Some New Generalizations of Reverse Hilbert-Type Inequalities on Time Scales

Abstract

This manuscript develops the study of reverse Hilbert-type inequalities by applying reverse Hölder inequalities on $\mathbb{T}$. We generalize the reverse inequality of Hilbert-type with power two by replacing the power with a new power $\beta ,$ $\beta >1.$ The main results are proved by using Specht’s ratio, chain rule and Jensen’s inequality. Our results (when $\mathbb{T}=\mathbb{N}$) are essentially new. Symmetrical properties play an essential role in determining the correct methods to solve inequalities.

1. Introduction

In [1], Hardy established that

$${\displaystyle \sum _{i=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{{\mathrm{\Xi }}_i{ϝ}_m}{i+m}\le \frac{\pi }{\sin \frac{\pi }{l}}{\left({\displaystyle \sum _{i=1}^{\infty }}{\mathrm{\Xi }}_i^l\right)}^{{}^{\frac{1}{l}}}{\left({\displaystyle \sum _{m=1}^{\infty }}{ϝ}_m^q\right)}^{{}^{\frac{1}{q}}},$$

(1)

where ${\mathrm{\Xi }}_i,$ ${ϝ}_m\ge 0$ with $0<{\displaystyle \sum _{i=1}^{\infty }}{\mathrm{\Xi }}_i^l<\infty ,$ $0<{\displaystyle \sum _{m=1}^{\infty }}{ϝ}_m^q<\infty$ and $l>1,$ $1/l+1/q=1$. The continuous form (see [2]) of (1) is

$${\int }_0^{\infty }{\int }_0^{\infty }\frac{\phi \left(\vartheta \right)\psi \left(y\right)}{\vartheta +y}d\vartheta dy\le \frac{\pi }{\sin \frac{\pi }{l}}{\left({\int }_0^{\infty }{\phi }^l\left(\vartheta \right)d\vartheta \right)}^{{}^{\frac{1}{l}}}{\left({\int }_0^{\infty }{\psi }^q\left(y\right)dy\right)}^{{}^{\frac{1}{q}}},$$

(2)

where $\phi ,\psi \ge 0$ are measurable functions such that $0<{\int }_0^{\infty }{\phi }^l\left(\vartheta \right)d\vartheta <\infty$ and $0<{\int }_0^{\infty }{\psi }^q\left(y\right)dy<\infty .$ The constant $\pi /\sin (\pi /l)$ in both (1) and (2) is sharp. In [2], Hardy showed that if $d>1,$ $q>1,$ $1/d+1/q\ge 1$ and $0<\lambda =2-(1/d+1/q)\le 1,$ then

$${\displaystyle \sum _{i=1}^{\infty }}{\displaystyle \sum _{n=1}^{\infty }}\frac{{\mathrm{\Xi }}_i{ϝ}_n}{{(i+n)}^{\lambda }}\le K(d,q){\left({\displaystyle \sum _{i=1}^{\infty }}{\mathrm{\Xi }}_i^d\right)}^{\frac{1}{d}}{\left({\displaystyle \sum _{n=1}^{\infty }}{ϝ}_n^q\right)}^{\frac{1}{q}}.$$

In [3], Hölder proved that

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

(3)

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 continuous form of (3) is

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

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

In [4], Zhao and Cheung proved that if $\psi \left(\zeta \right),\varpi \left(\zeta \right)\ge 0$ are continuous functions and ${\psi }^{1/\alpha }\left(\zeta \right){\varpi }^{1/\beta }\left(\zeta \right)$ is integrable on $[\varrho ,c],$ then

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

with

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

where $S(.)$ is Specht’s ratio function (see [5]) and defined as

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

In [4], the authors proved that if $\psi ,$ $\varpi \in C\left((\varrho ,c),{\mathbb{R}}^{+}\right)$ and $m>0,$ then

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

(4)

where

$$G={\displaystyle \underset{\varrho }{\overset{c}{\int }}}\varpi \left(\zeta \right)d\zeta \mathrm{and}F={\displaystyle \underset{\varrho }{\overset{c}{\int }}}\frac{{\psi }^{m+1}\left(\zeta \right)}{{\varpi }^m\left(\zeta \right)}d\zeta .$$

In addition, they proved the discrete case of (4) and established that

$${\displaystyle \sum _{i=1}^{\infty }}\frac{{\varrho }_i^{m+1}}{b_i^m}\le \frac{{\displaystyle \sum _{i=1}^{\infty }}S\left(\frac{B{\varrho }_i^{m+1}}{A{b}_i^{m+1}}\right){\varrho }_i}{{\left({\displaystyle \sum _{i=1}^{\infty }}{b}_i\right)}^m},$$

where $B={\displaystyle \sum _{i=1}^{\infty }}{b}_i$ and $A={\displaystyle \sum _{i=1}^{\infty }}{\varrho }_i^{m+1}/b_i^m.$

In 2019, Zhao and Cheung [6] studied the reverse Hilbert inequalities and proved that if $0\le d,q\le 1$ and ${\left\{{\lambda }_i\right\}}_1^k,$ ${\left\{{\psi }_n\right\}}_1^r$ are nonnegative and decreasing sequences of real numbers with $k,$ $r\in \mathbb{N},$ then

$$\begin{array}{c}{\displaystyle \sum _{i=1}^k}{\displaystyle \sum _{n=1}^r}\frac{S_{d,q,k,r,i,n}{\left({\displaystyle \sum _{s=1}^i}{\lambda }_s\right)}^d{\left({\displaystyle \sum _{t=1}^n}{\psi }_t\right)}^q}{{\left(in\right)}^{\frac{1}{2}}}\hfill \\ \ge 2C\left(d,q,k,r\right){\left({\displaystyle \sum _{i=1}^k}{\left[{\lambda }_i{\left({\displaystyle \sum _{s=1}^i}{\lambda }_s\right)}^{d-1}\right]}^2\left(k-i+1\right)\right)}^{\frac{1}{2}}\hfill \\ \times {\left({\displaystyle \sum _{n=1}^r}{\left[{\psi }_n{\left({\displaystyle \sum _{t=1}^n}{\psi }_t\right)}^{q-1}\right]}^2\left(r-n+1\right)\right)}^{\frac{1}{2}},\hfill \end{array}$$

(5)

where

$$C\left(d,q,r,s\right)=\frac{1}{2}dq{\left(kr\right)}^{\frac{1}{2}},$$

and

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

where

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

and

$$\begin{array}{cc}& S\left(\frac{n{\left[{\psi }_v{\left({\displaystyle \sum _{\tau =1}^v}{\psi }_{\tau }\right)}^{q-1}\right]}^2}{{\displaystyle \sum _{t=1}^n}{\left[{\psi }_t{\left({\displaystyle \sum _{\tau =1}^t}{\psi }_{\tau }\right)}^{q-1}\right]}^2}\right)\hfill \\ & =\max \left\{S\left(\frac{n{\left[{\psi }_1{\left({\displaystyle \sum _{\tau =1}^1}{\psi }_{\tau }\right)}^{q-1}\right]}^2}{{\displaystyle \sum _{t=1}^n}{\left[{\psi }_t{\left({\displaystyle \sum _{\tau =1}^t}{\psi }_{\tau }\right)}^{q-1}\right]}^2}\right)\right.\hfill \\ & \left.;S\left(\frac{n{\left[{\psi }_n{\left({\displaystyle \sum _{\tau =1}^n}{\psi }_{\tau }\right)}^{q-1}\right]}^2}{{\displaystyle \sum _{t=1}^n}{\left[{\psi }_t{\left({\displaystyle \sum _{\tau =1}^t}{\psi }_{\tau }\right)}^{q-1}\right]}^2}\right)\right\}.\hfill \end{array}$$

In addition, they proved that if ${\left\{{\lambda }_i\right\}}_1^k,$ ${\left\{{\varpi }_n\right\}}_1^r$ are nonnegative sequences and ${\left\{d_i\right\}}_1^k,$ ${\left\{q_n\right\}}_1^r$ are positive sequences with $k,r\in \mathbb{N}$, then

$$\begin{array}{c}{\displaystyle \sum _{i=1}^k}{\displaystyle \sum _{n=1}^r}\frac{S_{k,r,i,n}\varphi \left({\mathrm{\Lambda }}_i\right)\psi \left({\mathrm{\Omega }}_n\right)}{{\left(in\right)}^{\frac{1}{2}}}\hfill \\ \ge 2N\left(k,r\right){\left({\displaystyle \sum _{s=1}^k}{\left[d_s\varphi \left(\frac{{\lambda }_s}{d_s}\right)\right]}^2\left(k-s+1\right)\right)}^{\frac{1}{2}}\hfill \\ \times {\left({\displaystyle \sum _{t=1}^r}{\left[q_t\psi \left(\frac{{\varpi }_t}{q_t}\right)\right]}^2\left(r-t+1\right)\right)}^{\frac{1}{2}},\hfill \end{array}$$

(6)

with

$$N\left(k,r\right)=\frac{1}{2}{\left({\displaystyle \sum _{i=1}^k}{\left(\frac{\varphi \left(D_i\right)}{D_i}\right)}^2\right)}^{\frac{1}{2}}{\left({\displaystyle \sum _{n=1}^r}{\left(\frac{\psi \left(Q_n\right)}{Q_n}\right)}^2\right)}^{\frac{1}{2}},$$

$$\begin{array}{cc}\hfill S_{k,r,i,n}& =S\left(\frac{\left({\displaystyle \sum _{s=1}^k}{\left[d_s\varphi \left(\frac{{\lambda }_s}{d_s}\right)\right]}^2\left(k-s+1\right)\right){\left(\frac{\varphi \left(D_i\right)}{D_i}\right)}^2}{\left({\displaystyle \sum _{i=1}^k}{\left(\frac{\varphi \left(D_i\right)}{D_i}\right)}^2\right)\left({\displaystyle \sum _{s=1}^i}{\left[d_s\varphi \left(\frac{{\lambda }_s}{d_s}\right)\right]}^2\right)}\right)\hfill \\ & \times S\left(\frac{\left({\displaystyle \sum _{t=1}^r}{\left[q_t\psi \left(\frac{{\varpi }_t}{q_t}\right)\right]}^2\left(r-t+1\right)\right){\left(\frac{\psi \left(Q_n\right)}{Q_n}\right)}^2}{\left({\displaystyle \sum _{n=1}^r}{\left(\frac{\psi \left(Q_n\right)}{Q_n}\right)}^2\right)\left({\displaystyle \sum _{t=1}^n}{\left[q_t\psi \left(\frac{{\varpi }_t}{q_t}\right)\right]}^2\right)}\right),\hfill \end{array}$$

$${\mathrm{\Lambda }}_i={\displaystyle \sum _{s=1}^i}S\left(\frac{i{\left[d_s\varphi \left(\frac{{\lambda }_s}{d_s}\right)\right]}^2}{{\displaystyle \sum _{s=1}^i}{\left[d_s\varphi \left(\frac{{\lambda }_s}{d_s}\right)\right]}^2}\right){\lambda }_s,$$

$${\mathrm{\Omega }}_n={\displaystyle \sum _{t=1}^n}S\left(\frac{n{\left[q_t\psi \left(\frac{{\varpi }_t}{q_t}\right)\right]}^2}{{\displaystyle \sum _{t=1}^n}{\left[q_t\psi \left(\frac{{\varpi }_t}{q_t}\right)\right]}^2}\right){\varpi }_t;$$

$$D_i={\displaystyle \sum _{s=1}^i}S\left(\frac{i{\left[d_s\varphi \left(\frac{{\lambda }_s}{d_s}\right)\right]}^2}{{\displaystyle \sum _{s=1}^i}{\left[d_s\varphi \left(\frac{{\lambda }_s}{d_s}\right)\right]}^2}\right)d_s;$$

and

$$Q_n={\displaystyle \sum _{t=1}^n}S\left(\frac{n{\left[q_t\psi \left(\frac{{\varpi }_t}{q_t}\right)\right]}^2}{{\displaystyle \sum _{t=1}^n}{\left[q_t\psi \left(\frac{{\varpi }_t}{q_t}\right)\right]}^2}\right)q_t,$$

where $\varphi ,$ $\psi$ are nonnegative, concave and supermultiplicative functions.

In [6], the authors proved that if ${\left\{{\lambda }_i\right\}}_1^k,$ ${\left\{{\varpi }_n\right\}}_1^r$ are nonnegative sequenceswith $k,r\in \mathbb{N},$ then

$$\begin{array}{c}{\displaystyle \sum _{i=1}^k}{\displaystyle \sum _{n=1}^r}\frac{S_{k,r,i,n}{\mathrm{\Lambda }}_i{\mathrm{\Omega }}_n}{{\left(in\right)}^{\frac{1}{2}}}\hfill \\ \ge {\left(kr\right)}^{\frac{1}{2}}{\left({\displaystyle \sum _{i=1}^k}{\lambda }_i^2\left(k-i+1\right)\right)}^{\frac{1}{2}}{\left({\displaystyle \sum _{n=1}^r}{\varpi }_n^2\left(r-n+1\right)\right)}^{\frac{1}{2}},\hfill \end{array}$$

(7)

with

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

$${\mathrm{\Lambda }}_i={\displaystyle \sum _{s=1}^i}S\left(\frac{i{\lambda }_s^2}{{\displaystyle \sum _{s=1}^i}{\lambda }_s^2}\right){\lambda }_s\mathrm{and}{\mathrm{\Omega }}_n={\displaystyle \sum _{t=1}^n}S\left(\frac{n{\varpi }_t^2}{{\displaystyle \sum _{t=1}^n}{\varpi }_t^2}\right){\varpi }_t.$$

Furthermore, many authors studied the inequalities of Hilbert-type, see [7,8,9,10,11,12,13,14,15].

In the last decades, the time scale theory was discovered which is a unification of the continuous calculus and discrete calculus. A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset of the real numbers $\mathbb{R}$. Many authors established some dynamic inequalities of Hilbert-type on time scales. For example, in 2021, AlNemer et al. [16] studied some reversed dynamic inequalities of Hilbert-type and proved that if $a\in \mathbb{T}$, $0\le \alpha ,\beta \le 1$ and $\lambda ,$ $\psi$ are nonnegative and decreasing functions, then the inequality

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

(8)

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

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

Such that

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

and

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

where the function $S(.)$ is the Specht ratio (see [5]) which is defined as follows:

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

The aim of this manuscript is to use reverse Hölder inequalities with Specht’s ratio on time scales $\mathbb{T}$ to establish some new generalizations of reverse Hilbert-type inequalities. In particular, we generalize the inequality (8) by replacing the power 2 with a new power $\beta ,$ $\beta >1.$

The following is a breakdown of the paper’s structure. In Section 2, we cover some fundamentals of time scale theory as well as several time scale lemmas that will be useful in Section 3, where we prove our findings. As specific examples (when $\mathbb{T}=\mathbb{N}$), our major results yield (5)–(7) proven by Zhao and Cheung [6]. In addition, we obtain the inequality (8) proved by AlNemer et al. [16].

2. Definitions and Basic Lemmas

A time scale $\mathbb{T}$ is defined as an arbitrary nonempty closed subset of the real numbers $\mathbb{R}$ and the forward jump operator 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 $U:\mathbb{T}\to \mathbb{R}$, the notation $U^{\sigma }\left(\tau \right)$ denotes $U\left(\sigma \right(\tau \left)\right).$

The derivatives of $U\varpi$ and $U/\varpi$ (where $\varpi {\varpi }^{\sigma }\ne 0$) are given by

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

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

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

(9)

The time scales chain rule is

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

where it is supposed that $\varpi :\mathbb{R}\to \mathbb{R}$ is continuously differentiable and $\phi :\mathbb{T}\to \mathbb{R}$ is $\mathrm{\Delta }$-differentiable. For further information on the time scale calculus, see [17,18].

Definition 1

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

$$G\left(\varkappa s\right)\ge G\left(\varkappa \right)G\left(s\right),\forall \varkappa ,s\in J\subset \mathbb{R}.$$

(10)

Inequality (10) holds with equality if G is the identity map ($i.e.,$ $G\left(\varkappa \right)=\varkappa$). G is said to be a submultiplicative function if the last inequality has the opposite sign.

Lemma 1.

If $\varrho \in \mathbb{T},$ λ is a nonnegative rd-continuous function and $0<\gamma \le 1,$ then

$${\left({\int }_{\varrho }^{\sigma \left(s\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\gamma }\ge \gamma {\int }_{\varrho }^{\sigma \left(s\right)}{\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\gamma -1}\lambda \left(\vartheta \right)\mathrm{\Delta }\vartheta .$$

(11)

Proof. 

Using the time scales chain rule on the term ${\int }_{\varrho }^{\vartheta }\lambda \left(\tau \right)\mathrm{\Delta }\tau ,$ we obtain

$${\left[{\left({\int }_{\varrho }^{\vartheta }\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\gamma }\right]}^{\mathrm{\Delta }}=\gamma {\left({\int }_{\varrho }^{\zeta }\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\gamma -1}\lambda \left(\vartheta \right),\zeta \in [\vartheta ,\sigma \left(\vartheta \right)].$$

(12)

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

$${\left({\int }_{\varrho }^{\zeta }\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\gamma -1}\ge {\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\gamma -1},$$

(13)

Substituting (13) into (12), we see

$${\left[{\left({\int }_{\varrho }^{\vartheta }\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\gamma }\right]}^{\mathrm{\Delta }}\ge \gamma {\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\gamma -1}\lambda \left(\vartheta \right).$$

(14)

Integrating (14) over $\vartheta$ from $\varrho$ to $\sigma \left(s\right),$ we have

$${\int }_{\varrho }^{\sigma \left(s\right)}{\left[{\left({\int }_{\varrho }^{\vartheta }\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\gamma }\right]}^{\mathrm{\Delta }}\mathrm{\Delta }\vartheta \ge \gamma {\int }_{\varrho }^{\sigma \left(s\right)}{\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\gamma -1}\lambda \left(\vartheta \right)\mathrm{\Delta }\vartheta .$$

This means that

$${\left({\int }_{\varrho }^{\sigma \left(s\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\gamma }\ge \gamma {\int }_{\varrho }^{\sigma \left(s\right)}{\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\gamma -1}\lambda \left(\vartheta \right)\mathrm{\Delta }\vartheta ,$$

which is (11). □

Lemma 2

(Specht’s ratio [5]). Let $\alpha ,\beta$ be positive numbers, $d>1$ and $1/d+1/q=1.$ Then,

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

(15)

where

$$S\left(u\right)=\frac{u^{1/(u-1)}}{e\log u^{1/(u-1)}},u\ne 1.$$

Lemma 3

([5]). Let $S(.)$ be as defined in Lemma 2. Then, $S\left(l\right)$ is strictly decreasing for $0<l<1$ and strictly increasing for $l>1.$ In addition, the following equations are true

$$S\left(1\right)=1\mathit{and}S\left(l\right)=S\left(\frac{1}{l}\right)\forall l>0.$$

Lemma 4

([20], when $\alpha =1$). If $f,g\in C\left({[\varrho ,c]}_{\mathbb{T}},{\mathbb{R}}^{+}\right)$ such that $f^{\gamma },$ $g^{\nu }$ are Δ-integrable on ${[\varrho ,c]}_{\mathbb{T}}$ and let $\beta >1$ and $1/\beta +1/\nu =1$, then

$$\begin{array}{c}{\int }_{\varrho }^{c}S\left(\frac{Y{f}^{\beta }\left(\zeta \right)}{X{g}^{\nu }\left(\zeta \right)}\right)f\left(\zeta \right)g\left(\zeta \right)\mathrm{\Delta }\zeta \hfill \\ \ge {\left({\int }_{\varrho }^c{f}^{\beta }\left(\zeta \right)\mathrm{\Delta }\zeta \right)}^{\frac{1}{\beta }}{\left({\int }_{\varrho }^c{g}^{\nu }\left(\zeta \right)\mathrm{\Delta }\zeta \right)}^{\frac{1}{\nu }},\hfill \end{array}$$

(16)

where $X={\int }_{\varrho }^c{f}^{\beta }\left(\zeta \right)\mathrm{\Delta }\zeta$ and $Y={\int }_{\varrho }^c{g}^{\nu }\left(\zeta \right)\mathrm{\Delta }\zeta$.

Lemma 5

(Jensen’s inequality). Let ${\zeta }_0,\zeta \in \mathbb{T}$ and $r_0,d\in \mathbb{R}$. If $\lambda \in {\mathrm{C}}_{rd}({[{\zeta }_0,\zeta ]}_{\mathbb{T}},\mathbb{R}),$φ:${[{\zeta }_0,\zeta ]}_{\mathbb{T}}\to (r_0,d)$ is rd-continuous and Ψ:$(r_0,d)\to \mathbb{R}$ is continuous and convex, then

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

(17)

Lemma 6.

Let $\varrho \in \mathbb{T},$ $\lambda ,\psi \ge 0$ be decreasing functions and $0<d,q\le 1,$ $\beta >1.$ Then,

$$\begin{array}{c}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\zeta \right){\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\hfill \\ =\max \left\{S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\varrho \right){\left({\int }_{\varrho }^{\sigma \left(\varrho \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\right.\hfill \\ \left.;S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(t\right){\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\right\},\hfill \end{array}$$

(18)

and

$$\begin{array}{c}S\left(\frac{\left(\sigma \left(\xi \right)-\varrho \right){\left[\psi \left(\eta \right){\left({\int }_{\varrho }^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y}\right)\hfill \\ =\max \left\{S\left(\frac{\left(\sigma \left(\xi \right)-\varrho \right){\left[\psi \left(\varrho \right){\left({\int }_{\varrho }^{\sigma \left(\varrho \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y}\right)\right.\hfill \\ \left.;S\left(\frac{\left(\sigma \left(\xi \right)-\varrho \right){\left[\psi \left(\xi \right){\left({\int }_{\varrho }^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y}\right)\right\}.\hfill \end{array}$$

(19)

Proof. 

We have for $\vartheta \le y$ that

$${\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \le {\int }_{\varrho }^{\sigma \left(y\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau ,$$

and then (where $0<d\le 1$),

$${\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\ge {\left({\int }_{\varrho }^{\sigma \left(y\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}.$$

Since $\lambda$ is decreasing, we have

$${\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\ge {\left[\lambda \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta },$$

thus the function ${\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }$ is decreasing. Therefore, we have for $\varrho \le \vartheta$ that

$${\left[\lambda \left(\varrho \right){\left({\int }_{\varrho }^{\sigma \left(\varrho \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\ge {\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }.$$

(20)

Integrating (20) over $\vartheta$ from $\varrho$ to $\sigma \left(t\right),$ we obtain

$$\begin{array}{ccc}& & \left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\varrho \right){\left({\int }_{\varrho }^{\sigma \left(\varrho \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\hfill \\ & \ge & {\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta ,\hfill \end{array}$$

and then,

$$\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\varrho \right){\left({\int }_{\varrho }^{\sigma \left(\varrho \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\ge 1.$$

(21)

Since the function ${\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }$ is decreasing, we obtain that

$${\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\ge {\left[\lambda \left(t\right){\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }.$$

Integrating the last inequality over $\vartheta$ from $\varrho$ to $\sigma \left(t\right),$ we have

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta \hfill \\ \ge {\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(t\right){\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta \hfill \\ =\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(t\right){\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta },\hfill \end{array}$$

and then,

$$\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(t\right){\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\le 1.$$

(22)

From (21) and (22), we observe

$$\begin{array}{c}\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\varrho \right){\left({\int }_{\varrho }^{\sigma \left(\varrho \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\ge ...\ge 1\hfill \\ \ge ...\ge \frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(t\right){\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }.\hfill \end{array}$$

Since $S(.)$ is decreasing on $\left(0,1\right)$ and increasing on $\left(1,\infty \right)$, we find that one of

$$S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\varrho \right){\left({\int }_{\varrho }^{\sigma \left(\varrho \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right),$$

and

$$S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(t\right){\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right),$$

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

$$\begin{array}{c}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\zeta \right){\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\hfill \\ =\max \left\{S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\varrho \right){\left({\int }_{\varrho }^{\sigma \left(\varrho \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\right.\hfill \\ \left.;S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(t\right){\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\right\},\hfill \end{array}$$

which is (18). Similarly, with respect to the decreasing function $\psi$ when $0<q\le 1,$ we have

$$\begin{array}{c}S\left(\frac{\left(\sigma \left(\xi \right)-\varrho \right){\left[\psi \left(\eta \right){\left({\int }_{\varrho }^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y}\right)\hfill \\ =\max \left\{S\left(\frac{\left(\sigma \left(\xi \right)-\varrho \right){\left[\psi \left(\varrho \right){\left({\int }_{\varrho }^{\sigma \left(\varrho \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y}\right)\right.\hfill \\ \left.;S\left(\frac{\left(\sigma \left(\xi \right)-\varrho \right){\left[\psi \left(\xi \right){\left({\int }_{\varrho }^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y}\right)\right\},\hfill \end{array}$$

which is (19). □

3. Main Results

Theorem 1.

Let $\varrho \in \mathbb{T},$ $0\le d,q\le 1$ and $\lambda ,\psi$ be nonnegative and decreasing functions. If $\beta >1,$ $\nu >1$ with $1/\beta +1/\nu =1$, then

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(r\right)}\frac{S_{d,q,t,\xi ,r,s,\beta }{\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^d{\left({\int }_{\varrho }^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^q}{{\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}{\left(\sigma \left(\xi \right)-\varrho \right)}^{\frac{1}{\nu }}}\mathrm{\Delta }t\mathrm{\Delta }\xi \hfill \\ \ge \nu C\left(d,q,r,s\right){\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left[\lambda \left(t\right){\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\left(\sigma \left(r\right)-t\right)\mathrm{\Delta }t\right)}^{\frac{1}{\beta }}\hfill \\ \times {\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left[\psi \left(\xi \right){\left({\int }_{\varrho }^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\left(\sigma \left(s\right)-\xi \right)\mathrm{\Delta }\xi \right)}^{\frac{1}{\beta }},\hfill \end{array}$$

(23)

where

$$C\left(d,q,r,s,\nu \right)=\frac{1}{\nu }dq{\left(\sigma \left(r\right)-\varrho \right)}^{\frac{1}{\nu }}{\left(\sigma \left(s\right)-\varrho \right)}^{\frac{1}{\nu }},$$

and

$$\begin{array}{cc}\hfill S_{d,q,t,\xi ,r,s,\beta }& =S\left(\frac{\left(\sigma \left(r\right)-\varrho \right){\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\zeta \right){\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(r\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta }\right)\hfill \\ \hfill & \times S\left(\frac{\left(\sigma \left(s\right)-\varrho \right){\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(\eta \right){\left({\int }_{\varrho }^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(s\right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y}\right)\hfill \\ \hfill & \times S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\zeta \right){\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\hfill \\ \hfill & \times S\left(\frac{\left(\sigma \left(\xi \right)-\varrho \right){\left[\psi \left(\eta \right){\left({\int }_{\varrho }^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y}\right),\hfill \end{array}$$

such that

$$\begin{array}{c}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\zeta \right){\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\hfill \\ =\max \left\{S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\varrho \right){\left({\int }_{\varrho }^{\sigma \left(\varrho \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\right.\hfill \\ \left.;S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(t\right){\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\right\},\hfill \end{array}$$

and

$$\begin{array}{c}S\left(\frac{\left(\sigma \left(\xi \right)-\varrho \right){\left[\psi \left(\eta \right){\left({\int }_{\varrho }^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y}\right)\hfill \\ =\max \left\{S\left(\frac{\left(\sigma \left(\xi \right)-\varrho \right){\left[\psi \left(\varrho \right){\left({\int }_{\varrho }^{\sigma \left(\varrho \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y}\right)\right.\hfill \\ \left.;S\left(\frac{\left(\sigma \left(\xi \right)-\varrho \right){\left[\psi \left(\xi \right){\left({\int }_{\varrho }^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y}\right)\right\}.\hfill \end{array}$$

Proof. 

Applying (11) with $\gamma =d,$ we obtain

$${\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^d\ge d{\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\mathrm{\Delta }\vartheta .$$

(24)

Multiplying the last inequality by

$$S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\zeta \right){\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right),$$

we obtain

$$\begin{array}{c}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\zeta \right){\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right){\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^d\hfill \\ \ge d{\int }_{\varrho }^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\zeta \right){\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\hfill \\ \times \lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\mathrm{\Delta }\vartheta .\hfill \end{array}$$

From Lemma 6, the last inequality becomes

$$\begin{array}{c}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\zeta \right){\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right){\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^d\hfill \\ \ge d{\int }_{\varrho }^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\hfill \\ \times \lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\mathrm{\Delta }\vartheta .\hfill \end{array}$$

(25)

Similarly, we have for $\psi$ and $0<q\le 1$ that

$$\begin{array}{cc}\hfill & S\left(\frac{\left(\sigma \left(\xi \right)-\varrho \right){\left[\psi \left(\eta \right){\left({\int }_{\varrho }^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y}\right){\left({\int }_{\varrho }^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^q\hfill \\ \hfill & \ge q{\int }_{\varrho }^{\sigma \left(\xi \right)}S\left(\frac{\left(\sigma \left(\xi \right)-\varrho \right){\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y}\right)\hfill \\ \hfill & \times \psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\mathrm{\Delta }y.\hfill \end{array}$$

(26)

From (25) and (26), we see that

$$\begin{array}{c}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\zeta \right){\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\hfill \\ \times S\left(\frac{\left(\sigma \left(\xi \right)-\varrho \right){\left[\psi \left(\eta \right){\left({\int }_{\varrho }^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y}\right)\hfill \\ \times {\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^d{\left({\int }_{\varrho }^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^q\hfill \\ \ge dq{\int }_{\varrho }^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\hfill \\ \times \lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\times 1\mathrm{\Delta }\vartheta \hfill \\ \times {\int }_{\varrho }^{\sigma \left(\xi \right)}S\left(\frac{\left(\sigma \left(\xi \right)-\varrho \right){\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y}\right)\hfill \\ \times \psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\times 1\mathrm{\Delta }y.\hfill \end{array}$$

(27)

Applying (16) on the right hand side of (27), we have

$$\begin{array}{c}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[\lambda \left(\zeta \right){\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\hfill \\ \times S\left(\frac{\left(\sigma \left(\xi \right)-\varrho \right){\left[\psi \left(\eta \right){\left({\int }_{\varrho }^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y}\right)\hfill \\ \times {\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^d{\left({\int }_{\varrho }^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^q\hfill \\ \ge dq{\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}{\left({\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\hfill \\ \times {\left(\sigma \left(\xi \right)-\varrho \right)}^{\frac{1}{\nu }}{\left({\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y\right)}^{\frac{1}{\beta }}.\hfill \end{array}$$

(28)

Multiplying (28) by

$$\begin{array}{c}S\left(\frac{\left(\sigma \left(r\right)-\varrho \right){\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\zeta \right){\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(r\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta }\right)\hfill \\ \times S\left(\frac{\left(\sigma \left(s\right)-\varrho \right){\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(\eta \right){\left({\int }_{\varrho }^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(s\right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y}\right),\hfill \end{array}$$

we obtain

$$\begin{array}{c}{S}_{d,q,t,\xi ,r,s,\beta }{\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^d{\left({\int }_{\varrho }^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^q\hfill \\ \ge dq{\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}S\left(\frac{\left(\sigma \left(r\right)-\varrho \right){\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\zeta \right){\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(r\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta }\right)\hfill \\ \times {\left({\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\hfill \\ \times S\left(\frac{\left(\sigma \left(s\right)-\varrho \right){\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(\eta \right){\left({\int }_{\varrho }^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(s\right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y}\right)\hfill \\ \times {\left(\sigma \left(\xi \right)-\varrho \right)}^{\frac{1}{\nu }}{\left({\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y\right)}^{\frac{1}{\beta }}.\hfill \end{array}$$

(29)

Dividing the two sides of (29) by ${\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}{\left(\sigma \left(\xi \right)-\varrho \right)}^{\frac{1}{\nu }}$ and then taking the integration over t from $\varrho$ to $\sigma \left(r\right)$ and the integration over $\xi$ from $\varrho$ to $\sigma \left(s\right),$ we have

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(r\right)}\frac{S_{d,q,t,\xi ,r,s,\beta }{\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^d{\left({\int }_{\varrho }^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^q}{{\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}{\left(\sigma \left(\xi \right)-\varrho \right)}^{\frac{1}{\nu }}}\mathrm{\Delta }t\mathrm{\Delta }\xi \hfill \\ \ge dq{\int }_{\varrho }^{\sigma \left(r\right)}S\left(\frac{\left(\sigma \left(r\right)-\varrho \right){\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\zeta \right){\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(r\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta }\right)\hfill \\ \times {\left({\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\mathrm{\Delta }t\hfill \\ \times {\int }_{\varrho }^{\sigma \left(s\right)}S\left(\frac{\left(\sigma \left(s\right)-\varrho \right){\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(\eta \right){\left({\int }_{\varrho }^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(s\right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y}\right)\hfill \\ \times {\left({\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y\right)}^{\frac{1}{\beta }}\mathrm{\Delta }\xi .\hfill \end{array}$$

(30)

Applying (9) on the term

$${\int }_{\varrho }^{\sigma \left(r\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta ,$$

with $u\left(\vartheta \right)=\left(\sigma \left(r\right)-\vartheta \right)$ and $v^{\mathrm{\Delta }}\left(\vartheta \right)={\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta },$ we obtain

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(r\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \hfill \\ ={\left.\left(\sigma \left(r\right)-\vartheta \right)v\left(\vartheta \right)\right|}_{\varrho }^{\sigma \left(r\right)}+{\int }_{\varrho }^{\sigma \left(r\right)}{v}^{\sigma }\left(\vartheta \right)\mathrm{\Delta }\vartheta ,\hfill \end{array}$$

where $v\left(\vartheta \right)={\int }_{\varrho }^{\vartheta }{\left[\lambda \left(\theta \right){\left({\int }_{\varrho }^{\sigma \left(\theta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\theta ,$ and then (where $v\left(\varrho \right)=0$),

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(r\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \hfill \\ ={\int }_{\varrho }^{\sigma \left(r\right)}{\int }_{\varrho }^{\sigma \left(\vartheta \right)}{\left[\lambda \left(\theta \right){\left({\int }_{\varrho }^{\sigma \left(\theta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\theta \mathrm{\Delta }\vartheta .\hfill \end{array}$$

(31)

Similarly, we see that

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(s\right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y\hfill \\ ={\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(y\right)}{\left[\psi \left(\theta \right){\left({\int }_{\varrho }^{\sigma \left(\theta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }\theta \mathrm{\Delta }y.\hfill \end{array}$$

(32)

Substituting (31) and (32) into (30) and, then, by applying (16), we observe that

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(r\right)}\frac{S_{d,q,t,\xi ,r,s,\beta }{\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^d{\left({\int }_{\varrho }^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^q}{{\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}{\left(\sigma \left(\xi \right)-\varrho \right)}^{\frac{1}{\nu }}}\mathrm{\Delta }t\mathrm{\Delta }\xi \hfill \\ \ge dq{\int }_{\varrho }^{\sigma \left(r\right)}S\left(\frac{\left(\sigma \left(r\right)-\varrho \right){\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\zeta \right){\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\zeta }{{\int }_{\varrho }^{\sigma \left(r\right)}{\int }_{\varrho }^{\sigma \left(\vartheta \right)}{\left[\lambda \left(\theta \right){\left({\int }_{\varrho }^{\sigma \left(\theta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\theta \mathrm{\Delta }\vartheta }\right)\hfill \\ \times {\left({\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\times 1\mathrm{\Delta }t\hfill \\ \times {\int }_{\varrho }^{\sigma \left(s\right)}S\left(\frac{\left(\sigma \left(s\right)-\varrho \right){\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(\eta \right){\left({\int }_{\varrho }^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }\eta }{{\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(y\right)}{\left[\psi \left(\theta \right){\left({\int }_{\varrho }^{\sigma \left(\theta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }\theta \mathrm{\Delta }y}\right)\hfill \\ \times {\left({\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y\right)}^{\frac{1}{\beta }}\times 1\mathrm{\Delta }\xi \hfill \\ \ge dq{\left(\sigma \left(r\right)-\varrho \right)}^{\frac{1}{\nu }}{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\int }_{\varrho }^{\sigma \left(t\right)}{\left[\lambda \left(\vartheta \right){\left({\int }_{\varrho }^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\mathrm{\Delta }\vartheta \mathrm{\Delta }t\right)}^{\frac{1}{\beta }}\hfill \\ \times {\left(\sigma \left(s\right)-\varrho \right)}^{\frac{1}{\nu }}{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(\xi \right)}{\left[\psi \left(y\right){\left({\int }_{\varrho }^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\mathrm{\Delta }y\mathrm{\Delta }\xi \right)}^{\frac{1}{\beta }}.\hfill \end{array}$$

(33)

From (31)–(33), the last inequality becomes

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(r\right)}\frac{S_{d,q,t,\xi ,r,s,\beta }{\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^d{\left({\int }_{\varrho }^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^q}{{\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}{\left(\sigma \left(\xi \right)-\varrho \right)}^{\frac{1}{\nu }}}\mathrm{\Delta }t\mathrm{\Delta }\xi \hfill \\ \ge dq{\left(\sigma \left(r\right)-\varrho \right)}^{\frac{1}{\nu }}{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left[\lambda \left(t\right){\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\left(\sigma \left(r\right)-t\right)\mathrm{\Delta }t\right)}^{\frac{1}{\beta }}\hfill \\ \times {\left(\sigma \left(s\right)-\varrho \right)}^{\frac{1}{\nu }}{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left[\psi \left(\xi \right){\left({\int }_{\varrho }^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\left(\sigma \left(s\right)-\xi \right)\mathrm{\Delta }\xi \right)}^{\frac{1}{\beta }}\hfill \\ =\nu C\left(d,q,r,s\right){\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left[\lambda \left(t\right){\left({\int }_{\varrho }^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{d-1}\right]}^{\beta }\left(\sigma \left(r\right)-t\right)\mathrm{\Delta }t\right)}^{\frac{1}{\beta }}\hfill \\ \times {\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left[\psi \left(\xi \right){\left({\int }_{\varrho }^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{q-1}\right]}^{\beta }\left(\sigma \left(s\right)-\xi \right)\mathrm{\Delta }\xi \right)}^{\frac{1}{\beta }},\hfill \end{array}$$

which is (23). □

Remark 1.

If $\nu =\beta =2,$ we obtain (8) proved by AlNemer et al. [16].

Remark 2.

When $\mathbb{T}=\mathbb{N},$ $\varrho =1$ and $\nu =\beta =2,$ in Theorem 1, we obtain (5) as demonstrated in [6].

Remark 3.

As a special case of Theorem 1 (when $\mathbb{T}=\mathbb{R}$), we have that if $0\le d,q\le 1$ and $\lambda ,\psi$ are nonnegative and decreasing functions and assume that $\beta >1,$ $\nu >1$ with $1/\beta +1/\nu =1$, then

$$\begin{array}{c}{\int }_0^s{\int }_0^r\frac{S_{d,q,t,\xi ,r,s,\beta }{\left({\int }_0^t\lambda \left(\tau \right)d\tau \right)}^d{\left({\int }_0^{\xi }\psi \left(\tau \right)d\tau \right)}^q}{t^{\frac{1}{\nu }}{\xi }^{\frac{1}{\nu }}}dtd\xi \hfill \\ \ge \nu C\left(d,q,r,s\right){\left({\int }_0^r{\left[\lambda \left(t\right){\left({\int }_0^t\lambda \left(\tau \right)d\tau \right)}^{d-1}\right]}^{\beta }\left(r-t\right)dt\right)}^{\frac{1}{\beta }}\hfill \\ \times {\left({\int }_0^s{\left[\psi \left(\xi \right){\left({\int }_0^{\xi }\psi \left(\tau \right)d\tau \right)}^{q-1}\right]}^{\beta }\left(s-\xi \right)d\xi \right)}^{\frac{1}{\beta }},\hfill \end{array}$$

where

$$C\left(d,q,r,s,\nu \right)=\frac{1}{\nu }dq{r}^{\frac{1}{\nu }}{s}^{\frac{1}{\nu }},$$

and

$$\begin{array}{cc}\hfill S_{d,q,t,\xi ,r,s,\beta }& =S\left(\frac{r{\int }_0^t{\left[\lambda \left(\zeta \right){\left({\int }_0^{\zeta }\lambda \left(\tau \right)d\tau \right)}^{d-1}\right]}^{\beta }d\zeta }{{\int }_0^r{\left[\lambda \left(\vartheta \right){\left({\int }_0^{\vartheta }\lambda \left(\tau \right)d\tau \right)}^{d-1}\right]}^{\beta }\left(r-\vartheta \right)d\vartheta }\right)\hfill \\ \hfill & \times S\left(\frac{s{\int }_0^{\xi }{\left[\psi \left(\eta \right){\left({\int }_0^{\eta }\psi \left(\tau \right)d\tau \right)}^{q-1}\right]}^{\beta }d\eta }{{\int }_0^s{\left[\psi \left(y\right){\left({\int }_0^y\psi \left(\tau \right)d\tau \right)}^{q-1}\right]}^{\beta }\left(s-y\right)dy}\right)\hfill \\ \hfill & \times S\left(\frac{t{\left[\lambda \left(\zeta \right){\left({\int }_0^{\zeta }\lambda \left(\tau \right)d\tau \right)}^{d-1}\right]}^{\beta }}{{\int }_0^t{\left[\lambda \left(\vartheta \right){\left({\int }_0^{\vartheta }\lambda \left(\tau \right)d\tau \right)}^{d-1}\right]}^{\beta }d\vartheta }\right)\hfill \\ \hfill & \times S\left(\frac{\xi {\left[\psi \left(\eta \right){\left({\int }_0^{\eta }\psi \left(\tau \right)d\tau \right)}^{q-1}\right]}^{\beta }}{{\int }_0^{\xi }{\left[\psi \left(y\right){\left({\int }_0^y\psi \left(\tau \right)d\tau \right)}^{q-1}\right]}^{\beta }dy}\right),\hfill \end{array}$$

Theorem 2.

Let $\varrho \in \mathbb{T},$ $\lambda ,\varpi$ be nonnegative and $d,q$ be positive functions. If $\varphi ,\psi \ge 0$ are concave and supermultiplicative functions and $\beta >1,$ $\nu >1$ with $1/\beta +1/\nu =1,$ then

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(r\right)}\frac{S_{t,r,s,\zeta ,\nu ,\beta }\varphi \left(\mathrm{\Lambda }\left(t\right)\right)\psi \left(\mathrm{\Omega }\left(\zeta \right)\right)}{{\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}{\left(\sigma \left(\zeta \right)-\varrho \right)}^{\frac{1}{\nu }}}\mathrm{\Delta }t\mathrm{\Delta }\zeta \hfill \\ \ge \nu M\left(r,s,\nu \right){\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\hfill \\ \times {\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y\right)}^{\frac{1}{\beta }},\hfill \end{array}$$

(34)

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

$$M\left(r,s,\nu \right)=\frac{1}{\nu }{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left(\frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}\right)}^{\nu }\mathrm{\Delta }t\right)}^{\frac{1}{\nu }}{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left(\frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}\right)}^{\nu }\mathrm{\Delta }\zeta \right)}^{\frac{1}{\nu }},$$

$$\begin{array}{cc}\hfill S_{t,r,s,\zeta ,\nu ,\beta }& =S\left(\frac{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \right){\left(\frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}\right)}^{\nu }}{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left(\frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}\right)}^{\nu }\mathrm{\Delta }t\right)\left({\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta \right)}\right)\hfill \\ \hfill & \times S\left(\frac{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y\right){\left(\frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}\right)}^{\nu }}{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left(\frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}\right)}^{\nu }\mathrm{\Delta }\zeta \right)\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\mathrm{\Delta }y\right)}\right),\hfill \end{array}$$

$$\mathrm{\Lambda }\left(t\right)={\int }_{\varrho }^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)\lambda \left(\vartheta \right)\mathrm{\Delta }\vartheta ,$$

$$\mathrm{\Omega }\left(\zeta \right)={\int }_{\varrho }^{\sigma \left(\zeta \right)}S\left(\frac{\left(\sigma \left(\zeta \right)-\varrho \right){\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\zeta \right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\mathrm{\Delta }y}\right)\varpi \left(y\right)\mathrm{\Delta }y,$$

$$D\left(t\right)={\int }_{\varrho }^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)d\left(\vartheta \right)\mathrm{\Delta }\vartheta ,$$

and

$$Q\left(\zeta \right)={\int }_{\varrho }^{\sigma \left(\zeta \right)}S\left(\frac{\left(\sigma \left(\zeta \right)-\varrho \right){\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(\zeta \right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\mathrm{\Delta }y}\right)q\left(y\right)\mathrm{\Delta }y.$$

Proof. 

Using the fact that $\varphi$ is a supermultiplicative function, applying Jensen’s inequality and then applying (16), we find

$$\begin{array}{cc}\hfill \varphi \left(\mathrm{\Lambda }\right(t\left)\right)& =\varphi \left(\frac{D\left(t\right){\int }_{\varrho }^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)d\left(\vartheta \right)\lambda \left(\vartheta \right)/d\left(\vartheta \right)\mathrm{\Delta }\vartheta }{{\int }_{\varrho }^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)d\left(\vartheta \right)\mathrm{\Delta }\vartheta }\right)\hfill \\ \hfill & \ge \varphi \left(D\left(t\right)\right)\varphi \left(\frac{{\int }_{\varrho }^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)d\left(\vartheta \right)\left[\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right]\mathrm{\Delta }\vartheta }{{\int }_{\varrho }^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)d\left(\vartheta \right)\mathrm{\Delta }\vartheta }\right)\hfill \\ \hfill & \ge \frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}{\int }_{\varrho }^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }}{{\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta }\right)d\left(\vartheta \right)\varphi \left[\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right]\times 1\mathrm{\Delta }\vartheta \hfill \\ \hfill & \ge \frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}{\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}{\left({\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}.\hfill \end{array}$$

(35)

Similarly, we can obtain

$$\psi \left(\mathrm{\Omega }\left(\zeta \right)\right)\ge \frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}{\left(\sigma \left(\zeta \right)-\varrho \right)}^{\frac{1}{\nu }}{\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\mathrm{\Delta }y\right)}^{\frac{1}{\beta }}.$$

(36)

Multiplying both sides of (35) and (36), respectively, by

$$S\left(\frac{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \right){\left(\frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}\right)}^{\nu }}{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left(\frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}\right)}^{\nu }\mathrm{\Delta }t\right)\left({\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta \right)}\right),$$

and

$$S\left(\frac{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y\right){\left(\frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}\right)}^{\nu }}{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left(\frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}\right)}^{\nu }\mathrm{\Delta }\zeta \right)\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\mathrm{\Delta }y\right)}\right),$$

and then multiplying these inequalities, we obtain

$$\begin{array}{c}S\left(\frac{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \right){\left(\frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}\right)}^{\nu }}{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left(\frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}\right)}^{\nu }\mathrm{\Delta }t\right)\left({\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta \right)}\right)\varphi \left(\mathrm{\Lambda }\left(t\right)\right)\hfill \\ \times S\left(\frac{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y\right){\left(\frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}\right)}^{\nu }}{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left(\frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}\right)}^{\nu }\mathrm{\Delta }\zeta \right)\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\mathrm{\Delta }y\right)}\right)\psi \left(\mathrm{\Omega }\left(\zeta \right)\right)\hfill \\ \ge \frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}{\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}{\left({\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\hfill \\ \times S\left(\frac{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \right){\left(\frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}\right)}^{\nu }}{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left(\frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}\right)}^{\nu }\mathrm{\Delta }t\right)\left({\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta \right)}\right)\hfill \\ \times \frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}{\left(\sigma \left(\zeta \right)-\varrho \right)}^{\frac{1}{\nu }}{\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\mathrm{\Delta }y\right)}^{\frac{1}{\beta }}\hfill \\ \times S\left(\frac{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y\right){\left(\frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}\right)}^{\nu }}{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left(\frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}\right)}^{\nu }\mathrm{\Delta }\zeta \right)\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\mathrm{\Delta }y\right)}\right).\hfill \end{array}$$

(37)

By dividing the two sides of (37) on ${\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}{\left(\sigma \left(\zeta \right)-\varrho \right)}^{\frac{1}{\nu }}$ and then taking the integration over $\zeta$ from $\varrho$ to $\sigma \left(s\right)$ and, then, the integration over t from $\varrho$ to $\sigma \left(r\right)$, we obtain

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(r\right)}\frac{S_{t,r,s,\zeta ,\nu ,\beta }\varphi \left(\mathrm{\Lambda }\left(t\right)\right)\psi \left(\mathrm{\Omega }\left(\zeta \right)\right)}{{\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}{\left(\sigma \left(\zeta \right)-\varrho \right)}^{\frac{1}{\nu }}}\mathrm{\Delta }t\mathrm{\Delta }\zeta \hfill \\ \ge {\int }_{\varrho }^{\sigma \left(r\right)}S\left(\frac{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \right){\left(\frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}\right)}^{\nu }}{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left(\frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}\right)}^{\nu }\mathrm{\Delta }t\right)\left({\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta \right)}\right)\hfill \\ \times \frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}{\left({\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\mathrm{\Delta }t\hfill \\ \times {\int }_{\varrho }^{\sigma \left(s\right)}S\left(\frac{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y\right){\left(\frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}\right)}^{\nu }}{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left(\frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}\right)}^{\nu }\mathrm{\Delta }\zeta \right)\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\mathrm{\Delta }y\right)}\right)\hfill \\ \times \frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}{\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\mathrm{\Delta }y\right)}^{\frac{1}{\beta }}\mathrm{\Delta }\zeta .\hfill \end{array}$$

(38)

By using the integration by parts, we can see that

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(r\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \hfill \\ ={\int }_{\varrho }^{\sigma \left(r\right)}{\int }_{\varrho }^{\sigma \left(\vartheta \right)}{\left[d\left(\theta \right)\varphi \left(\frac{\lambda \left(\theta \right)}{d\left(\theta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\theta \mathrm{\Delta }\vartheta .\hfill \end{array}$$

(39)

In addition, we can obtain that

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(s\right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y\hfill \\ ={\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(y\right)}{\left[q\left(\theta \right)\psi \left(\frac{\varpi \left(\theta \right)}{q\left(\theta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\theta \mathrm{\Delta }y.\hfill \end{array}$$

(40)

Substituting (39) and (40) into (38), we have

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(r\right)}\frac{S_{t,r,s,\zeta ,\nu ,\beta }\varphi \left(\mathrm{\Lambda }\left(t\right)\right)\psi \left(\mathrm{\Omega }\left(\zeta \right)\right)}{{\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}{\left(\sigma \left(\zeta \right)-\varrho \right)}^{\frac{1}{\nu }}}\mathrm{\Delta }t\mathrm{\Delta }\zeta \hfill \\ \ge {\int }_{\varrho }^{\sigma \left(r\right)}S\left(\frac{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\int }_{\varrho }^{\sigma \left(\vartheta \right)}{\left[d\left(\theta \right)\varphi \left(\frac{\lambda \left(\theta \right)}{d\left(\theta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\theta \mathrm{\Delta }\vartheta \right){\left(\frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}\right)}^{\nu }}{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left(\frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}\right)}^{\nu }\mathrm{\Delta }t\right)\left({\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta \right)}\right)\hfill \\ \times \frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}{\left({\int }_{\varrho }^{\sigma \left(t\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\mathrm{\Delta }t\hfill \\ \times {\int }_{\varrho }^{\sigma \left(s\right)}S\left(\frac{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(y\right)}{\left[q\left(\theta \right)\psi \left(\frac{\varpi \left(\theta \right)}{q\left(\theta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\theta \mathrm{\Delta }y\right){\left(\frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}\right)}^{\nu }}{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left(\frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}\right)}^{\nu }\mathrm{\Delta }\zeta \right)\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\mathrm{\Delta }y\right)}\right)\hfill \\ \times \frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}{\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\mathrm{\Delta }y\right)}^{\frac{1}{\beta }}\mathrm{\Delta }\zeta .\hfill \end{array}$$

(41)

Applying (16) with $\gamma =\nu =\nu$ on the R.H.S. of (41), we have

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(r\right)}\frac{S_{t,r,s,\zeta ,\nu ,\beta }\varphi \left(\mathrm{\Lambda }\left(t\right)\right)\psi \left(\mathrm{\Omega }\left(\zeta \right)\right)}{{\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}{\left(\sigma \left(\zeta \right)-\varrho \right)}^{\frac{1}{\nu }}}\mathrm{\Delta }t\mathrm{\Delta }\zeta \hfill \\ \ge {\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left(\frac{\varphi \left(D\right(t\left)\right)}{D\left(t\right)}\right)}^{\nu }\mathrm{\Delta }t\right)}^{\frac{1}{\nu }}{\left({\int }_{\varrho }^{\sigma \left(r\right)}{\int }_{\varrho }^{\sigma \left(\vartheta \right)}{\left[d\left(\theta \right)\varphi \left(\frac{\lambda \left(\theta \right)}{d\left(\theta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\theta \mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\hfill \\ \times {\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left(\frac{\psi \left(Q\right(\zeta \left)\right)}{Q\left(\zeta \right)}\right)}^{\nu }\mathrm{\Delta }\zeta \right)}^{\frac{1}{\nu }}{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(y\right)}{\left[q\left(\theta \right)\psi \left(\frac{\varpi \left(\theta \right)}{q\left(\theta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\theta \mathrm{\Delta }y\right)}^{\frac{1}{\beta }}\hfill \\ =\nu M\left(r,s,\nu \right){\left({\int }_{\varrho }^{\sigma \left(r\right)}{\int }_{\varrho }^{\sigma \left(\vartheta \right)}{\left[d\left(\theta \right)\varphi \left(\frac{\lambda \left(\theta \right)}{d\left(\theta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\theta \mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\hfill \\ \times {\left({\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(y\right)}{\left[q\left(\theta \right)\psi \left(\frac{\varpi \left(\theta \right)}{q\left(\theta \right)}\right)\right]}^{\beta }\mathrm{\Delta }\theta \mathrm{\Delta }y\right)}^{\frac{1}{\beta }}.\hfill \end{array}$$

(42)

From (39) and (40), the Inequality (42) becomes

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(r\right)}\frac{S_{t,r,s,\zeta ,\nu ,\beta }\varphi \left(\mathrm{\Lambda }\left(t\right)\right)\psi \left(\mathrm{\Omega }\left(\zeta \right)\right)}{{\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}{\left(\sigma \left(\zeta \right)-\varrho \right)}^{\frac{1}{\nu }}}\mathrm{\Delta }t\mathrm{\Delta }\zeta \hfill \\ \ge \nu M\left(r,s,\nu \right){\left({\int }_{\varrho }^{\sigma \left(r\right)}{\left[d\left(\vartheta \right)\varphi \left(\frac{\lambda \left(\vartheta \right)}{d\left(\vartheta \right)}\right)\right]}^{\beta }\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\hfill \\ \times {\left({\int }_{\varrho }^{\sigma \left(s\right)}{\left[q\left(y\right)\psi \left(\frac{\varpi \left(y\right)}{q\left(y\right)}\right)\right]}^{\beta }\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y\right)}^{\frac{1}{\beta }},\hfill \end{array}$$

which is (34). □

Remark 4.

If $\mathbb{T}=\mathbb{N},$ $\varrho =1$ and $\nu =\beta =2,$ in Theorem 2, then we obtain (6) as demonstrated in [6].

By putting $\varphi \left(\vartheta \right)=\vartheta$ and $\psi \left(y\right)=y$ in Theorem 2, we have the following theorem.

Theorem 3.

Assume that $\varrho \in \mathbb{T}$ and $\lambda ,\varpi$ are nonnegative functions and $\beta >1,$ $\nu >1$ with $1/\beta +1/\nu =1.$ Then, for all $r,s\in {[\varrho ,\infty ]}_{\mathbb{T}},$ we have

$$\begin{array}{c}{\int }_{\varrho }^{\sigma \left(s\right)}{\int }_{\varrho }^{\sigma \left(r\right)}\frac{S_{t,r,s,\zeta ,\nu ,\beta }\mathrm{\Lambda }\left(t\right)\mathrm{\Omega }\left(\zeta \right)}{{\left(\sigma \left(t\right)-\varrho \right)}^{\frac{1}{\nu }}{\left(\sigma \left(\zeta \right)-\varrho \right)}^{\frac{1}{\nu }}}\mathrm{\Delta }t\mathrm{\Delta }\zeta \hfill \\ \ge \nu M\left(r,s,\nu \right){\left({\int }_{\varrho }^{\sigma \left(r\right)}{\lambda }^{\beta }\left(\vartheta \right)\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\hfill \\ \times {\left({\int }_{\varrho }^{\sigma \left(s\right)}{\varpi }^{\beta }\left(y\right)\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y\right)}^{\frac{1}{\beta }},\hfill \end{array}$$

where

$$M\left(r,s,\nu \right)=\frac{1}{\nu }{\left(\sigma \left(r\right)-\varrho \right)}^{\frac{1}{\nu }}{\left(\sigma \left(s\right)-\varrho \right)}^{\frac{1}{\nu }},$$

$$S_{t,r,s,\zeta ,\nu ,\beta }=S\left(\frac{{\int }_{\varrho }^{\sigma \left(r\right)}{\lambda }^{\beta }\left(\vartheta \right)\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta }{\left(\sigma \left(r\right)-\varrho \right)\left({\int }_{\varrho }^{\sigma \left(t\right)}{\lambda }^{\beta }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right)}\right)S\left(\frac{\left({\int }_{\varrho }^{\sigma \left(s\right)}{\varpi }^{\beta }\left(y\right)\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y\right)}{\left(\sigma \left(s\right)-\varrho \right)\left({\int }_{\varrho }^{\sigma \left(\zeta \right)}{\varpi }^{\beta }\left(y\right)\mathrm{\Delta }y\right)}\right),$$

$$\mathrm{\Lambda }\left(t\right)={\int }_{\varrho }^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\lambda }^{\beta }\left(\vartheta \right)}{{\int }_{\varrho }^{\sigma \left(t\right)}{\lambda }^{\beta }\left(\vartheta \right)\mathrm{\Delta }\vartheta }\right)\lambda \left(\vartheta \right)\mathrm{\Delta }\vartheta ,$$

$$\mathrm{\Omega }\left(\zeta \right)={\int }_{\varrho }^{\sigma \left(\zeta \right)}S\left(\frac{\left(\sigma \left(\zeta \right)-\varrho \right){\varpi }^{\beta }\left(y\right)}{{\int }_{\varrho }^{\sigma \left(\zeta \right)}{\varpi }^{\beta }\left(y\right)\mathrm{\Delta }y}\right)\varpi \left(y\right)\mathrm{\Delta }y,$$

$$D\left(t\right)={\int }_{\varrho }^{\sigma \left(t\right)}S\left(\frac{\left(\sigma \left(t\right)-\varrho \right){\lambda }^{\beta }\left(\vartheta \right)}{{\int }_{\varrho }^{\sigma \left(t\right)}{\lambda }^{\beta }\left(\vartheta \right)\mathrm{\Delta }\vartheta }\right)d\left(\vartheta \right)\mathrm{\Delta }\vartheta ,$$

and

$$Q\left(\zeta \right)={\int }_{\varrho }^{\sigma \left(\zeta \right)}S\left(\frac{\left(\sigma \left(\zeta \right)-\varrho \right){\varpi }^{\beta }\left(y\right)}{{\int }_{\varrho }^{\sigma \left(\zeta \right)}{\varpi }^{\beta }\left(y\right)\mathrm{\Delta }y}\right)q\left(y\right)\mathrm{\Delta }y.$$

Remark 5.

As a special case of Theorem 3, when $\mathbb{T}=\mathbb{N}$, $\varrho =1$ and $\nu =\beta =2,$ we obtain (7) as was proved by Zhao and Cheung [6].

4. Conclusions

In this paper, we establish some new generalizations of reverse Hilbert-type inequalities by applying reverse Hölder inequalities with the Specht ratio function on time scales. We generalize a number of those inequalities to a general time-scale measure space. In addition to this, in order to obtain some new inequalities as special cases, we also extend our inequalities to a discrete and continuous calculus. In future work, we will continue to generalize more fractional dynamic inequalities by using Specht’s ratio, Kantorovich’s ratio and n-tuple fractional integral. In particular, such inequalities can be introduced by using fractional integrals and fractional derivatives of the Riemann–Liouville-type on time scales. It will also be very interesting to introduce such inequalities in quantum calculations.