},\mathbf{z})}{\Lambda \left(\mathbf{y}\right)}\right)}^{\eta }\frac{\varkappa \left(\mathbf{y}\right)}{\left(\rho \left(y_1\right)-{\epsilon }_1\right)...\left(\rho \left(y_m\right)-{\epsilon }_m\right)}\nabla \mathbf{y}\right)}^{\frac{1}{\eta }}\nabla \mathbf{z}\right]}^{\eta }\hfill \\ \hfill & =B^{\eta }{\left[{\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\xi \left(\varpi \left(\mathbf{z}\right)\right)\frac{1}{\left(\rho \left(z_1\right)-{\epsilon }_1\right)...\left(\rho \left(z_m\right)-{\epsilon }_m\right)}\omega \left(\mathbf{z}\right)\nabla \mathbf{z}\right]}^{\eta },\hfill \end{array}$$

and then we have from (29) that

$$\begin{array}{cc}\hfill & {\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}{\chi }^{\eta }\left(A_{\varrho }\varpi \left(\mathbf{y}\right)\right)\varkappa \left(\mathbf{y}\right)\frac{\nabla \mathbf{y}}{\left(\rho \left(y_1\right)-{\epsilon }_1\right)...\left(\rho \left(y_m\right)-{\epsilon }_m\right)}\hfill \\ \hfill & \le {\left(\frac{B}{A}\right)}^{\eta }{\left[{\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\chi \left(\varpi \left(\mathbf{z}\right)\right)\frac{1}{\left(\rho \left(z_1\right)-{\epsilon }_1\right)...\left(\rho \left(z_m\right)-{\epsilon }_m\right)}\omega \left(\mathbf{z}\right)\nabla \mathbf{z}\right]}^{\eta },\hfill \end{array}$$

which is (30). □

Remark 5.

If $\mathbb{T}=\mathbb{R},$$\rho \left(\vartheta \right)=\vartheta$ and $A=B=1,$ then

$$\begin{array}{cc}\hfill & {\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}{\chi }^{\eta }\left(A_{\varrho }\varpi \left(\mathbf{y}\right)\right)\frac{\varkappa \left(\mathbf{y}\right)}{\left(y_1-{\epsilon }_1\right)...\left(y_m-{\epsilon }_m\right)}d\mathbf{y}\hfill \\ \hfill & \le {\left({\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\chi \left(\varpi \left(\mathbf{z}\right)\right)\frac{\omega \left(\mathbf{z}\right)}{\left(z_1-{\epsilon }_1\right)...\left(z_m-{\epsilon }_m\right)}d\mathbf{z}\right)}^{\eta },\hfill \end{array}$$

where $\varpi \left(\mathbf{z}\right)=\varpi (z_1,z_2,...,z_m),$$\varrho (\mathbf{y},\mathbf{z})=\varrho (y_1,...,y_m,z_1,...,z_m)$ and

$$A_{\varrho }\varpi \left(\mathbf{y}\right)=\frac{1}{\Lambda \left(\mathbf{y}\right)}{\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\varrho (\mathbf{y},\mathbf{z})\varpi \left(\mathbf{z}\right)d\mathbf{z},\Lambda \left(\mathbf{y}\right)={\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}\varrho (\mathbf{y},\mathbf{z})d\mathbf{z},$$

with

$$\begin{array}{cc}\hfill \omega \left(\mathbf{z}\right)& =\left(z_1-{\epsilon }_1\right)...\left(z_m-{\epsilon }_m\right)\hfill \\ \hfill & \times {\left({\int }_{{\epsilon }_1}^{{ϵ}_1}...{\int }_{{\epsilon }_m}^{{ϵ}_m}{\left(\frac{\varrho (\mathbf{y},\mathbf{z})}{\Lambda \left(\mathbf{y}\right)}\right)}^{\eta }\frac{\varkappa \left(\mathbf{y}\right)}{\left(y_1-{\epsilon }_1\right)...\left(y_m-{\epsilon }_m\right)}d\mathbf{y}\right)}^{\frac{1}{\eta }}.\hfill \end{array}$$

Remark 6.

If $\mathbb{T}=\mathbb{N},$$\rho \left(\vartheta \right)=\vartheta -1$ and $A=B=1,$ then

$$\begin{array}{cc}\hfill & \sum _{{\epsilon }_1}^{{ϵ}_1}...\sum _{{\epsilon }_m}^{{ϵ}_m}{\chi }^{\eta }\left(A_{\varrho }\varpi \left(\mathbf{y}\right)\right)\frac{\varkappa \left(\mathbf{y}\right)}{\left(y_1-{\epsilon }_1-1\right)...\left(y_m-{\epsilon }_m-1\right)}\hfill \\ \hfill & \le {\left[\sum _{{\epsilon }_1}^{{ϵ}_1}...\sum _{{\epsilon }_m}^{{ϵ}_m}\chi \left(\varpi \left(\mathbf{z}\right)\right)\frac{1}{\left(z_1-{\epsilon }_1-1\right)...\left(z_m-{\epsilon }_m-1\right)}\omega \left(\mathbf{z}\right)\right]}^{\eta },\hfill \end{array}$$

where $\varpi \left(\mathbf{z}\right)=\varpi (z_1,z_2,...,z_m),$$\varrho (\mathbf{y},\mathbf{z})=\varrho (y_1,...,y_m,z_1,...,z_m)$ and

$$A_{\varrho }\varpi \left(\mathbf{y}\right)=\frac{1}{\Lambda \left(\mathbf{y}\right)}\sum _{{\epsilon }_1}^{{ϵ}_1}...\sum _{{\epsilon }_m}^{{ϵ}_m}\varrho (\mathbf{y},\mathbf{z})\varpi \left(\mathbf{z}\right),\Lambda \left(\mathbf{y}\right)=\sum _{{\epsilon }_1}^{{ϵ}_1}...\sum _{{\epsilon }_m}^{{ϵ}_m}\varrho (\mathbf{y},\mathbf{z}),$$

with

$$\begin{array}{cc}\hfill \omega \left(\mathbf{z}\right)& =\left(z_1-{\epsilon }_1-1\right)...\left(z_m-{\epsilon }_m-1\right)\hfill \\ \hfill & \times {\left(\sum _{{\epsilon }_1}^{{ϵ}_1}...\sum _{{\epsilon }_m}^{{ϵ}_m}{\left(\frac{\varrho (\mathbf{y},\mathbf{z})}{\Lambda \left(\mathbf{y}\right)}\right)}^{\eta }\frac{\varkappa \left(\mathbf{y}\right)}{\left(y_1-{\epsilon }_1-1\right)...\left(y_m-{\epsilon }_m-1\right)}\right)}^{\frac{1}{\eta }}.\hfill \end{array}$$

4. Conclusions

In this research, we generalize some new inequalities on time-scale nabla calculus. We will also establish some dynamic inequalities for several functions. Furthermore, we will establish these inequalities in multiple dimensions on time-scales nabla calculus. All of these inequalities can be proved by applying Minkowski’s inequality, Jensen’s inequality and Arithmetic Mean–Geometric Mean inequality. In the future, we hope to study these dynamic inequalities via conformable nabla fractional calculus on time scales.