Some New Reverse Hilbert’s Inequalities on Time Scales

Abstract

This paper is interested in establishing some new reverse Hilbert-type inequalities, by using chain rule on time scales, reverse Jensen’s, and reverse Hölder’s with Specht’s ratio and mean inequalities. To get the results, we used the Specht’s ratio function and its applications for reverse inequalities of Hilbert-type. Symmetrical properties play an essential role in determining the correct methods to solve inequalities. The new inequalities in special cases yield some recent relevance, which also provide new estimates on inequalities of these type.

1. Introduction

In [1], Hardy established that

$$\sum _{j=1}^{\infty }\sum _{i=1}^{\infty }\frac{{\phi }_i{\psi }_j}{i+j}\le \frac{\pi }{sin\frac{\pi }{\alpha }}{\left(\sum _{i=1}^{\infty }{\phi }_i^{\alpha }\right)}^{{}^{\frac{1}{\alpha }}}{\left(\sum _{j=1}^{\infty }{\psi }_j^{\beta }\right)}^{{}^{\frac{1}{\beta }}},$$

(1)

where ${\phi }_i,$${\psi }_j\ge 0$ with $0<{\sum }_{i=1}^{\infty }{\phi }_i^{\alpha }<\infty$, $0<{\sum }_{j=1}^{\infty }{\psi }_j^{\beta }<\infty$ and $\alpha >1$, $1/\alpha +1/\beta =1$. Hardy and Reisz [2] established the continuous form of (1) in the following

$${\int }_0^{\infty }{\int }_0^{\infty }\frac{\delta \left(\chi \right)\omega \left(z\right)}{\chi +z}d\chi dz\le \frac{\pi }{sin\frac{\pi }{\alpha }}{\left({\int }_0^{\infty }{\delta }^{\alpha }\left(\chi \right)d\chi \right)}^{{}^{\frac{1}{\alpha }}}{\left({\int }_0^{\infty }{\omega }^{\beta }\left(z\right)dz\right)}^{{}^{\frac{1}{\beta }}},$$

(2)

where $\delta$ and $\omega$ are measurable nonnegative functions such that $0<{\int }_0^{\infty }{\delta }^{\alpha }\left(\chi \right)d\chi <\infty$, $0<{\int }_0^{\infty }{\omega }^{\beta }\left(z\right)dz<\infty$ and $\pi /sin(\pi /\alpha )$ in (1), and (2) is the best value. In [2], Hardy showed that, if $\alpha >1$, $\beta >1$, $1/\alpha +1/\beta \ge 1$ and $0<\lambda =2-(1/\alpha +1/\beta )\le 1,$ then

$$\sum _{j=1}^{\infty }\sum _{i=1}^{\infty }\frac{{\phi }_i{\psi }_j}{{(i+j)}^{\lambda }}\le K{\left(\sum _{i=1}^{\infty }{\phi }_i^{\alpha }\right)}^{\frac{1}{\alpha }}{\left(\sum _{j=1}^{\infty }{\psi }_j^{\beta }\right)}^{\frac{1}{\beta }},$$

(3)

where $K=K(\alpha$, $\beta )$ relates to $\alpha$, $\beta ,$ only for $1/\alpha +1/\beta =1,$$\lambda =2-(1/\alpha +1/\beta )=1$ and the constant factor K is optimal. For more details about the Hilbert-type inequalities, see the papers [3,4,5,6]. In [7], Hölder proved that

$$\sum _{k=1}^n{\zeta }_k{y}_k\le {\left(\sum _{k=1}^n{\zeta }_k^{\alpha }\right)}^{\frac{1}{\alpha }}{\left(\sum _{k=1}^n{y}_k^{\beta }\right)}^{\frac{1}{\beta }},$$

(4)

where $\left({\zeta }_k\right)$ and $\left(y_k\right)$ are positive sequences and $\alpha$, $\beta >1$ such that $1/\alpha +1/\beta =1.$ The integral form of (4) is

$$\underset{a}{\overset{b}{\int }}\psi \left(\tau \right)\varpi \left(\tau \right)d\tau \le {\left(\underset{a}{\overset{b}{\int }}{\psi }^{\alpha }\left(\tau \right)d\tau \right)}^{\frac{1}{\alpha }}{\left(\underset{a}{\overset{b}{\int }}{\varpi }^{\beta }\left(\tau \right)d\tau \right)}^{\frac{1}{\beta }},$$

(5)

where $\alpha$, $\beta >1$ such that $1/\alpha +1/\beta =1$ and $\psi$, $\varpi \in C\left((a,b),{\mathbb{R}}^{+}\right).$

Some authors established the reverse Hölder inequalities, the reverse Young inequalities, and the reverse Hilbert inequalities by using the Specht’s ratio function, see [8,9,10,11,12]. In particular, Zhao and Cheung [11] established the reverse Hölder inequalities by using the Specht’s ratio function and proved that if $\psi \left(\zeta \right)$ and $\varpi \left(\zeta \right)$ are nonnegative continuous functions and ${\psi }^{1/\alpha }\left(\zeta \right){\varpi }^{1/\beta }\left(\zeta \right)$ is integrable on $[a,b],$ then

$${\left(\underset{a}{\overset{b}{\int }}{\psi }^{\alpha }\left(\zeta \right)d\zeta \right)}^{\frac{1}{\alpha }}{\left(\underset{a}{\overset{b}{\int }}{\varpi }^{\beta }\left(\zeta \right)d\zeta \right)}^{\frac{1}{\beta }}\le \underset{a}{\overset{b}{\int }}S\left(\frac{Y{\psi }^{\alpha }\left(\zeta \right)}{X{\varpi }^{\beta }\left(\zeta \right)}\right).\psi \left(\zeta \right)\varpi \left(\zeta \right)d\zeta ,$$

(6)

with

$$X=\underset{a}{\overset{b}{\int }}{\psi }^{\alpha }\left(\zeta \right)d\zeta ,Y=\underset{a}{\overset{b}{\int }}{\varpi }^{\beta }\left(\zeta \right)d\zeta ,\alpha >1\mathrm{and}\frac{1}{\alpha }+\frac{1}{\beta }=1,$$

where the function $S(.)$ is called the Specht’s ratio function (see [10]) and defined as follows:

$$S\left(h\right)=\frac{h^{1/(h-1)}}{elog{h}^{1/(h-1)}},h\ne 1\mathrm{and}S\left(1\right)=1.$$

In [11], the authors proved that if $\psi$, $\varpi \in C\left((a,b),{\mathbb{R}}^{+}\right)$ and $m>0,$ then

$$\underset{a}{\overset{b}{\int }}\frac{{\psi }^{m+1}\left(\zeta \right)}{{\varpi }^m\left(\zeta \right)}d\zeta \le \frac{{\left(\underset{a}{\overset{b}{\int }}S\left(\frac{G{\psi }^{m+1}\left(\zeta \right)}{F{\varpi }^{m+1}\left(\zeta \right)}\right)\psi \left(\zeta \right)d\zeta \right)}^{m+1}}{{\left(\underset{a}{\overset{b}{\int }}\varpi \left(\zeta \right)d\zeta \right)}^m},$$

(7)

where

$$G=\underset{a}{\overset{b}{\int }}\varpi \left(\zeta \right)d\zeta \mathrm{and}F=\underset{a}{\overset{b}{\int }}{\psi }^{m+1}\left(\zeta \right)/{\varpi }^m\left(\zeta \right)d\zeta .$$

In addition, they proved the discrete case of (7) as follows:

$$\sum \frac{a_i^{m+1}}{b_i^m}\le \frac{\sum S\left(\frac{B{a}_i^{m+1}}{A{b}_i^{m+1}}\right)a_i}{{\left(\sum b_i\right)}^m},$$

(8)

where $B=\sum b_i$ and $A=\sum a_i^{m+1}/b_i^m.$

In [12], Zhao and Cheung established the reverse Hilbert inequalities by using the Specht’s ratio and proved that if $0\le \alpha ,\beta \le 1$, $\left\{{\lambda }_i\right\}$, $\left\{{\psi }_j\right\}$ are nonnegative and decreasing sequences of real numbers for $i=1,2,\dots ,k$ and $j=1,2,\dots ,r$ with $k,$$r\in \mathbb{N},$ then

$$\begin{array}{ccc}& & \sum _{i=1}^k\sum _{j=1}^r\frac{S_{\alpha ,\beta ,k,r,i,j}{\left({\sum }_{s=1}^i{\lambda }_s\right)}^{\alpha }{\left({\sum }_{t=1}^j{\psi }_t\right)}^{\beta }}{{\left(ij\right)}^{\frac{1}{2}}}\hfill \\ & \ge & 2D\left(\alpha ,\beta ,k,r\right){\left(\sum _{i=1}^k{\left[{\lambda }_i{\left(\sum _{s=1}^i{\lambda }_s\right)}^{\alpha -1}\right]}^2\left(k-i+1\right)\right)}^{\frac{1}{2}}\hfill \\ & & \times {\left(\sum _{j=1}^r{\left[{\psi }_j{\left(\sum _{t=1}^j{\psi }_t\right)}^{\beta -1}\right]}^2\left(r-j+1\right)\right)}^{\frac{1}{2}},\hfill \end{array}$$

(9)

where

$$D\left(\alpha ,\beta ,k,r\right)=\frac{1}{2}\alpha \beta {\left(kr\right)}^{\frac{1}{2}},$$

and

$$\begin{array}{cc}\hfill S_{\alpha ,\beta ,k,r,i,j}& =S\left(\frac{k{\sum }_{s=1}^i{\left[{\lambda }_s{\left({\sum }_{\tau =1}^s{\lambda }_{\tau }\right)}^{\alpha -1}\right]}^2}{{\sum }_{s=1}^k\left(k-s+1\right){\left[{\lambda }_s{\left({\sum }_{\tau =1}^s{\lambda }_{\tau }\right)}^{\alpha -1}\right]}^2}\right)\hfill \\ \hfill & \times S\left(\frac{r{\sum }_{t=1}^j{\left[{\psi }_t{\left({\sum }_{\tau =1}^t{\psi }_{\tau }\right)}^{\beta -1}\right]}^2}{{\sum }_{t=1}^r\left(r-t+1\right){\left[{\psi }_t{\left({\sum }_{\tau =1}^t{\psi }_{\tau }\right)}^{\beta -1}\right]}^2}\right)\hfill \\ \hfill & \times S\left(\frac{i{\left[{\lambda }_u{\left({\sum }_{\tau =1}^u{\lambda }_{\tau }\right)}^{\alpha -1}\right]}^2}{{\sum }_{s=1}^i{\left[{\lambda }_s{\left({\sum }_{\tau =1}^s{\lambda }_{\tau }\right)}^{\alpha -1}\right]}^2}\right)S\left(\frac{j{\left[{\psi }_v{\left({\sum }_{\tau =1}^v{\psi }_{\tau }\right)}^{\beta -1}\right]}^2}{{\sum }_{t=1}^j{\left[{\psi }_t{\left({\sum }_{\tau =1}^t{\psi }_{\tau }\right)}^{\beta -1}\right]}^2}\right),\hfill \end{array}$$

where

$$\begin{array}{ccc}& & S\left(\frac{i{\left[{\lambda }_u{\left({\sum }_{\tau =1}^u{\lambda }_{\tau }\right)}^{\alpha -1}\right]}^2}{{\sum }_{s=1}^i{\left[{\lambda }_s{\left({\sum }_{\tau =1}^s{\lambda }_{\tau }\right)}^{\alpha -1}\right]}^2}\right)\hfill \\ & =& max\left\{S\left(\frac{i{\left[{\lambda }_1{\left({\sum }_{\tau =1}^1{\lambda }_{\tau }\right)}^{\alpha -1}\right]}^2}{{\sum }_{s=1}^i{\left[{\lambda }_s{\left({\sum }_{\tau =1}^s{\lambda }_{\tau }\right)}^{\alpha -1}\right]}^2}\right)\right.\left.;S\left(\frac{i{\left[{\lambda }_i{\left({\sum }_{\tau =1}^i{\lambda }_{\tau }\right)}^{\alpha -1}\right]}^2}{{\sum }_{s=1}^i{\left[{\lambda }_s{\left({\sum }_{\tau =1}^s{\lambda }_{\tau }\right)}^{\alpha -1}\right]}^2}\right)\right\},\hfill \end{array}$$

and

$$\begin{array}{ccc}& & S\left(\frac{j{\left[{\psi }_v{\left({\sum }_{\tau =1}^v{\psi }_{\tau }\right)}^{\beta -1}\right]}^2}{{\sum }_{t=1}^j{\left[{\psi }_t{\left({\sum }_{\tau =1}^t{\psi }_{\tau }\right)}^{\beta -1}\right]}^2}\right)\hfill \\ & =& max\left\{S\left(\frac{j{\left[{\psi }_1{\left({\sum }_{\tau =1}^1{\psi }_{\tau }\right)}^{\beta -1}\right]}^2}{{\sum }_{t=1}^j{\left[{\psi }_t{\left({\sum }_{\tau =1}^t{\psi }_{\tau }\right)}^{\beta -1}\right]}^2}\right)\right.\left.;S\left(\frac{j{\left[{\psi }_j{\left({\sum }_{\tau =1}^j{\psi }_{\tau }\right)}^{\beta -1}\right]}^2}{{\sum }_{t=1}^j{\left[{\psi }_t{\left({\sum }_{\tau =1}^t{\psi }_{\tau }\right)}^{\beta -1}\right]}^2}\right)\right\},\hfill \end{array}$$

In addition, they proved that, if $\left\{{\lambda }_i\right\}$, $\left\{{\omega }_j\right\}$ are nonnegative sequences for $i=1,2,\dots ,k,$ and $j=1,2,\dots ,r$ with $k$, $r\in \mathbb{N}$ and $\left\{{\alpha }_i\right\}$, $\left\{{\beta }_j\right\}$ are positive sequences. Let $\varphi$, $\psi$ are nonnegative, concave and supermultiplicative functions. Then,

$$\begin{array}{ccc}& & \sum _{i=1}^k\sum _{j=1}^r\frac{S_{k,r,i,j}\varphi \left({\mathsf{\Lambda }}_i\right)\psi \left({\mathsf{\Omega }}_j\right)}{{\left(ij\right)}^{\frac{1}{2}}}\hfill \\ & \ge & 2M\left(k,r\right){\left(\sum _{s=1}^k{\left[{\alpha }_s\varphi \left(\frac{{\lambda }_s}{{\alpha }_s}\right)\right]}^2\left(k-s+1\right)\right)}^{\frac{1}{2}}\hfill \\ & & \times {\left(\sum _{t=1}^r{\left[{\beta }_t\psi \left(\frac{{\omega }_t}{{\beta }_t}\right)\right]}^2\left(r-t+1\right)\right)}^{\frac{1}{2}},\hfill \end{array}$$

(10)

with

$$M\left(k,r\right)=\frac{1}{2}{\left(\sum _{i=1}^k{\left(\frac{\varphi \left({\alpha }_i\right)}{{\alpha }_i}\right)}^2\right)}^{\frac{1}{2}}{\left(\sum _{j=1}^r{\left(\frac{\psi \left(Q_j\right)}{Q_j}\right)}^2\right)}^{\frac{1}{2}},$$

$$S_{k,r,i,j}=S\left(\frac{\left({\sum }_{s=1}^k{\left[{\alpha }_s\varphi \left(\frac{{\lambda }_s}{{\alpha }_s}\right)\right]}^2\left(k-s+1\right)\right){\left(\frac{\varphi \left({\alpha }_i\right)}{{\alpha }_i}\right)}^2}{\left({\sum }_{i=1}^k{\left(\frac{\varphi \left({\alpha }_i\right)}{{\alpha }_i}\right)}^2\right)\left({\sum }_{s=1}^i{\left[{\alpha }_s\varphi \left(\frac{{\lambda }_s}{{\alpha }_s}\right)\right]}^2\right)}\right)S\left(\frac{\left({\sum }_{t=1}^r{\left[{\beta }_t\psi \left(\frac{{\omega }_t}{{\beta }_t}\right)\right]}^2\left(r-t+1\right)\right){\left(\frac{\psi \left(Q_j\right)}{Q_j}\right)}^2}{\left({\sum }_{j=1}^r{\left(\frac{\psi \left(Q_j\right)}{Q_j}\right)}^2\right)\left({\sum }_{t=1}^j{\left[{\beta }_t\psi \left(\frac{{\omega }_t}{{\beta }_t}\right)\right]}^2\right)}\right),$$

$${\mathsf{\Lambda }}_i=\sum _{s=1}^{i}S\left(\frac{i{\left[{\alpha }_s\varphi \left(\frac{{\lambda }_s}{{\alpha }_s}\right)\right]}^2}{{\sum }_{s=1}^i{\left[{\alpha }_s\varphi \left(\frac{{\lambda }_s}{{\alpha }_s}\right)\right]}^2}\right){\lambda }_s,{\mathsf{\Omega }}_j=\sum _{t=1}^{j}S\left(\frac{j{\left[{\beta }_t\psi \left(\frac{{\omega }_t}{{\beta }_t}\right)\right]}^2}{{\sum }_{t=1}^j{\left[{\beta }_t\psi \left(\frac{{\omega }_t}{{\beta }_t}\right)\right]}^2}\right){\omega }_t,$$

and

$${\alpha }_i=\sum _{s=1}^{i}S\left(\frac{i{\left[{\alpha }_s\varphi \left(\frac{{\lambda }_s}{{\alpha }_s}\right)\right]}^2}{{\sum }_{s=1}^i{\left[{\alpha }_s\varphi \left(\frac{{\lambda }_s}{{\alpha }_s}\right)\right]}^2}\right){\alpha }_s,Q_j=\sum _{t=1}^{j}S\left(\frac{j{\left[{\beta }_t\psi \left(\frac{{\omega }_t}{{\beta }_t}\right)\right]}^2}{{\sum }_{t=1}^j{\left[{\beta }_t\psi \left(\frac{{\omega }_t}{{\beta }_t}\right)\right]}^2}\right){\beta }_t,$$

where the function $S(.)$ is the Specht’s ratio. In [12], the authors proved that, if $\left\{{\lambda }_i\right\}$, $\left\{{\omega }_j\right\}$ are nonnegative sequences for $i=1,2,\dots ,k$ and $j=1,2,\dots ,r$ with $k,$$r\in \mathbb{N},$ then

$$\begin{array}{ccc}& & \sum _{i=1}^k\sum _{j=1}^r\frac{S_{k,r,i,j}{\mathsf{\Lambda }}_i{\mathsf{\Omega }}_j}{{\left(ij\right)}^{\frac{1}{2}}}\hfill \\ & \ge & {\left(kr\right)}^{\frac{1}{2}}{\left(\sum _{i=1}^k{\lambda }_i^2\left(k-i+1\right)\right)}^{\frac{1}{2}}{\left(\sum _{j=1}^r{\omega }_j^2\left(r-j+1\right)\right)}^{\frac{1}{2}},\hfill \end{array}$$

(11)

with

$$S_{k,r,i,j}=S\left(\frac{{\sum }_{s=1}^k{\lambda }_s^2\left(k-s+1\right)}{k\left({\sum }_{s=1}^i{\lambda }_s^2\right)}\right)S\left(\frac{{\sum }_{t=1}^r{\omega }_t^2\left(r-t+1\right)}{r\left({\sum }_{t=1}^j{\omega }_t^2\right)}\right),$$

$${\mathsf{\Lambda }}_i=\sum _{s=1}^{i}S\left(\frac{i{\lambda }_s^2}{{\sum }_{s=1}^i{\lambda }_s^2}\right){\lambda }_s\mathrm{and}{\mathsf{\Omega }}_j=\sum _{t=1}^{j}S\left(\frac{j{\omega }_t^2}{{\sum }_{t=1}^j{\omega }_t^2}\right){\omega }_t.$$

In the last few decades, a new theory has been discovered to unify the continuous calculus and discrete calculus. It is called a time scale theory. A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset of the real numbers $\mathbb{R}$. Many authors established dynamic inequalities and generalized them on time scales. For more details, see ([13,14,15,16,17]).

In particular, El-Deeb, Elsennary, and Wing-Sum Cheung [15] proved the reverse Hölder inequality on time scales by using the Specht’s ratio function and proved that, if $\psi ,\varpi \in C\left({[a,b]}_{\mathbb{T}},{\mathbb{R}}^{+}\right)$ such that ${\psi }^{\alpha }$, ${\varpi }^{\beta }$ are ${◊}_{\alpha }-$ integrable on ${[a,b]}_{\mathbb{T}}$. If $\alpha >1$ and $1/\alpha +1/\beta =1,$ then

$$\begin{array}{ccc}& & {\int }_a^{b}S\left(\frac{Y{\psi }^{\alpha }\left(\zeta \right)}{X{\varpi }^{\beta }\left(\zeta \right)}\right)\psi \left(\zeta \right)\varpi \left(\zeta \right){◊}_{\alpha }\zeta \hfill \\ & \ge & {\left({\int }_a^b{\psi }^{\alpha }\left(\zeta \right){◊}_{\alpha }\zeta \right)}^{\frac{1}{\alpha }}{\left({\int }_a^b{\varpi }^{\beta }\left(\zeta \right){◊}_{\alpha }\zeta \right)}^{\frac{1}{\beta }},\hfill \end{array}$$

(12)

where $X={\int }_a^b{\psi }^{\alpha }\left(\zeta \right){◊}_{\alpha }\zeta ,$$Y={\int }_a^b{\varpi }^{\beta }\left(\zeta \right){◊}_{\alpha }\zeta$ and $S(.)$ is the Specht’s ratio (see [11]). In addition, they proved (12) with weighted functions and proved that if $\psi ,\varpi ,w\in C\left({[a,b]}_{\mathbb{T}},{\mathbb{R}}^{+}\right)$ such that ${\psi }^{\alpha }$, ${\varpi }^{\beta }$ are ${◊}_{\alpha }-$ integrable on ${[a,b]}_{\mathbb{T}}$. If $\alpha >1$ and $1/\alpha +1/\beta =1,$ then

$$\begin{array}{ccc}& & {\int }_a^{b}S\left(\frac{Y{\psi }^{\alpha }\left(\zeta \right)}{X{\varpi }^{\beta }\left(\zeta \right)}\right).w\left(\zeta \right)\psi \left(\zeta \right)\varpi \left(\zeta \right){◊}_{\alpha }\zeta \hfill \\ & \ge & {\left({\int }_a^{b}w\left(\zeta \right){\psi }^{\alpha }\left(\zeta \right){◊}_{\alpha }\zeta \right)}^{\frac{1}{\alpha }}{\left({\int }_a^{b}w\left(\zeta \right){\varpi }^{\beta }\left(\zeta \right){◊}_{\alpha }\zeta \right)}^{\frac{1}{\beta }},\hfill \end{array}$$

(13)

where $X={\int }_a^{b}w\left(\zeta \right){\psi }^{\alpha }\left(\zeta \right){◊}_{\alpha }\zeta$ and $Y={\int }_a^{b}w\left(\zeta \right){\varpi }^{\beta }\left(\zeta \right){◊}_{\alpha }\zeta .$

The authors [15] proved that if $\psi ,\varpi \in C\left({[a,b]}_{\mathbb{T}},{\mathbb{R}}^{+}\right)$ such that $0<m\le \psi \left(t\right)/\varpi \left(t\right)\le M<\infty$ for all $t\in {[a,b]}_{\mathbb{T}}.$ If $\alpha >1$ and $1/\alpha +1/\beta =1,$ then

$$\begin{array}{ccc}& & {\int }_a^{b}S\left(\frac{Y\psi \left(\zeta \right)}{X\varpi \left(\zeta \right)}\right){\psi }^{\frac{1}{\alpha }}\left(\zeta \right){\varpi }^{\frac{1}{\beta }}\left(\zeta \right){◊}_{\alpha }\zeta \hfill \\ & \ge & \frac{m^{\frac{1}{{\alpha }^2}}}{M^{\frac{1}{{\beta }^2}}}{\int }_a^b{\psi }^{\frac{1}{\beta }}\left(\zeta \right){\varpi }^{\frac{1}{\alpha }}\left(\zeta \right){◊}_{\alpha }\zeta ,\hfill \end{array}$$

(14)

where $X={\int }_a^b\psi \left(\zeta \right){◊}_{\alpha }\zeta$ and $Y={\int }_a^b\varpi \left(\zeta \right){◊}_{\alpha }\zeta .$

The aim of this paper is to establish some new reverse Hilbert-type inequalities on time scales by using the Specht’s ratio function and applying reverse Hölder inequalities on time scales.

The organization of the paper is as follows: in Section 2, we show some basics of the time scale theory and some lemmas on time scales needed in Section 3 where we prove our results. Our main results (when $\mathbb{T}=\mathbb{R}$) give the inequalities (9)–(11) proved by Zhao and Cheung [12].

2. Preliminaries and Basic Lemmas

A forward jump operator on time scales is defined by: $\sigma \left(\tau \right):=inf\{r\in \mathbb{T}:r>\tau \}.$ The set of all such rd–continuous functions is ushered by $C_{rd}(\mathbb{T},\mathbb{R})$ and for any function $\mathsf{\Phi }:\mathbb{T}\to \mathbb{R}$ the notation ${\mathsf{\Phi }}^{\sigma }\left(\tau \right)$ denotes $\mathsf{\Phi }\left(\sigma \right(\tau \left)\right).$ To learn more about the time scale calculus, see ([18,19]).

The derivative of the product $\mathsf{\Phi }\varpi$ and the quotient $\mathsf{\Phi }/\varpi$ (where $\varpi {\varpi }^{\sigma }\ne 0$) of two differentiable functions $\mathsf{\Phi }$ and $\varpi$ are given by

$${\left(\mathsf{\Phi }\varpi \right)}^{\mathsf{\Delta }}={\mathsf{\Phi }}^{\mathsf{\Delta }}\varpi +{\mathsf{\Phi }}^{\sigma }{\varpi }^{\mathsf{\Delta }}=\mathsf{\Phi }{\varpi }^{\mathsf{\Delta }}+{\mathsf{\Phi }}^{\mathsf{\Delta }}{\varpi }^{\sigma },{\left(\frac{\mathsf{\Phi }}{\varpi }\right)}^{\mathsf{\Delta }}=\frac{{\mathsf{\Phi }}^{\mathsf{\Delta }}\varpi -\mathsf{\Phi }{\varpi }^{\mathsf{\Delta }}}{\varpi {\varpi }^{\sigma }}.$$

(15)

The integration by parts formula on time scales is given by

$${\int }_{{\upsilon }_0}^{\upsilon }\lambda \left(\tau \right){\phi }^{\mathsf{\Delta }}\left(\tau \right)\mathsf{\Delta }\tau ={\left[\lambda \left(\tau \right)\phi \left(\tau \right)\right]}_{{\upsilon }_0}^{\upsilon }-{\int }_{{\upsilon }_0}^{\upsilon }{\lambda }^{\mathsf{\Delta }}\left(\tau \right){\phi }^{\sigma }\left(\tau \right)\mathsf{\Delta }\tau .$$

(16)

The time scales chain rule (see Theorem 1.87 [18]) is as follows:

$${(\varpi \circ \phi )}^{\mathsf{\Delta }}\left(\tau \right)={\varpi }^{{}^{\prime }}\left(\phi \left(\chi \right)\right){\phi }^{\mathsf{\Delta }}\left(\tau \right),\mathrm{where}\chi \in \left[\tau ,\sigma \left(\tau \right)\right],$$

(17)

and $\varpi :\mathbb{R}\to \mathbb{R}$ is continuously differentiable, $\phi :\mathbb{T}\to \mathbb{R}$ is $\mathsf{\Delta }-$ differentiable.

Definition 1

([20]). A function $h:J\subset \mathbb{R}\to {\mathbb{R}}^{+}$ is supermultiplicative if

$$h\left(\chi z\right)\ge h\left(\chi \right)h\left(z\right),\forall \chi ,z\in J.$$

(18)

The inequality (18) holds with equality when h is the identity map ($i.e.,$$h\left(\chi \right)=\chi$). If the last inequality has a reversed sign, then $h$ is said to be a submultiplicative function.

Lemma 1.

Let $\mathbb{T}$ be a time scale with $a\in \mathbb{T}$, λ is nonnegative rd-continuous function and $0<\gamma \le 1,$ then

$${\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\gamma }\ge \gamma {\int }_a^{\sigma \left(t\right)}{\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\gamma -1}\lambda \left(\chi \right)\mathsf{\Delta }\chi .$$

(19)

Proof. 

By applying (17) on the term ${\int }_a^{\chi }\lambda \left(\tau \right)\mathsf{\Delta }\tau ,$ we get

$${\left[{\left({\int }_a^{\chi }\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\gamma }\right]}^{\mathsf{\Delta }}=\gamma {\left({\int }_a^{\zeta }\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\gamma -1}\lambda \left(\chi \right),\zeta \in [\chi ,\sigma \left(\chi \right)].$$

(20)

Since $\zeta \le \sigma \left(\chi \right)$, then we have (note $0<\gamma \le 1$) that

$${\left({\int }_a^{\zeta }\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\gamma -1}\ge {\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\gamma -1}.$$

(21)

Substituting (21) into (20), we see that

$${\left[{\left({\int }_a^{\chi }\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\gamma }\right]}^{\mathsf{\Delta }}\ge \gamma {\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\gamma -1}\lambda \left(\chi \right).$$

Integrating the last inequality over $\chi$ from a to $\sigma \left(t\right),$ we observe that

$${\int }_a^{\sigma \left(t\right)}{\left[{\left({\int }_a^{\chi }\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\gamma }\right]}^{\mathsf{\Delta }}\mathsf{\Delta }\chi \ge \gamma {\int }_a^{\sigma \left(t\right)}{\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\gamma -1}\lambda \left(\chi \right)\mathsf{\Delta }\chi .$$

i.e.,

$${\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\gamma }\ge \gamma {\int }_a^{\sigma \left(t\right)}{\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\gamma -1}\lambda \left(\chi \right)\mathsf{\Delta }\chi ,$$

which is (19). The proof is complete. □

Lemma 2

(Specht’s ratio [10]). If α and β are positive numbers, $p>1$ and $1/p+1/q=1,$ then

$$S\left(\frac{\alpha }{\beta }\right){\alpha }^{1/p}{\beta }^{1/q}\ge \frac{\alpha }{p}+\frac{\beta }{q},$$

(22)

where

$$S\left(h\right)=\frac{h^{1/(h-1)}}{elog{h}^{1/(h-1)}},h\ne 1.$$

Lemma 3

([10]). Let $S(.)$ be a Specht’s ratio function which is defined in Lemma 2, then the function $S\left(t\right)$ is strictly decreasing for $0<t<1$ and strictly increasing for $t>1.$ Furthermore, the following equations hold:

$$S\left(1\right)=1\mathit{and}S\left(t\right)=S\left(\frac{1}{t}\right)\mathit{for}\mathit{all}t>0.$$

In [15] for $\alpha =1$, we get the following lemma.

Lemma 4.

If $\delta ,\omega \in C\left({[a,b]}_{\mathbb{T}},{\mathbb{R}}^{+}\right)$ such that ${\delta }^{\gamma }$, ${\omega }^{\nu }$ are $\mathsf{\Delta }-$ integrable on ${[a,b]}_{\mathbb{T}}$. If $\gamma >1$ and $1/\gamma +1/\nu =1,$ then

$$\begin{array}{ccc}& & {\int }_a^{b}S\left(\frac{Y{\delta }^{\gamma }\left(\zeta \right)}{X{\omega }^{\nu }\left(\zeta \right)}\right)\delta \left(\zeta \right)\omega \left(\zeta \right)\mathsf{\Delta }\zeta \hfill \\ & \ge & {\left({\int }_a^b{\delta }^{\gamma }\left(\zeta \right)\mathsf{\Delta }\zeta \right)}^{\frac{1}{\gamma }}{\left({\int }_a^b{\omega }^{\nu }\left(\zeta \right)\mathsf{\Delta }\zeta \right)}^{\frac{1}{\nu }},\hfill \end{array}$$

(23)

where $X={\int }_a^b{\delta }^{\gamma }\left(\zeta \right)\mathsf{\Delta }\zeta ,$$Y={\int }_a^b{\omega }^{\nu }\left(\zeta \right)\mathsf{\Delta }\zeta$ and $S(.)$ is the Specht’s ratio.

Theorem 1

(Jensen’s inequality). Assume that $\mathbb{T}$ is a time scale with ${\zeta }_0,\zeta \in \mathbb{T}$ and $r_0,r\in \mathbb{R}$. If $\lambda \in {\mathrm{C}}_{rd}({[{\zeta }_0,\zeta ]}_{\mathbb{T}},\mathbb{R})$, $\phi :{[{\zeta }_0,\zeta ]}_{\mathbb{T}}\to (r_0,r)$ is rd-continuous and $\mathsf{\Psi }:(r_0,r)\to \mathbb{R}$ is continuous and convex, then

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

(24)

Lemma 5.

Let $a\in \mathbb{T},$$\lambda$, ψ be nonnegative and decreasing functions and $0<\alpha$, $\beta \le 1.$ Then,

$$\begin{array}{ccc}& & S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)\hfill \\ & =& max\left\{S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(a\right){\left({\int }_a^{\sigma \left(a\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)\right.\hfill \\ & & \left.;S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(t\right){\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)\right\},\hfill \end{array}$$

(25)

and

$$\begin{array}{ccc}& & S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(\eta \right){\left({\int }_a^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right)\hfill \\ & =& max\left\{S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(a\right){\left({\int }_a^{\sigma \left(a\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right)\right.\hfill \\ & & \left.;S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(\xi \right){\left({\int }_a^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right)\right\}.\hfill \end{array}$$

(26)

Proof. 

We have for $\chi \le z,$ that

$${\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \le {\int }_a^{\sigma \left(z\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau ,$$

and then (where $0<\alpha \le 1$)

$${\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\ge {\left({\int }_a^{\sigma \left(z\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}.$$

Since $\lambda$ is decreasing, we have that

$${\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\ge {\left[\lambda \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2,$$

thus the function ${\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2$ is decreasing. Therefore, we have for $a\le \chi$ that

$${\left[\lambda \left(a\right){\left({\int }_a^{\sigma \left(a\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\ge {\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2.$$

Integrate the last inequality over $\chi$ from a to $\sigma \left(t\right),$ to get

$$\left(\sigma \left(t\right)-a\right){\left[\lambda \left(a\right){\left({\int }_a^{\sigma \left(a\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\ge {\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi ,$$

and then

$$\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(a\right){\left({\int }_a^{\sigma \left(a\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\ge 1.$$

(27)

Since the function ${\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2$ is decreasing, we have for $\chi \le t$ that

$${\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\ge {\left[\lambda \left(t\right){\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2,$$

and by integrating the last inequality over $\chi$ from a to $\sigma \left(t\right),$ we get that

$$\begin{array}{ccc}& & {\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi \hfill \\ & \ge & {\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(t\right){\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi \hfill \\ & =& \left(\sigma \left(t\right)-a\right){\left[\lambda \left(t\right){\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2,\hfill \end{array}$$

and then

$$\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(t\right){\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\le 1.$$

(28)

From (27) and (28), we observe that

$$\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(a\right){\left({\int }_a^{\sigma \left(a\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\ge \dots \ge 1\ge \dots \ge \frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(t\right){\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }.$$

Since the function (Specht’s ratio $S(.)$) is decreasing on $\left(0,1\right)$ and increasing on $\left(1,\infty \right),$ we observe that one of

$$S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(a\right){\left({\int }_a^{\sigma \left(a\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)\mathrm{and}S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(t\right){\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)$$

is maximum (where $S\left(1\right)=1$), and it is in the form

$$\begin{array}{ccc}& & S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)\hfill \\ & =& max\left\{S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(a\right){\left({\int }_a^{\sigma \left(a\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)\right.\hfill \\ & & \left.;S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(t\right){\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)\right\},\hfill \end{array}$$

which is (25). Similarly, with respect to the decreasing function $\psi$ when $0<\beta \le 1,$ we have

$$\begin{array}{ccc}& & S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(\eta \right){\left({\int }_a^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right)\hfill \\ & =& max\left\{S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(a\right){\left({\int }_a^{\sigma \left(a\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right)\right.\hfill \\ & & \left.;S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(\xi \right){\left({\int }_a^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right)\right\},\hfill \end{array}$$

which is (26). □

3. Main Results

Throughout the paper, we will assume that the functions are nonnegative rd-continuous functions on ${[a,b]}_{\mathbb{T}}$ and the integrals considered are assumed to exist. We define the time scale interval ${[a,b]}_{\mathbb{T}}$ by ${[a,b]}_{\mathbb{T}}:=[a,b]\cap \mathbb{T}$.

Now, we can present and prove the first result of this section.

Theorem 2.

Let $a\in \mathbb{T}$, $0\le \alpha ,\beta \le 1$, $\lambda$, ψ be nonnegative and decreasing functions. Then, the inequality

$$\begin{array}{ccc}& & {\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(r\right)}\frac{S_{\alpha ,\beta ,t,\xi ,r,s}{\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha }{\left({\int }_a^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta }}{{\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}{\left(\sigma \left(\xi \right)-a\right)}^{\frac{1}{2}}}\mathsf{\Delta }t\mathsf{\Delta }\xi \hfill \\ & \ge & 2C\left(\alpha ,\beta ,r,s\right){\left({\int }_a^{\sigma \left(r\right)}{\left[\lambda \left(t\right){\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\left(\sigma \left(r\right)-t\right)\mathsf{\Delta }t\right)}^{\frac{1}{2}}\hfill \\ & & \times {\left({\int }_a^{\sigma \left(s\right)}{\left[\psi \left(\xi \right){\left({\int }_a^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\left(\sigma \left(s\right)-\xi \right)\mathsf{\Delta }\xi \right)}^{\frac{1}{2}},\hfill \end{array}$$

(29)

holds for all $r,s\in {[a,\infty ]}_{\mathbb{T}}$, with

$$C\left(\alpha ,\beta ,r,s\right)=\frac{1}{2}\alpha \beta {\left(\sigma \left(r\right)-a\right)}^{\frac{1}{2}}{\left(\sigma \left(s\right)-a\right)}^{\frac{1}{2}},$$

and

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta ,t,\xi ,r,s}& =& S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)\hfill \\ & & \times S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(\eta \right){\left({\int }_a^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right)\hfill \\ & & \times S\left(\frac{\left(\sigma \left(r\right)-a\right){\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(r\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi }\right)\hfill \\ & & \times S\left(\frac{\left(\sigma \left(s\right)-a\right){\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(\eta \right){\left({\int }_a^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(s\right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z}\right),\hfill \end{array}$$

such that

$$\begin{array}{ccc}& & S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)\hfill \\ & =& max\left\{S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(a\right){\left({\int }_a^{\sigma \left(a\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)\right.\hfill \\ & & \left.;S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(t\right){\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)\right\},\hfill \end{array}$$

and

$$\begin{array}{ccc}& & S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(\eta \right){\left({\int }_a^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right)\hfill \\ & =& max\left\{S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(a\right){\left({\int }_a^{\sigma \left(a\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right)\right.\hfill \\ & & \left.;S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(\xi \right){\left({\int }_a^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right)\right\}.\hfill \end{array}$$

Proof. 

Applying (19) with $\gamma =\alpha ,$ we have

$${\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha }\ge \alpha {\int }_a^{\sigma \left(t\right)}\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\mathsf{\Delta }\chi .$$

(30)

Multiplying the last inequality by

$$S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right),$$

we get

$$\begin{array}{ccc}& & S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right){\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha }\hfill \\ & \ge & \alpha {\int }_a^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\mathsf{\Delta }\chi .\hfill \end{array}$$

From Lemma (5), the last inequality becomes

$$\begin{array}{cc}\hfill & S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right){\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha }\hfill \\ \hfill & \ge \alpha {\int }_a^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\mathsf{\Delta }\chi .\hfill \end{array}$$

(31)

Similarly, we have for the decreasing function $\psi$ and $0<\beta \le 1$ that

$$\begin{array}{cc}\hfill & S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(\eta \right){\left({\int }_a^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right){\left({\int }_a^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta }\hfill \\ \hfill & \ge \beta {\int }_a^{\sigma \left(\xi \right)}S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right)\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\mathsf{\Delta }z.\hfill \end{array}$$

(32)

From (31) and (32), we see that

$$\begin{array}{cc}\hfill & S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(\eta \right){\left({\int }_a^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right)\hfill \\ \hfill & \times {\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha }{\left({\int }_a^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta }\hfill \\ \hfill & \ge \alpha \beta \left({\int }_a^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\times 1.\mathsf{\Delta }\chi \right)\hfill \\ \hfill & \times \left({\int }_a^{\sigma \left(\xi \right)}S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right)\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\times 1.\mathsf{\Delta }z\right).\hfill \end{array}$$

(33)

Applying (23) when $\gamma =\nu =2,$ we have

$$\begin{array}{cc}\hfill & S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(\eta \right){\left({\int }_a^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right)\hfill \\ \hfill & \times {\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha }{\left({\int }_a^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta }\hfill \\ \hfill & \ge \alpha \beta {\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}{\left({\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi \right)}^{\frac{1}{2}}\hfill \\ \hfill & \times {\left(\sigma \left(\xi \right)-a\right)}^{\frac{1}{2}}{\left({\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z\right)}^{\frac{1}{2}}.\hfill \end{array}$$

(34)

Multiplying (34) by

$$S\left(\frac{\left(\sigma \left(r\right)-a\right){\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(r\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi }\right)S\left(\frac{\left(\sigma \left(s\right)-a\right){\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(\eta \right){\left({\int }_a^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(s\right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z}\right),$$

we see that

$$\begin{array}{cc}\hfill & S_{\alpha ,\beta ,t,\xi ,r,s}{\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha }\hfill \\ \hfill & ={\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha }S\left(\frac{\left(\sigma \left(r\right)-a\right){\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(r\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi }\right)\hfill \\ \hfill & \times {\left({\int }_a^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta }S\left(\frac{\left(\sigma \left(s\right)-a\right){\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(\eta \right){\left({\int }_a^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(s\right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z}\right)\hfill \\ \hfill & \times S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi }\right)S\left(\frac{\left(\sigma \left(\xi \right)-a\right){\left[\psi \left(\eta \right){\left({\int }_a^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z}\right)\hfill \\ \hfill & \ge \alpha \beta {\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}S\left(\frac{\left(\sigma \left(r\right)-a\right){\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(r\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi }\right)\hfill \\ \hfill & \times {\left(\sigma \left(\xi \right)-a\right)}^{\frac{1}{2}}S\left(\frac{\left(\sigma \left(s\right)-a\right){\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(\eta \right){\left({\int }_a^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(s\right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z}\right)\hfill \\ \hfill & \times {\left(\begin{array}{c}\left({\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi \right)\\ ·\left({\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z\right)\end{array}\right)}^{\frac{1}{2}}.\hfill \end{array}$$

(35)

Dividing the two sides of (35) by ${\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}{\left(\sigma \left(\xi \right)-a\right)}^{\frac{1}{2}}$ and then taking the integration over t from a to $\sigma \left(r\right)$ and the integration over $\xi$ from a to $\sigma \left(s\right),$ we get

$$\begin{array}{cc}\hfill & {\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(r\right)}\frac{S_{\alpha ,\beta ,t,\xi ,r,s}{\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha }{\left({\int }_a^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta }}{{\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}{\left(\sigma \left(\xi \right)-a\right)}^{\frac{1}{2}}}\mathsf{\Delta }t\mathsf{\Delta }\xi \hfill \\ \hfill & \ge \alpha \beta {\int }_a^{\sigma \left(r\right)}S\left(\frac{\left(\sigma \left(r\right)-a\right){\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2}{{\int }_a^{\sigma \left(r\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi }\right)\hfill \\ \hfill & \times {\left({\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi \right)}^{\frac{1}{2}}\mathsf{\Delta }t\hfill \\ \hfill & \times {\int }_a^{\sigma \left(s\right)}S\left(\frac{\left(\sigma \left(s\right)-a\right){\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(\eta \right){\left({\int }_a^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2}{{\int }_a^{\sigma \left(s\right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z}\right)\hfill \\ \hfill & \times {\left({\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z\right)}^{\frac{1}{2}}\mathsf{\Delta }\xi .\hfill \end{array}$$

(36)

By applying (16) on the term

$${\int }_a^{\sigma \left(r\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi ,$$

with $u\left(\chi \right)=\left(\sigma \left(r\right)-\chi \right)$ and $v^{\mathsf{\Delta }}\left(\chi \right)={\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2,$ we get

$$\begin{array}{ccc}& & {\int }_a^{\sigma \left(r\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi \hfill \\ & =& {\left.\left(\sigma \left(r\right)-\chi \right)v\left(\chi \right)\right|}_a^{\sigma \left(r\right)}+{\int }_a^{\sigma \left(r\right)}{v}^{\sigma }\left(\chi \right)\mathsf{\Delta }\chi ,\hfill \end{array}$$

where $v\left(\chi \right)={\int }_a^{\chi }{\left[\lambda \left(\theta \right){\left({\int }_a^{\sigma \left(\theta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\theta$ and then (where $v\left(a\right)=0$)

$$\begin{array}{ccc}& & {\int }_a^{\sigma \left(r\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi \hfill \\ & =& {\int }_a^{\sigma \left(r\right)}{\int }_a^{\sigma \left(\chi \right)}{\left[\lambda \left(\theta \right){\left({\int }_a^{\sigma \left(\theta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\theta \mathsf{\Delta }\chi .\hfill \end{array}$$

(37)

Similarly, we see that

$$\begin{array}{ccc}& & {\int }_a^{\sigma \left(s\right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z\hfill \\ & =& {\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(z\right)}{\left[\psi \left(\theta \right){\left({\int }_a^{\sigma \left(\theta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }\theta \mathsf{\Delta }z.\hfill \end{array}$$

(38)

Substituting (37) and (38) into (36) and then by applying (23) when $\gamma =\nu =2,$ we observe that

$$\begin{array}{cc}\hfill & {\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(r\right)}\frac{S_{\alpha ,\beta ,t,\xi ,r,s}{\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha }{\left({\int }_a^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta }}{{\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}{\left(\sigma \left(\xi \right)-a\right)}^{\frac{1}{2}}}\mathsf{\Delta }t\mathsf{\Delta }\xi \hfill \\ \hfill & \ge \alpha \beta {\int }_a^{\sigma \left(r\right)}S\left(\frac{\left(\sigma \left(r\right)-a\right){\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\zeta \right){\left({\int }_a^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\zeta }{{\int }_a^{\sigma \left(r\right)}{\int }_a^{\sigma \left(\chi \right)}{\left[\lambda \left(\theta \right){\left({\int }_a^{\sigma \left(\theta \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\theta \mathsf{\Delta }\chi }\right)\hfill \\ \hfill & \times {\left({\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi \right)}^{\frac{1}{2}}\times 1\mathsf{\Delta }t\hfill \\ \hfill & \times {\int }_a^{\sigma \left(s\right)}S\left(\frac{\left(\sigma \left(s\right)-a\right){\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(\eta \right){\left({\int }_a^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }\eta }{{\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(z\right)}{\left[\psi \left(\theta \right){\left({\int }_a^{\sigma \left(\theta \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }\theta \mathsf{\Delta }z}\right)\hfill \\ \hfill & \times {\left({\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z\right)}^{\frac{1}{2}}\times 1\mathsf{\Delta }\xi \hfill \\ \hfill & \ge \alpha \beta {\left(\sigma \left(r\right)-a\right)}^{\frac{1}{2}}{\left({\int }_a^{\sigma \left(r\right)}{\int }_a^{\sigma \left(t\right)}{\left[\lambda \left(\chi \right){\left({\int }_a^{\sigma \left(\chi \right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\mathsf{\Delta }\chi \mathsf{\Delta }t\right)}^{\frac{1}{2}}\hfill \\ \hfill & \times {\left(\sigma \left(s\right)-a\right)}^{\frac{1}{2}}{\left({\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(\xi \right)}{\left[\psi \left(z\right){\left({\int }_a^{\sigma \left(z\right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\mathsf{\Delta }z\mathsf{\Delta }\xi \right)}^{\frac{1}{2}}.\hfill \end{array}$$

(39)

From (37)–(39), the last inequality becomes

$$\begin{array}{ccc}& & {\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(r\right)}\frac{S_{\alpha ,\beta ,t,\xi ,r,s}{\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha }{\left({\int }_a^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta }}{{\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}{\left(\sigma \left(\xi \right)-a\right)}^{\frac{1}{2}}}\mathsf{\Delta }t\mathsf{\Delta }\xi \hfill \\ & \ge & \alpha \beta {\left(\sigma \left(r\right)-a\right)}^{\frac{1}{2}}{\left({\int }_a^{\sigma \left(r\right)}{\left[\lambda \left(t\right){\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\left(\sigma \left(r\right)-t\right)\mathsf{\Delta }t\right)}^{\frac{1}{2}}\hfill \\ & & \times {\left(\sigma \left(s\right)-a\right)}^{\frac{1}{2}}{\left({\int }_a^{\sigma \left(s\right)}{\left[\psi \left(\xi \right){\left({\int }_a^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\left(\sigma \left(s\right)-\xi \right)\mathsf{\Delta }\xi \right)}^{\frac{1}{2}}\hfill \\ & =& 2C\left(\alpha ,\beta ,r,s\right){\left({\int }_a^{\sigma \left(r\right)}{\left[\lambda \left(t\right){\left({\int }_a^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\alpha -1}\right]}^2\left(\sigma \left(r\right)-t\right)\mathsf{\Delta }t\right)}^{\frac{1}{2}}\hfill \\ & & \times {\left({\int }_a^{\sigma \left(s\right)}{\left[\psi \left(\xi \right){\left({\int }_a^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathsf{\Delta }\tau \right)}^{\beta -1}\right]}^2\left(\sigma \left(s\right)-\xi \right)\mathsf{\Delta }\xi \right)}^{\frac{1}{2}},\hfill \end{array}$$

which is (29). □

Remark 1.

As a special case of Theorem 2, when $\mathbb{T}=\mathbb{N}$, we can get (9) proved by Zhao and Cheung [12].

Theorem 3.

Let $a\in \mathbb{T}$ and $\lambda$, ω be nonnegative functions. In addition, we assume that $\alpha$, β are positive functions. If $\varphi$, ψ are nonnegative, concave, and supermultiplicative functions, then the inequality

$$\begin{array}{ccc}& & {\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(r\right)}\frac{S_{t,r,s,\zeta }\varphi \left(\mathsf{\Lambda }\left(t\right)\right)\psi \left(\mathsf{\Omega }\left(\zeta \right)\right)}{{\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}{\left(\sigma \left(\zeta \right)-a\right)}^{\frac{1}{2}}}\mathsf{\Delta }t\mathsf{\Delta }\zeta \hfill \\ & \ge & 2M\left(r,s\right){\left({\int }_a^{\sigma \left(r\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi \right)}^{\frac{1}{2}}\hfill \\ & & \times {\left({\int }_a^{\sigma \left(s\right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z\right)}^{\frac{1}{2}},\hfill \end{array}$$

(40)

holds for all $r,s\in {[a,\infty ]}_{\mathbb{T}}$, with

$$M\left(r,s\right)=\frac{1}{2}{\left({\int }_a^{\sigma \left(r\right)}{\left(\frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}\right)}^2\mathsf{\Delta }t\right)}^{\frac{1}{2}}{\left({\int }_a^{\sigma \left(s\right)}{\left(\frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}\right)}^2\mathsf{\Delta }\zeta \right)}^{\frac{1}{2}},$$

$$\begin{array}{ccc}\hfill S_{t,r,s,\zeta }& =& S\left(\frac{\left({\int }_a^{\sigma \left(r\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi \right){\left(\frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}\right)}^2}{\left({\int }_a^{\sigma \left(r\right)}{\left(\frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}\right)}^2\mathsf{\Delta }t\right)\left({\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi \right)}\right)\hfill \\ & & \times S\left(\frac{\left({\int }_a^{\sigma \left(s\right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z\right){\left(\frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}\right)}^2}{\left({\int }_a^{\sigma \left(s\right)}{\left(\frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}\right)}^2\mathsf{\Delta }\zeta \right)\left({\int }_a^{\sigma \left(\zeta \right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\mathsf{\Delta }z\right)}\right),\hfill \end{array}$$

$$\mathsf{\Lambda }\left(t\right)={\int }_a^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi }\right)\lambda \left(\chi \right)\mathsf{\Delta }\chi ,$$

$$\begin{array}{ccc}\hfill \mathsf{\Omega }\left(\zeta \right)& =& {\int }_a^{\sigma \left(\zeta \right)}S\left(\frac{\left(\sigma \left(\zeta \right)-a\right){\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2}{{\int }_a^{\sigma \left(\zeta \right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\mathsf{\Delta }z}\right)\omega \left(z\right)\mathsf{\Delta }z,\hfill \\ \hfill \mathsf{\Phi }\left(t\right)& =& {\int }_a^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi }\right)\alpha \left(\chi \right)\mathsf{\Delta }\chi ,\hfill \end{array}$$

and

$$\mathsf{\Psi }\left(\zeta \right)={\int }_a^{\sigma \left(\zeta \right)}S\left(\frac{\left(\sigma \left(\zeta \right)-a\right){\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2}{{\int }_a^{\sigma \left(\zeta \right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\mathsf{\Delta }z}\right)\beta \left(z\right)\mathsf{\Delta }z.$$

Proof. 

Using the fact that $\varphi$ is a supermultiplicative function, applying Jensen’s inequality and then applying (23) with $\gamma =\nu =2.$ Then,

$$\begin{array}{ccc}\hfill \varphi \left(\mathsf{\Lambda }\right(t\left)\right)& =& \varphi \left(\frac{\mathsf{\Phi }\left(t\right){\int }_a^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi }\right)\alpha \left(\chi \right)\lambda \left(\chi \right)/\alpha \left(\chi \right)\mathsf{\Delta }\chi }{{\int }_a^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi }\right)\alpha \left(\chi \right)\mathsf{\Delta }\chi }\right)\hfill \\ & \ge & \varphi \left(\mathsf{\Phi }\left(t\right)\right)\varphi \left(\frac{{\int }_a^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi }\right)\alpha \left(\chi \right)\left[\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right]\mathsf{\Delta }\chi }{{\int }_a^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi }\right)\alpha \left(\chi \right)\mathsf{\Delta }\chi }\right)\hfill \\ & \ge & \frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}{\int }_a^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-a\right){\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2}{{\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi }\right)\alpha \left(\chi \right)\varphi \left[\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right]\times 1\mathsf{\Delta }\chi \hfill \\ & \ge & \frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}{\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}{\left({\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi \right)}^{\frac{1}{2}}.\hfill \end{array}$$

(41)

Similarly, we can get

$$\psi \left(\mathsf{\Omega }\left(\zeta \right)\right)\ge \frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}{\left(\sigma \left(\zeta \right)-a\right)}^{\frac{1}{2}}{\left({\int }_a^{\sigma \left(\zeta \right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\mathsf{\Delta }z\right)}^{\frac{1}{2}}.$$

(42)

Multiplying the both sides of (41) and (42), respectively, by

$$S\left(\frac{\left({\int }_a^{\sigma \left(r\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi \right){\left(\frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}\right)}^2}{\left({\int }_a^{\sigma \left(r\right)}{\left(\frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}\right)}^2\mathsf{\Delta }t\right)\left({\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi \right)}\right),$$

and

$$S\left(\frac{\left({\int }_a^{\sigma \left(s\right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z\right){\left(\frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}\right)}^2}{\left({\int }_a^{\sigma \left(s\right)}{\left(\frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}\right)}^2\mathsf{\Delta }\zeta \right)\left({\int }_a^{\sigma \left(\zeta \right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\mathsf{\Delta }z\right)}\right),$$

and then multiplying these inequalities, we get

$$\begin{array}{ccc}& & S\left(\frac{\left({\int }_a^{\sigma \left(s\right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z\right){\left(\frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}\right)}^2}{\left({\int }_a^{\sigma \left(s\right)}{\left(\frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}\right)}^2\mathsf{\Delta }\zeta \right)\left({\int }_a^{\sigma \left(\zeta \right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\mathsf{\Delta }z\right)}\right)\varphi \left(\mathsf{\Lambda }\left(t\right)\right)\psi \left(\mathsf{\Omega }\left(\zeta \right)\right)\hfill \\ & & \times S\left(\frac{\left({\int }_a^{\sigma \left(r\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi \right){\left(\frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}\right)}^2}{\left({\int }_a^{\sigma \left(r\right)}{\left(\frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}\right)}^2\mathsf{\Delta }t\right)\left({\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi \right)}\right)\hfill \\ & \ge & \frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}{\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}{\left({\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi \right)}^{\frac{1}{2}}\hfill \\ & & \times S\left(\frac{\left({\int }_a^{\sigma \left(r\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi \right){\left(\frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}\right)}^2}{\left({\int }_a^{\sigma \left(r\right)}{\left(\frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}\right)}^2\mathsf{\Delta }t\right)\left({\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi \right)}\right)\hfill \\ & & \times \frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}{\left(\sigma \left(\zeta \right)-a\right)}^{\frac{1}{2}}{\left({\int }_a^{\sigma \left(\zeta \right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\mathsf{\Delta }z\right)}^{\frac{1}{2}}\hfill \\ & & \times S\left(\frac{\left({\int }_a^{\sigma \left(s\right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z\right){\left(\frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}\right)}^2}{\left({\int }_a^{\sigma \left(s\right)}{\left(\frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}\right)}^2\mathsf{\Delta }\zeta \right)\left({\int }_a^{\sigma \left(\zeta \right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\mathsf{\Delta }z\right)}\right).\hfill \end{array}$$

(43)

By dividing the two sides of (43) on ${\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}{\left(\sigma \left(\zeta \right)-a\right)}^{\frac{1}{2}}$ and then taking the integration over $\zeta$ from a to $\sigma \left(s\right)$ and then the integration over t from a to $\sigma \left(r\right)$, we observe that

$$\begin{array}{ccc}& & {\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(r\right)}\frac{S_{t,r,s,\zeta }\varphi \left(\mathsf{\Lambda }\left(t\right)\right)\psi \left(\mathsf{\Omega }\left(\zeta \right)\right)}{{\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}{\left(\sigma \left(\zeta \right)-a\right)}^{\frac{1}{2}}}\mathsf{\Delta }t\mathsf{\Delta }\zeta \hfill \\ & \ge & {\int }_a^{\sigma \left(r\right)}S\left(\frac{\left({\int }_a^{\sigma \left(r\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi \right){\left(\frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}\right)}^2}{\left({\int }_a^{\sigma \left(r\right)}{\left(\frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}\right)}^2\mathsf{\Delta }t\right)\left({\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi \right)}\right)\hfill \\ & & \times \frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}{\left({\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi \right)}^{\frac{1}{2}}\mathsf{\Delta }t\hfill \\ & & \times {\int }_a^{\sigma \left(s\right)}S\left(\frac{\left({\int }_a^{\sigma \left(s\right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z\right){\left(\frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}\right)}^2}{\left({\int }_a^{\sigma \left(s\right)}{\left(\frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}\right)}^2\mathsf{\Delta }\zeta \right)\left({\int }_a^{\sigma \left(\zeta \right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\mathsf{\Delta }z\right)}\right)\hfill \\ & & \times \frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}{\left({\int }_a^{\sigma \left(\zeta \right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\mathsf{\Delta }z\right)}^{\frac{1}{2}}\mathsf{\Delta }\zeta .\hfill \end{array}$$

(44)

By using (16), we can see that

$$\begin{array}{ccc}& & {\int }_a^{\sigma \left(r\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi \hfill \\ & =& {\int }_a^{\sigma \left(r\right)}{\int }_a^{\sigma \left(\chi \right)}{\left[\alpha \left(\theta \right)\varphi \left(\frac{\lambda \left(\theta \right)}{\alpha \left(\theta \right)}\right)\right]}^2\mathsf{\Delta }\theta \mathsf{\Delta }\chi .\hfill \end{array}$$

(45)

In addition, we can obtain that

$$\begin{array}{ccc}& & {\int }_a^{\sigma \left(s\right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z\hfill \\ & =& {\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(z\right)}{\left[\beta \left(\theta \right)\psi \left(\frac{\omega \left(\theta \right)}{\beta \left(\theta \right)}\right)\right]}^2\mathsf{\Delta }\theta \mathsf{\Delta }z.\hfill \end{array}$$

(46)

Substituting (45) and (46) into (44), we have

$$\begin{array}{ccc}& & {\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(r\right)}\frac{S_{t,r,s,\zeta }\varphi \left(\mathsf{\Lambda }\left(t\right)\right)\psi \left(\mathsf{\Omega }\left(\zeta \right)\right)}{{\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}{\left(\sigma \left(\zeta \right)-a\right)}^{\frac{1}{2}}}\mathsf{\Delta }t\mathsf{\Delta }\zeta \hfill \\ & \ge & {\int }_a^{\sigma \left(r\right)}S\left(\frac{\left({\int }_a^{\sigma \left(r\right)}{\int }_a^{\sigma \left(\chi \right)}{\left[\alpha \left(\theta \right)\varphi \left(\frac{\lambda \left(\theta \right)}{\alpha \left(\theta \right)}\right)\right]}^2\mathsf{\Delta }\theta \mathsf{\Delta }\chi \right){\left(\frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}\right)}^2}{\left({\int }_a^{\sigma \left(r\right)}{\left(\frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}\right)}^2\mathsf{\Delta }t\right)\left({\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi \right)}\right)\hfill \\ & & \times \frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}{\left({\int }_a^{\sigma \left(t\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\mathsf{\Delta }\chi \right)}^{\frac{1}{2}}\mathsf{\Delta }t\hfill \\ & & \times {\int }_a^{\sigma \left(s\right)}S\left(\frac{\left({\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(z\right)}{\left[\beta \left(\theta \right)\psi \left(\frac{\omega \left(\theta \right)}{\beta \left(\theta \right)}\right)\right]}^2\mathsf{\Delta }\theta \mathsf{\Delta }z\right){\left(\frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}\right)}^2}{\left({\int }_a^{\sigma \left(s\right)}{\left(\frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}\right)}^2\mathsf{\Delta }\zeta \right)\left({\int }_a^{\sigma \left(\zeta \right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\mathsf{\Delta }z\right)}\right)\hfill \\ & & \times \frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}{\left({\int }_a^{\sigma \left(\zeta \right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\mathsf{\Delta }z\right)}^{\frac{1}{2}}\mathsf{\Delta }\zeta .\hfill \end{array}$$

(47)

Applying (23) with $\gamma =\nu =2$ on the right-hand side of (47), we get

$$\begin{array}{ccc}& & {\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(r\right)}\frac{S_{t,r,s,\zeta }\varphi \left(\mathsf{\Lambda }\left(t\right)\right)\psi \left(\mathsf{\Omega }\left(\zeta \right)\right)}{{\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}{\left(\sigma \left(\zeta \right)-a\right)}^{\frac{1}{2}}}\mathsf{\Delta }t\mathsf{\Delta }\zeta \hfill \\ & \ge & {\left({\int }_a^{\sigma \left(r\right)}{\left(\frac{\varphi \left(\mathsf{\Phi }\right(t\left)\right)}{\mathsf{\Phi }\left(t\right)}\right)}^2\mathsf{\Delta }t\right)}^{\frac{1}{2}}{\left({\int }_a^{\sigma \left(r\right)}{\int }_a^{\sigma \left(\chi \right)}{\left[\alpha \left(\theta \right)\varphi \left(\frac{\lambda \left(\theta \right)}{\alpha \left(\theta \right)}\right)\right]}^2\mathsf{\Delta }\theta \mathsf{\Delta }\chi \right)}^{\frac{1}{2}}\hfill \\ & & \times {\left({\int }_a^{\sigma \left(s\right)}{\left(\frac{\psi \left(\mathsf{\Psi }\right(\zeta \left)\right)}{\mathsf{\Psi }\left(\zeta \right)}\right)}^2\mathsf{\Delta }\zeta \right)}^{\frac{1}{2}}{\left({\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(z\right)}{\left[\beta \left(\theta \right)\psi \left(\frac{\omega \left(\theta \right)}{\beta \left(\theta \right)}\right)\right]}^2\mathsf{\Delta }\theta \mathsf{\Delta }z\right)}^{\frac{1}{2}}\hfill \\ & =& 2M\left(r,s\right){\left({\int }_a^{\sigma \left(r\right)}{\int }_a^{\sigma \left(\chi \right)}{\left[\alpha \left(\theta \right)\varphi \left(\frac{\lambda \left(\theta \right)}{\alpha \left(\theta \right)}\right)\right]}^2\mathsf{\Delta }\theta \mathsf{\Delta }\chi \right)}^{\frac{1}{2}}\hfill \\ & & \times {\left({\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(z\right)}{\left[\beta \left(\theta \right)\psi \left(\frac{\omega \left(\theta \right)}{\beta \left(\theta \right)}\right)\right]}^2\mathsf{\Delta }\theta \mathsf{\Delta }z\right)}^{\frac{1}{2}}.\hfill \end{array}$$

(48)

From (45) and (46), the inequality (48) becomes

$$\begin{array}{ccc}& & {\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(r\right)}\frac{S_{t,r,s,\zeta }\varphi \left(\mathsf{\Lambda }\left(t\right)\right)\psi \left(\mathsf{\Omega }\left(\zeta \right)\right)}{{\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}{\left(\sigma \left(\zeta \right)-a\right)}^{\frac{1}{2}}}\mathsf{\Delta }t\mathsf{\Delta }\zeta \hfill \\ & \ge & 2M\left(r,s\right){\left({\int }_a^{\sigma \left(r\right)}{\left[\alpha \left(\chi \right)\varphi \left(\frac{\lambda \left(\chi \right)}{\alpha \left(\chi \right)}\right)\right]}^2\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi \right)}^{\frac{1}{2}}\hfill \\ & & \times {\left({\int }_a^{\sigma \left(s\right)}{\left[\beta \left(z\right)\psi \left(\frac{\omega \left(z\right)}{\beta \left(z\right)}\right)\right]}^2\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z\right)}^{\frac{1}{2}},\hfill \end{array}$$

which is (40). □

Remark 2.

As a special case of Theorem 3, when $\mathbb{T}=\mathbb{N}$, we can get (10) proved by Zhao and Cheung [12].

By applying Theorem 3 with $\varphi \left(\chi \right)=\chi$ and $\psi \left(z\right)=z,$ we can obtain the following theorem.

Theorem 4.

Let $a\in \mathbb{T}$ and $\lambda$, ω be nonnegative functions. Then, the inequality

$$\begin{array}{ccc}& & {\int }_a^{\sigma \left(s\right)}{\int }_a^{\sigma \left(r\right)}\frac{S_{t,r,s,\zeta }\mathsf{\Lambda }\left(t\right)\mathsf{\Omega }\left(\zeta \right)}{{\left(\sigma \left(t\right)-a\right)}^{\frac{1}{2}}{\left(\sigma \left(\zeta \right)-a\right)}^{\frac{1}{2}}}\mathsf{\Delta }t\mathsf{\Delta }\zeta \hfill \\ & \ge & {\left(\sigma \left(r\right)-a\right)}^{\frac{1}{2}}{\left(\sigma \left(s\right)-a\right)}^{\frac{1}{2}}{\left({\int }_a^{\sigma \left(r\right)}{\lambda }^2\left(\chi \right)\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi \right)}^{\frac{1}{2}}\hfill \\ & & \times {\left({\int }_a^{\sigma \left(s\right)}{\omega }^2\left(z\right)\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z\right)}^{\frac{1}{2}},\hfill \end{array}$$

holds for all $r,s\in {[a,\infty ]}_{\mathbb{T}}$, with

$$S_{t,r,s,\zeta }=S\left(\frac{{\int }_a^{\sigma \left(r\right)}{\lambda }^2\left(\chi \right)\left(\sigma \left(r\right)-\chi \right)\mathsf{\Delta }\chi }{\left(\sigma \left(r\right)-a\right)\left({\int }_a^{\sigma \left(t\right)}{\lambda }^2\left(\chi \right)\mathsf{\Delta }\chi \right)}\right)S\left(\frac{{\int }_a^{\sigma \left(s\right)}{\omega }^2\left(z\right)\left(\sigma \left(s\right)-z\right)\mathsf{\Delta }z}{\left(\sigma \left(s\right)-a\right)\left({\int }_a^{\sigma \left(\zeta \right)}{\omega }^2\left(z\right)\mathsf{\Delta }z\right)}\right),$$

and

$$\mathsf{\Lambda }\left(t\right)={\int }_a^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-a\right){\lambda }^2\left(\chi \right)}{{\int }_a^{\sigma \left(t\right)}{\lambda }^2\left(\chi \right)\mathsf{\Delta }\chi }\right)\lambda \left(\chi \right)\mathsf{\Delta }\chi ,\mathsf{\Omega }\left(\zeta \right)={\int }_a^{\sigma \left(\zeta \right)}S\left(\frac{\left(\sigma \left(\zeta \right)-a\right){\omega }^2\left(z\right)}{{\int }_a^{\sigma \left(\zeta \right)}{\omega }^2\left(z\right)\mathsf{\Delta }z}\right)\omega \left(z\right)\mathsf{\Delta }z.$$

Remark 3.

As a special case of Theorem 4, when $\mathbb{T}=\mathbb{N}$, we can get (11) proved by Zhao and Cheung [12].

4. Conclusions

In this article, we first proved the reverse Hilbert-type inequalities on time scales which involving nonnegative and decreasing functions. After that, we proved the reverse Hilbert-type inequalities on time scales which involve nonnegative, concave, and supermultiplicative functions. All of these results are proved by using the Specht’s ratio function. In future work, we will continue to generalize more the reverse Hilbert-type inequalities by using Kantorovich’s ratio.