Generalized Inequalities of Hilbert-Type on Time Scales Nabla Calculus

Abstract

In this paper, we prove some new generalized inequalities of Hilbert-type on time scales nabla calculus by applying Hölder’s inequality, Young’s inequality, and Jensen’s inequality. Symmetrical properties play an essential role in determining the correct methods to solve inequalities.

1. Introduction

In the time (1862–1943), David Hilbert proved Hilbert’s double series inequality without an exact determination of the constant in his lectures on integral equations. If $\left\{{\beta }_m\right\}$ and $\left\{d_n\right\}$ are two real sequences such that $0<{\sum }_{m=1}^{\infty }{\beta }_m^2<\infty$ and $0<{\sum }_{n=1}^{\infty }{d}_n^2<\infty ,$ then

$$\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }\frac{{\beta }_m{d}_n}{m+n}\le \pi {\left(\sum _{m=1}^{\infty }{\beta }_m^2\right)}^{\frac{1}{2}}{\left(\sum _{n=1}^{\infty }{d}_n^2\right)}^{\frac{1}{2}}.$$

(1)

In 1911, Schur [1] proved that $\pi$ in (1) is sharp and also discovered the integral analogue of (1), which became known as the Hilbert integral inequality in the form

$${\int }_0^{\infty }{\int }_0^{\infty }\frac{\mathrm{\Xi }\left(\tau \right)\mathrm{{\rm Y}}\left(y\right)}{\tau +y}d\tau dy\le \pi {\left({\int }_0^{\infty }{\mathrm{\Xi }}^2\left(\tau \right)d\tau \right)}^{\frac{1}{2}}{\left({\int }_0^{\infty }{\mathrm{{\rm Y}}}^2\left(y\right)dy\right)}^{\frac{1}{2}},$$

(2)

where $\mathrm{\Xi }$ and $\mathrm{{\rm Y}}$ are measurable functions such that, $0<{\int }_0^{\infty }{\mathrm{\Xi }}^2\left(\tau \right)d\tau <\infty$ and $0<{\int }_0^{\infty }{\mathrm{{\rm Y}}}^2\left(y\right)dy<\infty$.

In 1925, by introducing one pair of conjugate exponents $(p,$ $q)$ with $1/p+1/q=1,$ Hardy [2] gave an extension of (1) as follows. If $p,q>1,$ ${\beta }_m,d_n\ge 0$ such that $0<{\sum }_{m=1}^{\infty }{\beta }_m^p<\infty$ and $0<{\sum }_{n=1}^{\infty }{d}_n^q<\infty ,$ then

$$\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }\frac{{\beta }_m{d}_n}{m+n}\le \frac{\pi }{sin\frac{\pi }{p}}{\left(\sum _{m=1}^{\infty }{\beta }_m^p\right)}^{{}^{\frac{1}{p}}}{\left(\sum _{n=1}^{\infty }{d}_n^q\right)}^{{}^{\frac{1}{q}}},$$

(3)

where the constant $\pi /sin(\pi /p)$ in (3) is sharp. In 1934, Hardy et al. [3] proved the equivalent integral analog of (3) in the form

$${\int }_0^{\infty }{\int }_0^{\infty }\frac{\mathrm{\Xi }\left(\tau \right)\mathrm{{\rm Y}}\left(y\right)}{\tau +y}d\tau dy\le \frac{\pi }{sin\frac{\pi }{p}}{\left({\int }_0^{\infty }{\mathrm{\Xi }}^p\left(\tau \right)d\tau \right)}^{{}^{\frac{1}{p}}}{\left({\int }_0^{\infty }{\mathrm{{\rm Y}}}^q\left(y\right)dy\right)}^{{}^{\frac{1}{q}}},$$

(4)

where $\mathrm{\Xi }$ and $\mathrm{{\rm Y}}$ are measurable non-negative functions such that $0<{\int }_0^{\infty }{\mathrm{\Xi }}^p\left(\tau \right)d\tau <\infty$ and $0<{\int }_0^{\infty }{\mathrm{{\rm Y}}}^q\left(y\right)dy<\infty$.

In 1998, Pachpatte [4] gave a new inequality closed to that of Hilbert as follows: Let $\beta \left(\iota \right):\left\{0,1,2,\dots ,p\right\}\subset \mathbb{N}\to \mathbb{R}$ and $d\left(\vartheta \right):\left\{0,1,2,\dots ,q\right\}\subset \mathbb{N}\to \mathbb{R}$ with $\beta \left(0\right)=d\left(0\right)=0.$ Then

$$\begin{array}{c}{\displaystyle \sum _{\iota =1}^p}{\displaystyle \sum _{\vartheta =1}^q}\frac{\left|{\beta }_{\iota }\right|\left|d_{\vartheta }\right|}{\iota +\vartheta }\le C(p,q){\left({\displaystyle \sum _{\iota =1}^p}(p-\iota +1){\left|\nabla {\beta }_{\iota }\right|}^2\right)}^{\frac{1}{2}}\\ \times {\left({\displaystyle \sum _{\vartheta =1}^q}(q-\vartheta +1){\left|\nabla d_{\vartheta }\right|}^2\right)}^{\frac{1}{2}},\end{array}$$

(5)

where $\nabla {\beta }_{\iota }={\beta }_{\iota }-{\beta }_{\iota -1},$ $\nabla d_{\vartheta }=d_{\vartheta }-d_{\vartheta -1}$ and

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

In 2000, Pachpatte [5] generalized (5) by introducing one pair of conjugate exponents $(\lambda ,\mu )$, such that $\lambda ,\mu >1$ with $1/\lambda +1/\mu =1$. Then, it is established that if $\beta \left(\iota \right):\left\{0,1,2,\dots ,p\right\}\subset \mathbb{N}\to \mathbb{R}$ and $d\left(\vartheta \right):\left\{0,1,2,\dots ,q\right\}\subset \mathbb{N}\to \mathbb{R}$ with $\beta \left(0\right)=d\left(0\right)=0,$ then

$$\begin{array}{c}{\displaystyle \sum _{\iota =1}^p\sum _{\vartheta =1}^q}\frac{\left|{\beta }_{\iota }\right|\left|d_{\vartheta }\right|}{\mu {\iota }^{\lambda -1}+\lambda {\vartheta }^{\mu -1}}\le D(\lambda ,\mu ,p,q){\left({\displaystyle \sum _{\iota =1}^p}(p-\iota +1){\left|\nabla {\beta }_{\iota }\right|}^{\lambda }\right)}^{\frac{1}{\lambda }}\\ \times {\left({\displaystyle \sum _{\vartheta =1}^q}(q-\vartheta +1){\left|\nabla d_{\vartheta }\right|}^{\mu }\right)}^{\frac{1}{\mu }},\end{array}$$

(6)

where $\nabla {\beta }_{\iota }={\beta }_{\iota }-{\beta }_{\iota -1},$ $\nabla d_{\vartheta }=d_{\vartheta }-d_{\vartheta -1}$ and

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

In 2002, Kim et al. [6] generalized (6) and proved that if $\lambda ,$ $\mu >1,$$\beta \left(\iota \right):\left\{0,1,2,\dots ,p\right\}\subset \mathbb{N}\to \mathbb{R}$ and $d\left(\vartheta \right):\left\{0,1,2,\dots ,q\right\}\subset \mathbb{N}\to \mathbb{R}$ with $\beta \left(0\right)=d\left(0\right)=0,$ then

$$\begin{array}{c}{\displaystyle \sum _{\iota =1}^p\sum _{\vartheta =1}^q}\frac{\left|{\beta }_{\iota }\right|\left|d_{\vartheta }\right|}{\mu {\iota }^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\vartheta }^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}}\le D^{*}(\lambda ,\mu ,p,q){\left({\displaystyle \sum _{\iota =1}^p}(p-\iota +1){\left|\nabla {\beta }_{\iota }\right|}^{\lambda }\right)}^{\frac{1}{\lambda }}\\ \times {\left({\displaystyle \sum _{\vartheta =1}^q}(q-\vartheta +1){\left|\nabla d_{\vartheta }\right|}^{\mu }\right)}^{\frac{1}{\mu }},\end{array}$$

(7)

where $\nabla {\beta }_{\iota }={\beta }_{\iota }-{\beta }_{\iota -1},$ $\nabla d_{\vartheta }=d_{\vartheta }-d_{\vartheta -1}$ and

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

Furthermore, the researchers [6] proved the continuous analog of (7) and proved that if $\lambda ,$ $\mu >1,$ and $\mathrm{\Xi }\left(\iota \right),$ $\mathrm{{\rm Y}}\left(t\right)$ are real continuous functions on the intervals $\left(0,\tau \right),$ $\left(0,y\right),$ respectively, and let $\mathrm{\Xi }\left(0\right)=\mathrm{{\rm Y}}\left(0\right)=0.$ Then

$$\begin{array}{c}{\int }_0^{\tau }{\int }_0^y\frac{\left|\mathrm{\Xi }\left(\iota \right)\right|\left|\mathrm{{\rm Y}}\left(t\right)\right|}{\mu {\iota }^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\vartheta }^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}}d\iota dt\\ \le M(\lambda ,\mu ,\tau ,y){\left({\int }_0^{\tau }\left(\tau -\iota \right){\left|{\mathrm{\Xi }}^{\prime }\left(\iota \right)\right|}^{\lambda }d\iota \right)}^{\frac{1}{\lambda }}{\left({\int }_0^y\left(y-t\right){\left|{\mathrm{{\rm Y}}}^{\prime }\left(t\right)\right|}^{\mu }dt\right)}^{\frac{1}{\mu }},\end{array}$$

(8)

for $\tau ,$$y\in \left(0,\infty \right)$, where

$$M(\lambda ,\mu ,\tau ,y)=\frac{1}{\lambda +\mu }{\tau }^{\frac{\lambda -1}{\lambda }}{y}^{\frac{\mu -1}{\mu }}.$$

In recent decades, a new theory has been discovered to unify the continuous calculus and discrete calculus. Many authors proved some dynamic inequalities of Hilbert type, its generalizations and also the reversed forms, see the papers [7,8,9].

The goal of this article is to use time scales nabla calculus to prove various dynamic inequalities for Hilbert-type. Furthermore, we establish some generalized inequalities of Hilbert type by using submuliplicative and bounded functions.

The organization of the paper as follows. In Section 2, we show some basics of the time scale theory and some lemmas needed in Section 3 where we prove our results. Our results as special cases give the inequalities (7) and (8) proved by Kim et al. [6].

2. Preliminaries

For a time scale $\mathbb{T}$, we define the backward jump operator as following $\rho \left(\tau \right):=sup\{\iota \in \mathbb{T}:\iota <\tau \}.$ Let $\mathrm{\Xi }:\mathbb{T}\to \mathbb{R}$ be a function, we say that $\mathrm{\Xi }$ is ld-continuous if it is continuous at each left dense point in $\mathbb{T}$ and the right limit exists as a finite number for all right dense points $t\in \mathbb{T}.$ The set of all such ld–continuous functions is ushered by $C_{ld}(\mathbb{T},\mathbb{R})$ and for any function $\mathrm{\Xi }:\mathbb{T}\to \mathbb{R}$, the notation ${\mathrm{\Xi }}^{\rho }\left(\tau \right)$ denotes $\mathrm{\Xi }\left(\rho \right(\tau \left)\right).$ For more information about the time scale calculus, see [10,11].

The nabla derivative of $uv$ and $u/v$ (where $v\left(\tau \right)v^{\rho }\left(\tau \right)\ne 0$) are

$$\begin{array}{ccc}\hfill {\left(uv\right)}^{\nabla }\left(\tau \right)& =& u^{\nabla }\left(\tau \right)v\left(\tau \right)+u^{\rho }\left(\tau \right)v^{\nabla }\left(\tau \right)\hfill \\ & =& u\left(\tau \right)v^{\nabla }\left(\tau \right)+u^{\nabla }\left(\tau \right)v^{\rho }\left(\tau \right),\hfill \end{array}$$

and

$${\left(\frac{u}{v}\right)}^{\nabla }\left(\tau \right)=\frac{u^{\nabla }\left(\tau \right)v\left(\tau \right)-u\left(\tau \right)v^{\nabla }\left(\tau \right)}{v\left(\tau \right)v^{\rho }\left(\tau \right)}.$$

Definition 1

([10]). A function $\mathrm{\Xi }:\mathbb{T}\to \mathbb{R}$ is called a nabla antiderivative of $\mathrm{\Xi }:\mathbb{T}\to \mathbb{R}$ provided ${\mathrm{\Xi }}^{\nabla }\left(t\right)=\mathrm{\Xi }\left(t\right)$ holds $\forall t\in \mathbb{T}$. We then define the integral of Ξ by

$${\int }_{\beta }^t\mathrm{\Xi }\left(\tau \right)\nabla \tau =\mathrm{\Xi }\left(t\right)-\mathrm{\Xi }\left(\beta \right)\forall t\in \mathbb{T}.$$

Theorem 1

([10]). If $\beta ,$ $d\in \mathbb{T},$ $\alpha \in \mathbb{R}$ and $\mathrm{\Xi },\mathrm{{\rm Y}}\in C_{ld}(\mathbb{T},\mathbb{R})$, then

(1)

${\int }_{\beta }^d\left[\mathrm{\Xi }\left(\tau \right)+\mathrm{{\rm Y}}\left(\tau \right)\right]\nabla \tau ={\int }_{\beta }^d\mathrm{\Xi }\left(\tau \right)\nabla \tau +{\int }_{\beta }^d\mathrm{{\rm Y}}\left(\tau \right)\nabla \tau ;$

(2)

${\int }_{\beta }^d\alpha \mathrm{\Xi }\left(\tau \right)\nabla \tau =\alpha {\int }_{\beta }^d\mathrm{\Xi }\left(\tau \right)\nabla \tau ;$

(3)

${\int }_{\beta }^{\beta }\mathrm{\Xi }\left(\tau \right)\nabla \tau =0.$

The integration by parts formula on $\mathbb{T}$ is

$${\int }_{\beta }^{d}u\left(\tau \right)v^{\nabla }\left(\tau \right)\nabla \tau ={\left[u\left(\tau \right)v\left(\tau \right)\right]}_{\beta }^d-{\int }_{\beta }^d{u}^{\nabla }\left(\tau \right)v^{\rho }\left(\tau \right)\nabla \tau .$$

(9)

The Hölder inequality on $\mathbb{T}$ is

$${\int }_{\beta }^d\left|\mathrm{\Xi }\left(\tau \right)\mathrm{{\rm Y}}\left(\tau \right)\right|\nabla \tau \le {\left[{\int }_{\beta }^d{\left|\mathrm{\Xi }\left(\tau \right)\right|}^{\gamma }\nabla \tau \right]}^{\frac{1}{\gamma }}{\left[{\int }_{\beta }^d{\left|\mathrm{{\rm Y}}\left(\tau \right)\right|}^{\nu }\nabla \tau \right]}^{\frac{1}{\nu }},$$

(10)

where $\beta ,$ $d\in \mathbb{T},$ $\mathrm{\Xi },\mathrm{{\rm Y}}\in {\mathrm{C}}_{ld}(\mathbb{I},\mathbb{R}),$ $\gamma >1$ and $1/\gamma +1/\nu =1.$

Definition 2

([12]). A function $G:J\to {\mathbb{R}}^{+}$ is sub-multiplicative if

$$G\left(\tau \eta \right)\le G\left(\tau \right)G\left(\eta \right),\forall \tau ,\eta \in J\subset \mathbb{R}.$$

(11)

The inequality (11) holds with equality when G is the identity map (i.e., $G\left(\tau \right)=\tau$).

Lemma 1

(Young’s inequality [13]). Let $\beta ,d\ge 0$ be real numbers. Then we have for $p,q>1$ and $1/p+1/q=1,$ that

$$\beta d\le \frac{{\beta }^p}{p}+\frac{d^q}{q}.$$

Lemma 2.

Let ${\omega }_1,{\omega }_2,$ ${\beta }_1,$ ${\beta }_2>0.$ Then

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

(12)

Proof. 

Applying Lemma 1 with $p=\left({\omega }_1+{\omega }_2\right)/{\omega }_1>1,$ $q=\left({\omega }_1+{\omega }_2\right)/{\omega }_2>1,$ $\beta ={\beta }_1^{{\omega }_1}$ and $d={\beta }_2^{{\omega }_2},$ we obtain that

$$\begin{array}{ccc}\hfill {\beta }_1^{{\omega }_1}{\beta }_2^{{\omega }_2}& \le & \frac{{\omega }_1}{{\omega }_1+{\omega }_2}{\beta }_1^{{\omega }_1+{\omega }_2}+\frac{{\omega }_2}{{\omega }_1+{\omega }_2}{\beta }_2^{{\omega }_1+{\omega }_2}\hfill \\ & =& \frac{1}{{\omega }_1+{\omega }_2}\left({\omega }_1{\beta }_1^{{\omega }_1+{\omega }_2}+{\omega }_2{\beta }_2^{{\omega }_1+{\omega }_2}\right),\hfill \end{array}$$

which is (12). □

In the following, we present Jensen’s inequality in the time scale nabla calculus which is a special case of ([14] Theorem 3.3) by taking $\alpha =0.$

Lemma 3.

Let ${\zeta }_0,\zeta \in \mathbb{T}$ and $r_0,\delta \in \mathbb{R}$. If $\lambda \in {\mathrm{C}}_{ld}({[{\zeta }_0,\zeta ]}_{\mathbb{T}},\mathbb{R}),$ $\phi :{[{\zeta }_0,\zeta ]}_{\mathbb{T}}\to (r_0,\delta )$ is ld-continuous and $\mathrm{\Psi }:(r_0,r)\to \mathbb{R}$ is continuous and convex, then

$$\mathrm{\Psi }\left(\frac{1}{{\int }_{{\zeta }_0}^{\zeta }\lambda \left(\tau \right)\nabla \tau }{\int }_{{\zeta }_0}^{\zeta }\lambda \left(\tau \right)\phi \left(\tau \right)\nabla \tau \right)\le \frac{1}{{\int }_{{\zeta }_0}^{\zeta }\lambda \left(\tau \right)\nabla \tau }{\int }_{{\zeta }_0}^{\zeta }\lambda \left(\tau \right)\mathrm{\Psi }\left(\phi \left(\tau \right)\right)\nabla \tau .$$

(13)

3. Main Results

Throughout the article, we will assume that the functions are ld-continuous functions on ${[\beta ,d]}_{\mathbb{T}}:=[\beta ,d]\cap \mathbb{T}$ and the integrals considered are assumed to exist.

Theorem 2.

Let $\iota ,$ $\vartheta ,$ $r_0\in \mathbb{T},$ $\lambda ,$ $\mu >1,$ $\mathrm{\Xi }\in C_{ld}^1\left({\left[r_0,\tau \right)}_{\mathbb{T}},\mathbb{R}\right),$ $\mathrm{{\rm Y}}\in C_{ld}^1\left({\left[r_0,y\right)}_{\mathbb{T}},\mathbb{R}\right)$ and $\mathrm{\Xi }\left(r_0\right)=\mathrm{{\rm Y}}\left(r_0\right)=0.$ Assume that $h\in C_{ld}^1\left({\left[r_0,\tau \right)}_{\mathbb{T}},\mathbb{R}\right),$ $l\in C_{ld}^1\left({\left[r_0,y\right)}_{\mathbb{T}},\mathbb{R}\right)$ and ${\mathrm{\Omega }}_1,$ ${\mathrm{\Omega }}_2,$ $\mathrm{\Phi },$ $\mathrm{\Psi }$ are non-negative functions such that $\mathrm{\Phi },\mathrm{\Psi }$ are increasing, convex and submultiplicative functions with

$$\beta \mathrm{\Phi }\le {\mathrm{\Omega }}_1\le d\mathrm{\Phi }and\beta \mathrm{\Psi }\le {\mathrm{\Omega }}_2\le d\mathrm{\Psi },$$

(14)

where $\beta ,d$ are positive constants such that $\beta \le d.$ If

$$H\left(\iota \right)={\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\nabla \tau andL\left(\vartheta \right)={\int }_{r_0}^{\vartheta }\left|l\left(\zeta \right)\right|\nabla \zeta ,$$

then we have for $\iota \in {\left[r_0,\tau \right)}_{\mathbb{T}},$ $\vartheta \in {\left[r_0,y\right)}_{\mathbb{T}}$ that

$$\begin{array}{cc}\hfill & {\int }_{r_0}^y{\int }_{r_0}^{\tau }\frac{{\mathrm{\Omega }}_1\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right){\mathrm{\Omega }}_2\left(\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\right)}{\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}}\nabla \iota \nabla \vartheta \hfill \\ \hfill & \le \frac{d^2}{{\beta }^2}\mathrm{{\rm Y}}(\lambda ,\mu ,\tau ,y){\left[{\int }_{r_0}^{\tau }{\left[\left|h\left(\iota \right)\right|{\mathrm{\Omega }}_1\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\iota \right)}{h\left(\iota \right)}\right|\right)\right]}^{\lambda }\left(\tau -\rho \left(\iota \right)\right)\nabla \iota \right]}^{\frac{1}{\lambda }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^y{\left[\left|l\left(\vartheta \right)\right|{\mathrm{\Omega }}_2\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\vartheta \right)}{l\left(\vartheta \right)}\right|\right)\right]}^{\mu }\left(y-\rho \left(\vartheta \right)\right)\nabla \vartheta \right]}^{\frac{1}{\mu }},\hfill \end{array}$$

(15)

where $\tau ,$ $y\in {\left(r_0,\infty \right)}_{\mathbb{T}}$ and

$$\begin{array}{ccc}\hfill \mathrm{{\rm Y}}(\lambda ,\mu ,\tau ,y)& =& \frac{1}{\lambda +\mu }{\left[{\int }_{r_0}^{\tau }{\left[\frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}\right]}^{\frac{\lambda }{\lambda -1}}\nabla \iota \right]}^{\frac{\lambda -1}{\lambda }}\hfill \\ & & \times {\left[{\int }_{r_0}^y{\left[\frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}\right]}^{\frac{\mu }{\mu -1}}\nabla \vartheta \right]}^{\frac{\mu -1}{\mu }}.\hfill \end{array}$$

(16)

Proof. 

From (14) and (13) and using the fact that $\mathrm{\Phi }$ is a non-negative, increasing, submultiplicative and convex function, we have

$$\begin{array}{ccc}\hfill {\mathrm{\Omega }}_1\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right)& \le & d\mathrm{\Phi }\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right)\hfill \\ & \le & d\mathrm{\Phi }\left({\int }_{r_0}^{\iota }\left|{\mathrm{\Xi }}^{\nabla }\left(\tau \right)\right|\nabla \tau \right)\hfill \\ & =& d\mathrm{\Phi }\left(H\left(\iota \right)\frac{{\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\nabla \tau }{{\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\nabla \tau }\right)\hfill \\ & \le & d\mathrm{\Phi }\left(H\left(\iota \right)\right)\mathrm{\Phi }\left(\frac{{\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\nabla \tau }{{\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\nabla \tau }\right)\hfill \\ & \le & d\frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}\left({\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\nabla \tau \right).\hfill \end{array}$$

(17)

Applying (10) on the term

$${\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\nabla \tau ,$$

with indices $\lambda >1$ and $\lambda /\left(\lambda -1\right),$ we observe that

$$\begin{array}{cc}\hfill & {\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\nabla \tau \hfill \\ \hfill & \le {\left[{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \right]}^{\frac{1}{\lambda }}{\left[{\int }_{r_0}^{\iota }1\nabla \tau \right]}^{\frac{\lambda -1}{\lambda }}\hfill \\ \hfill & ={\left(\iota -r_0\right)}^{\frac{\lambda -1}{\lambda }}{\left[{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \right]}^{\frac{1}{\lambda }}.\hfill \end{array}$$

(18)

From (17) and (18), we obtain

$$\begin{array}{ccc}\hfill {\mathrm{\Omega }}_1\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right)& \le & d{\left(\iota -r_0\right)}^{\frac{\lambda -1}{\lambda }}\frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}\hfill \\ & & \times {\left[{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \right]}^{\frac{1}{\lambda }}.\hfill \end{array}$$

(19)

Similarly, we have that

$$\begin{array}{ccc}\hfill {\mathrm{\Omega }}_2\left(\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\right)& \le & d{\left(\vartheta -r_0\right)}^{\frac{\mu -1}{\mu }}\frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}\hfill \\ & & \times {\left[{\int }_{r_0}^{\vartheta }{\left[\left|l\left(\tau \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)}{l\left(\tau \right)}\right|\right)\right]}^{\mu }\nabla \tau \right]}^{\frac{1}{\mu }}.\hfill \end{array}$$

(20)

From (19) and (20), we observe that

$$\begin{array}{cc}\hfill & {\mathrm{\Omega }}_1\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right){\mathrm{\Omega }}_2\left(\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\right)\hfill \\ \hfill & \le d^2{\left(\iota -r_0\right)}^{\frac{\lambda -1}{\lambda }}{\left(\vartheta -r_0\right)}^{\frac{\mu -1}{\mu }}\hfill \\ \hfill & \times \frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}{\left[{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \right]}^{\frac{1}{\lambda }}\hfill \\ \hfill & \times \frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}{\left[{\int }_{r_0}^{\vartheta }{\left[\left|l\left(\tau \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)}{l\left(\tau \right)}\right|\right)\right]}^{\mu }\nabla \tau \right]}^{\frac{1}{\mu }}.\hfill \end{array}$$

(21)

From Lemma 2, the inequality (21) becomes

$$\begin{array}{cc}\hfill & {\mathrm{\Omega }}_1\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right){\mathrm{\Omega }}_2\left(\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\right)\hfill \\ \hfill & \le d^2\frac{\lambda \mu }{\lambda +\mu }\left(\frac{1}{\lambda }{\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\frac{1}{\mu }{\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}\right)\hfill \\ \hfill & \times \frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}{\left[{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \right]}^{\frac{1}{\lambda }}\hfill \\ \hfill & \times \frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}{\left[{\int }_{r_0}^{\vartheta }{\left[\left|l\left(\tau \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)}{l\left(\tau \right)}\right|\right)\right]}^{\mu }\nabla \tau \right]}^{\frac{1}{\mu }}\hfill \\ \hfill & =d^2\frac{1}{\lambda +\mu }\left(\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}\right)\hfill \\ \hfill & \times \frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}{\left[{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \right]}^{\frac{1}{\lambda }}\hfill \\ \hfill & \times \frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}{\left[{\int }_{r_0}^{\vartheta }{\left[\left|l\left(\tau \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)}{l\left(\tau \right)}\right|\right)\right]}^{\mu }\nabla \tau \right]}^{\frac{1}{\mu }}.\hfill \end{array}$$

(22)

Dividing the two sides of (22) on the term

$$\left(\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}\right),$$

and then integrating with respect to $\iota$ from $r_0$ to $\tau$ and for $\vartheta$ from $r_0$ to $y,$ to obtain

$$\begin{array}{cc}\hfill & {\int }_{r_0}^y{\int }_{r_0}^{\tau }\frac{{\mathrm{\Omega }}_1\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right){\mathrm{\Omega }}_2\left(\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\right)}{\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}}\nabla \iota \nabla \vartheta \hfill \\ \hfill & \le \frac{d^2}{\lambda +\mu }{\int }_{r_0}^{\tau }\frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}{\left[{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \right]}^{\frac{1}{\lambda }}\nabla \iota \hfill \\ \hfill & \times {\int }_{r_0}^y\frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}{\left[{\int }_{r_0}^{\vartheta }{\left[\left|l\left(\tau \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)}{l\left(\tau \right)}\right|\right)\right]}^{\mu }\nabla \tau \right]}^{\frac{1}{\mu }}\nabla \vartheta .\hfill \end{array}$$

(23)

Applying (10) on the R.H.S of (23), we see that

$$\begin{array}{cc}\hfill & {\int }_{r_0}^y{\int }_{r_0}^{\tau }\frac{{\mathrm{\Omega }}_1\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right){\mathrm{\Omega }}_2\left(\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\right)}{\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}}\nabla \iota \nabla \vartheta \hfill \\ \hfill & \le \frac{d^2}{\lambda +\mu }{\left[{\int }_{r_0}^{\tau }{\left[\frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}\right]}^{\frac{\lambda }{\lambda -1}}\nabla \iota \right]}^{\frac{\lambda -1}{\lambda }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^{\tau }{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \nabla \iota \right]}^{\frac{1}{\lambda }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^y{\left[\frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}\right]}^{\frac{\mu }{\mu -1}}\nabla \vartheta \right]}^{\frac{\mu -1}{\mu }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^y{\int }_{r_0}^{\vartheta }{\left[\left|l\left(\tau \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)}{l\left(\tau \right)}\right|\right)\right]}^{\mu }\nabla \tau \nabla \vartheta \right]}^{\frac{1}{\mu }}.\hfill \end{array}$$

(24)

By applying the Formula (9), we have that

$$\begin{array}{ccc}& & {\int }_{r_0}^{\tau }{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \nabla \iota \hfill \\ & =& {\int }_{r_0}^{\tau }{\left[\left|h\left(\iota \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\iota \right)}{h\left(\iota \right)}\right|\right)\right]}^{\lambda }\left(\tau -\rho \left(\iota \right)\right)\nabla \iota ,\hfill \end{array}$$

and also,

$$\begin{array}{ccc}& & {\int }_{r_0}^y{\int }_{r_0}^{\vartheta }{\left[\left|l\left(\tau \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)}{l\left(\tau \right)}\right|\right)\right]}^{\mu }\nabla \tau \nabla \vartheta \hfill \\ & =& {\int }_{r_0}^y{\left[\left|l\left(\vartheta \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\vartheta \right)}{l\left(\vartheta \right)}\right|\right)\right]}^{\mu }\left(y-\rho \left(\vartheta \right)\right)\nabla \vartheta \hfill \end{array}$$

and then substituting into (24), we obtain

$$\begin{array}{cc}\hfill & {\int }_{r_0}^y{\int }_{r_0}^{\tau }\frac{{\mathrm{\Omega }}_1\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right){\mathrm{\Omega }}_2\left(\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\right)}{\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}}\nabla \iota \nabla \vartheta \hfill \\ \hfill & \le \frac{d^2}{\lambda +\mu }{\left[{\int }_{r_0}^{\tau }{\left[\frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}\right]}^{\frac{\lambda }{\lambda -1}}\nabla \iota \right]}^{\frac{\lambda -1}{\lambda }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^{\tau }{\left[\left|h\left(\iota \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\iota \right)}{h\left(\iota \right)}\right|\right)\right]}^{\lambda }\left(\tau -\rho \left(\iota \right)\right)\nabla \iota \right]}^{\frac{1}{\lambda }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^y{\left[\frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}\right]}^{\frac{\mu }{\mu -1}}\nabla \vartheta \right]}^{\frac{\mu -1}{\mu }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^y{\left[\left|l\left(\vartheta \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\vartheta \right)}{l\left(\vartheta \right)}\right|\right)\right]}^{\mu }\left(y-\rho \left(\vartheta \right)\right)\nabla \vartheta \right]}^{\frac{1}{\mu }}.\hfill \end{array}$$

From (14) and (16), the last inequality becomes

$$\begin{array}{cc}\hfill & {\int }_{r_0}^y{\int }_{r_0}^{\tau }\frac{{\mathrm{\Omega }}_1\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right){\mathrm{\Omega }}_2\left(\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\right)}{\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}}\nabla \iota \nabla \vartheta \hfill \\ \hfill & \le d^2\mathrm{{\rm Y}}(\lambda ,\mu ,\tau ,y){\left[{\int }_{r_0}^{\tau }{\left[\left|h\left(\iota \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\iota \right)}{h\left(\iota \right)}\right|\right)\right]}^{\lambda }\left(\tau -\rho \left(\iota \right)\right)\nabla \iota \right]}^{\frac{1}{\lambda }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^y{\left[\left|l\left(\vartheta \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\vartheta \right)}{l\left(\vartheta \right)}\right|\right)\right]}^{\mu }\left(y-\rho \left(\vartheta \right)\right)\nabla \vartheta \right]}^{\frac{1}{\mu }}\hfill \\ \hfill & \le \frac{d^2}{{\beta }^2}\mathrm{{\rm Y}}(\lambda ,\mu ,\tau ,y){\left[{\int }_{r_0}^{\tau }{\left[\left|h\left(\iota \right)\right|{\mathrm{\Omega }}_1\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\iota \right)}{h\left(\iota \right)}\right|\right)\right]}^{\lambda }\left(\tau -\rho \left(\iota \right)\right)\nabla \iota \right]}^{\frac{1}{\lambda }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^y{\left[\left|l\left(\vartheta \right)\right|{\mathrm{\Omega }}_2\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\vartheta \right)}{l\left(\vartheta \right)}\right|\right)\right]}^{\mu }\left(y-\rho \left(\vartheta \right)\right)\nabla \vartheta \right]}^{\frac{1}{\mu }},\hfill \end{array}$$

which is (15). □

Corollary 1.

Let $\iota ,$ $\vartheta ,$ $r_0\in \mathbb{T},$ $\lambda ,$ $\mu >1,$ $\mathrm{\Xi }\in C_{ld}^1\left({\left[r_0,\tau \right)}_{\mathbb{T}},\mathbb{R}\right),$ $\mathrm{{\rm Y}}\in C_{ld}^1\left({\left[r_0,y\right)}_{\mathbb{T}},\mathbb{R}\right)$ and $\mathrm{\Xi }\left(r_0\right)=\mathrm{{\rm Y}}\left(r_0\right)=0.$ Then we have for $\iota \in {\left[r_0,\tau \right)}_{\mathbb{T}},$ $\vartheta \in {\left[r_0,y\right)}_{\mathbb{T}}$ that

$$\begin{array}{ccc}& & {\int }_{r_0}^y{\int }_{r_0}^{\tau }\frac{\left|\mathrm{\Xi }\left(\iota \right)\right|\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|}{\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}}\nabla \iota \nabla \vartheta \hfill \\ & \le & \frac{1}{\lambda +\mu }{\left(\tau -r_0\right)}^{\frac{\lambda -1}{\lambda }}{\left(y-r_0\right)}^{\frac{\mu -1}{\mu }}\hfill \\ & & \times {\left[{\int }_{r_0}^y{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\vartheta \right)\right|}^{\mu }\left(y-\rho \left(\vartheta \right)\right)\nabla \vartheta \right]}^{\frac{1}{\mu }}{\left[{\int }_{r_0}^{\tau }{\left|{\mathrm{\Xi }}^{\nabla }\left(\iota \right)\right|}^{\lambda }\left(\tau -\rho \left(\iota \right)\right)\nabla \iota \right]}^{\frac{1}{\lambda }}.\hfill \end{array}$$

(25)

for $\tau ,y\in {\left(r_0,\infty \right)}_{\mathbb{T}}.$

Proof. 

Since $\mathrm{\Xi }\left(r_0\right)=\mathrm{{\rm Y}}\left(r_0\right)=0,$ we have

$${\int }_{r_0}^{\iota }\left|{\mathrm{\Xi }}^{\nabla }\left(\tau \right)\right|\nabla \tau \ge \left|{\int }_{r_0}^{\iota }{\mathrm{\Xi }}^{\nabla }\left(\tau \right)\nabla \tau \right|=\left|\mathrm{\Xi }\left(\iota \right)-\mathrm{\Xi }\left(r_0\right)\right|=\left|\mathrm{\Xi }\left(\iota \right)\right|,$$

(26)

and

$${\int }_{r_0}^{\vartheta }\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)\right|\nabla \tau \ge \left|{\int }_{r_0}^{\vartheta }{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)\nabla \tau \right|=\left|\mathrm{{\rm Y}}\left(\vartheta \right)-\mathrm{{\rm Y}}\left(r_0\right)\right|=\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|$$

(27)

By applying (10) on L.H.S of (26) and (27), we obtain

$$\begin{array}{ccc}\hfill \left|\mathrm{\Xi }\left(\iota \right)\right|& \le & {\left({\int }_{r_0}^{\iota }{\left|{\mathrm{\Xi }}^{\nabla }\left(\tau \right)\right|}^{\lambda }\nabla \tau \right)}^{\frac{1}{\lambda }}{\left({\int }_{r_0}^{\iota }{\left(1\right)}^{\frac{\lambda }{\lambda -1}}\nabla \tau \right)}^{\frac{\lambda -1}{\lambda }}\hfill \\ & =& {\left({\int }_{r_0}^{\iota }{\left|{\mathrm{\Xi }}^{\nabla }\left(\tau \right)\right|}^{\lambda }\nabla \tau \right)}^{\frac{1}{\lambda }}{\left(\iota -r_0\right)}^{\frac{\lambda -1}{\lambda }},\hfill \end{array}$$

(28)

and also,

$$\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\le {\left({\int }_{r_0}^{\vartheta }{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)\right|}^{\mu }\nabla \tau \right)}^{\frac{1}{\mu }}{\left(\vartheta -r_0\right)}^{\frac{\mu -1}{\mu }}.$$

(29)

From (28) and (29), we see that

$$\left|\mathrm{\Xi }\left(\iota \right)\right|\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\le {\left(\iota -r_0\right)}^{\frac{\lambda -1}{\lambda }}{\left(\vartheta -r_0\right)}^{\frac{\mu -1}{\mu }}{\left({\int }_{r_0}^{\vartheta }{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)\right|}^{\mu }\nabla \tau \right)}^{\frac{1}{\mu }}{\left({\int }_{r_0}^{\iota }{\left|{\mathrm{\Xi }}^{\nabla }\left(\tau \right)\right|}^{\lambda }\nabla \tau \right)}^{\frac{1}{\lambda }}.$$

(30)

Applying (12) with ${\beta }_1={\left(\iota -r_0\right)}^{\lambda -1},$ ${\beta }_2={\left(\vartheta -r_0\right)}^{\mu -1},$ ${\omega }_1=1/\lambda ,$ ${\omega }_2=1/\mu ,$ we observe that

$$\begin{array}{ccc}& & {\left(\iota -r_0\right)}^{\frac{\lambda -1}{\lambda }}{\left(\vartheta -r_0\right)}^{\frac{\mu -1}{\mu }}\hfill \\ & \le & \frac{1}{\frac{1}{\lambda }+\frac{1}{\mu }}\left(\frac{1}{\lambda }{\left(\iota -r_0\right)}^{\left(\lambda -1\right)\left(\frac{1}{\lambda }+\frac{1}{\mu }\right)}+\frac{1}{\mu }{\left(\vartheta -r_0\right)}^{\left(\mu -1\right)\left(\frac{1}{\lambda }+\frac{1}{\mu }\right)}\right)\hfill \\ & =& \frac{\lambda \mu }{\lambda +\mu }\left(\frac{1}{\lambda }{\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\frac{1}{\mu }{\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}\right).\hfill \end{array}$$

(31)

Substituting (31) into (30), we obtain

$$\begin{array}{ccc}\hfill \left|\mathrm{\Xi }\left(\iota \right)\right|\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|& \le & \frac{\lambda \mu }{\lambda +\mu }\left(\frac{1}{\lambda }{\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\frac{1}{\mu }{\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}\right)\hfill \\ & & \times {\left({\int }_{r_0}^{\vartheta }{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)\right|}^{\mu }\nabla \tau \right)}^{\frac{1}{\mu }}{\left({\int }_{r_0}^{\iota }{\left|{\mathrm{\Xi }}^{\nabla }\left(\tau \right)\right|}^{\lambda }\nabla \tau \right)}^{\frac{1}{\lambda }}\hfill \\ & =& \frac{1}{\lambda +\mu }\left(\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}\right)\hfill \\ & & \times {\left({\int }_{r_0}^{\vartheta }{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)\right|}^{\mu }\nabla \tau \right)}^{\frac{1}{\mu }}{\left({\int }_{r_0}^{\iota }{\left|{\mathrm{\Xi }}^{\nabla }\left(\tau \right)\right|}^{\lambda }\nabla \tau \right)}^{\frac{1}{\lambda }}.\hfill \end{array}$$

(32)

By dividing (32) on the term

$$\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }},$$

and then integrating with respect to $\iota$ from $r_0$ to $\tau$ and for $\vartheta$ from $r_0$ to $y,$ to obtain

$$\begin{array}{ccc}& & {\int }_{r_0}^y{\int }_{r_0}^{\tau }\frac{\left|\mathrm{\Xi }\left(\iota \right)\right|\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|}{\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}}\nabla \iota \nabla \vartheta \hfill \\ & \le & \frac{1}{\lambda +\mu }\left[{\int }_{r_0}^y{\left({\int }_{r_0}^{\vartheta }{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)\right|}^{\mu }\nabla \tau \right)}^{\frac{1}{\mu }}\nabla \vartheta \right]\left[{\int }_{r_0}^{\tau }{\left({\int }_{r_0}^{\iota }{\left|{\mathrm{\Xi }}^{\nabla }\left(\tau \right)\right|}^{\lambda }\nabla \tau \right)}^{\frac{1}{\lambda }}\nabla \iota \right].\hfill \end{array}$$

(33)

Applying (10) on the two parts of the right term of (33), we see

$$\begin{array}{ccc}& & {\int }_{r_0}^y{\int }_{r_0}^{\tau }\frac{\left|\mathrm{\Xi }\left(\iota \right)\right|\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|}{\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}}\nabla \iota \nabla \vartheta \hfill \\ & \le & \frac{1}{\lambda +\mu }{\left[{\int }_{r_0}^{\tau }{\int }_{r_0}^{\iota }{\left|{\mathrm{\Xi }}^{\nabla }\left(\tau \right)\right|}^{\lambda }\nabla \tau \nabla \iota \right]}^{\frac{1}{\lambda }}{\left(\tau -r_0\right)}^{\frac{\lambda -1}{\lambda }}\hfill \\ & & \times {\left[{\int }_{r_0}^y{\int }_{r_0}^{\vartheta }{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)\right|}^{\mu }\nabla \tau \nabla \vartheta \right]}^{\frac{1}{\mu }}{\left(y-r_0\right)}^{\frac{\mu -1}{\mu }}.\hfill \end{array}$$

(34)

Applying (9) on the term

$${\int }_{r_0}^y{\int }_{r_0}^{\vartheta }{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)\right|}^{\mu }\nabla \tau \nabla \vartheta ,$$

with $u\left(\vartheta \right)={\int }_{r_0}^{\vartheta }{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)\right|}^{\mu }\nabla \tau$ and $v^{\nabla }\left(\vartheta \right)=1,$ we obtain

$$\begin{array}{cc}\hfill & {\int }_{r_0}^y{\int }_{r_0}^{\vartheta }{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)\right|}^{\mu }\nabla \tau \nabla \vartheta \hfill \\ \hfill & ={\left.v\left(\vartheta \right)\left({\int }_{r_0}^{\vartheta }{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)\right|}^{\mu }\nabla \tau \right)\right|}_{r_0}^y-{\int }_{r_0}^y{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\vartheta \right)\right|}^{\mu }{v}^{\rho }\left(\vartheta \right)\nabla \vartheta ,\hfill \end{array}$$

where $v\left(\vartheta \right)=\vartheta -y,$ and then

$$\begin{array}{cc}\hfill & {\int }_{r_0}^y{\int }_{r_0}^{\vartheta }{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)\right|}^{\mu }\nabla \tau \nabla \vartheta \hfill \\ \hfill & ={\left.(\vartheta -y)\left({\int }_{r_0}^{\vartheta }{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)\right|}^{\mu }\nabla \tau \right)\right|}_{r_0}^y+{\int }_{r_0}^y{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\vartheta \right)\right|}^{\mu }\left(y-\rho \left(\vartheta \right)\right)\nabla \vartheta \hfill \\ \hfill & ={\int }_{r_0}^y{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\vartheta \right)\right|}^{\mu }\left(y-\rho \left(\vartheta \right)\right)\nabla \vartheta .\hfill \end{array}$$

(35)

Similarly, by applying (9) on the term

$${\int }_{r_0}^{\tau }{\int }_{r_0}^{\iota }{\left|{\mathrm{\Xi }}^{\nabla }\left(\tau \right)\right|}^{\lambda }\nabla \tau \nabla \iota ,$$

we observe that

$${\int }_{r_0}^{\tau }{\int }_{r_0}^{\iota }{\left|{\mathrm{\Xi }}^{\nabla }\left(\tau \right)\right|}^{\lambda }\nabla \tau \nabla \iota ={\int }_{r_0}^{\tau }{\left|{\mathrm{\Xi }}^{\nabla }\left(\iota \right)\right|}^{\lambda }\left(\tau -\rho \left(\iota \right)\right)\nabla \iota .$$

(36)

Substituting (35) and (36) into (34), to obtain

$$\begin{array}{ccc}& & {\int }_{r_0}^y{\int }_{r_0}^{\tau }\frac{\left|\mathrm{\Xi }\left(\iota \right)\right|\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|}{\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}}\nabla \iota \nabla \vartheta \hfill \\ & \le & \frac{1}{\lambda +\mu }{\left(\tau -r_0\right)}^{\frac{\lambda -1}{\lambda }}{\left(y-r_0\right)}^{\frac{\mu -1}{\mu }}\hfill \\ & & \times {\left[{\int }_{r_0}^y{\left|{\mathrm{{\rm Y}}}^{\nabla }\left(\vartheta \right)\right|}^{\mu }\left(y-\rho \left(\vartheta \right)\right)\nabla \vartheta \right]}^{\frac{1}{\mu }}{\left[{\int }_{r_0}^{\tau }{\left|{\mathrm{\Xi }}^{\nabla }\left(\iota \right)\right|}^{\lambda }\left(\tau -\rho \left(\iota \right)\right)\nabla \iota \right]}^{\frac{1}{\lambda }},\hfill \end{array}$$

which is (25). □

Remark 1.

As particular cases of Corollary 1, (when $\mathbb{T}=\mathbb{N}$, $r_0=0$), we obtain the inequality (7) and (when $\mathbb{T}=\mathbb{R}$, $r_0=0$), we obtain the inequality (8).

In what follows, we generalize Corollary 1 by using a submultiplicative function.

Corollary 2.

Let $\iota ,$ $\vartheta ,$ $r_0\in \mathbb{T},$ $\lambda ,$ $\mu >1,$ $\mathrm{\Xi }\in C_{ld}^1\left({\left[r_0,\tau \right)}_{\mathbb{T}},\mathbb{R}\right),$ $\mathrm{{\rm Y}}\in C_{ld}^1\left({\left[r_0,y\right)}_{\mathbb{T}},\mathbb{R}\right)$ with $\mathrm{\Xi }\left(r_0\right)=\mathrm{{\rm Y}}\left(r_0\right)=0.$ Assume that $h\in C_{ld}^1\left({\left[r_0,\tau \right)}_{\mathbb{T}},\mathbb{R}\right),$ $l\in C_{ld}^1\left({\left[r_0,y\right)}_{\mathbb{T}},\mathbb{R}\right)$ and $\mathrm{\Phi },\mathrm{\Psi }\ge 0$ are increasing, convex and submultiplicative functions. If

$$H\left(\iota \right)={\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\nabla \tau andL\left(\vartheta \right)={\int }_{r_0}^{\vartheta }\left|l\left(\zeta \right)\right|\nabla \zeta ,$$

for $\iota \in {\left[r_0,\tau \right)}_{\mathbb{T}},$ $\vartheta \in {\left[r_0,y\right)}_{\mathbb{T}},$ then

$$\begin{array}{cc}\hfill & {\int }_{r_0}^y{\int }_{r_0}^{\tau }\frac{\mathrm{\Phi }\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right)\mathrm{\Psi }\left(\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\right)}{\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}}\nabla \iota \nabla \vartheta \hfill \\ \hfill & \le \mathrm{{\rm Y}}(\tau ,y,\lambda ,\mu ){\left[{\int }_{r_0}^{\tau }{\left[\left|h\left(\iota \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\iota \right)}{h\left(\iota \right)}\right|\right)\right]}^{\lambda }\left(\tau -\rho \left(\iota \right)\right)\nabla \iota \right]}^{\frac{1}{\lambda }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^y{\left[\left|l\left(\vartheta \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\vartheta \right)}{l\left(\vartheta \right)}\right|\right)\right]}^{\mu }\left(y-\rho \left(\vartheta \right)\right)\nabla \vartheta \right]}^{\frac{1}{\mu }},\hfill \end{array}$$

(37)

where $\tau ,y\in {\left(r_0,\infty \right)}_{\mathbb{T}}$ and

$$\begin{array}{ccc}\hfill \mathrm{{\rm Y}}(\tau ,y,\lambda ,\mu )& =& \frac{1}{\lambda +\mu }{\left[{\int }_{r_0}^{\tau }{\left[\frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}\right]}^{\frac{\lambda }{\lambda -1}}\nabla \iota \right]}^{\frac{\lambda -1}{\lambda }}\hfill \\ & & \times {\left[{\int }_{r_0}^y{\left[\frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}\right]}^{\frac{\mu }{\mu -1}}\nabla \vartheta \right]}^{\frac{\mu -1}{\mu }}.\hfill \end{array}$$

Proof. 

Using the fact that $\mathrm{\Phi }$ is a non-negative, increasing and submultiplicative function, we have

$$\begin{array}{ccc}\hfill \mathrm{\Phi }\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right)& \le & \mathrm{\Phi }\left({\int }_{r_0}^{\iota }\left|{\mathrm{\Xi }}^{\nabla }\left(\tau \right)\right|\nabla \tau \right)\hfill \\ & =& \mathrm{\Phi }\left(H\left(\iota \right)\frac{{\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\nabla \tau }{{\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\nabla \tau }\right)\hfill \\ & \le & \mathrm{\Phi }\left(H\left(\iota \right)\right)\mathrm{\Phi }\left(\frac{{\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\nabla \tau }{{\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\nabla \tau }\right).\hfill \end{array}$$

(38)

Applying (13) on the R.H.S of (38), we observe that

$$\mathrm{\Phi }\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right)\le \frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}\left({\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\nabla \tau \right).$$

(39)

Applying (10) on the term

$${\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\nabla \tau ,$$

with indices $\lambda >1$ and $\lambda /\left(\lambda -1\right),$ we obtain

$$\begin{array}{cc}\hfill & {\int }_{r_0}^{\iota }\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\nabla \tau \hfill \\ \hfill & \le {\left[{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \right]}^{\frac{1}{\lambda }}{\left[{\int }_{r_0}^{\iota }1\nabla \tau \right]}^{\frac{\lambda -1}{\lambda }}\hfill \\ \hfill & ={\left(\iota -r_0\right)}^{\frac{\lambda -1}{\lambda }}{\left[{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \right]}^{\frac{1}{\lambda }}.\hfill \end{array}$$

(40)

From (39) and (40), we obtain

$$\begin{array}{ccc}\hfill \mathrm{\Phi }\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right)& \le & \frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}{\left(\iota -r_0\right)}^{\frac{\lambda -1}{\lambda }}\hfill \\ & & \times {\left[{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \right]}^{\frac{1}{\lambda }}.\hfill \end{array}$$

(41)

Similarly, we have that

$$\begin{array}{ccc}\hfill \mathrm{\Psi }\left(\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\right)& \le & \frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}{\left(\vartheta -r_0\right)}^{\frac{\mu -1}{\mu }}\hfill \\ & & \times {\left[{\int }_{r_0}^{\vartheta }{\left[\left|l\left(\tau \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)}{l\left(\tau \right)}\right|\right)\right]}^{\mu }\nabla \tau \right]}^{\frac{1}{\mu }}.\hfill \end{array}$$

(42)

From (41) and (42), we observe that

$$\begin{array}{cc}\hfill & \mathrm{\Phi }\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right)\mathrm{\Psi }\left(\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\right)\hfill \\ \hfill & \le {\left(\iota -r_0\right)}^{\frac{\lambda -1}{\lambda }}{\left(\vartheta -r_0\right)}^{\frac{\mu -1}{\mu }}\hfill \\ \hfill & \times \frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}{\left[{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \right]}^{\frac{1}{\lambda }}\hfill \\ \hfill & \times \frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}{\left[{\int }_{r_0}^{\vartheta }{\left[\left|l\left(\tau \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)}{l\left(\tau \right)}\right|\right)\right]}^{\mu }\nabla \tau \right]}^{\frac{1}{\mu }}.\hfill \end{array}$$

(43)

From Lemma 2, the inequality (43) becomes

$$\begin{array}{cc}\hfill & \mathrm{\Phi }\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right)\mathrm{\Psi }\left(\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\right)\hfill \\ \hfill & \le \frac{\lambda \mu }{\lambda +\mu }\left(\frac{1}{\lambda }{\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\frac{1}{\mu }{\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}\right)\hfill \\ \hfill & \times \frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}{\left[{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \right]}^{\frac{1}{\lambda }}\hfill \\ \hfill & \times \frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}{\left[{\int }_{r_0}^{\vartheta }{\left[\left|l\left(\tau \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)}{l\left(\tau \right)}\right|\right)\right]}^{\mu }\nabla \tau \right]}^{\frac{1}{\mu }}\hfill \\ \hfill & =\frac{1}{\lambda +\mu }\left(\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}\right)\hfill \\ \hfill & \times \frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}{\left[{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \right]}^{\frac{1}{\lambda }}\hfill \\ \hfill & \times \frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}{\left[{\int }_{r_0}^{\vartheta }{\left[\left|l\left(\tau \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)}{l\left(\tau \right)}\right|\right)\right]}^{\mu }\nabla \tau \right]}^{\frac{1}{\mu }}.\hfill \end{array}$$

(44)

Dividing the two sides of (44) on the term

$$\left(\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}\right),$$

and then integrating with respect to $\iota$ from $r_0$ to $\tau$ and for $\vartheta$ from $r_0$ to $y,$ to obtain

$$\begin{array}{cc}\hfill & {\int }_{r_0}^y{\int }_{r_0}^{\tau }\frac{\mathrm{\Phi }\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right)\mathrm{\Psi }\left(\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\right)}{\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}}\nabla \iota \nabla \vartheta \hfill \\ \hfill & \le \frac{1}{\lambda +\mu }{\int }_{r_0}^{\tau }\frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}{\left[{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \right]}^{\frac{1}{\lambda }}\nabla \iota \hfill \\ \hfill & \times {\int }_{r_0}^y\frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}{\left[{\int }_{r_0}^{\vartheta }{\left[\left|l\left(\tau \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)}{l\left(\tau \right)}\right|\right)\right]}^{\mu }\nabla \tau \right]}^{\frac{1}{\mu }}\nabla \vartheta .\hfill \end{array}$$

(45)

Applying (10) on the R.H.S of (45), we see that

$$\begin{array}{cc}\hfill & {\int }_{r_0}^y{\int }_{r_0}^{\tau }\frac{\mathrm{\Phi }\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right)\mathrm{\Psi }\left(\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\right)}{\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}}\nabla \iota \nabla \vartheta \hfill \\ \hfill & \le \frac{1}{\lambda +\mu }{\left[{\int }_{r_0}^{\tau }{\left[\frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}\right]}^{\frac{\lambda }{\lambda -1}}\nabla \iota \right]}^{\frac{\lambda -1}{\lambda }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^{\tau }{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \nabla \iota \right]}^{\frac{1}{\lambda }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^y{\left[\frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}\right]}^{\frac{\mu }{\mu -1}}\nabla \vartheta \right]}^{\frac{\mu -1}{\mu }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^y{\int }_{r_0}^{\vartheta }{\left[\left|l\left(\tau \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)}{l\left(\tau \right)}\right|\right)\right]}^{\mu }\nabla \tau \nabla \vartheta \right]}^{\frac{1}{\mu }}.\hfill \end{array}$$

(46)

By applying (9), we have that

$$\begin{array}{ccc}& & {\int }_{r_0}^{\tau }{\int }_{r_0}^{\iota }{\left[\left|h\left(\tau \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\tau \right)}{h\left(\tau \right)}\right|\right)\right]}^{\lambda }\nabla \tau \nabla \iota \hfill \\ & =& {\int }_{r_0}^{\tau }{\left[\left|h\left(\iota \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\iota \right)}{h\left(\iota \right)}\right|\right)\right]}^{\lambda }\left(\tau -\rho \left(\iota \right)\right)\nabla \iota ,\hfill \end{array}$$

and also,

$$\begin{array}{ccc}& & {\int }_{r_0}^y{\int }_{r_0}^{\vartheta }{\left[\left|l\left(\tau \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\tau \right)}{l\left(\tau \right)}\right|\right)\right]}^{\mu }\nabla \tau \nabla \vartheta \hfill \\ & =& {\int }_{r_0}^y{\left[\left|l\left(\vartheta \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\vartheta \right)}{l\left(\vartheta \right)}\right|\right)\right]}^{\mu }\left(y-\rho \left(\vartheta \right)\right)\nabla \vartheta \hfill \end{array}$$

and then substituting into (46), we obtain

$$\begin{array}{cc}\hfill & {\int }_{r_0}^y{\int }_{r_0}^{\tau }\frac{\mathrm{\Phi }\left(\left|\mathrm{\Xi }\left(\iota \right)\right|\right)\mathrm{\Psi }\left(\left|\mathrm{{\rm Y}}\left(\vartheta \right)\right|\right)}{\mu {\left(\iota -r_0\right)}^{\frac{\left(\lambda -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}+\lambda {\left(\vartheta -r_0\right)}^{\frac{\left(\mu -1\right)\left(\lambda +\mu \right)}{\lambda \mu }}}\nabla \iota \nabla \vartheta \hfill \\ \hfill & \le \frac{1}{\lambda +\mu }{\left[{\int }_{r_0}^{\tau }{\left[\frac{\mathrm{\Phi }\left(H\left(\iota \right)\right)}{H\left(\iota \right)}\right]}^{\frac{\lambda }{\lambda -1}}\nabla \iota \right]}^{\frac{\lambda -1}{\lambda }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^{\tau }{\left[\left|h\left(\iota \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\iota \right)}{h\left(\iota \right)}\right|\right)\right]}^{\lambda }\left(\tau -\rho \left(\iota \right)\right)\nabla \iota \right]}^{\frac{1}{\lambda }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^y{\left[\frac{\mathrm{\Psi }\left(L\left(\vartheta \right)\right)}{L\left(\vartheta \right)}\right]}^{\frac{\mu }{\mu -1}}\nabla \vartheta \right]}^{\frac{\mu -1}{\mu }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^y{\left[\left|l\left(\vartheta \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\vartheta \right)}{l\left(\vartheta \right)}\right|\right)\right]}^{\mu }\left(y-\rho \left(\vartheta \right)\right)\nabla \vartheta \right]}^{\frac{1}{\mu }}\hfill \\ \hfill & =\mathrm{{\rm Y}}(\tau ,y,\lambda ,\mu ){\left[{\int }_{r_0}^{\tau }{\left[\left|h\left(\iota \right)\right|\mathrm{\Phi }\left(\left|\frac{{\mathrm{\Xi }}^{\nabla }\left(\iota \right)}{h\left(\iota \right)}\right|\right)\right]}^{\lambda }\left(\tau -\rho \left(\iota \right)\right)\nabla \iota \right]}^{\frac{1}{\lambda }}\hfill \\ \hfill & \times {\left[{\int }_{r_0}^y{\left[\left|l\left(\vartheta \right)\right|\mathrm{\Psi }\left(\left|\frac{{\mathrm{{\rm Y}}}^{\nabla }\left(\vartheta \right)}{l\left(\vartheta \right)}\right|\right)\right]}^{\mu }\left(y-\rho \left(\vartheta \right)\right)\nabla \vartheta \right]}^{\frac{1}{\mu }},\hfill \end{array}$$

which is (37). □

Remark 2.

As a specific case of Corollary 2, when $\beta =d=1,$ we obtain Corollary 1.

4. Conclusions and Future Work

In this work, we explored some new generalization inequalities of Hilbert type on time scales by using nabla calculus, which are used in various problems involving symmetry. Further, we also applied our inequalities to discrete and continuous calculus to obtain some new Hilbert type inequalities as special cases. Moreover, some new inequalities as special cases are discussed. In future work, we will continue to generalize more dynamic inequalities by conformable fractional calculus on time scales by using Specht’s ratio, Kantorovich’s ratio and n-tuple fractional integral. It will also be very enjoyable to introduce such inequalities on quantum calculus.