Some New Generalizations of Reverse Hilbert-Type Inequalities via Supermultiplicative Functions

Abstract

Our work in this paper is based on the reverse Hölder-type dynamic inequalities illustrated by El-Deeb in 2018 and the reverse Hilbert-type dynamic inequalities illustrated by Rezk in 2021 and 2022. With the help of Specht’s ratio, the concept of supermultiplicative functions, chain rule, and Jensen’s inequality on time scales, we can establish some comprehensive and generalize a number of classical reverse Hilbert-type inequalities to a general time scale space. In time scale calculus, results are unified and extended. At the same time, the theory of time scale calculus is applied to unify discrete and continuous analysis and to combine them in one comprehensive form. This hybrid theory is also widely applied on symmetrical properties which play an essential role in determining the correct methods to solve inequalities. As a special case of our results when the supermultiplicative function represents the identity map, we obtain some results that have been recently published.

1. Introduction

In [1] (p. 253), Hardy established that

$$\sum _{l=1}^{\infty }\sum _{r=1}^{\infty }\frac{z_r{ϝ}_l}{r+l}\le \frac{\pi }{sin\frac{\pi }{p}}{\left(\sum _{r=1}^{\infty }{z}_r^p\right)}^{{}^{\frac{1}{p}}}{\left(\sum _{l=1}^{\infty }{ϝ}_l^{\delta }\right)}^{{}^{\frac{1}{\delta }}},$$

(1)

where $z_r,$ ${ϝ}_l\ge 0$ with $0<{\sum }_{r=1}^{\infty }{z}_r^p<\infty ,$ $0<{\sum }_{l=1}^{\infty }{ϝ}_l^{\delta }<\infty$ and $p>1,$ $p^{-1}+{\delta }^{-1}=1$. The continuous shape (see [2]) of (1) is called Hardy–Hilbert’s inequality and given by

$${\int }_0^{\infty }{\int }_0^{\infty }\frac{Z\left(\vartheta \right)ϝ\left(y\right)}{\vartheta +y}d\vartheta dy\le \frac{\pi }{sin\frac{\pi }{p}}{\left({\int }_0^{\infty }{Z}^p\left(\vartheta \right)d\vartheta \right)}^{{}^{\frac{1}{p}}}{\left({\int }_0^{\infty }{ϝ}^{\delta }\left(y\right)dy\right)}^{{}^{\frac{1}{\delta }}},$$

(2)

where $p>1,$ $p^{-1}+{\delta }^{-1}=1$ and $Z,$ $ϝ$ are measurable nonnegative functions such that $0<{\int }_0^{\infty }{Z}^p\left(\vartheta \right)d\vartheta <\infty$ and $0<{\int }_0^{\infty }{ϝ}^{\delta }\left(y\right)dy<\infty$. The constant $\frac{\pi }{sin\frac{\pi }{p}}$ in (1) and (2) sharp. In particular, when $p=\delta =2,$ the inequality (2) is reduced to the classical Hilbert integral inequality:

$${\int }_0^{\infty }{\int }_0^{\infty }\frac{Z\left(\vartheta \right)ϝ\left(y\right)}{\vartheta +y}d\vartheta dy\le \pi {\left({\int }_0^{\infty }{Z}^2\left(\vartheta \right)d\vartheta \right)}^{{}^{\frac{1}{2}}}{\left({\int }_0^{\infty }{ϝ}^2\left(y\right)dy\right)}^{{}^{\frac{1}{2}}}.$$

In [2] (p. 253), the author proved the following extension of Hilber’s double-series (1). Let $p,\delta >1,$ $p^{-1}+{\delta }^{-1}\ge 1$ and $0<\gamma =2-(p^{-1}+{\delta }^{-1})=p^{-1}+{\delta }^{-1}\le 1.$ Then,

$$\sum _{l=1}^{\infty }\sum _{r=1}^{\infty }\frac{z_r{b}_l}{{(r+l)}^{\gamma }}\le K(p,\delta ){\left(\sum _{r=1}^{\infty }{z}_r^p\right)}^{\frac{1}{p}}{\left(\sum _{l=1}^{\infty }{b}_l^{\delta }\right)}^{\frac{1}{\delta }},$$

(3)

The following continuous shape of (3) is also given in [2] (p. 254). Under the same condition with (3), we have

$${\int }_0^{\infty }{\int }_0^{\infty }\frac{Z\left(\vartheta \right)ϝ\left(y\right)}{{(\vartheta +y)}^{\gamma }}d\vartheta dy\le K(p,\delta ){\left({\int }_0^{\infty }{Z}^p\left(\vartheta \right)d\vartheta \right)}^{{}^{\frac{1}{p}}}{\left({\int }_0^{\infty }{ϝ}^{\delta }\left(y\right)dy\right)}^{{}^{\frac{1}{\delta }}},$$

(4)

where $K(p,\delta )$ in (3) and (4) depends on p and $\delta$ only.

As we all know, the classic Hölder inequality plays a very important and basic role in many areas of pure and applied mathematics. It is also a bridge to help solve problems in depth. In [3], Hölder established that

$$\sum _{l=1}^r{\zeta }_l{y}_l\le {\left(\sum _{l=1}^r{\zeta }_l^{\mu }\right)}^{\frac{1}{\mu }}{\left(\sum _{l=1}^r{y}_l^{\beta }\right)}^{\frac{1}{\beta }},$$

(5)

where ${\zeta }_l\ge 0,$ $y_l\ge 0$ $(l=1,2,...,r),$ $\mu \ge \beta >0$ and ${\mu }^{-1}+{\beta }^{-1}=1.$ The continuous shape of (5) is

$$\underset{z}{\overset{r}{\int }}\varphi \left(x\right)\varpi \left(x\right)dx\le {\left(\underset{z}{\overset{r}{\int }}{\varphi }^{\mu }\left(x\right)dx\right)}^{\frac{1}{\mu }}{\left(\underset{z}{\overset{r}{\int }}{\varpi }^{\beta }\left(x\right)dx\right)}^{\frac{1}{\beta }},$$

(6)

where $\mu ,$ $\beta >1$ s.t ${\mu }^{-1}+{\beta }^{-1}=1$ and $\varphi ,$ $\varpi \in C\left((z,r),{\mathbb{R}}^{+}\right).$

In [4], the researchers proved that, if $\varphi \left(x\right)$ and $\varpi \left(x\right)$ are nonnegative continuous functions on $[z,r]$, then

$${\left(\underset{z}{\overset{r}{\int }}{\varphi }^{\mu }\left(x\right)dx\right)}^{\frac{1}{\mu }}{\left(\underset{z}{\overset{r}{\int }}{\varpi }^{\beta }\left(x\right)dx\right)}^{\frac{1}{\beta }}\le \underset{z}{\overset{r}{\int }}S\left(\frac{Y{\varphi }^{\mu }\left(x\right)}{\vartheta {\varpi }^{\beta }\left(x\right)}\right)\varphi \left(x\right)\varpi \left(x\right)dx,$$

(7)

with

$$\vartheta =\underset{z}{\overset{r}{\int }}{\varphi }^{\mu }\left(x\right)dx,Y=\underset{z}{\overset{r}{\int }}{\varpi }^{\beta }\left(x\right)dx,\alpha >1\mathrm{and}{\mu }^{-1}+{\beta }^{-1}=1,$$

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

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

In [4], the researchers established that, if $\psi ,$ $\varpi \in C\left((z,r),{\mathbb{R}}^{+}\right)$ and $q>0,$ then

$$\underset{z}{\overset{r}{\int }}\frac{{\psi }^{q+1}\left(x\right)}{{\varpi }^q\left(x\right)}dx\le \frac{{\left(\underset{z}{\overset{r}{\int }}S\left(\frac{G{\psi }^{q+1}\left(x\right)}{F{\varpi }^{q+1}\left(x\right)}\right)\psi \left(x\right)dx\right)}^{q+1}}{{\left(\underset{z}{\overset{r}{\int }}\varpi \left(x\right)dx\right)}^q},$$

(8)

where

$$G=\underset{z}{\overset{r}{\int }}\varpi \left(x\right)dx\mathrm{and}F=\underset{z}{\overset{r}{\int }}\frac{{\psi }^{q+1}\left(x\right)}{{\varpi }^q\left(x\right)}dx.$$

In addition, they established the discrete form of (8) as follows:

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

(9)

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

In [6], the researchers proved that, if $0<p,\delta \le 1$, and ${\left\{{\lambda }_i\right\}}_{i=1}^k,$ ${\left\{{\omega }_j\right\}}_{j=1}^r$ are nonnegative and decreasing sequences of real numberswith $k,r\in \mathbb{N},$ then

$$\begin{array}{cc}\hfill & \sum _{i=1}^k\sum _{j=1}^r\frac{S_{p,\delta ,k,r,i,j}{\left({\sum }_{\mu =1}^i{\lambda }_{\mu }\right)}^p{\left({\sum }_{t=1}^j{\psi }_t\right)}^{\delta }}{{\left(ij\right)}^{\frac{1}{2}}}\hfill \\ \hfill & \ge p\delta {\left(kr\right)}^{\frac{1}{2}}{\left(\sum _{i=1}^k{\left[{\lambda }_i{\left(\sum _{\mu =1}^i{\lambda }_{\mu }\right)}^{p-1}\right]}^2\left(k-i+1\right)\right)}^{\frac{1}{2}}\hfill \\ \hfill & \times {\left(\sum _{j=1}^r{\left[{\psi }_j{\left(\sum _{t=1}^j{\psi }_t\right)}^{\delta -1}\right]}^2\left(r-j+1\right)\right)}^{\frac{1}{2}},\hfill \end{array}$$

(10)

where

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

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

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

In addition, they proved that

$$\begin{array}{cc}\hfill & \sum _{i=1}^k\sum _{j=1}^r\frac{S_{k,r,i,j}\varphi \left({\mathrm{\Lambda }}_i\right)\psi \left({\mathrm{\Omega }}_j\right)}{{\left(ij\right)}^{\frac{1}{2}}}\hfill \\ \hfill & \ge {\left(\sum _{i=1}^k{\left(\frac{\varphi \left(P_i\right)}{P_i}\right)}^2\right)}^{\frac{1}{2}}{\left(\sum _{j=1}^r{\left(\frac{\psi \left(W_j\right)}{W_j}\right)}^2\right)}^{\frac{1}{2}}\hfill \\ \hfill & \times {\left(\sum _{\mu =1}^k{\left[p_{\mu }\varphi \left(\frac{{\lambda }_{\mu }}{p_{\mu }}\right)\right]}^2\left(k-\mu +1\right)\right)}^{\frac{1}{2}}\hfill \\ \hfill & \times {\left(\sum _{t=1}^r{\left[{\delta }_t\psi \left(\frac{{\omega }_t}{{\delta }_t}\right)\right]}^2\left(r-t+1\right)\right)}^{\frac{1}{2}},\hfill \end{array}$$

(11)

where

$$\begin{array}{cc}\hfill S_{k,r,i,j}& =S\left(\frac{\left({\sum }_{\mu =1}^k{\left[p_{\mu }\varphi \left(\frac{{\lambda }_{\mu }}{p_{\mu }}\right)\right]}^2\left(k-\mu +1\right)\right){\left(\frac{\varphi \left(P_i\right)}{P_i}\right)}^2}{\left({\sum }_{i=1}^k{\left(\frac{\varphi \left(P_i\right)}{P_i}\right)}^2\right)\left({\sum }_{\mu =1}^i{\left[p_{\mu }\varphi \left(\frac{{\lambda }_{\mu }}{p_{\mu }}\right)\right]}^2\right)}\right)\hfill \\ \hfill & \times S\left(\frac{\left({\sum }_{t=1}^r{\left[{\delta }_t\psi \left(\frac{{\omega }_t}{{\delta }_t}\right)\right]}^2\left(r-t+1\right)\right){\left(\frac{\psi \left(W_j\right)}{W_j}\right)}^2}{\left({\sum }_{j=1}^r{\left(\frac{\psi \left(W_j\right)}{W_j}\right)}^2\right)\left({\sum }_{t=1}^j{\left[{\delta }_t\psi \left(\frac{{\omega }_t}{{\delta }_t}\right)\right]}^2\right)}\right),\hfill \end{array}$$

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

$${\mathrm{\Omega }}_j=\sum _{t=1}^{j}S\left(\frac{j{\left[{\delta }_t\psi \left(\frac{{\omega }_t}{{\delta }_t}\right)\right]}^2}{{\sum }_{t=1}^j{\left[{\delta }_t\psi \left(\frac{{\omega }_t}{{\delta }_t}\right)\right]}^2}\right){\omega }_t,$$

$$P_i=\sum _{\mu =1}^{i}S\left(\frac{i{\left[p_{\mu }\varphi \left(\frac{{\lambda }_{\mu }}{p_{\mu }}\right)\right]}^2}{{\sum }_{\mu =1}^i{\left[p_{\mu }\varphi \left(\frac{{\lambda }_{\mu }}{p_{\mu }}\right)\right]}^2}\right)p_s,$$

$$W_j=\sum _{t=1}^{j}S\left(\frac{j{\left[{\delta }_t\psi \left(\frac{{\omega }_t}{{\delta }_t}\right)\right]}^2}{{\sum }_{t=1}^j{\left[{\delta }_t\psi \left(\frac{{\omega }_t}{{\delta }_t}\right)\right]}^2}\right){\delta }_t,$$

and ${\left\{{\lambda }_i\right\}}_{i=1}^k,$ ${\left\{{\omega }_j\right\}}_{j=1}^r$ are nonnegative sequences with $k,$ $r\in \mathbb{N},$ $\left\{p_i\right\},$ $\left\{{\delta }_j\right\}$ are positive sequences $\varphi ,\psi$ are nonnegative, concave and supermultiplicative functions.

In [6], the authors proved that

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

(12)

where

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

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

For some generalizations and extensions of reversed inequalities of Hilbert-type and Hölder-type on time scales, see ([7,8,9,10,11,12,13,14]).

The primary objective of this article is to develop some new generalisations of reverse Hilbert-type inequalities via supermultiplicative functions by using reverse Hölder inequalities with Specht’s ratio on $\mathbb{T}$ (a time scale $\mathbb{T}$ is defined as an arbitrary nonempty closed subset of the real numbers $\mathbb{R}$).

The structure of the paper is summarised below. Section 2 covers some of the 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 particular examples (when $\mathbb{T}=\mathbb{N}$), our major findings are (10), as demonstrated by Zhao and Cheung [6].

2. Preliminaries

The forward jump operator is defined as

$$\sigma \left(c\right)≔inf\{u\in \mathbb{T}:u>c\}.$$

The set of all such rd-continuous functions is denoted by the space $C_{rd}(\mathbb{T},\mathbb{R})$, and for any function $Z:\mathbb{T}\to \mathbb{R}$, the notation $Z^{\sigma }\left(c\right)$ denotes $Z\left(\sigma \right(c\left)\right).$

The derivatives of $Z\mathrm{\Omega }$ and $Z/\mathrm{\Omega }$ of two differentiable functions Z and $\mathrm{\Omega }$ are given by

$${\left(Z\mathrm{\Omega }\right)}^{\mathrm{\Delta }}=Z^{\mathrm{\Delta }}\mathrm{\Omega }+Z^{\sigma }{\mathrm{\Omega }}^{\mathrm{\Delta }}=Z{\mathrm{\Omega }}^{\mathrm{\Delta }}+Z^{\mathrm{\Delta }}{\mathrm{\Omega }}^{\sigma },{\left(\frac{Z}{\mathrm{\Omega }}\right)}^{\mathrm{\Delta }}=\frac{Z^{\mathrm{\Delta }}\mathrm{\Omega }-Z{\mathrm{\Omega }}^{\mathrm{\Delta }}}{\mathrm{\Omega }{\mathrm{\Omega }}^{\sigma }},\mathrm{\Omega }{\mathrm{\Omega }}^{\sigma }\ne 0.$$

(13)

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

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

(14)

The time scales chain rule ([10] (Theorem 1.87)) is

$${(\mathrm{\Omega }\circ Z)}^{\mathrm{\Delta }}\left(\tau \right)={\mathrm{\Omega }}^{{}^{\prime }}\left(Z\left(c\right)\right)Z^{\mathrm{\Delta }}\left(x\right),\mathrm{where}c\in \left[x,\sigma \left(x\right)\right],$$

(15)

where $\mathrm{\Omega }:\mathbb{R}\to \mathbb{R}$ is continuously differentiable, and $\mathbb{Z}:\mathbb{T}\to \mathbb{R}$ is $\mathrm{\Delta }-$differentiable. More information on time scale calculus can be found at ([10,11]).

Now, we will give some properties of multiplicative and supermultiplicative functions.

Definition 1.

A function $L:I\to {\mathbb{R}}^{+}$ is multiplicative if

$$L\left(\varkappa \zeta \right)=L\left(\varkappa \right)L\left(\zeta \right),\forall \varkappa ,\zeta \in I\subset \mathbb{R}.$$

(16)

Definition 2

([15]). A function $L:I\to {\mathbb{R}}^{+}$ is supermultiplicative if

$$L\left(\varkappa \zeta \right)\ge L\left(\varkappa \right)L\left(\zeta \right),\forall \varkappa ,\zeta \in I\subset \mathbb{R}.$$

(17)

where L is the identity map (i.e., $L\left(\zeta \right)=\zeta$) and represents the multiplicative function. L is said to be a submultiplicative function if the last inequality has the opposite sign.

Lemma 1.

Let $z\in \mathbb{T},$ $\lambda \in C_{rd}(\mathbb{T},\mathbb{R})$ be nonnegative and $0\le \gamma \le 1.$ Then,

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

(18)

Proof. 

By using (15) on ${\int }_z^{\vartheta }\lambda \left(\tau \right)\mathrm{\Delta }\tau ,$ we obtain

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

(19)

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

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

(20)

By substituting (20) into (19), we can observe that

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

(21)

By integrating (21) from z to $\sigma \left(t\right),$ we obtain

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

i.e.,

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

which is (18). □

Lemma 2

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

$$S\left(\frac{c}{d}\right)c^{1/\gamma }{d}^{1/\delta }\ge \frac{c}{\gamma }+\frac{d}{\delta },$$

where

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

Lemma 3

([5]). Let $S(.)$ be defined as 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

([12], when α = 1). Let $g,h\in C\left({[z,w]}_{\mathbb{T}},{\mathbb{R}}^{+}\right)$ s.t. $g^{\beta },h^{\nu }$ be $\mathrm{\Delta }-$integrable on ${[z,w]}_{\mathbb{T}}$. If $\mu >1$ and $1/\mu +1/\nu =1,$ then

$$\begin{array}{ccc}& & {\int }_z^{w}S\left(\frac{Y{g}^{\mu }\left(\zeta \right)}{X{h}^{\nu }\left(\zeta \right)}\right)g\left(\zeta \right)h\left(\zeta \right)\mathrm{\Delta }\zeta \hfill \\ & \ge & {\left({\int }_z^w{g}^{\mu }\left(\zeta \right)\mathrm{\Delta }\zeta \right)}^{\frac{1}{\mu }}{\left({\int }_z^w{h}^{\nu }\left(\zeta \right)\mathrm{\Delta }\zeta \right)}^{\frac{1}{\nu }},\hfill \end{array}$$

(22)

where $X={\int }_z^w{g}^{\mu }\left(\zeta \right)\mathrm{\Delta }\zeta$ and $Y={\int }_z^w{h}^{\nu }\left(\zeta \right)\mathrm{\Delta }\zeta$.

Lemma 5

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

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

(23)

The inequality (23) is reversed when Ψ is continuous and concave.

Lemma 6.

Let $z\in \mathbb{T}$, $\lambda ,$ ψ be positive and decreasing functions, $f,$ g are positive and nondecreasing functions and $0\le p,$ $\delta \le 1.$ Furthermore, assume that $\varphi ,$φ are positive, increasing, concave and supermultiplicative functions. If $\beta >1,$ $\nu >1$ with $1/\beta +1/\nu =1,$ then

$$\begin{array}{ccc}& & S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\hfill \\ & =& max\left\{S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(z\right){\left({\int }_z^{\sigma \left(z\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(z\right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\right.\hfill \\ & & \left.;S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(t\right){\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(t\right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\right\},\hfill \end{array}$$

(24)

and

$$\begin{array}{ccc}& & S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(\eta \right){\left({\int }_z^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(\eta \right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)\hfill \\ & =& max\left\{S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(z\right){\left({\int }_z^{\sigma \left(z\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(z\right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)\right.\hfill \\ & & \left.;S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(\xi \right){\left({\int }_z^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(\xi \right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)\right\}.\hfill \end{array}$$

(25)

Proof. 

For $\vartheta \le y,$ we have

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

and then (where $0\le p\le 1$)

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

(26)

Because $\lambda$ is decreasing and $\vartheta \le y$, we can deduce from (26) that

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

(27)

Based on the knowledge $\beta >1,$ $\varphi$ is an increasing function and (27), we can conclude that

$${\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\ge {\varphi }^{\beta }\left[\lambda \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right].$$

Then, we obtain (where $\vartheta \le y$ and $f$ is nondecreasing) that

$$\begin{array}{cc}\hfill & \frac{1}{f^{\nu }\left(\vartheta \right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\hfill \\ \hfill & \ge \frac{1}{f^{\nu }\left(y\right)}{\varphi }^{\beta }\left[\lambda \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right],\hfill \end{array}$$

thus the function $\frac{1}{f^{\nu }\left(\vartheta \right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]$ is decreasing. Therefore, we have for $z\le \vartheta$ that

$$\begin{array}{ccc}& & \frac{1}{f^{\nu }\left(z\right)}{\varphi }^{\beta }\left[\lambda \left(z\right){\left({\int }_z^{\sigma \left(z\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\hfill \\ & \ge & \frac{1}{f^{\nu }\left(\vartheta \right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\hfill \end{array}$$

and then

$$\begin{array}{cc}\hfill & f^{\nu }\left(\vartheta \right){\varphi }^{\beta }\left[\lambda \left(z\right){\left({\int }_z^{\sigma \left(z\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\hfill \\ \hfill & \ge f^{\nu }\left(z\right){\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right].\hfill \end{array}$$

(28)

Integrating (28) over $\vartheta$ from z to $\sigma \left(t\right),$ we obtain

$$\begin{array}{cc}\hfill & {\varphi }^{\beta }\left[\lambda \left(z\right){\left({\int }_z^{\sigma \left(z\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]{\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \hfill \\ \hfill & \ge f^{\nu }\left(z\right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta ,\hfill \end{array}$$

and then

$$\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(z\right){\left({\int }_z^{\sigma \left(z\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(z\right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\ge 1.$$

(29)

Since the function $\frac{1}{f^{\nu }\left(\vartheta \right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]$ is decreasing and $\vartheta \le t$, we obtain

$$\begin{array}{cc}\hfill & \frac{1}{f^{\nu }\left(\vartheta \right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\hfill \\ \hfill & \ge \frac{1}{f^{\nu }\left(t\right)}{\varphi }^{\beta }\left[\lambda \left(t\right){\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right],\hfill \end{array}$$

and then

$$f^{\nu }\left(t\right){\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\ge f^{\nu }\left(\vartheta \right){\varphi }^{\beta }\left[\lambda \left(t\right){\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right].$$

(30)

Integrating (30) over $\vartheta$ from z to $\sigma \left(t\right),$ we obtain

$$\begin{array}{cc}\hfill & f^{\nu }\left(t\right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta \hfill \\ \hfill & \ge {\varphi }^{\beta }\left[\lambda \left(t\right){\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]{\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta ,\hfill \end{array}$$

thus

$$\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(t\right){\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(t\right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\le 1.$$

(31)

Based on (29) and (31), we can see that

$$\begin{array}{cc}\hfill & \frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(z\right){\left({\int }_z^{\sigma \left(z\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(z\right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\ge ...\ge 1\hfill \\ \hfill & \ge ...\ge \frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(t\right){\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(t\right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }.\hfill \end{array}$$

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

$$S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(z\right){\left({\int }_z^{\sigma \left(z\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(z\right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)$$

and

$$S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(t\right){\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(t\right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)$$

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

$$\begin{array}{cc}\hfill & S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\hfill \\ \hfill & =max\left\{S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(z\right){\left({\int }_z^{\sigma \left(z\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(z\right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\right.\hfill \end{array}$$

$$;S\left.\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(t\right){\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(t\right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\right\},$$

and that is (24). In a similar manner, for $\psi$ and $0\le \delta \le 1,$ we obtain

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

which is (25). □

Throughout the article, we will assume that the functions are nonnegative rd-continuous functions on ${[z,\infty )}_{\mathbb{T}}≔[z,\infty )\cap \mathbb{T}$.

3. Principal Findings

Theorem 1.

Let $z\in \mathbb{T}$, $0\le p,\delta \le 1,$ $\lambda ,$ ψ be positive and decreasing functions and $\varphi ,$ φ are positive, increasing, concave and supermultiplicative functions. If $f,$ g are positive and nondecreasing functions and $\beta >1,$ $\nu >1$ with $1/\beta +1/\nu =1,$ then

$$\begin{array}{cc}\hfill & {\int }_z^{\sigma \left(s\right)}{\int }_z^{\sigma \left(r\right)}\mathrm{\Phi }(t,\xi )P\left(t\right)W\left(\xi \right)\hfill \\ \hfill & \times \varphi \left[\frac{1}{P\left(t\right)}S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\right.\hfill \\ \hfill & \left.\times {\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p}f\left(t\right)\right]\hfill \\ \hfill & \times \phi \left[\frac{1}{W\left(\xi \right)}S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(\eta \right){\left({\int }_z^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(\eta \right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)\right.\hfill \\ \hfill & \left.{\left({\int }_z^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta }g\left(\xi \right)\right]\mathrm{\Delta }t\mathrm{\Delta }\xi \hfill \\ \hfill & \ge \nu C\left(p,\delta ,r,s,\nu \right){\left({\int }_z^{\sigma \left(r\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\hfill \\ \hfill & \times {\left({\int }_z^{\sigma \left(s\right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y\right)}^{\frac{1}{\beta }},\hfill \end{array}$$

(32)

holds for all $r,s\in {[z,\infty )}_{\mathbb{T}}$, where

$$\begin{array}{ccc}\hfill \mathrm{\Phi }(t,\xi )& =& \frac{f\left(t\right)g\left(\xi \right)}{{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\nu }}{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right)}^{\frac{1}{\nu }}}\hfill \\ & & \times S\left(\frac{\left({\int }_z^{\sigma \left(r\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(t\right){\int }_z^{\sigma \left(r\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta }\right)\hfill \\ & & \times S\left(\frac{\left({\int }_z^{\sigma \left(s\right)}{g}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(\eta \right){\left({\int }_z^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(\xi \right){\int }_z^{\sigma \left(s\right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y}\right),\hfill \end{array}$$

with

$$C\left(p,\delta ,r,s,\nu \right)=\frac{\varphi \left(p\right)\phi \left(\delta \right)}{\nu }{\left({\int }_z^{\sigma \left(r\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\nu }}{\left({\int }_z^{\sigma \left(s\right)}{g}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\nu }},$$

$$P\left(t\right)={\int }_z^{\sigma \left(t\right)}S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\vartheta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)f\left(\vartheta \right)\mathrm{\Delta }\vartheta ,$$

$$W\left(\xi \right)={\int }_z^{\sigma \left(\xi \right)}S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(y\right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)g\left(y\right)\mathrm{\Delta }y,$$

such that

$$\begin{array}{ccc}& & S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\hfill \\ & =& max\left\{S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(z\right){\left({\int }_z^{\sigma \left(z\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(z\right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\right.\hfill \\ & & \left.;S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(t\right){\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(t\right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\right\},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(\eta \right){\left({\int }_z^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(\eta \right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)\hfill \\ \hfill =& max\left\{S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(z\right){\left({\int }_z^{\sigma \left(z\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(z\right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)\right.;\hfill \end{array}$$

$$\left.S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(\xi \right){\left({\int }_z^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(\xi \right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)\right\}.$$

Proof. 

Denote

$$\begin{array}{cc}\hfill & \mathrm{\Lambda }\left(t,\xi \right)\hfill \\ \hfill & =P\left(t\right)\varphi \left[S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\right.\hfill \\ \hfill & \times \left.\frac{1}{P\left(t\right)}{\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p}f\left(t\right)\right]\hfill \\ \hfill & \times W\left(\xi \right)\phi \left[S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(\eta \right){\left({\int }_z^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(\eta \right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)\right.\hfill \\ \hfill & \times \left.\frac{1}{W\left(\xi \right)}{\left({\int }_z^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta }g\left(\xi \right)\right],\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathrm{\Omega }\left(t,\xi \right)& =P\left(t\right)\varphi \left[S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\right.\hfill \\ \hfill & \times \left.\frac{1}{P\left(t\right)}{\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p}f\left(t\right)\right]\hfill \\ \hfill & \times W\left(\xi \right)\phi \left[S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(\eta \right){\left({\int }_z^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(\eta \right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)\right.\hfill \\ \hfill & \times \left.\frac{1}{W\left(\xi \right)}{\left({\int }_z^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta }g\left(\xi \right)\right].\hfill \end{array}$$

Applying (18) with $\gamma =p$ gives us

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

(33)

By multiplying the previous inequality by

$$S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)f\left(t\right),$$

we have

$$\begin{array}{ccc}& & S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\hfill \\ & & \times f\left(t\right){\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^p\hfill \\ & \ge & p{\int }_z^{\sigma \left(t\right)}S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\hfill \\ & & \times f\left(t\right)\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\mathrm{\Delta }\vartheta .\hfill \end{array}$$

Using the fact that f is nondecreasing and $t\le \vartheta$, we obtain

$$\begin{array}{cc}\hfill & S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\hfill \\ \hfill & \times f\left(t\right){\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^p\hfill \\ \hfill & \ge p{\int }_z^{\sigma \left(t\right)}S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\hfill \\ \hfill & \times f\left(\vartheta \right)\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\mathrm{\Delta }\vartheta .\hfill \end{array}$$

(34)

By Lemma 6, inequality (34) is

$$\begin{array}{cc}\hfill & S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\hfill \\ \hfill & \times f\left(t\right){\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^p\hfill \\ \hfill & \ge p{\int }_z^{\sigma \left(t\right)}S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\vartheta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\hfill \\ \hfill & \times f\left(\vartheta \lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right)\mathrm{\Delta }\vartheta .\hfill \end{array}$$

(35)

Likewise, for the decreasing function $\psi ,$ the nondecreasing function g and $0\le \delta \le 1,$ we obtain

$$\begin{array}{cc}\hfill & S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(\eta \right){\left({\int }_z^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(\eta \right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \times {\left({\int }_z^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta }g\left(\xi \right)\hfill \\ \hfill & \ge \delta {\int }_z^{\sigma \left(\xi \right)}S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(y\right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)\hfill \\ \hfill & \times \psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}g\left(y\right)\mathrm{\Delta }y.\hfill \end{array}$$

(36)

From (35), we deduce that

$$\begin{array}{ccc}& & \varphi \left[S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\right.\hfill \\ & & \times \left.\frac{1}{P\left(t\right)}{\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p}f\left(t\right)\right]\hfill \\ & \ge & \varphi \left[\frac{p}{P\left(t\right)}{\int }_z^{\sigma \left(t\right)}S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\vartheta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\right.\hfill \\ & & \times \left.\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}f\left(\vartheta \right)\mathrm{\Delta }\vartheta \right].\hfill \end{array}$$

Since $\varphi$ is a positive, increasing and super-multiplicative function, we have

$$\begin{array}{cc}\hfill & \varphi \left[S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\right.\hfill \\ \hfill & \times \left.\frac{1}{P\left(t\right)}{\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p}f\left(t\right)\right]\hfill \\ \hfill & \ge \varphi \left(p\right)\varphi \left[\frac{1}{P\left(t\right)}{\int }_z^{\sigma \left(t\right)}S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\vartheta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\right.\hfill \\ \hfill & \times \left.\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}f\left(\vartheta \right)\mathrm{\Delta }\vartheta \right].\hfill \end{array}$$

(37)

Then, by using the Jensen inequality on the right-hand side of (37) (where $\varphi$ is a concave function), we obtain

$$\begin{array}{cc}\hfill & \varphi \left[S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\right.\hfill \\ \hfill & \times \left.\frac{1}{P\left(t\right)}{\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p}f\left(t\right)\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \ge \frac{\varphi \left(p\right)}{P\left(t\right)}{\int }_z^{\sigma \left(t\right)}S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\vartheta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\hfill \\ \hfill & \times f\left(\vartheta \right)\varphi \left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta .\hfill \end{array}$$

(38)

Analogously, in the case of (36), we can see (where $\phi$ is a positive, increasing, concave and supermultiplicative function) that

$$\begin{array}{cc}\hfill & \phi \left[S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(\eta \right){\left({\int }_z^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(\eta \right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)\right.\hfill \\ \hfill & \times \left.\frac{1}{W\left(\xi \right)}{\left({\int }_z^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta }g\left(\xi \right)\right]\hfill \\ \hfill & \ge \frac{\phi \left(\delta \right)}{W\left(\xi \right)}{\int }_z^{\sigma \left(\xi \right)}S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(y\right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)\hfill \\ \hfill & \times g\left(y\right)\phi \left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y.\hfill \end{array}$$

(39)

Multiplying (38) and (39), we obtain

$$\begin{array}{cc}\hfill \mathrm{\Lambda }\left(t,\xi \right)& \ge \varphi \left(p\right){\int }_z^{\sigma \left(t\right)}S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\vartheta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\hfill \\ \hfill & \times f\left(\vartheta \right)\varphi \left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta \hfill \\ \hfill & \times \phi \left(\delta \right){\int }_z^{\sigma \left(\xi \right)}S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(y\right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)\hfill \\ \hfill & \times g\left(y\right)\phi \left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y.\hfill \end{array}$$

(40)

Applying (22) on the right-hand side of (40), we obtain

$$\begin{array}{cc}\hfill \mathrm{\Omega }\left(t,\xi \right)& \ge \varphi \left(p\right){\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\nu }}{\left({\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\hfill \\ \hfill & \times \phi \left(\delta \right){\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right)}^{\frac{1}{\nu }}{\left({\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y\right)}^{\frac{1}{\beta }}.\hfill \end{array}$$

(41)

Multiplying (41) by

$$\begin{array}{cc}\hfill \mathrm{\Phi }(t,\xi )& =\frac{f\left(t\right)g\left(\xi \right)}{{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\nu }}{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right)}^{\frac{1}{\nu }}}\hfill \\ \hfill & \times S\left(\frac{\left({\int }_z^{\sigma \left(r\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(t\right){\int }_z^{\sigma \left(r\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta }\right)\hfill \\ \hfill & \times S\left(\frac{\left({\int }_z^{\sigma \left(s\right)}{g}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(\eta \right){\left({\int }_z^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(\xi \right){\int }_z^{\sigma \left(s\right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y}\right),\hfill \end{array}$$

we have

$$\begin{array}{ccc}\hfill \mathrm{\Phi }(t,\xi )\mathrm{\Omega }\left(t,\xi \right)& \ge & \varphi \left(p\right)\phi \left(\delta \right){\left({\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}f\left(t\right)\hfill \\ & & \times S\left(\frac{\left({\int }_z^{\sigma \left(r\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(t\right){\int }_z^{\sigma \left(r\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta }\right)\hfill \\ & & \times {\left({\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y\right)}^{\frac{1}{\beta }}g\left(\xi \right)\hfill \\ & & \times S\left(\frac{\left({\int }_z^{\sigma \left(s\right)}{g}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(\eta \right){\left({\int }_z^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(\xi \right){\int }_z^{\sigma \left(s\right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y}\right).\hfill \end{array}$$

Using the integration over t from z to $\sigma \left(r\right)$ and the integration over $\xi$ from z to $\sigma \left(s\right),$ respectively, we arrive at

$$\begin{array}{ccc}& & {\int }_z^{\sigma \left(s\right)}{\int }_z^{\sigma \left(r\right)}\mathrm{\Phi }(t,\xi )\mathrm{\Omega }\left(t,\xi \right)\mathrm{\Delta }t\mathrm{\Delta }\xi \hfill \\ & \ge & \varphi \left(p\right)\phi \left(\delta \right){\int }_z^{\sigma \left(r\right)}{\left({\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}f\left(t\right)\hfill \\ & & \times S\left(\frac{\left({\int }_z^{\sigma \left(r\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(t\right){\int }_z^{\sigma \left(r\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta }\right)\mathrm{\Delta }t\hfill \\ & & \times {\int }_z^{\sigma \left(s\right)}{\left({\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y\right)}^{\frac{1}{\beta }}g\left(\xi \right)\hfill \\ & & \times S\left(\frac{\left({\int }_z^{\sigma \left(s\right)}{g}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(\eta \right){\left({\int }_z^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(\xi \right){\int }_z^{\sigma \left(s\right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y}\right)\mathrm{\Delta }\xi .\hfill \end{array}$$

(42)

Applying Formula (14) on the term

$${\int }_z^{\sigma \left(r\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta ,$$

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

$$\begin{array}{cc}\hfill & {\int }_z^{\sigma \left(r\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \hfill \\ \hfill & ={\left.\left(\sigma \left(r\right)-\vartheta \right)v\left(\vartheta \right)\right|}_z^{\sigma \left(r\right)}+{\int }_z^{\sigma \left(r\right)}{v}^{\sigma }\left(\vartheta \right)\mathrm{\Delta }\vartheta ,\hfill \end{array}$$

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

$$\begin{array}{cc}\hfill & {\int }_z^{\sigma \left(r\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \hfill \\ \hfill & ={\int }_z^{\sigma \left(r\right)}{\int }_z^{\sigma \left(\vartheta \right)}{\varphi }^{\beta }\left[\lambda \left(\theta \right){\left({\int }_z^{\sigma \left(\theta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\theta \mathrm{\Delta }\vartheta .\hfill \end{array}$$

(43)

In a similar vein, we note

$$\begin{array}{cc}\hfill & {\int }_z^{\sigma \left(s\right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y\hfill \\ \hfill & ={\int }_z^{\sigma \left(s\right)}{\int }_z^{\sigma \left(y\right)}{\phi }^{\beta }\left[\psi \left(\theta \right){\left({\int }_z^{\sigma \left(\theta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }\theta \mathrm{\Delta }y.\hfill \end{array}$$

(44)

Substituting (43) and (44) into (42), we have

$$\begin{array}{ccc}& & {\int }_z^{\sigma \left(s\right)}{\int }_z^{\sigma \left(r\right)}\mathrm{\Phi }(t,\xi )\mathrm{\Omega }\left(t,\xi \right)\mathrm{\Delta }t\mathrm{\Delta }\xi \hfill \\ & \ge & \varphi \left(p\right)\phi \left(\delta \right){\int }_z^{\sigma \left(r\right)}{\left({\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}f\left(t\right)\hfill \\ & & \times S\left(\frac{\left({\int }_z^{\sigma \left(r\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\zeta }{f^{\nu }\left(t\right){\int }_z^{\sigma \left(r\right)}{\int }_z^{\sigma \left(\vartheta \right)}{\varphi }^{\beta }\left[\lambda \left(\theta \right){\left({\int }_z^{\sigma \left(\theta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\theta \mathrm{\Delta }\vartheta }\right)\mathrm{\Delta }t\hfill \\ & & \times {\int }_z^{\sigma \left(s\right)}{\left({\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y\right)}^{\frac{1}{\beta }}g\left(\xi \right)\hfill \\ & & \times S\left(\frac{\left({\int }_z^{\sigma \left(s\right)}{g}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(\eta \right){\left({\int }_z^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(\xi \right){\int }_z^{\sigma \left(s\right)}{\int }_z^{\sigma \left(y\right)}{\phi }^{\beta }\left[\psi \left(\theta \right){\left({\int }_z^{\sigma \left(\theta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }\theta \mathrm{\Delta }y}\right)\mathrm{\Delta }\xi .\hfill \end{array}$$

Then, by applying (22) on the previous inequality, using (43) and (44), we obtain

$$\begin{array}{ccc}& & {\int }_z^{\sigma \left(s\right)}{\int }_z^{\sigma \left(r\right)}\mathrm{\Phi }(t,\xi )\mathrm{\Omega }\left(t,\xi \right)\mathrm{\Delta }t\mathrm{\Delta }\xi \hfill \\ & \ge & \varphi \left(p\right)\phi \left(\delta \right){\left({\int }_z^{\sigma \left(r\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\nu }}{\left({\int }_z^{\sigma \left(s\right)}{g}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\nu }}\hfill \\ & & \times {\left({\int }_z^{\sigma \left(r\right)}{\int }_z^{\sigma \left(\vartheta \right)}{\varphi }^{\beta }\left[\lambda \left(\theta \right){\left({\int }_z^{\sigma \left(\theta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\theta \mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\hfill \\ & & \times {\left({\int }_z^{\sigma \left(s\right)}{\int }_z^{\sigma \left(y\right)}{\phi }^{\beta }\left[\psi \left(\theta \right){\left({\int }_z^{\sigma \left(\theta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }\theta \mathrm{\Delta }y\right)}^{\frac{1}{\beta }}\hfill \\ & =& \varphi \left(p\right)\phi \left(\delta \right){\left({\int }_z^{\sigma \left(r\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\nu }}{\left({\int }_z^{\sigma \left(s\right)}{g}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\nu }}\hfill \\ & & \times {\left({\int }_z^{\sigma \left(r\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\hfill \\ & & \times {\left({\int }_z^{\sigma \left(s\right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y\right)}^{\frac{1}{\beta }},\hfill \end{array}$$

and thus we obtain

$$\begin{array}{cc}\hfill & {\int }_z^{\sigma \left(s\right)}{\int }_z^{\sigma \left(r\right)}\mathrm{\Phi }(t,\xi )P\left(t\right)W\left(\xi \right)\hfill \\ \hfill & \times \varphi S\left(\frac{\left({\int }_z^{\sigma \left(t\right)}{f}^{\nu }\left(\vartheta \right)\mathrm{\Delta }\vartheta \right){\varphi }^{\beta }\left[\lambda \left(\zeta \right){\left({\int }_z^{\sigma \left(\zeta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]}{f^{\nu }\left(\zeta \right){\int }_z^{\sigma \left(t\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\mathrm{\Delta }\vartheta }\right)\frac{1}{P\left(t\right)}{\left({\int }_z^{\sigma \left(t\right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p}f\left(t\right)\hfill \\ \hfill & \times \phi S\left(\frac{\left({\int }_z^{\sigma \left(\xi \right)}{g}^{\nu }\left(y\right)\mathrm{\Delta }y\right){\phi }^{\beta }\left[\psi \left(\eta \right){\left({\int }_z^{\sigma \left(\eta \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]}{g^{\nu }\left(\eta \right){\int }_z^{\sigma \left(\xi \right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\mathrm{\Delta }y}\right)\frac{g\left(\xi \right)}{W\left(\xi \right)}{\left({\int }_z^{\sigma \left(\xi \right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta }\mathrm{\Delta }t\mathrm{\Delta }\xi \hfill \\ \hfill & \ge \nu C\left(p,\delta ,r,s,\nu \right){\left({\int }_z^{\sigma \left(r\right)}{\varphi }^{\beta }\left[\lambda \left(\vartheta \right){\left({\int }_z^{\sigma \left(\vartheta \right)}\lambda \left(\tau \right)\mathrm{\Delta }\tau \right)}^{p-1}\right]\left(\sigma \left(r\right)-\vartheta \right)\mathrm{\Delta }\vartheta \right)}^{\frac{1}{\beta }}\hfill \\ \hfill & \times {\left({\int }_z^{\sigma \left(s\right)}{\phi }^{\beta }\left[\psi \left(y\right){\left({\int }_z^{\sigma \left(y\right)}\psi \left(\tau \right)\mathrm{\Delta }\tau \right)}^{\delta -1}\right]\left(\sigma \left(s\right)-y\right)\mathrm{\Delta }y\right)}^{\frac{1}{\beta }},\hfill \end{array}$$

which is (32). The proof is complete. □

Remark 1.

As a special case of Theorem 1, when $\mathbb{T}=\mathbb{N}$, $\phi \left(\vartheta \right)=\varphi \left(\vartheta \right)=\vartheta ,$ $f\left(\vartheta \right)=g\left(\vartheta \right)=1$ and $\beta =\nu =2,$ we can obtain (10) demonstrated in [6].

Remark 2.

As a special case of Theorem 1, when $\phi \left(\vartheta \right)=\varphi \left(\vartheta \right)=\vartheta ,$ and $\beta =\nu =2,$ we can obtain the results demonstrated in [8].

Remark 3.

As a special case of Theorem 1, when $\phi \left(\vartheta \right)=\varphi \left(\vartheta \right)=\vartheta ,$ we can obtain the results demonstrated in [16].

4. Conclusions

In this paper, we use reverse Hölder inequalities with Specht’s ratio on time scales to develop the study of reversed Hilbert-type inequalities. This aim holds by a study on some new generalizations of reversed Hilbert-type inequalities via supermultiplicative functions. In the future work, we can generalize dynamic inequalities of this article using a fractional Riemann–Liouville integral on time scale calculus, and we can present some of these dynamic inequalities on quantum calculus. It will be interesting to present dynamic inequalities in two or more dimensions.