Novel Integral Inequalities on Nabla Time Scales with C-Monotonic Functions

Abstract

Through the paper, we present several inequalities involving C-monotonic functions with $C\ge 1,$ on nabla calculus via time scales. It is known that dynamic inequalities generate many different inequalities in different calculus. The main results will be proved by applying the chain rule formula on nabla calculus. As a special case for our results, when $\mathbb{T}=\mathbb{R},$ we obtain the continuous analouges of inequalities that had previously been proved in the literature. When $\mathbb{T}=\mathbb{N}$, the results, to the best of the authors’ knowledge, are essentially new. Symmetrical properties of $C$-monotonic functions are critical in determining the best way to solve inequalities.

1. Introduction

In [1], the authors proved that if $-\infty \le \alpha <\gamma \le \infty ,$ $\epsilon \in (0,1],$ $\psi ,\omega$ are non-negative and differentiable functions, such that $\psi$ is a decreasing function and $\omega$ is an increasing function with $\omega \left(\alpha \right)=0,$ then

$${\int }_{\alpha }^{\gamma }\psi \left(z\right)d\omega \left(z\right)\le {\left({\int }_{\alpha }^{\gamma }{\psi }^{\epsilon }\left(z\right)d\left[{\omega }^{\epsilon }\left(z\right)\right]\right)}^{\frac{1}{\epsilon }}.$$

(1)

For $1\le \epsilon <\infty ,$ (1) is reversed. Furthermore, they established that if $\psi ,\omega$ are non-negative and differentiable functions, such that $\psi$ is increasing and $\omega$ is decreasing with $\omega \left(\gamma \right)=0,$ then

$${\int }_{\alpha }^{\gamma }\psi \left(z\right)d\left[-\omega \left(z\right)\right]\le {\left({\int }_{\alpha }^{\gamma }{\psi }^{\epsilon }\left(z\right)d\left[-{\omega }^{\epsilon }\left(z\right)\right]\right)}^{\frac{1}{\epsilon }}.$$

(2)

In [2], the authors proved the following generalization of (1) and (2), respectively, as follows, let $0<p<r<\infty ,$ $\psi$ be $C$-decreasing with $C\ge 1,$ $\omega$ be increasing and differentiable with $\omega \left(\alpha \right)=0.$ Then

$${\left({\int }_{\alpha }^{\gamma }{\psi }^r\left(z\right)d\left[{\omega }^r\left(z\right)\right]\right)}^{\frac{1}{r}}\le C^{1-(p/r)}{\left({\int }_{\alpha }^{\gamma }{\psi }^p\left(z\right)d\left[{\omega }^p\left(z\right)\right]\right)}^{\frac{1}{p}}.$$

(3)

Additionally, they showed that let $\psi$ be $C$-increasing with $C\ge 1$. Then

$${\left({\int }_{\alpha }^{\gamma }{\psi }^r\left(z\right)d\left[{\omega }^r\left(z\right)\right]\right)}^{\frac{1}{r}}\ge C^{(p/r)-1}{\left({\int }_{\alpha }^{\gamma }{\psi }^p\left(z\right)d\left[{\omega }^p\left(z\right)\right]\right)}^{\frac{1}{p}}.$$

(4)

In the same paper [2], they established that let $0<p<r<\infty ,$ $\psi$ be $C$-increasing with $C\ge 1,$ $\omega$ be decreasing and differentiable with $\omega \left(\gamma \right)=0$. Then

$${\left({\int }_{\alpha }^{\gamma }{\psi }^r\left(z\right)d\left[-{\omega }^r\left(z\right)\right]\right)}^{\frac{1}{r}}\le C^{1-(p/r)}{\left({\int }_{\alpha }^{\gamma }{\psi }^p\left(z\right)d\left[-{\omega }^p\left(z\right)\right]\right)}^{\frac{1}{p}},$$

(5)

and also, they showed that by letting $\psi$ be $C$-decreasing for $C\ge 1,$ $\omega$ be decreasing and differentiable with $\omega \left(\gamma \right)=0$. Then

$${\left({\int }_{\alpha }^{\gamma }{\psi }^r\left(z\right)d\left[-{\omega }^r\left(z\right)\right]\right)}^{\frac{1}{r}}\ge C^{p/r-1}{\left({\int }_{\alpha }^{\gamma }{\psi }^p\left(z\right)d\left[-{\omega }^p\left(z\right)\right]\right)}^{\frac{1}{p}}.$$

(6)

In the last few decades, much attention has been devoted to establishing discrete analogues of the corresponding continuous results in various yields of analysis. One of the reasons for the increased interest in the discrete case lies in the fact that discrete operators can behave significantly differently from the corresponding continuous counterparts.

A time scale $\mathbb{T}$ is defined as an arbitrary non-empty closed subset of the real numbers $\mathbb{R}$. The results in this paper contain the classical continuous and discrete inequalities as special cases when $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{N}.$ In addition, we can obtain some extended inequalities on various time scales, such as $\mathbb{T}=h\mathbb{N},$ $h>0$ and $\mathbb{T}=q^{\mathbb{N}}$ for $q>1,$ etc.

In [3], the authors established that if $\phi :[0,\infty )\to \mathbb{R}$ is a concave function, $\psi$ is $C$-decreasing with $C\ge 1$ and $\omega$ is increasing, such that $\omega \left(\alpha \right)=0,$ then

$$\phi \left(C{\int }_{\alpha }^{\gamma }\psi \left(z\right){\omega }^{\Delta }\left(z\right)\Delta z\right)\le C{\int }_{\alpha }^{\gamma }\psi \left(z\right){\omega }^{\Delta }\left(z\right){\phi }^{\prime }\left[\psi \left(z\right)\omega \left(z\right)\right]\Delta z,$$

and if $\psi$ is $C$-increasing with $C\ge 1$, then

$$\phi \left(\frac{1}{C}{\int }_{\alpha }^{\gamma }\psi \left(z\right){\omega }^{\Delta }\left(z\right)\Delta z\right)\ge \frac{1}{C}{\int }_{\alpha }^{\gamma }\psi \left(z\right){\omega }^{\Delta }\left(z\right){\phi }^{\prime }\left[{\psi }^{\sigma }\left(z\right){\omega }^{\sigma }\left(z\right)\right]\Delta z.$$

Additionally, they established that if $\psi$ is $C$-increasing with $C\ge 1,$ $\omega$ is decreasing such that $\omega \left(\gamma \right)=0,$ then

$$\phi \left(C{\int }_{\alpha }^{\gamma }\psi \left(z\right){\left[-\omega \left(z\right)\right]}^{\Delta }\Delta z\right)\le C{\int }_{\alpha }^{\gamma }\psi \left(z\right){\left[-\omega \left(z\right)\right]}^{\Delta }{\phi }^{\prime }\left[{\psi }^{\sigma }\left(z\right){\omega }^{\sigma }\left(z\right)\right]\Delta z,$$

and if $\psi$ is $C$-decreasing with $C\ge 1,$ $\omega$ is decreasing such that $\omega \left(\gamma \right)=0,$ then

$$\phi \left(\frac{1}{C}{\int }_{\alpha }^{\gamma }\psi \left(z\right){\left[-\omega \left(z\right)\right]}^{\Delta }\Delta z\right)\ge \frac{1}{C}{\int }_{\alpha }^{\gamma }\psi \left(z\right){\left[-\omega \left(z\right)\right]}^{\Delta }{\phi }^{\prime }\left[\psi \left(z\right)\omega \left(z\right)\right]\Delta z.$$

In [4], the authors generalized (1) to an arbitrary time scale using the delta-integral and proved that if $r>0,$ $0<\delta <1,$ $\psi$ is $C^r$-decreasing for $C\ge 1,$ $\omega$ is increasing with $\omega \left(\alpha \right)=0$ and

$${\omega }^{\delta -1}\left(\sigma \left(\eta \right)\right){\omega }^{\delta -1}\left(\eta \right)\ge 1,$$

then

$${\int }_{\alpha }^{\gamma }\psi \left(z\right){\left[\omega \left(z\right)\right]}^{\Delta }\Delta z\le C^{r\left(1-\delta \right)}{\left({\int }_{\alpha }^{\gamma }{\psi }^{\delta }\left(z\right){\left[{\omega }^{\delta }\left(z\right)\right]}^{\Delta }\Delta z\right)}^{\frac{1}{\delta }}.$$

Additionally, they proved that if $r>0,$ $0<\delta <1,$ $\psi$ is $C^r$-increasing for $C\ge 1,$ $\omega$ is increasing with $\omega \left(\alpha \right)=0$ and

$${\omega }^{\delta -1}\left(\eta \right){\omega }^{1-\delta }\left(\sigma \left(\eta \right)\right){\psi }^{\delta }\left(\eta \right){\psi }^{1-\delta }\left(\sigma \left(\eta \right)\right)\le \psi \left(\eta \right),$$

then

$${\int }_{\alpha }^{\gamma }\psi \left(z\right){\left[\omega \left(z\right)\right]}^{\Delta }\Delta z\ge C^{r\left(\delta -1\right)}{\left({\int }_{\alpha }^{\gamma }{\psi }^{\delta }\left(z\right){\left[{\omega }^{\delta }\left(z\right)\right]}^{\Delta }\Delta z\right)}^{\frac{1}{\delta }}.$$

For more details about the dynamic inequalities on time scales, see [5,6,7,8,9,10,11,12].

The organization of the paper as follows. In Section 2, we offer some preliminary information about nabla calculus on time scales. In Section 3, we apply some features of $C$-monotonic functions for $C\ge 1$ and the chain rule on nabla calculus to show the key result. These results as special cases when $\mathbb{T}=\mathbb{R}$ give the inequalities (1)–(6). When $\mathbb{T}=\mathbb{N}$ are fundamentally novel.

2. Preliminaries

In 2001, Bohner and Peterson [13] defined the backward jump operator by $\rho \left(\zeta \right):=sup\{s\in \mathbb{T}:s<\zeta \}.$ For $\psi :\mathbb{T}\to \mathbb{R}$, the notation ${\psi }^{\rho }\left(\zeta \right)$ denotes $\psi \left(\rho \right(\zeta \left)\right).$ The time scale interval ${[\alpha ,\gamma ]}_{\mathbb{T}}$ is ${[\alpha ,\gamma ]}_{\mathbb{T}}:=[\alpha ,\gamma ]\cap \mathbb{T}$ and $\nu :\mathbb{T}\to {\mathbb{R}}^{+}$ is $\nu \left(\zeta \right)=\zeta -\rho \left(\zeta \right).$

Definition 1

([14]). A function $\psi :\mathbb{T}\to \mathbb{R}$ is a nabla differentiable at $\zeta \in \mathbb{T}$, if ψ is defined in a neighborhood U of ζ and there exists a unique real number ${\psi }^{\nabla }\left(\zeta \right)$, called the nabla derivative of ψ at ζ, such that for any $ϵ>0$, there exists a neighborhood N of ζ with $N\subseteq U$ and

$$\left|\psi \left(\rho \left(\zeta \right)\right)-\psi \left(s\right)-{\psi }^{\nabla }\left(\zeta \right)[\rho \left(\zeta \right)-s]\right|\le ϵ\left|\rho \left(\zeta \right)-s\right|,\forall s\in N.$$

Theorem 1

([14]). Let $\psi ,$ ω be nabla differentiable at $\zeta \in \mathbb{T}$. Then

1. 

$\psi \omega :\mathbb{T}\to \mathbb{R}$ is nabla differentiable at ζ, and

$${\left(\psi \omega \right)}^{\nabla }\left(\zeta \right)={\psi }^{\nabla }\left(\zeta \right)\omega \left(\zeta \right)+{\psi }^{\rho }\left(\zeta \right){\omega }^{\nabla }\left(\zeta \right)=\psi \left(\zeta \right){\omega }^{\nabla }\left(\zeta \right)+{\psi }^{\nabla }\left(\zeta \right){\omega }^{\rho }\left(\zeta \right).$$

2. 

If $\omega \left(\zeta \right){\omega }^{\rho }\left(\zeta \right)\ne 0$, then $\psi /\omega$ is nabla differentiable at ζ, and

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

Lemma 1

([15]). Assume $\psi :\mathbb{R}\to \mathbb{R}$ is continuously differentiable and $\omega :\mathbb{R}\to \mathbb{R},$ $\omega :\mathbb{T}\to \mathbb{R}$ is continuous and nabla differentiable, respectively. Then

$${\left(\psi \circ \omega \right)}^{\nabla }\left(\zeta \right)={\psi }^{\prime }\left(\omega \left(d\right)\right){\omega }^{\nabla }\left(\zeta \right),d\in [\rho \left(\zeta \right),\zeta ].$$

(7)

Definition 2

([14]). Assume $\psi :\mathbb{T}\to \mathbb{R}$ is a continuous function. If the continuity of ψ at all left-dense points in $\mathbb{T}$ and the existence of the limits of the right-sided at all right-dense points in $\mathbb{T}$ are achieved. The space of $ld$-continuous functions is represented by ${\mathrm{C}}_{ld}(\mathbb{T},\mathbb{R})$.

Definition 3

([14]). A function F is referred to as a nabla antiderivative of ψ if $F^{\nabla }\left(\zeta \right)=\psi \left(\zeta \right)$ holds for every $\zeta \in \mathbb{T}$. As a result, the nabla integral of ψ is

$${\int }_{\alpha }^{\zeta }\psi \left(s\right)\nabla s=F\left(\zeta \right)-F\left(\alpha \right),\forall \zeta \in \mathbb{T}.$$

Theorem 2

([14]). If $\alpha ,\gamma ,c\in \mathbb{T},$ $\beta ,\lambda \in \mathbb{R}$ and $\psi ,\omega :\mathbb{T}\to \mathbb{R}$ are $ld$-continuous, then

1. 

${\int }_{\alpha }^{\gamma }\left[\beta \psi \left(\zeta \right)+\lambda \omega \left(\zeta \right)\right]\Delta \zeta =\beta {\int }_{\alpha }^{\gamma }\psi \left(\zeta \right)\nabla \zeta +\lambda {\int }_{\alpha }^{\gamma }\omega \left(\zeta \right)\nabla \zeta$,

2. 

${\int }_{\alpha }^{\gamma }\psi \left(\zeta \right)\nabla \zeta =-{\int }_{\gamma }^{\alpha }\psi \left(\zeta \right)\nabla \zeta$,

3. 

${\int }_{\alpha }^c\psi \left(\zeta \right)\nabla \zeta ={\int }_{\alpha }^{\gamma }\psi \left(\zeta \right)\nabla \zeta +{\int }_{\gamma }^c\psi \left(\zeta \right)\nabla \zeta$,

4. 

$\left|{\int }_{\alpha }^{\gamma }\psi \left(\zeta \right)\nabla \zeta \right|\le {\int }_{\alpha }^{\gamma }\left|\psi \left(\zeta \right)\right|\nabla \zeta ,$

5. 

${\int }_{\alpha }^{\gamma }\psi \left(\zeta \right){\omega }^{\nabla }\left(\zeta \right)\nabla \zeta ={\left.\psi \left(\zeta \right)\omega \left(\zeta \right)\right|}_{\alpha }^{\gamma }-{\int }_{\alpha }^{\gamma }{\psi }^{\nabla }\left(\zeta \right){\omega }^{\rho }\left(\zeta \right)\nabla \zeta ,$

6. 

${\int }_{\alpha }^{\gamma }{\psi }^{\rho }\left(\zeta \right){\omega }^{\nabla }\left(\zeta \right)\nabla \zeta ={\left.\psi \left(\zeta \right)\omega \left(\zeta \right)\right|}_{\alpha }^{\gamma }-{\int }_{\alpha }^{\gamma }{\psi }^{\nabla }\left(\zeta \right)\omega \left(\zeta \right)\nabla \zeta .$

Definition 4

([3]). Let $\psi :\mathbb{T}\to \mathbb{R}$ be a function and $C\ge 1.$ If $s\le \zeta \Rightarrow$ $\psi \left(\zeta \right)\le C\psi \left(s\right),$ then ψ is $C$-decreasing. If $s\le \zeta$⇒$\psi \left(s\right)\le C\psi \left(\zeta \right),$ then ψ is $C$-increasing. When $C=1,$ we find that the $1$-decreasing is decreasing and the $1$-increasing is increasing.

Lemma 2.

If $0<r<\infty ,$ ψ is $C$-decreasing for $C\ge 1$, then ${\psi }^r$ is $C^r$-decreasing and if ω is $C$-increasing, then ${\omega }^r$ is $C^r$-increasing.

Proof. 

Because $\psi$ is $C$-decreasing, then for $s\le \zeta$, we obtain $\psi \left(\zeta \right)\le C\psi \left(s\right)$. As a result, we have (where $r>0$) that

$${\psi }^r\left(\zeta \right)\le C^r{\psi }^r\left(s\right).$$

Thus, ${\psi }^r$ is $C^r$-decreasing.

Now, because $\omega$ is $C$-increasing, then for $s\le \zeta$, we obtain $\omega \left(s\right)\le C\omega \left(\zeta \right).$ Therefore, we find (where $r>0$) that

$${\omega }^r\left(s\right)\le C^r{\omega }^r\left(\zeta \right),$$

This shows that ${\omega }^r$ is $C^r$-increasing. □

3. Main Results

In this section, the functions in the theorems below are ld-continuous non-negative and $\nabla$—differentiable functions, locally $\nabla$—integrable on ${[\alpha ,\gamma ]}_{\mathbb{T}}$ and the integrals are assumed to be exist.

Theorem 3.

Assume that $\alpha ,\gamma \in \mathbb{T},$ $r>0,$ $0<\epsilon <1.$ In addition, let χ be $C^r$-decreasing on ${[\alpha ,\gamma ]}_{\mathbb{T}},$ $C\ge 1$ and ϑ be increasing on ${[\alpha ,\gamma ]}_{\mathbb{T}}$, such that $\vartheta \left(\alpha \right)=0$ and

$${\vartheta }^{\epsilon -1}\left(\zeta \right){\vartheta }^{1-\epsilon }\left(\rho \left(\zeta \right)\right){\chi }^{1-\epsilon }(\rho \left(\zeta \right)\ge {\chi }^{1-\epsilon }\left(\zeta \right).$$

(8)

Then

$$\left({\int }_{\alpha }^{\gamma }\chi \left(z\right){\left[\vartheta \left(z\right)\right]}^{\nabla }\nabla z\right)\le C^{r\left(1-\epsilon \right)}{\left({\int }_{\alpha }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}.$$

(9)

Proof. 

From the hypotheses of the Theorem, we find for $z\le \zeta$ that $\chi \left(\zeta \right)\le C^r\chi \left(z\right),$ and

$$\begin{array}{ccc}\hfill {\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z& \ge & {\left(\frac{1}{C}\right)}^{r\epsilon }{\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(\zeta \right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\hfill \\ & =& {\left(\frac{1}{C}\right)}^{r\epsilon }{\chi }^{\epsilon }\left(\zeta \right){\int }_{\alpha }^{\zeta }{\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\hfill \\ & =& {\left(\frac{1}{C}\right)}^{r\epsilon }{\chi }^{\epsilon }\left(\zeta \right)\left[{\vartheta }^{\epsilon }\left(\zeta \right)-{\vartheta }^{\epsilon }\left(\alpha \right)\right]\hfill \\ & =& {\left(\frac{1}{C}\right)}^{r\epsilon }{\chi }^{\epsilon }\left(\zeta \right){\vartheta }^{\epsilon }\left(\zeta \right).\hfill \end{array}$$

(10)

Hence,

$${\chi }^{\epsilon }\left(\zeta \right){\vartheta }^{\epsilon }\left(\zeta \right)\le C^{r\epsilon }{\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z.$$

(11)

Consider

$$F\left(\zeta \right)=C^{r\left(1-\epsilon \right)}{\left({\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}-{\int }_{\alpha }^{\zeta }\chi \left(z\right){\left[\vartheta \left(z\right)\right]}^{\nabla }\nabla z.$$

(12)

By applying (7) on

$${\left({\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }},$$

we find, for $\xi \in \left[\rho \right(\zeta ),\zeta ]$, that

$$\begin{array}{cc}\hfill & {\left[{\left({\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}\right]}^{\nabla }\hfill \\ \hfill & =\frac{1}{\epsilon }{\left({\int }_{\alpha }^{\xi }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right){\left[{\vartheta }^{\epsilon }\left(\zeta \right)\right]}^{\nabla }.\hfill \end{array}$$

(13)

Again by applying (7) on ${\vartheta }^{\epsilon }\left(\zeta \right),$ we obtain

$${\left[{\vartheta }^{\epsilon }\left(\zeta \right)\right]}^{\nabla }=\epsilon {\vartheta }^{\epsilon -1}\left(\xi \right){\vartheta }^{\nabla }\left(\zeta \right),$$

(14)

where $\xi \in \left[\rho \right(\zeta ),\zeta ].$ From (12), we observe

$$F^{\nabla }\left(\zeta \right)=C^{r\left(1-\epsilon \right)}{\left[{\left({\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}\right]}^{\nabla }-\chi \left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right).$$

(15)

Using (13) and (14) in (15), we obtain

$$\begin{array}{cc}\hfill F^{\nabla }\left(\zeta \right)& =\frac{1}{\epsilon }{C}^{r\left(1-\epsilon \right)}{\left({\int }_{\alpha }^{\xi }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right){\left[{\vartheta }^{\epsilon }\left(\zeta \right)\right]}^{\nabla }-\chi \left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right)\hfill \\ \hfill & =C^{r\left(1-\epsilon \right)}{\vartheta }^{\epsilon -1}\left(\xi \right){\left({\int }_{\alpha }^{\xi }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right)-\chi \left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right)\hfill \\ \hfill & ={\vartheta }^{\nabla }\left(\zeta \right)\left[C^{r\left(1-\epsilon \right)}{\vartheta }^{\epsilon -1}\left(\xi \right){\left({\int }_{\alpha }^{\xi }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right)-\chi \left(\zeta \right)\right].\hfill \end{array}$$

(16)

Since $\xi \in \left[\rho \right(\zeta ),\zeta ],$ $0<\epsilon <1$ and $\vartheta$ is an increasing, then

$$\begin{array}{cc}\hfill & {\vartheta }^{\epsilon -1}\left(\xi \right){\left({\int }_{\alpha }^{\xi }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}\hfill \\ \hfill & \ge {\vartheta }^{\epsilon -1}\left(\zeta \right){\left({\int }_{\alpha }^{\rho \left(\zeta \right)}{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}.\hfill \end{array}$$

(17)

Substituting (17) into (16), we observe

$$F^{\nabla }\left(\zeta \right)\ge {\vartheta }^{\nabla }\left(\zeta \right)\left[C^{r\left(1-\epsilon \right)}{\vartheta }^{\epsilon -1}\left(\zeta \right){\left({\int }_{\alpha }^{\rho \left(\zeta \right)}{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right)-\chi \left(\zeta \right)\right].$$

(18)

Substituting (11) into (18), we obtain

$$\begin{array}{ccc}\hfill F^{\nabla }\left(\zeta \right)& \ge & {\vartheta }^{\nabla }\left(\zeta \right)\left[{\vartheta }^{\epsilon -1}\left(\zeta \right){\left[{\chi }^{\epsilon }\left(\rho \left(\zeta \right)\right){\vartheta }^{\epsilon }\left(\rho \left(\zeta \right)\right)\right]}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right)-\chi \left(\zeta \right)\right]\hfill \\ & =& {\vartheta }^{\nabla }\left(\zeta \right)\left[{\vartheta }^{\epsilon -1}\left(\zeta \right){\vartheta }^{1-\epsilon }\left(\rho \left(\zeta \right)\right){\chi }^{1-\epsilon }\left(\rho \left(\zeta \right)\right){\chi }^{\epsilon }\left(\zeta \right)-\chi \left(\zeta \right)\right].\hfill \end{array}$$

(19)

Using (8) and $\vartheta$ is an increasing, we obtain from (19) that

$$F^{\nabla }\left(\zeta \right)\ge 0.$$

Thus, F is increasing on ${[\alpha ,\gamma ]}_{\mathbb{T}}.$

Now, because F is an increasing, we have $F\left(\gamma \right)\ge F\left(\alpha \right)$ for $\gamma >\alpha$ and then (note that $F\left(\alpha \right)=0$)

$${\int }_{\alpha }^{\gamma }\chi \left(z\right){\left[\vartheta \left(z\right)\right]}^{\nabla }\nabla z\le C^{r\left(1-\epsilon \right)}{\left({\int }_{\alpha }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }},$$

which is (9). □

Remark 1.

If $\mathbb{T}=\mathbb{R},$ $C=1,$ then $\rho \left(\zeta \right)=\zeta$ and (8) holds with equality. Thus, we obtain the inequality (1).

As a specific instance of Theorem 3, if $0<\epsilon =p/r<1,$ $\chi \left(z\right)={\psi }^r\left(z\right)$ and $\vartheta \left(z\right)={\omega }^r\left(z\right)$, we obtain the result below.

Corollary 1.

Assume that $0<p<r<\infty ,$ ψ is $C$-decreasing on ${[\alpha ,\gamma ]}_{\mathbb{T}},$ $C\ge 1$ and ω is increasing on ${[\alpha ,\gamma ]}_{\mathbb{T}},$ with $\omega \left(\alpha \right)=0.$ Let

$${\omega }^{p-r}\left(\zeta \right){\omega }^{r-p}\left(\rho \left(\zeta \right)\right){\psi }^{r-p}(\rho \left(\zeta \right)\ge {\psi }^{r-p}\left(\zeta \right).$$

(20)

Then

$${\left({\int }_{\alpha }^{\gamma }{\psi }^r\left(z\right){\left[{\omega }^r\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{r}}\le C^{1-(p/r)}{\left({\int }_{\alpha }^{\gamma }{\psi }^p\left(z\right){\left[{\omega }^p\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{p}}.$$

(21)

Remark 2.

In Corollary 1, if $\mathbb{T}=\mathbb{R},$ then $\rho \left(\zeta \right)=\zeta$ and (20) holds with equality. Thus, we obtain the inequality (3).

Remark 3.

In Corollary 1, if $\mathbb{T}=\mathbb{N},$ $\alpha ,\gamma \in \mathbb{N},$ $0<p<r<\infty ,$ ψ is $C$-decreasing sequence for $C\ge 1,$ ω is increasing with $\omega \left(\alpha \right)=0,$ then $\rho \left(i\right)=i-1$ and

$${\omega }^{p-r}\left(i\right){\omega }^{r-p}(i-1){\psi }^{r-p}(i-1)\ge {\psi }^{r-p}\left(i\right).$$

Thus,

$${\left(\sum _{i=\alpha }^{\gamma }{\psi }^r\left(i\right)\nabla \left[{\omega }^r\left(i\right)\right]\right)}^{\frac{1}{r}}\le C^{1-(p/r)}{\left(\sum _{i=\alpha }^{\gamma }{\psi }^p\left(i\right)\nabla \left[{\omega }^p\left(i\right)\right]\right)}^{\frac{1}{p}}.$$

Theorem 4.

Assume that $\alpha ,\gamma \in \mathbb{T},$ $r>0,$ $0<\epsilon <1.$ In addition, let χ be $C^r$-increasing on ${[\alpha ,\gamma ]}_{\mathbb{T}},$ $C\ge 1,$ ϑ be increasing on ${[\alpha ,\gamma ]}_{\mathbb{T}},$ with $\vartheta \left(\alpha \right)=0,$ and

$${\vartheta }^{\epsilon -1}\left(\rho \left(\zeta \right)\right){\vartheta }^{1-\epsilon }\left(\zeta \right)\le 1.$$

(22)

Then

$$\left({\int }_{\alpha }^{\gamma }\chi \left(z\right){\left[\vartheta \left(z\right)\right]}^{\nabla }\nabla z\right)\ge C^{r\left(\epsilon -1\right)}{\left({\int }_{\alpha }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}.$$

(23)

Proof. 

From the hypotheses of the Theorem, we have $\chi \left(z\right)\le C^r\chi \left(\zeta \right)$ for $z\le \zeta$ and

$$\begin{array}{ccc}\hfill {\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z& \le & C^{r\epsilon }{\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(\zeta \right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\hfill \\ & =& C^{r\epsilon }{\chi }^{\epsilon }\left(\zeta \right){\int }_{\alpha }^{\zeta }{\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\hfill \\ & =& C^{r\epsilon }{\chi }^{\epsilon }\left(\zeta \right)\left[{\vartheta }^{\epsilon }\left(\zeta \right)-{\vartheta }^{\epsilon }\left(\alpha \right)\right]\hfill \\ & =& C^{r\epsilon }{\chi }^{\epsilon }\left(\zeta \right){\vartheta }^{\epsilon }\left(\zeta \right).\hfill \end{array}$$

(24)

Hence

$${\chi }^{\epsilon }\left(\zeta \right){\vartheta }^{\epsilon }\left(\zeta \right)\ge \frac{1}{C^{r\epsilon }}{\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z.$$

(25)

Consider the function

$$F\left(\zeta \right)=C^{r\left(\epsilon -1\right)}{\left({\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}-{\int }_{\alpha }^{\zeta }\chi \left(z\right){\left[\vartheta \left(z\right)\right]}^{\nabla }\nabla z.$$

(26)

By applying (7) on

$${\left({\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }},$$

we have for $\xi \in \left[\rho \right(\zeta ),\zeta ]$ that

$$\begin{array}{cc}\hfill & {\left[{\left({\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}\right]}^{\nabla }\hfill \\ \hfill & =\frac{1}{\epsilon }{\left({\int }_{\alpha }^{\xi }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right){\left[{\vartheta }^{\epsilon }\left(\zeta \right)\right]}^{\nabla }.\hfill \end{array}$$

(27)

Again, by applying (7) on ${\vartheta }^{\epsilon }\left(\zeta \right),$ we obtain

$${\left[{\vartheta }^{\epsilon }\left(\zeta \right)\right]}^{\nabla }=\epsilon {\vartheta }^{\epsilon -1}\left(\xi \right){\vartheta }^{\nabla }\left(\zeta \right),$$

(28)

where $\xi \in \left[\rho \right(\zeta ),\zeta ].$ From (26), we observe

$$F^{\nabla }\left(\zeta \right)=C^{r\left(\epsilon -1\right)}{\left[{\left({\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}\right]}^{\nabla }-\chi \left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right).$$

(29)

Substituting (27) and (28) into (29), we obtain

$$\begin{array}{cc}\hfill F^{\nabla }\left(\zeta \right)& =\frac{1}{\epsilon }{C}^{r\left(\epsilon -1\right)}{\left({\int }_{\alpha }^{\xi }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right){\left[{\vartheta }^{\epsilon }\left(\zeta \right)\right]}^{\nabla }-\chi \left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right)\hfill \\ \hfill & =C^{r\left(\epsilon -1\right)}{\vartheta }^{\epsilon -1}\left(\xi \right){\left({\int }_{\alpha }^{\xi }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right)-\chi \left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right)\hfill \\ \hfill & ={\vartheta }^{\nabla }\left(\zeta \right)\left[C^{r\left(\epsilon -1\right)}{\vartheta }^{\epsilon -1}\left(\xi \right){\left({\int }_{\alpha }^{\xi }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right)-\chi \left(\zeta \right)\right].\hfill \end{array}$$

(30)

Since $\xi \in \left[\rho \right(\zeta ),\zeta ],$ $0<\epsilon <1$ and $\vartheta$ is an increasing, then

$$\begin{array}{cc}\hfill & {\vartheta }^{\epsilon -1}\left(\xi \right){\left({\int }_{\alpha }^{\xi }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}\hfill \\ \hfill & \le {\vartheta }^{\epsilon -1}\left(\rho \left(\zeta \right)\right){\left({\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}.\hfill \end{array}$$

(31)

Substituting (31) into (30), we see

$$F^{\nabla }\left(\zeta \right)\le {\vartheta }^{\nabla }\left(\zeta \right)\left[C^{r\left(\epsilon -1\right)}{\vartheta }^{\epsilon -1}\left(\rho \left(\zeta \right)\right){\left({\int }_{\alpha }^{\zeta }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right)-\chi \left(\zeta \right)\right].$$

(32)

Substituting (25) into (32), we have

$$\begin{array}{ccc}\hfill F^{\nabla }\left(\zeta \right)& \le & {\vartheta }^{\nabla }\left(\zeta \right)\left[{\vartheta }^{\epsilon -1}\left(\rho \left(\zeta \right)\right){\left[{\chi }^{\epsilon }\left(\zeta \right){\vartheta }^{\epsilon }\left(\zeta \right)\right]}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right)-\chi \left(\zeta \right)\right]\hfill \\ & =& {\vartheta }^{\nabla }\left(\zeta \right)\left[{\vartheta }^{\epsilon -1}\left(\rho \left(\zeta \right)\right){\vartheta }^{1-\epsilon }\left(\zeta \right)\chi \left(\zeta \right)-\chi \left(\zeta \right)\right].\hfill \end{array}$$

(33)

By (22) and $\vartheta$ is an increasing, (33) is

$$F^{\nabla }\left(\zeta \right)\le 0.$$

Thus, F is decreasing on ${[\alpha ,\gamma ]}_{\mathbb{T}}.$

Now, because F is a decreasing, we have for $\gamma >\alpha$ that $F\left(\gamma \right)\le F\left(\alpha \right)$ and then (note that $F\left(\alpha \right)=0$)

$$C^{r\left(\epsilon -1\right)}{\left({\int }_{\alpha }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}\le {\int }_{\alpha }^{\gamma }\chi \left(z\right){\left[\vartheta \left(z\right)\right]}^{\nabla }\nabla z,$$

which is (23). □

In Theorem 4, if $0<p<r<\infty ,$ $0<\epsilon =p/r<1,$ $\chi \left(z\right)={\psi }^r\left(z\right)$ and $\vartheta \left(z\right)={\omega }^r\left(z\right)$, we obtain the result below.

Corollary 2.

Assume that $\alpha ,\gamma \in \mathbb{T}$ and $0<p<r<\infty .$ In addition, let ψ be $C$-increasing on ${[\alpha ,\gamma ]}_{\mathbb{T}}$ for $C\ge 1,$ ω be increasing on ${[\alpha ,\gamma ]}_{\mathbb{T}},$ with $\omega \left(\alpha \right)=0,$ and

$${\left[{\omega }^{\rho }\left(\zeta \right)\right]}^{p-r}{\omega }^{r-p}\left(\zeta \right)\le 1.$$

(34)

Then

$${\left({\int }_{\alpha }^{\gamma }{\psi }^r\left(z\right){\left[{\omega }^r\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{r}}\ge C^{(p/r)-1}{\left({\int }_{\alpha }^{\gamma }{\psi }^p\left(z\right){\left[{\omega }^p\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{p}}.$$

Remark 4.

In Corollary 2, if $\mathbb{T}=\mathbb{R},$ then $\rho \left(\zeta \right)=\zeta$ and (34) holds with equality. Thus, we obtain the inequality (4).

Remark 5.

In Corollary 2, if $\mathbb{T}=\mathbb{N},$ $\alpha ,\gamma \in \mathbb{N},$ $0<p<r<\infty ,$ ψ is $C$-increasing sequence for $C\ge 1,$ ω is increasing with $\omega \left(\alpha \right)=0,$ then $\rho \left(i\right)=i-1$ and

$${\left[\omega (i-1)\right]}^{p-r}{\omega }^{r-p}\left(i\right)\le 1.$$

Thus,

$${\left(\sum _{i=\alpha }^{\gamma }{\psi }^r\left(i\right)\nabla {\omega }^r\left(i\right)\right)}^{\frac{1}{r}}\ge C^{(p/r)-1}{\left(\sum _{i=\alpha }^{\gamma }{\int }_{\alpha }^{\gamma }{\psi }^p\left(i\right)\nabla {\omega }^p\left(i\right)\right)}^{\frac{1}{p}}.$$

Theorem 5.

Assume $\alpha ,\gamma \in \mathbb{T},$ $r>0$ and $0<\epsilon <1.$ In addition, let χ be $C^r$-increasing on ${[\alpha ,\gamma ]}_{\mathbb{T}},$ $C\ge 1,$ ϑ be decreasing on ${[\alpha ,\gamma ]}_{\mathbb{T}},$ such that $\vartheta \left(\gamma \right)=0$ and

$${\vartheta }^{\epsilon -1}\left(\rho \left(\zeta \right)\right){\vartheta }^{1-\epsilon }\left(\zeta \right)\ge 1.$$

(35)

Then

$${\int }_{\alpha }^{\gamma }\chi \left(z\right){\left[-\vartheta \left(z\right)\right]}^{\nabla }\nabla z\le C^{r\left(1-\epsilon \right)}{\left({\int }_{\alpha }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}.$$

(36)

Proof. 

From the hypotheses of the Theorem, we have $\chi \left(\zeta \right)\le C^r\chi \left(z\right),$ for $z\ge \zeta$ and

$$\begin{array}{ccc}\hfill {\int }_{\zeta }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z& \ge & \frac{1}{C^{r\epsilon }}{\int }_{\zeta }^{\gamma }{\chi }^{\epsilon }\left(\zeta \right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\hfill \\ & =& \frac{1}{C^{r\epsilon }}{\chi }^{\epsilon }\left(\zeta \right){\int }_{\zeta }^{\gamma }{\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\hfill \\ & =& \frac{1}{C^{r\epsilon }}{\chi }^{\epsilon }\left(\zeta \right)\left[{\vartheta }^{\epsilon }\left(\zeta \right)-{\vartheta }^{\epsilon }\left(\gamma \right)\right]\hfill \\ & =& \frac{1}{C^{r\epsilon }}{\chi }^{\epsilon }\left(\zeta \right){\vartheta }^{\epsilon }\left(\zeta \right).\hfill \end{array}$$

(37)

Consider the function

$$F\left(\zeta \right)=C^{r\left(1-\epsilon \right)}{\left({\int }_{\zeta }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}-{\int }_{\zeta }^{\gamma }\chi \left(z\right){\left[-\vartheta \left(z\right)\right]}^{\nabla }\nabla z.$$

(38)

By applying (7) on

$${\left({\int }_{\zeta }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }},$$

we obtain, for $\xi \in \left[\rho \right(\zeta ),\zeta ]$, that

$$\begin{array}{cc}\hfill & {\left[{\left({\int }_{\zeta }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}\right]}^{\nabla }\hfill \\ \hfill & =\frac{1}{\epsilon }{\left({\int }_{\xi }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right){\left[{\vartheta }^{\epsilon }\left(\zeta \right)\right]}^{\nabla }.\hfill \end{array}$$

(39)

Again by applying (7) on ${\vartheta }^{\epsilon }\left(\zeta \right),$ we obtain

$${\left[{\vartheta }^{\epsilon }\left(\zeta \right)\right]}^{\nabla }=\epsilon {\vartheta }^{\epsilon -1}\left(\xi \right){\vartheta }^{\nabla }\left(\zeta \right),$$

(40)

where $\xi \in \left[\rho \right(\zeta ),\zeta ].$ From (38), we find

$$F^{\nabla }\left(\zeta \right)=C^{r\left(1-\epsilon \right)}{\left[{\left({\int }_{\zeta }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}\right]}^{\nabla }-\chi \left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right).$$

(41)

Substituting (39) and (40) into (41), we obtain

$$\begin{array}{cc}\hfill F^{\nabla }\left(\zeta \right)& =\frac{1}{\epsilon }{C}^{r\left(1-\epsilon \right)}{\left({\int }_{\xi }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right){\left[{\vartheta }^{\epsilon }\left(\zeta \right)\right]}^{\nabla }-\chi \left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right)\hfill \\ \hfill & =C^{r\left(1-\epsilon \right)}{\vartheta }^{\epsilon -1}\left(\xi \right){\left({\int }_{\xi }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right)-\chi \left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right)\hfill \\ \hfill & =\left[-{\vartheta }^{\nabla }\left(\zeta \right)\right]\left[-C^{r\left(1-\epsilon \right)}{\vartheta }^{\epsilon -1}\left(\xi \right){\left({\int }_{\xi }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right)+\chi \left(\zeta \right)\right].\hfill \end{array}$$

(42)

Since $\xi \in \left[\rho \right(\zeta ),\zeta ],$ $0<\epsilon <1$ and $\vartheta$ is a decreasing, then

$$\begin{array}{ccc}& & {\vartheta }^{\epsilon -1}\left(\xi \right){\left({\int }_{\xi }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}\hfill \\ & \ge & {\vartheta }^{\epsilon -1}\left(\rho \left(\zeta \right)\right){\left({\int }_{\zeta }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}.\hfill \end{array}$$

(43)

Substituting (43) into (42), we see

$$F^{\nabla }\left(\zeta \right)\le \left[-{\vartheta }^{\nabla }\left(\zeta \right)\right]\left[-C^{r\left(1-\epsilon \right)}{\vartheta }^{\epsilon -1}\left(\rho \left(\zeta \right)\right){\left({\int }_{\zeta }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right)+\chi \left(\zeta \right)\right].$$

(44)

Substituting (37) into (44), we obtain

$$F^{\nabla }\left(\zeta \right)\le \left[-{\vartheta }^{\nabla }\left(\zeta \right)\right]\left[-{\vartheta }^{\epsilon -1}\left(\rho \left(\zeta \right)\right){\vartheta }^{1-\epsilon }\left(\zeta \right)\chi \left(\zeta \right)+\chi \left(\zeta \right)\right].$$

(45)

By (35) and using the fact $\vartheta$ is a decreasing, we have

$$F^{\nabla }\left(\zeta \right)\le 0.$$

Thus, F is decreasing on ${[\alpha ,\gamma ]}_{\mathbb{T}}.$

Now, because F is a decreasing, we obtain $F\left(\gamma \right)\le F\left(\alpha \right)$ for $\gamma >\alpha$ and then (note that $F\left(\gamma \right)=0$)

$${\int }_{\alpha }^{\gamma }\chi \left(z\right){\left[-\vartheta \left(z\right)\right]}^{\nabla }\nabla z\le C^{r\left(1-\epsilon \right)}{\left({\int }_{\alpha }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }},$$

which is (36). □

Remark 6.

In Theorem 5, when $\mathbb{T}=\mathbb{R},$ $\rho \left(\zeta \right)=\zeta$ and $C=1,$ then (35) holds with equality. Thus, we obtain the inequality (2).

As a specific instance of Theorem 5, if $0<\epsilon =p/r<1,$ $\chi \left(x\right)={\psi }^r\left(x\right)$ and $\vartheta \left(x\right)={\omega }^r\left(x\right)$, we obtain the result below.

Corollary 3.

Assume $0<p<r<\infty ,$ ψ is $C$-increasing on ${[\alpha ,\gamma ]}_{\mathbb{T}},$ $C\ge 1,$ ω is decreasing on ${[\alpha ,\gamma ]}_{\mathbb{T}}$ with $\omega \left(\gamma \right)=0$ and

$${\left[{\omega }^{\rho }\left(\zeta \right)\right]}^{p-r}{\omega }^{r-p}\left(\zeta \right)\ge 1.$$

(46)

Then

$${\left({\int }_{\alpha }^{\gamma }{\psi }^r\left(z\right){\left[-{\omega }^r\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{r}}\le C^{1-(p/r)}{\left({\int }_{\alpha }^{\gamma }{\psi }^p\left(z\right){\left[-{\omega }^p\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{p}}.$$

Remark 7.

In Corollary 3, if $\mathbb{T}=\mathbb{R},$ then $\rho \left(\zeta \right)=\zeta$ and (46) holds with equality. Thus, we obtain the inequality (5).

Remark 8.

In Corollary 3, if $\mathbb{T}=\mathbb{N},$ $\alpha ,\gamma \in \mathbb{N},$ $0<p<r<\infty ,$ ψ is $C$-increasing sequence for $C\ge 1,$ ω is decreasing, such that $\omega \left(\gamma \right)=0,$ then $\rho \left(i\right)=i-1$ and

$${\omega }^{p-r}(i-1){\omega }^{r-p}\left(i\right)\ge 1.$$

Thus,

$${\left(\sum _{i=\alpha }^{\gamma }{\psi }^r\left(i\right)\nabla \left[-{\omega }^r\left(i\right)\right]\right)}^{\frac{1}{r}}\le C^{1-(p/r)}{\left(\sum _{i=\alpha }^{\gamma }{\psi }^p\left(i\right)\nabla \left[-{\omega }^p\left(i\right)\right]\right)}^{\frac{1}{p}}.$$

Theorem 6.

Assume that $\alpha ,\gamma \in \mathbb{T},$ $r>0$, and $0<\epsilon <1.$ In addition, let χ be $C^r$-decreasing on ${[\alpha ,\gamma ]}_{\mathbb{T}},$ $C\ge 1,$ ϑ be decreasing on ${[\alpha ,\gamma ]}_{\mathbb{T}}$, such that $\vartheta \left(\gamma \right)=0$ and

$${\chi }^{1-\epsilon }\left(\zeta \right){\chi }^{\epsilon -1}\left(\rho \left(\zeta \right)\right)\ge {\vartheta }^{\epsilon -1}\left(\zeta \right){\vartheta }^{1-\epsilon }\left(\rho \left(\zeta \right)\right).$$

(47)

Then

$${\int }_{\alpha }^{\gamma }\chi \left(z\right){\left[-\vartheta \left(z\right)\right]}^{\nabla }\nabla z\ge C^{r\left(\epsilon -1\right)}{\left({\int }_{\alpha }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}.$$

(48)

Proof. 

From the hypotheses of the Theorem, we obtain $\chi \left(z\right)\le C^r\chi \left(\zeta \right),$ for $z\ge \zeta$ and

$$\begin{array}{ccc}\hfill {\int }_{\zeta }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z& \le & C^{r\epsilon }{\int }_{\zeta }^{\gamma }{\chi }^{\epsilon }\left(\zeta \right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\hfill \\ & =& C^{r\epsilon }{\chi }^{\epsilon }\left(\zeta \right){\int }_{\zeta }^{\gamma }{\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\hfill \\ & =& C^{r\epsilon }{\chi }^{\epsilon }\left(\zeta \right)\left[{\vartheta }^{\epsilon }\left(\zeta \right)-{\vartheta }^{\epsilon }\left(\gamma \right)\right]\hfill \\ & =& C^{r\epsilon }{\chi }^{\epsilon }\left(\zeta \right){\vartheta }^{\epsilon }\left(\zeta \right).\hfill \end{array}$$

(49)

Consider the function

$$F\left(\zeta \right)=C^{r\left(\epsilon -1\right)}{\left({\int }_{\zeta }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}-{\int }_{\zeta }^{\gamma }\chi \left(z\right){\left[-\vartheta \left(z\right)\right]}^{\nabla }\nabla z.$$

(50)

By applying (7) on

$${\left({\int }_{\zeta }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }},$$

we obtain for $\xi \in \left[\rho \right(\zeta ),\zeta ]$ that

$$\begin{array}{cc}\hfill & {\left[{\left({\int }_{\zeta }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}\right]}^{\nabla }\hfill \\ \hfill & =\frac{1}{\epsilon }{\left({\int }_{\xi }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right){\left[{\vartheta }^{\epsilon }\left(\zeta \right)\right]}^{\nabla }.\hfill \end{array}$$

(51)

Again, by applying on ${\vartheta }^{\epsilon }\left(\zeta \right),$ we obtain

$${\left[{\vartheta }^{\epsilon }\left(\zeta \right)\right]}^{\nabla }=\epsilon {\vartheta }^{\epsilon -1}\left(\xi \right){\vartheta }^{\nabla }\left(\zeta \right),$$

(52)

where $\xi \in \left[\rho \right(\zeta ),\zeta ].$ From (50), we have

$$F^{\nabla }\left(\zeta \right)=C^{r\left(\epsilon -1\right)}{\left[{\left({\int }_{\zeta }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }}\right]}^{\nabla }-\chi \left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right).$$

(53)

Substituting (51) and (52) into (53), we obtain

$$\begin{array}{cc}\hfill F^{\nabla }\left(\zeta \right)& =\frac{1}{\epsilon }{C}^{r\left(\epsilon -1\right)}{\left({\int }_{\xi }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right){\left[{\vartheta }^{\epsilon }\left(\zeta \right)\right]}^{\nabla }-\chi \left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right)\hfill \\ \hfill & =C^{r\left(\epsilon -1\right)}{\vartheta }^{\epsilon -1}\left(\xi \right){\left({\int }_{\xi }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right)-\chi \left(\zeta \right){\vartheta }^{\nabla }\left(\zeta \right)\hfill \\ \hfill & =\left[-{\vartheta }^{\nabla }\left(\zeta \right)\right]\left[-C^{r\left(\epsilon -1\right)}{\vartheta }^{\epsilon -1}\left(\xi \right){\left({\int }_{\xi }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right)+\chi \left(\zeta \right)\right].\hfill \end{array}$$

(54)

Since $\xi \in \left[\rho \right(\zeta ),\zeta ],$ $0<\epsilon <1$ and $\vartheta$ is a decreasing, we have

$$\begin{array}{cc}\hfill & {\vartheta }^{\epsilon -1}\left(\xi \right){\left({\int }_{\xi }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}\hfill \\ \hfill & \le {\vartheta }^{\epsilon -1}\left(\zeta \right){\left({\int }_{\rho \left(\zeta \right)}^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}.\hfill \end{array}$$

(55)

Substituting (55) into (54), we observe

$$F^{\nabla }\left(\zeta \right)\ge \left[-{\vartheta }^{\nabla }\left(\zeta \right)\right]\left[-C^{r\left(\epsilon -1\right)}{\vartheta }^{\epsilon -1}\left(\zeta \right){\left({\int }_{\rho \left(\zeta \right)}^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }-1}{\chi }^{\epsilon }\left(\zeta \right)+\chi \left(\zeta \right)\right].$$

(56)

Substituting (49) into (56), we obtain

$$F^{\nabla }\left(\zeta \right)\ge \left[-{\vartheta }^{\nabla }\left(\zeta \right)\right]\left[-{\vartheta }^{\epsilon -1}\left(\zeta \right){\chi }^{1-\epsilon }\left(\rho \left(\zeta \right)\right){\vartheta }^{1-\epsilon }\left(\rho \left(\zeta \right)\right){\chi }^{\epsilon }\left(\zeta \right)+\chi \left(\zeta \right)\right].$$

(57)

By (47) and using the fact $\vartheta$ is a decreasing, we have

$$F^{\nabla }\left(\zeta \right)\ge 0.$$

Thus, F is increasing on ${[\alpha ,\gamma ]}_{\mathbb{T}}.$

Now, because F is an increasing, we obtain $F\left(\gamma \right)\ge F\left(\alpha \right)$ for $\gamma >\alpha$ and then (note that $F\left(\gamma \right)=0$)

$${\int }_{\alpha }^{\gamma }\chi \left(z\right){\left[-\vartheta \left(z\right)\right]}^{\nabla }\nabla z\ge C^{r\left(\epsilon -1\right)}{\left({\int }_{\alpha }^{\gamma }{\chi }^{\epsilon }\left(z\right){\left[-{\vartheta }^{\epsilon }\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{\epsilon }},$$

which is (48). □

As a specific instance of Theorem 6, if $0<\epsilon =p/r<1,$ $\chi \left(z\right)={\psi }^r\left(z\right)$ and $\vartheta \left(z\right)={\omega }^r\left(z\right)$, we obtain the result below.

Corollary 4.

Assume that $\alpha ,\gamma \in \mathbb{T}$ and $0<p<r<\infty .$ In addition, let ψ be $C$-decreasing on ${[\alpha ,\gamma ]}_{\mathbb{T}},$ $C\ge 1,$ ω be decreasing on ${[\alpha ,\gamma ]}_{\mathbb{T}}$ with $\omega \left(\gamma \right)=0$ and

$${\psi }^{r-p}\left(\zeta \right){\left[{\psi }^{\rho }\left(\zeta \right)\right]}^{p-r}\ge {\omega }^{p-r}\left(\zeta \right){\left[{\omega }^{\rho }\left(\zeta \right)\right]}^{r-p}.$$

(58)

Then

$${\left({\int }_{\alpha }^{\gamma }{\psi }^r\left(z\right){\left[-{\omega }^r\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{r}}\ge C^{(p/r)-1}{\left({\int }_{\alpha }^{\gamma }{\psi }^p\left(z\right){\left[-{\omega }^p\left(z\right)\right]}^{\nabla }\nabla z\right)}^{\frac{1}{p}}.$$

Remark 9.

In Corollary 4, let $\mathbb{T}=\mathbb{R}.$ Then $\rho \left(\zeta \right)=\zeta$ and (58) holds with equality. Thus, we obtain the inequality (6).

Remark 10.

In Corollary 4, let $\mathbb{T}=\mathbb{N},$ $\alpha ,\gamma \in \mathbb{N},$ $0<p<r<\infty ,$ ψ be $C$-decreasing sequence for $C\ge 1,$ ω be decreasing with $\omega \left(\gamma \right)=0.$ Then $\rho \left(i\right)=i-1$ and

$${\psi }^{r-p}\left(i\right){\psi }^{p-r}(i-1)\ge {\omega }^{p-r}\left(i\right){\omega }^{r-p}(i-1).$$

Thus,

$${\left(\sum _{i=\alpha }^{\gamma }{\psi }^r\left(i\right)\nabla \left[-{\omega }^r\left(i\right)\right]\right)}^{\frac{1}{r}}\ge C^{(p/r)-1}{\left(\sum _{i=\alpha }^{\gamma }{\psi }^p\left(i\right)\nabla \left[-{\omega }^p\left(i\right)\right]\right)}^{\frac{1}{p}}.$$