Generalized Dynamic Inequalities of Copson Type on Time Scales

Abstract

This paper introduces novel generalizations of dynamic inequalities of Copson type within the framework of time scales delta calculus. The proposed generalizations leverage mathematical tools such as Hölder’s inequality, Minkowski’s inequality, the chain rule on time scales, and the properties of power rules on time scales. As special cases of our results, particularly when the time scale $\mathbb{T}$ equals the real line ($\mathbb{T}=\mathbb{R}$), we derive some classical continuous analogs of previous inequalities. Furthermore, when $\mathbb{T}$ corresponds to the set of natural numbers including zero ($\mathbb{T}={\mathbb{N}}_0$), the obtained results, to the best of the authors’ knowledge, represent innovative contributions to the field.

1. Introduction

Hardy, known primarily as a problem-solver, contributed significantly to the development of inequalities. His initial systematic treatment of inequalities laid the groundwork for subsequent work, much of which was influenced by his collaboration with Littlewood and Pólya on the book Inequalities, published in 1934 [1]. This seminal work spurred widespread study in the field, with notable contributions from Copson [2,3] and Knopp [4].

One of Hardy’s key contributions is the classical discrete inequality established in 1920 [5], expressed as follows:

$$\sum _{\varkappa =1}^{\infty }{\left(\frac{1}{\varkappa }\sum _{i=1}^{\varkappa }a\left(i\right)\right)}^{\mathsf{\ell }}\le {\left(\frac{\mathsf{\ell }}{\mathsf{\ell }-1}\right)}^{\mathsf{\ell }}\sum _{\varkappa =1}^{\infty }{\left[a\left(\varkappa \right)\right]}^{\mathsf{\ell }},\mathsf{\ell }>1,$$

(1)

where $a\left(\varkappa \right)\ge 0$ for $\varkappa \ge 1$ and ${\sum }_{\varkappa =1}^{\infty }{\left[a\left(\varkappa \right)\right]}^{\mathsf{\ell }}<\infty$. Hardy later extended this to an integral inequality [6], providing conditions under which a function $\Xi \ge 0$ is integrable over $(0,\lambda ),$ $\lambda \in (0,\infty ),$ given by:

$${\int }_0^{\infty }{\left(\frac{1}{\lambda }{\int }_0^{\lambda }\Xi \left(\tau \right)d\tau \right)}^{\mathsf{\ell }}d\lambda \le {\left(\frac{\mathsf{\ell }}{\mathsf{\ell }-1}\right)}^{\mathsf{\ell }}{\int }_0^{\infty }{\left[\Xi \left(\lambda \right)\right]}^{\mathsf{\ell }}d\lambda ,\mathsf{\ell }>1,$$

(2)

where ${\left(\mathsf{\ell }/\left(\mathsf{\ell }-1\right)\right)}^{\mathsf{\ell }}$ in (1) and (2) is the optimal constant. The authors of [7] showed that if $\Xi \left(\lambda \right)\ge 0$ and ${\int }_0^{\infty }{\left[\Xi \left(\lambda \right)\right]}^{\mathsf{\ell }}d\lambda <\infty ,$ then

$${\int }_0^{\infty }{\left(\frac{1}{\lambda }{\int }_{\lambda }^{\infty }\Xi \left(\tau \right)d\tau \right)}^{\mathsf{\ell }}d\lambda \ge {\left(\frac{\mathsf{\ell }}{1-\mathsf{\ell }}\right)}^{\mathsf{\ell }}{\int }_0^{\infty }{\left[\Xi \left(\lambda \right)\right]}^{\mathsf{\ell }}d\lambda ,0<\mathsf{\ell }<1.$$

Also, the constant ${\left(\mathsf{\ell }/\left(1-\mathsf{\ell }\right)\right)}^{\mathsf{\ell }}$ is the best possible.

Building upon Hardy’s work, Copson [8] provided a generalized form of the integral inequality (2), introducing parameters $\mathsf{\ell }\ge 1$ and $\alpha >1,$ expressed as follows:

$${\int }_0^{\epsilon }\xi \left(\vartheta \right)\frac{{\left({\rm Y}\left(\vartheta \right)\right)}^{\mathsf{\ell }}}{{\left(\Lambda \left(\vartheta \right)\right)}^{\alpha }}d\vartheta \le {\left(\frac{\mathsf{\ell }}{\alpha -1}\right)}^{\mathsf{\ell }}{\int }_0^{\epsilon }\xi \left(\vartheta \right){\left(\Lambda \left(\vartheta \right)\right)}^{\mathsf{\ell }-\alpha }{\left(\Xi \left(\vartheta \right)\right)}^{\mathsf{\ell }}d\vartheta ,$$

(3)

where

$${\rm Y}\left(\vartheta \right)={\int }_0^{\vartheta }\xi \left(x\right)\Xi \left(x\right)dx\mathrm{and}\Lambda \left(\vartheta \right)={\int }_0^{\vartheta }\xi \left(x\right)dx.$$

Mohapatra and Vajravelu [9] showed that if $\xi \left(\vartheta \right)\in C[0,\infty )$ is a positive function, $\Xi \left(\vartheta \right)\ge 0,$ $\mathsf{\ell }>1$ and ∃$G,H>0$ such that

$$\vartheta \left|{\xi }^{\prime }\left(\vartheta \right)\right|\le G\xi \left(\vartheta \right)\mathrm{and}\vartheta \xi \left(\vartheta \right)\le H\Lambda \left(\vartheta \right),\forall \vartheta >0,$$

(4)

then

$${\int }_0^{\infty }\xi \left(\vartheta \right){\left(\frac{{\rm Y}\left(\vartheta \right)}{\Lambda \left(\vartheta \right)}\right)}^{\mathsf{\ell }}d\vartheta \le {\left(G+H\right)}^{\mathsf{\ell }}{\int }_0^{\infty }{\left(\frac{1}{\vartheta }{\int }_0^{\vartheta }{\left(\xi \left(y\right)\right)}^{\frac{1}{\mathsf{\ell }}}\Xi \left(y\right)dy\right)}^{\mathsf{\ell }}d\vartheta ,$$

(5)

and

$${\int }_0^{\infty }\xi \left(\vartheta \right){\left({\int }_{\vartheta }^{\infty }\frac{\xi \left(\kappa \right)\Xi \left(\kappa \right)}{\Lambda \left(\kappa \right)}d\kappa \right)}^{\mathsf{\ell }}d\vartheta \le {\left(H+\mathsf{\ell }C\right)}^{\mathsf{\ell }}{\int }_0^{\infty }{\left({\int }_{\vartheta }^{\infty }\frac{{\left(\xi \left(y\right)\right)}^{\frac{1}{\mathsf{\ell }}}\Xi \left(y\right)}{y}dy\right)}^{\mathsf{\ell }}d\vartheta ,$$

(6)

where ${\rm Y}\left(\vartheta \right)={\int }_0^{\vartheta }\xi \left(\kappa \right)\Xi \left(\kappa \right)d\kappa ,$ $\Lambda \left(\vartheta \right)={\int }_0^{\vartheta }\xi \left(\kappa \right)d\kappa$ and $C=G+H+1.$

Also, some authors presented the discrete Formulas (5) and (6) (see [10]).

The purpose of this study is to establish some new generalizations of (5) and (6) on time scales by leveraging Hölder’s inequality, Minkowski’s inequality and chain rule formula. The paper’s exploration of novel inequalities within the framework of time scales delta calculus hints at potential applications of symmetry, suggesting a deeper understanding of the inherent balance and invariance present in the mathematical structures under consideration. The concept of symmetry in this context may be connected to exploring patterns, invariances or relationships within the established dynamic inequalities on time scales. For more results about the dynamic inequalities, see the book [11] and the papers [12,13].

In [14], El-Deeb et al. proved that if $0\le x\in \mathbb{T},$ $\delta \ge a,$ then $\Omega$ and $\varphi$ are non-negative $(\gamma ,a)$-nabla fractional differentiable and locally integrable functions on ${[x,\infty )}_{\mathbb{T}}$. Let ∃$\Lambda >0$ such that $\frac{{\omega }^{\rho }\left(\delta \right)}{\omega \left(\delta \right)}\le \Lambda$ for $\delta \in {[x,\infty )}_{\mathbb{T}}.$ If $k\ge 0,$ $\mathsf{\ell }>1,$ $k+\theta >1$ and $0<\gamma <1$ are real constants, then

$${\int }_x^{\infty }\frac{\varphi \left(\delta \right){\left[{\Psi }^{\rho }\left(\delta \right)\right]}^{k+\mathsf{\ell }-\gamma +1}}{{\left[{\omega }^{\rho }\left(\delta \right)\right]}^{k+\theta -\gamma +1}}{\nabla }_a^{\gamma }\delta \le \frac{k+\mathsf{\ell }-\gamma +1}{k+\theta -\gamma }{\int }_x^{\infty }\frac{\varphi \left(\delta \right)\Omega \left(\delta \right){\left[{\Psi }^{\rho }\left(\delta \right)\right]}^{k+\mathsf{\ell }-\gamma }}{{\left[\omega \left(\delta \right)\right]}^{k+\theta -\gamma }}{\nabla }_a^{\gamma }\delta ,$$

where $\omega \left(\delta \right)={\int }_{\delta }^{\infty }\varphi \left(t\right){\nabla }_a^{\gamma }t,$ $\Psi \left(\delta \right)={\int }_{\delta }^{\infty }\varphi \left(t\right)\Omega \left(t\right){\nabla }_a^{\gamma }t,$ $\Psi \left(x\right)<\infty$ and ${\int }_x^{\infty }\frac{\varphi \left(\delta \right)}{{\omega }^{\rho }\left(\delta \right)}{\nabla }_a^{\gamma }\delta <\infty .$

In [15], Saker et al. proved that if $\lambda ,\Omega \ge 0$ functions, $\Lambda \left(\vartheta \right)={\int }_{\epsilon }^{\vartheta }\lambda \left(x\right)\Delta x,$ $\Lambda (\infty )=\infty$ and $\Psi \left(\vartheta \right)={\int }_{\epsilon }^{\vartheta }\lambda \left(x\right)\Omega \left(x\right)\Delta x$ for any $\vartheta \in {[\epsilon ,\infty )}_{\mathbb{T}},$ then

$${\int }_{\epsilon }^{\infty }\frac{\lambda \left(\vartheta \right){\left[{\Psi }^{\sigma }\left(\vartheta \right)\right]}^{\mathsf{\ell }}}{{\left[{\Lambda }^{\sigma }\left(\vartheta \right)\right]}^{\mathsf{\ell }}}\Delta \vartheta \le \frac{\mathsf{\ell }}{\mathsf{\ell }-1}{\int }_{\epsilon }^{\infty }\frac{\lambda \left(\vartheta \right)\Omega \left(\vartheta \right){\Psi }^{\mathsf{\ell }-1}\left(\vartheta \right)}{{\Lambda }^{\mathsf{\ell }-1}\left(\vartheta \right)}\Delta \vartheta ,\mathsf{\ell }>1.$$

In [16], Saker et al. showed that if $\lambda ,\Omega \ge 0$ functions, $\Lambda \left(\vartheta \right)={\int }_{\epsilon }^{\vartheta }\lambda \left(x\right)\Delta x,$ $\Phi \left(\vartheta \right)={\int }_{\epsilon }^{\vartheta }\lambda \left(x\right)\Omega \left(x\right)\Delta x$ for any $\vartheta \in {[\epsilon ,\infty )}_{\mathbb{T}},$ $1<c\le \mathsf{\ell },$ then

$${\int }_{\epsilon }^{\infty }\frac{\lambda \left(\vartheta \right){\left[{\Phi }^{\sigma }\left(\vartheta \right)\right]}^{\mathsf{\ell }}}{{\left[{\Lambda }^{\sigma }\left(\vartheta \right)\right]}^c}\Delta \vartheta \le \frac{\mathsf{\ell }}{c-1}{\int }_{\epsilon }^{\infty }{\Lambda }^{1-c}\left(\vartheta \right)\lambda \left(\vartheta \right)\Omega \left(\vartheta \right){\Psi }^{\mathsf{\ell }-1}\left(\vartheta \right)\Delta \vartheta ,$$

and

$${\int }_{\epsilon }^{\infty }\frac{\lambda \left(\vartheta \right){\left[{\Phi }^{\sigma }\left(\vartheta \right)\right]}^{\mathsf{\ell }}}{{\left[{\Lambda }^{\sigma }\left(\vartheta \right)\right]}^c}\Delta \vartheta \le {\left(\frac{\mathsf{\ell }}{c-1}\right)}^{\mathsf{\ell }}{\int }_{\epsilon }^{\infty }\frac{{\left({\Lambda }^{\sigma }\left(\vartheta \right)\right)}^{\left(\mathsf{\ell }-1\right)c}}{{\left(\Lambda \left(\vartheta \right)\right)}^{\mathsf{\ell }\left(c-1\right)}}\lambda \left(\vartheta \right){\Omega }^{\mathsf{\ell }}\left(\vartheta \right)\Delta \vartheta .$$

In [17], Ashraf et al. proved that ${\mathbb{T}}_1,\dots ,{\mathbb{T}}_n$ denote time scales. For $\mathsf{\ell }>1,$ $j=1,\dots ,n,$ consider $b_j\in {[0,\infty )}_{\mathbb{T}},$ $\Omega :{\mathbb{T}}_1\times \dots \times {\mathbb{T}}_n\to {\mathbb{R}}^{+},$ ${\xi }_j:{\mathbb{T}}_j\to {\mathbb{R}}^{+},$ ${\Lambda }_j\left(y_j\right)={\int }_{b_j}^{y_i}{\xi }_i\left({\vartheta }_i\right)\Delta {\vartheta }_i$ and ${\Lambda }_j\left(\infty \right)=0.$ If

$$\varphi \left(y_1,\dots ,y_n\right)={\int }_{y_1}^{\infty }\dots {\int }_{y_n}^{\infty }\prod _{j=1}^n\frac{{\xi }_j\left({\vartheta }_j\right)}{{\Lambda }_j^{{\sigma }_j}\left({\vartheta }_j\right)}\Omega \left({\vartheta }_1,\dots ,{\vartheta }_n\right)\Delta {\vartheta }_1\dots \Delta {\vartheta }_n,$$

then

$$\begin{array}{ccc}& & {\int }_{b_1}^{\infty }\dots {\int }_{b_n}^{\infty }\prod _{j=1}^n{\xi }_j\left(y_j\right){\varphi }^{\mathsf{\ell }}\left(y_1,\dots ,y_n\right)\Delta y_1\dots \Delta y_n\hfill \\ & \le & {\mathsf{\ell }}^{n\mathsf{\ell }}{\int }_{b_1}^{\infty }\dots {\int }_{b_n}^{\infty }\prod _{j=1}^n{\xi }_j\left(y_j\right){\Omega }^{\mathsf{\ell }}\left(y_1,\dots ,y_n\right)\Delta y_1\dots \Delta y_n.\hfill \end{array}$$

In [18], Kayar and Kaymakçalan observed that $\omega ,$ $\Xi$ are non-negative, along with ld-continuous, ∇-differentiable and locally nabla integrable functions. In addition, if $G\left(\vartheta \right)={\int }_{\epsilon }^{\vartheta }\omega \left(y\right)\nabla y$ and $H\left(\vartheta \right)={\int }_{\epsilon }^{\vartheta }\omega \left(y\right)\Xi \left(y\right)\nabla y$ with $H\left(\infty \right)<\infty$, then ${\int }_{\epsilon }^{\infty }\frac{\omega \left(\vartheta \right)}{G^{l+r}\left(\vartheta \right)}\nabla \vartheta <\infty .$ If there exists $L>0$ such that $\frac{G\left(\vartheta \right)}{G^{\rho }\left(\vartheta \right)}\le L$ for $\vartheta \in {(\epsilon ,\infty )}_{\mathbb{T}},$ $l\ge 0,$ $\varsigma >1,$ $l+r>1$ are real constants, then

$${\int }_{\epsilon }^{\infty }\frac{\omega \left(\vartheta \right)H^{l+\varsigma }\left(\vartheta \right)}{{\left[G\left(\vartheta \right)\right]}^{l+r}}\nabla \vartheta \le \frac{l+\varsigma }{l+r-1}{\int }_{\epsilon }^{\infty }\frac{\omega \left(\vartheta \right)\Xi \left(\vartheta \right)H^{l+\varsigma -1}\left(\vartheta \right)}{{\left[G^{\rho }\left(\vartheta \right)\right]}^{l+r-1}}\nabla \vartheta .$$

In [19], Saker et al. proved that if $u,$ h are positive rd-continuous functions on ${[\epsilon ,b]}_{\mathbb{T}},U\left(\kappa \right)={\int }_{\epsilon }^{\kappa }u\left(\vartheta \right)\Delta \vartheta$ and $1<\mathsf{\ell }<\infty ,$ then

$${\int }_{\epsilon }^{b}u\left(\kappa \right){\left({\int }_{\kappa }^{b}h\left(\vartheta \right)\Delta \vartheta \right)}^{\mathsf{\ell }}\Delta \kappa \le {\mathsf{\ell }}^{\mathsf{\ell }}{\int }_{\epsilon }^{b}u\left(\kappa \right){\left(\frac{h\left(\kappa \right)U^{\sigma }\left(\kappa \right)}{u\left(\kappa \right)}\right)}^{\mathsf{\ell }}\Delta \kappa .$$

(7)

The inequality (7) is reversed for $0<\mathsf{\ell }<1,$ while $\mathsf{\ell }=1$ holds equality.

The structure of this article unfolds as follows: In Section 2, we introduce several lemmas on time scales, which are essential for the proofs presented in Section 3. The outcomes we establish in Section 3 encompass the inequalities (5) and (6), previously demonstrated by Mohapatra and Vajravelu [9] in the special cases where $\mathbb{T}$ equals the real numbers ($\mathbb{T}=\mathbb{R}$). Additionally, when $\mathbb{T}$ is equivalent to the set of natural numbers ($\mathbb{T}=\mathbb{N}$), our results, to the best of the authors’ knowledge, represent novel findings.

2. Preliminaries

Bohner and Peterson [20] defined the time scale $\mathbb{T}$ as an arbitrary non-empty closed subset of the real numbers $\mathbb{R}$. Thus

$$\mathbb{R},{\mathbb{R}}^{+},\mathbb{Z},\mathbb{N},{\mathbb{N}}_0,$$

i.e., the real numbers, non-negative real numbers, the integers, the natural numbers, and the non-negative integers are examples of time scales. Also, they defined the forward jump operator by $\sigma \left(\epsilon \right):=inf\{w\in \mathbb{T}:w>\epsilon \}$ and the graininess function $\mu$ by $\mu \left(\epsilon \right):=\sigma \left(\epsilon \right)-\epsilon \ge 0$. For any function $\Xi :\mathbb{T}\to \mathbb{R}$, ${\Xi }^{\sigma }\left(\epsilon \right)$ denotes $\Xi \left(\sigma \right(\epsilon \left)\right).$

The set ${\mathbb{T}}^{\kappa }$ is derived from the time scale $\mathbb{T}$ as follows: If $\mathbb{T}$ has a left-scattered maximum m, then ${\mathbb{T}}^{\kappa }=\mathbb{T}-\left\{m\right\}.$ Otherwise, ${\mathbb{T}}^{\kappa }=\mathbb{T}$. In summary,

$${\mathbb{T}}^{\kappa }=\left\{\begin{array}{cc}\mathbb{T}\setminus \left(\rho \left(sup\mathbb{T}\right),sup\mathbb{T}\right]\hfill & \mathrm{if}sup\mathbb{T}<\infty \hfill \\ \mathbb{T}\hfill & \mathrm{if}sup\mathbb{T}=\infty .\hfill \end{array}\right.$$

Definition 1

([20]). $\Xi :\mathbb{T}\to \mathbb{R}$ is called $rd-$ continuous if it is continuous at right dense points in $\mathbb{T}$ and its left-sided limits exist (finite) at left dense points in $\mathbb{T}$. The set of $rd-$ continuous functions is denoted by ${\mathrm{C}}_{rd}(\mathbb{T},$ $\mathbb{R})$.

Definition 2

([20]). Assume that $\Xi :\mathbb{T}\to \mathbb{R}$ is a function and let $\xi \in \mathbb{T}.$ We define ${\Xi }^{\Delta }\left(\xi \right)$ to be the number, if it exists, as follows: for any $\epsilon >0$, there is a neighborhood $U=(\xi -\delta ,$ $\xi +\delta )\cap \mathbb{T}$ for some $\delta >0$, such that

$$|\Xi \left(\sigma \left(\xi \right)\right)-\Xi \left(y\right)-{\Xi }^{\Delta }\left(\xi \right)(\sigma \left(\xi \right)-y)|\le \epsilon |\sigma \left(\xi \right)-y|\forall y\in U,y\ne \sigma \left(\xi \right).$$

In this case, we say that ${\Xi }^{\Delta }\left(\xi \right)$ is the delta derivative of Ξ at ξ.

Theorem 1

([20]). Assume $\Xi ,$ $\Omega :\mathbb{T}\to \mathbb{R}$ are differentiable at $\xi \in \mathbb{T}$. Then

1. 

$\Xi +\Omega :\mathbb{T}\to \mathbb{R}$ is differentiable at $\xi ,$ and

$${(\Xi +\Omega )}^{\Delta }\left(\xi \right)={\Xi }^{\Delta }\left(\xi \right)+{\Omega }^{\Delta }\left(\xi \right).$$

2. 

$\Xi \Omega$$:\mathbb{T}\to \mathbb{R}$ is differentiable at ξ and the “product rule,” which is defined by

$${(\Xi \Omega )}^{\Delta }\left(\xi \right)={\Xi }^{\Delta }\left(\xi \right)\Omega \left(\xi \right)+\Xi \left(\sigma \left(\xi \right)\right){\Omega }^{\Delta }\left(\xi \right)=\Xi \left(\xi \right){\Omega }^{\Delta }\left(\xi \right)+{\Xi }^{\Delta }\left(\xi \right)\Omega \left(\sigma \left(\xi \right)\right),$$

holds.

3. 

If $\Omega \left(\xi \right)\Omega \left(\sigma \right(\xi \left)\right)\ne 0$, then we have the quotient $\Xi /\Omega :\mathbb{T}\to \mathbb{R}$ is differentiable at $\xi ,$ and the “quotient rule”

$${\left(\frac{\Xi }{\Omega }\right)}^{\Delta }\left(\xi \right)=\frac{{\Xi }^{\Delta }\left(\xi \right)\Omega \left(\xi \right)-\Xi \left(\xi \right){\Omega }^{\Delta }\left(\xi \right)}{\Omega \left(\xi \right)\Omega \left(\sigma \right(\xi \left)\right)},$$

holds.

Theorem 2

([20]). Suppose there is a continuous function $\Omega :\mathbb{R}\to \mathbb{R}$ whose restriction $\Omega :\mathbb{T}\to \mathbb{R}$ is delta differentiable on $\mathbb{T}$ and $\Xi :\mathbb{R}\to \mathbb{R}$ is continuously differentiable. Then

$${\left(\Xi \circ \Omega \right)}^{\Delta }\left(\xi \right)={\Xi }^{\prime }\left(\Omega \left(c\right)\right){\Omega }^{\Delta }\left(\xi \right);c\in [\xi ,\sigma \left(\xi \right)].$$

(8)

Definition 3

([20]). ${\rm Y}:\mathbb{T}\to \mathbb{R}$ is called an antiderivative of $\Xi :\mathbb{T}\to \mathbb{R}$ if

$${{\rm Y}}^{\Delta }\left(\xi \right)=\Xi \left(\xi \right),\forall \xi \in {\mathbb{T}}^k.$$

In this case, the Cauchy integral of Ξ is defined by

$${\int }_r^y\Xi \left(\xi \right)\Delta \xi ={\rm Y}\left(y\right)-{\rm Y}\left(r\right),\forall r,y\in \mathbb{T}.$$

Theorem 3

([20]). Every rd-continuous function $\Xi :\mathbb{T}\to \mathbb{R}$ has an antiderivative. In particular, if ${\xi }_0\in \mathbb{T}$, then

$${\left({\int }_{{\xi }_0}^{\xi }\Xi \left(\tau \right)\Delta \tau \right)}^{\Delta }=\Xi \left(\xi \right),\mathrm{for}\xi \in \mathbb{T}.$$

Lemma 1

([21]). If $r,s\in \mathbb{T}$ and $u,$ $v\in {\mathrm{C}}_{rd}{([r,s]}_{\mathbb{T}},$ $\mathbb{R}),$ then

$${\int }_r^{s}u\left(\xi \right)v^{\Delta }\left(\xi \right)\Delta \xi ={\left[u\left(\xi \right)v\left(\xi \right)\right]}_r^s-{\int }_r^s{u}^{\Delta }\left(\xi \right)v^{\sigma }\left(\xi \right)\Delta \xi .$$

(9)

Lemma 2

([21]). If $r,s\in \mathbb{T}$ and $\lambda ,$ $\omega \in {\mathrm{C}}_{rd}{([r,s]}_{\mathbb{T}},$ ${\mathbb{R}}^{+}),$ then

$${\int }_r^s\lambda \left(\xi \right)\omega \left(\xi \right)\Delta \xi \le {\left[{\int }_r^s{\left[\lambda \left(\xi \right)\right]}^{\gamma }\Delta \xi \right]}^{\frac{1}{\gamma }}{\left[{\int }_r^s{\left[\omega \left(\xi \right)\right]}^{\nu }\Delta \xi \right]}^{\frac{1}{\nu }},$$

(10)

where $\gamma >1$ and $1/\gamma +1/\nu =1.$

Lemma 3

(Weighted Minkowski’s Inequality [21]). Let $\chi ,$ $\lambda ,$ $\omega \in {\mathrm{C}}_{rd}{([r,s]}_{\mathbb{T}},$ ${\mathbb{R}}^{+})$ and $\mathsf{\ell }>1.$ Then

$$\begin{array}{cc}\hfill & {\left[{\int }_r^s\chi \left(\xi \right){\left[\lambda \left(\xi \right)+\omega \left(\xi \right)\right]}^{\mathsf{\ell }}\Delta \xi \right]}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & \le {\left[{\int }_r^s\chi \left(\xi \right){\left[\lambda \left(\xi \right)\right]}^{\mathsf{\ell }}\Delta \xi \right]}^{\frac{1}{\mathsf{\ell }}}+{\left[{\int }_r^s\chi \left(\xi \right){\left[\omega \left(\xi \right)\right]}^{\mathsf{\ell }}\Delta \xi \right]}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(11)

Lemma 4

([22]). Assume $r,\xi \in \mathbb{T}$ and $\Omega \in C_{rd}\left({[r,\infty )}_{\mathbb{T}},{\mathbb{R}}^{+}\right)$. If $\mathsf{\ell }\ge 1,$ then

$${\left({\int }_r^{\sigma \left(\xi \right)}\Omega \left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\le \mathsf{\ell }{\int }_r^{\sigma \left(\xi \right)}\Omega \left(\kappa \right){\left({\int }_r^{\sigma \left(\kappa \right)}\Omega \left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \kappa .$$

Lemma 5

([22]). Let $r\in \mathbb{T}$ and $\phi ,$ $\eta \in C_{rd}\left({[r,\infty )}_{\mathbb{T}},{\mathbb{R}}^{+}\right)$. Then

$${\int }_r^{\infty }\phi \left(\xi \right)\left({\int }_{\xi }^{\infty }\eta \left(\kappa \right)\Delta \kappa \right)\Delta \xi ={\int }_{\epsilon }^{\infty }\eta \left(\xi \right)\left({\int }_{\epsilon }^{\sigma \left(\xi \right)}\phi \left(\kappa \right)\Delta \kappa \right)\Delta \xi .$$

Lemma 6

([23]). Let $\Omega \in C_{rd}\left({[r,\infty )}_{\mathbb{T}},{\mathbb{R}}^{+}\right)$ and $\mathsf{\ell }\ge 1.$ Then

$${\left({\int }_{\xi }^{\infty }\Omega \left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\le \mathsf{\ell }{\int }_{\xi }^{\infty }\Omega \left(\kappa \right){\left({\int }_{\kappa }^{\infty }\Omega \left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \kappa .$$

Lemma 7.

Assume that $\epsilon \in \mathbb{T},$ $\mathsf{\ell }>1$ with $1/\mathsf{\ell }+1/{\mathsf{\ell }}^{*}=1$ and $\Xi :{[\epsilon ,\infty )}_{\mathbb{T}}\to {\mathbb{R}}^{+}.$ If $Z:{[\epsilon ,\infty )}_{\mathbb{T}}\to {\mathbb{R}}^{+}$ is a positive, differentiable and nonincreasing function with

$$\phi \left(\xi \right)={\int }_{\xi }^{\infty }\frac{{\left[Z\left(\delta \right)\right]}^{\frac{1}{\mathsf{\ell }}}\Xi \left(\delta \right)}{\delta -\epsilon }\Delta \delta ,\Lambda \left(\xi \right)={\int }_{\epsilon }^{\xi }Z\left(\delta \right)\Delta \delta ,$$

and there exists a positive constant H such that $\left(\xi -\epsilon \right)Z\left(\xi \right)\le H\Lambda \left(\xi \right),$ then

$$\underset{\kappa \to \infty }{lim}\frac{\left(\kappa -\epsilon \right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\phi \left(\kappa \right)=0.$$

Proof. 

Since $Z\left(\delta \right)$ is a positive and non-increasing function, we have for $\delta \ge \kappa ,$ that $Z\left(\delta \right)\le Z\left(\kappa \right)$ and then

$${\int }_{\kappa }^{\infty }\frac{{\left[Z\left(\delta \right)\right]}^{\frac{1}{\mathsf{\ell }}}\Xi \left(\delta \right)}{\delta -\epsilon }\Delta \delta \le {\left[Z\left(\kappa \right)\right]}^{\frac{1}{\mathsf{\ell }}}{\int }_{\kappa }^{\infty }\frac{\Xi \left(\delta \right)}{\delta -\epsilon }\Delta \delta .$$

(12)

Since

$$\frac{\left(\kappa -\epsilon \right)Z\left(\kappa \right)}{\Lambda \left(\kappa \right)}\le H,$$

we have from (12) that

$$\begin{array}{ccc}\hfill 0& \le & \frac{\left(\kappa -\epsilon \right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\phi \left(\kappa \right)=\frac{\left(\kappa -\epsilon \right)Z\left(\kappa \right)}{\Lambda \left(\kappa \right)}{\left[Z\left(\kappa \right)\right]}^{\frac{-1}{\mathsf{\ell }}}{\int }_{\kappa }^{\infty }\frac{{\left[Z\left(\delta \right)\right]}^{\frac{1}{\mathsf{\ell }}}\Xi \left(\delta \right)}{\delta -\epsilon }\Delta \delta \hfill \\ & \le & H{\left[Z\left(\kappa \right)\right]}^{\frac{-1}{\mathsf{\ell }}}{\int }_{\kappa }^{\infty }\frac{{\left[Z\left(\delta \right)\right]}^{\frac{1}{\mathsf{\ell }}}\Xi \left(\delta \right)}{\delta -\epsilon }\Delta \delta \le H{\int }_{\kappa }^{\infty }\frac{\Xi \left(\delta \right)}{\delta -\epsilon }\Delta \delta ,\hfill \end{array}$$

then

$$0\le \underset{\kappa \to \infty }{lim}\frac{\left(\kappa -\epsilon \right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\phi \left(\kappa \right)\le H\underset{\kappa \to \infty }{lim}{\int }_{\kappa }^{\infty }\frac{\Xi \left(\delta \right)}{\delta -\epsilon }\Delta \delta =0,$$

and then by using the Sandwich theorem, we obtain

$$\underset{\kappa \to \infty }{lim}\frac{\left(\kappa -\epsilon \right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\phi \left(\kappa \right)=0.$$

□

3. Main Results

Throughout the following results, we will assume that the functions are rd-continuous functions on ${[\epsilon ,\infty )}_{\mathbb{T}}$ and the integrals considered are assumed to exist.

Theorem 4.

Assume $\epsilon \in \mathbb{T}$ and $\mathsf{\ell }>1$ with $1/\mathsf{\ell }+1/{\mathsf{\ell }}^{*}=1.$ Let $Z\left(\xi \right)>0$ be a differentiable and non-increasing function on ${[\epsilon ,\infty )}_{\mathbb{T}}$ and there exists positive constants $G,$ H such that

$$\left(\sigma \left(\xi \right)-\epsilon \right)\left|Z^{\Delta }\left(\xi \right)\right|\le G{Z}^{\sigma }\left(\xi \right)\mathrm{and}\left(\sigma \left(\xi \right)-\epsilon \right)Z\left(\xi \right)\le H{\Lambda }^{\sigma }\left(\xi \right).$$

(13)

Then

$$\begin{array}{cc}\hfill & {\left({\int }_{\epsilon }^{\infty }Z\left(\xi \right){\left(\frac{{{\rm Y}}^{\sigma }\left(\xi \right)}{{\Lambda }^{\sigma }\left(\xi \right)}\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & \le G{\left({\int }_{\epsilon }^{\infty }{\left[\frac{{\Lambda }^{\sigma }\left(\xi \right)}{\Lambda \left(\xi \right)}\right]}^{\mathsf{\ell }\left(\mathsf{\ell }-1\right)}{\left(\frac{{\psi }^{\sigma }\left(\xi \right)}{\sigma \left(\xi \right)-\epsilon }\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}+H{\left({\int }_{\epsilon }^{\infty }{\left(\frac{{\psi }^{\sigma }\left(\xi \right)}{\sigma \left(\xi \right)-\epsilon }\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}},\hfill \end{array}$$

(14)

where

$${\rm Y}\left(\xi \right)={\int }_{\epsilon }^{\xi }Z\left(\kappa \right)\Xi \left(\kappa \right)\Delta \kappa ,\psi \left(\xi \right)={\int }_{\epsilon }^{\xi }{\left[Z\left(y\right)\right]}^{\frac{1}{\mathsf{\ell }}}\Xi \left(y\right)\Delta yand\Lambda \left(\xi \right)={\int }_{\epsilon }^{\xi }Z\left(\kappa \right)\Delta \kappa .$$

(15)

Proof. 

Denote

$$h\left(\kappa \right)={\left(Z^{\sigma }\left(\kappa \right)\right)}^{-\frac{1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|{\psi }^{\sigma }\left(\kappa \right).$$

(16)

Since $1/\mathsf{\ell }+1/{\mathsf{\ell }}^{*}=1,$ we obtain

$$\begin{array}{ccc}& & {\int }_{\epsilon }^{\infty }Z\left(\xi \right){\left(\frac{{{\rm Y}}^{\sigma }\left(\xi \right)}{{\Lambda }^{\sigma }\left(\xi \right)}\right)}^{\mathsf{\ell }}\Delta \xi \hfill \\ & =& {\int }_{\epsilon }^{\infty }Z\left(\xi \right){\left(\frac{{\int }_{\epsilon }^{\sigma \left(\xi \right)}Z\left(\kappa \right)\Xi \left(\kappa \right)\Delta \kappa }{{\Lambda }^{\sigma }\left(\xi \right)}\right)}^{\mathsf{\ell }}\Delta \xi \hfill \\ & =& {\int }_{\epsilon }^{\infty }Z\left(\xi \right){\left(\frac{{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\left[Z\left(\kappa \right)\right]}^{\frac{1}{\mathsf{\ell }}}\Xi \left(\kappa \right)\Delta \kappa }{{\Lambda }^{\sigma }\left(\xi \right)}\right)}^{\mathsf{\ell }}\Delta \xi \hfill \\ & =& {\int }_{\epsilon }^{\infty }Z\left(\xi \right){\left(\frac{{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\psi }^{\Delta }\left(\kappa \right)\Delta \kappa }{{\Lambda }^{\sigma }\left(\xi \right)}\right)}^{\mathsf{\ell }}\Delta \xi .\hfill \end{array}$$

(17)

Applying (9) on ${\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\psi }^{\Delta }\left(\kappa \right)\Delta \kappa ,$ with $u\left(\kappa \right)={\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}$ and $v\left(\kappa \right)=\psi \left(\kappa \right),$ we see that

$$\begin{array}{ccc}& & {\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\psi }^{\Delta }\left(\kappa \right)\Delta \kappa \hfill \\ & =& {\left.{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}\psi \left(\kappa \right)\right|}_{\epsilon }^{\sigma \left(\xi \right)}-{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left[{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}\right]}^{\Delta }{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa .\hfill \end{array}$$

(18)

Since $\psi \left(\epsilon \right)=0,$ we have that

$$\begin{array}{ccc}& & {\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\psi }^{\Delta }\left(\kappa \right)\Delta \kappa \hfill \\ & =& {\left(Z^{\sigma }\left(\xi \right)\right)}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\psi }^{\sigma }\left(\xi \right)+{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left[-{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}\right]}^{\Delta }{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa .\hfill \end{array}$$

(19)

Applying (8) on ${\left[-{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}\right]}^{\Delta },$ we observe that

$${\left[-{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}\right]}^{\Delta }=\frac{-1}{{\mathsf{\ell }}^{*}}{Z}^{-\frac{1}{\mathsf{\ell }}}\left(\xi \right)Z^{\Delta }\left(\kappa \right),\xi \in [\kappa ,\sigma \left(\kappa \right)].$$

(20)

Since $Z\left(\kappa \right)$ is a non-increasing function, then $Z^{\Delta }\left(\kappa \right)\le 0$ and then

$$\left|Z^{\Delta }\left(\kappa \right)\right|=-Z^{\Delta }\left(\kappa \right).$$

(21)

Since $\xi \le \sigma \left(\kappa \right)$ and $Z\left(\kappa \right)$ is a non-increasing function, we observe that

$$Z^{\frac{-1}{\mathsf{\ell }}}\left(\xi \right)\le {\left[Z^{\sigma }\left(\kappa \right)\right]}^{\frac{-1}{\mathsf{\ell }}}.$$

(22)

Substituting (21) and (22) into (20), we obtain

$${\left[-{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}\right]}^{\Delta }\le \frac{1}{{\mathsf{\ell }}^{*}}{\left[Z^{\sigma }\left(\kappa \right)\right]}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|,$$

and then we have from (19) that

$$\begin{array}{ccc}& & {\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\psi }^{\Delta }\left(\kappa \right)\Delta \kappa \hfill \\ & \le & {\left(Z^{\sigma }\left(\xi \right)\right)}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\psi }^{\sigma }\left(\xi \right)+\frac{1}{{\mathsf{\ell }}^{*}}{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left(Z^{\sigma }\left(\kappa \right)\right)}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa .\hfill \end{array}$$

(23)

Substituting (23) into (17), we observe that

$$\begin{array}{cc}\hfill & {\int }_{\epsilon }^{\infty }Z\left(\xi \right){\left(\frac{{{\rm Y}}^{\sigma }\left(\xi \right)}{{\Lambda }^{\sigma }\left(\xi \right)}\right)}^{\mathsf{\ell }}\Delta \xi \hfill \\ \hfill & \le {\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}\hfill \\ \hfill & \times {\left({\left(Z^{\sigma }\left(\xi \right)\right)}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\psi }^{\sigma }\left(\xi \right)+\frac{1}{{\mathsf{\ell }}^{*}}{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left(Z^{\sigma }\left(\kappa \right)\right)}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi ,\hfill \end{array}$$

and then

$$\begin{array}{cc}\hfill & {\left({\int }_{\epsilon }^{\infty }Z\left(\xi \right){\left(\frac{{{\rm Y}}^{\sigma }\left(\xi \right)}{{\Lambda }^{\sigma }\left(\xi \right)}\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & \le \left({\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}\right.\hfill \\ \hfill & \times {\left.{\left({\left(Z^{\sigma }\left(\xi \right)\right)}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\psi }^{\sigma }\left(\xi \right)+\frac{1}{{\mathsf{\ell }}^{*}}{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left(Z^{\sigma }\left(\kappa \right)\right)}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(24)

Applying (11) to

$$\begin{array}{cc}\hfill & \left({\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}\right.\hfill \\ \hfill & \times {\left.{\left({\left(Z^{\sigma }\left(\xi \right)\right)}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\psi }^{\sigma }\left(\xi \right)+\frac{1}{{\mathsf{\ell }}^{*}}{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left(Z^{\sigma }\left(\kappa \right)\right)}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}},\hfill \end{array}$$

with

$$\chi \left(\xi \right)=\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}},\lambda \left(\xi \right)={\left(Z^{\sigma }\left(\xi \right)\right)}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\psi }^{\sigma }\left(\xi \right),$$

and

$$\omega \left(\xi \right)=\frac{1}{{\mathsf{\ell }}^{*}}{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left(Z^{\sigma }\left(\kappa \right)\right)}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa ,$$

we see that

$$\begin{array}{cc}\hfill & \left({\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}\right.\hfill \\ \hfill & \times {\left.{\left({\left(Z^{\sigma }\left(\xi \right)\right)}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\psi }^{\sigma }\left(\xi \right)+\frac{1}{{\mathsf{\ell }}^{*}}{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left(Z^{\sigma }\left(\kappa \right)\right)}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & \le {\left({\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left[{\left(Z^{\sigma }\left(\xi \right)\right)}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\psi }^{\sigma }\left(\xi \right)\right]}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & +\frac{1}{{\mathsf{\ell }}^{*}}{\left({\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left[{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left(Z^{\sigma }\left(\kappa \right)\right)}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa \right]}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(25)

Since $Z\left(\xi \right)$ is a non-increasing function and $\sigma \left(\xi \right)\ge \xi ,$ we observe that $Z^{\sigma }\left(\xi \right)\le Z\left(\xi \right).$ Moreover, we then have from (13) that

$$\begin{array}{ccc}& & {\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left[{\left(Z^{\sigma }\left(\xi \right)\right)}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\psi }^{\sigma }\left(\xi \right)\right]}^{\mathsf{\ell }}\Delta \xi \hfill \\ & \le & {\int }_{\epsilon }^{\infty }\frac{{\left[Z\left(\xi \right)\right]}^{\mathsf{\ell }}}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left[{\psi }^{\sigma }\left(\xi \right)\right]}^{\mathsf{\ell }}\Delta \xi \hfill \\ & \le & H^{\mathsf{\ell }}{\int }_{\epsilon }^{\infty }{\left[\frac{{\psi }^{\sigma }\left(\xi \right)}{\sigma \left(\xi \right)-\epsilon }\right]}^{\mathsf{\ell }}\Delta \xi .\hfill \end{array}$$

(26)

From (16), we observe that

$$\begin{array}{ccc}& & {\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left[{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left(Z^{\sigma }\left(\kappa \right)\right)}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa \right]}^{\mathsf{\ell }}\Delta \xi \hfill \\ & =& {\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left[{\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\kappa \right)\Delta \kappa \right]}^{\mathsf{\ell }}\Delta \xi .\hfill \end{array}$$

(27)

Applying Lemma 4, we see that

$$\begin{array}{ccc}& & {\left[{\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\kappa \right)\Delta \kappa \right]}^{\mathsf{\ell }}\hfill \\ & \le & \mathsf{\ell }{\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\kappa \right){\left({\int }_{\epsilon }^{\sigma \left(\kappa \right)}h\left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \kappa ,\hfill \end{array}$$

(28)

and then we have from (27) that

$$\begin{array}{ccc}& & {\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left[{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left(Z^{\sigma }\left(\kappa \right)\right)}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa \right]}^{\mathsf{\ell }}\Delta \xi \hfill \\ & \le & \mathsf{\ell }{\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}\left({\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\kappa \right){\left({\int }_{\epsilon }^{\sigma \left(\kappa \right)}h\left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \kappa \right)\Delta \xi .\hfill \end{array}$$

(29)

Applying Lemma 5, we observe that

$$\begin{array}{ccc}& & {\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}\left({\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\kappa \right){\left({\int }_{\epsilon }^{\sigma \left(\kappa \right)}h\left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \kappa \right)\Delta \xi \hfill \\ & =& {\int }_{\epsilon }^{\infty }h\left(\xi \right){\left({\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\left({\int }_{\xi }^{\infty }\frac{Z\left(\kappa \right)}{{\left({\Lambda }^{\sigma }\left(\kappa \right)\right)}^{\mathsf{\ell }}}\Delta \kappa \right)\Delta \xi .\hfill \end{array}$$

(30)

Applying (8) on ${\Lambda }^{1-\mathsf{\ell }}\left(\kappa \right),$ we see that

$$\frac{1}{1-\mathsf{\ell }}{\left[{\left(\Lambda \left(\kappa \right)\right)}^{1-\mathsf{\ell }}\right]}^{\Delta }={\left[\Lambda \left(\xi \right)\right]}^{-\mathsf{\ell }}Z\left(\kappa \right),$$

where $\xi \in [\kappa ,\sigma (\kappa \left)\right].$ Since ${\Lambda }^{\Delta }\left(\kappa \right)=Z\left(\kappa \right)>0,$ $\Lambda$ is an increasing function on ${[\epsilon ,\infty )}_{\mathbb{T}};$ and then we have for $\xi \le \sigma \left(\kappa \right)$ that

$${\left[\Lambda \left(\xi \right)\right]}^{-\mathsf{\ell }}\ge {\left[{\Lambda }^{\sigma }\left(\kappa \right)\right]}^{-\mathsf{\ell }},$$

thus

$$\frac{1}{1-\mathsf{\ell }}{\left[{\left(\Lambda \left(\kappa \right)\right)}^{1-\mathsf{\ell }}\right]}^{\Delta }\ge \frac{Z\left(\kappa \right)}{{\left({\Lambda }^{\sigma }\left(\kappa \right)\right)}^{\mathsf{\ell }}},$$

(31)

Integrating (31) over $\kappa$ from $\xi$ to $\infty ,$ we have

$$\begin{array}{ccc}\hfill {\int }_{\xi }^{\infty }\frac{Z\left(\kappa \right)}{{\left({\Lambda }^{\sigma }\left(\kappa \right)\right)}^{\mathsf{\ell }}}\Delta \kappa & \le & \frac{1}{1-\mathsf{\ell }}{\int }_{\xi }^{\infty }{\left[{\left(\Lambda \left(\kappa \right)\right)}^{1-\mathsf{\ell }}\right]}^{\Delta }\Delta \kappa \hfill \\ & =& \frac{1}{\mathsf{\ell }-1}\left[{\left[\Lambda \left(\xi \right)\right]}^{1-\mathsf{\ell }}-{\left({\int }_{\epsilon }^{\infty }Z\left(\kappa \right)\Delta \kappa \right)}^{1-\mathsf{\ell }}\right]\hfill \\ & \le & \frac{1}{\mathsf{\ell }-1}{\left[\Lambda \left(\xi \right)\right]}^{1-\mathsf{\ell }}.\hfill \end{array}$$

(32)

Substituting (32) into (30), we observe that

$$\begin{array}{ccc}& & {\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}\left({\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\kappa \right){\left({\int }_{\epsilon }^{\sigma \left(\kappa \right)}h\left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \kappa \right)\Delta \xi \hfill \\ & \le & \frac{1}{\mathsf{\ell }-1}{\int }_{\epsilon }^{\infty }h\left(\xi \right){\left[\Lambda \left(\xi \right)\right]}^{1-\mathsf{\ell }}{\left({\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \xi .\hfill \end{array}$$

(33)

From (33) and (29), we have that

$$\begin{array}{ccc}& & {\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left[{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left(Z^{\sigma }\left(\kappa \right)\right)}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa \right]}^{\mathsf{\ell }}\Delta \xi \hfill \\ & \le & \frac{\mathsf{\ell }}{\mathsf{\ell }-1}{\int }_{\epsilon }^{\infty }h\left(\xi \right){\left[\Lambda \left(\xi \right)\right]}^{1-\mathsf{\ell }}{\left({\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \xi \hfill \\ & =& \frac{\mathsf{\ell }}{\mathsf{\ell }-1}{\int }_{\epsilon }^{\infty }{\left[\frac{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{Z^{\sigma }\left(\xi \right)}\right]}^{\frac{\mathsf{\ell }-1}{\mathsf{\ell }}}h\left(\xi \right){\left[\Lambda \left(\xi \right)\right]}^{1-\mathsf{\ell }}\hfill \\ & & \times {\left[\frac{Z^{\sigma }\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}\right]}^{\frac{\mathsf{\ell }-1}{\mathsf{\ell }}}{\left({\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \xi .\hfill \end{array}$$

(34)

Applying (10) to

$${\int }_{\epsilon }^{\infty }{\left[\frac{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{Z^{\sigma }\left(\xi \right)}\right]}^{\frac{\mathsf{\ell }-1}{\mathsf{\ell }}}h\left(\xi \right){\left[\Lambda \left(\xi \right)\right]}^{1-\mathsf{\ell }}{\left[\frac{Z^{\sigma }\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}\right]}^{\frac{\mathsf{\ell }-1}{\mathsf{\ell }}}{\left({\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \xi ,$$

with indices $\mathsf{\ell }>1$ and $\mathsf{\ell }/\left(\mathsf{\ell }-1\right),$ we obtain

$$\begin{array}{ccc}& & {\int }_{\epsilon }^{\infty }{\left[\frac{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{Z^{\sigma }\left(\xi \right)}\right]}^{\frac{\mathsf{\ell }-1}{\mathsf{\ell }}}h\left(\xi \right){\left[\Lambda \left(\xi \right)\right]}^{1-\mathsf{\ell }}\hfill \\ & & \times {\left[\frac{Z^{\sigma }\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}\right]}^{\frac{\mathsf{\ell }-1}{\mathsf{\ell }}}{\left({\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \xi \hfill \\ & \le & {\left({\int }_{\epsilon }^{\infty }\frac{Z^{\sigma }\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left({\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{\mathsf{\ell }-1}{\mathsf{\ell }}}\hfill \\ & & \times {\left({\int }_{\epsilon }^{\infty }{\left[\frac{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{Z^{\sigma }\left(\xi \right)}\right]}^{\mathsf{\ell }-1}{h}^{\mathsf{\ell }}\left(\xi \right){\left[\Lambda \left(\xi \right)\right]}^{\mathsf{\ell }\left(1-\mathsf{\ell }\right)}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(35)

Substituting (35) into (34), we observe that

$$\begin{array}{ccc}& & {\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left[{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left(Z^{\sigma }\left(\kappa \right)\right)}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa \right]}^{\mathsf{\ell }}\Delta \xi \hfill \\ & \le & \frac{\mathsf{\ell }}{\mathsf{\ell }-1}{\left({\int }_{\epsilon }^{\infty }\frac{Z^{\sigma }\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left({\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{\mathsf{\ell }-1}{\mathsf{\ell }}}\hfill \\ & & \times {\left({\int }_{\epsilon }^{\infty }{\left[\frac{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{Z^{\sigma }\left(\xi \right)}\right]}^{\mathsf{\ell }-1}{h}^{\mathsf{\ell }}\left(\xi \right){\left[\Lambda \left(\xi \right)\right]}^{\mathsf{\ell }\left(1-\mathsf{\ell }\right)}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(36)

Since Z is a non-increasing function, then $Z^{\sigma }\left(\xi \right)\le Z\left(\xi \right),$ and the inequality (36) becomes

$$\begin{array}{cc}\hfill & {\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left[{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left(Z^{\sigma }\left(\kappa \right)\right)}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa \right]}^{\mathsf{\ell }}\Delta \xi \hfill \\ \hfill & \le \frac{\mathsf{\ell }}{\mathsf{\ell }-1}{\left({\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left({\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{\mathsf{\ell }-1}{\mathsf{\ell }}}\hfill \\ \hfill & \times {\left({\int }_{\epsilon }^{\infty }{\left[\frac{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{Z^{\sigma }\left(\xi \right)}\right]}^{\mathsf{\ell }-1}{h}^{\mathsf{\ell }}\left(\xi \right){\left[\Lambda \left(\xi \right)\right]}^{\mathsf{\ell }\left(1-\mathsf{\ell }\right)}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}},\hfill \end{array}$$

and from (16), we have that

$$\begin{array}{ccc}& & {\left({\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left[{\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\kappa \right)\Delta \kappa \right]}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ & \le & \frac{\mathsf{\ell }}{\mathsf{\ell }-1}{\left({\int }_{\epsilon }^{\infty }{\left[\frac{{\Lambda }^{\sigma }\left(\xi \right)}{\Lambda \left(\xi \right)}\right]}^{\mathsf{\ell }\left(\mathsf{\ell }-1\right)}{h}^{\mathsf{\ell }}\left(\xi \right){\left[Z^{\sigma }\left(\xi \right)\right]}^{1-\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ & =& \frac{\mathsf{\ell }}{\mathsf{\ell }-1}{\left({\int }_{\epsilon }^{\infty }{\left[\frac{{\Lambda }^{\sigma }\left(\xi \right)}{\Lambda \left(\xi \right)}\right]}^{\mathsf{\ell }\left(\mathsf{\ell }-1\right)}\frac{1}{{\left(Z^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left|Z^{\Delta }\left(\xi \right)\right|}^{\mathsf{\ell }}{\left({\psi }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(37)

Substituting (13) into (37), we observe that

$$\begin{array}{ccc}& & {\left({\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left[{\int }_{\epsilon }^{\sigma \left(\xi \right)}h\left(\kappa \right)\Delta \kappa \right]}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ & \le & G{\mathsf{\ell }}^{*}{\left({\int }_{\epsilon }^{\infty }{\left[\frac{{\Lambda }^{\sigma }\left(\xi \right)}{\Lambda \left(\xi \right)}\right]}^{\mathsf{\ell }\left(\mathsf{\ell }-1\right)}\frac{{\left({\psi }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{{\left(\sigma \left(\xi \right)-\epsilon \right)}^{\mathsf{\ell }}}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(38)

From (27) and (38), we see that

$$\begin{array}{ccc}& & {\left({\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{\left[{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left(Z^{\sigma }\left(\kappa \right)\right)}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa \right]}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ & \le & G{\mathsf{\ell }}^{*}{\left({\int }_{\epsilon }^{\infty }{\left[\frac{{\Lambda }^{\sigma }\left(\xi \right)}{\Lambda \left(\xi \right)}\right]}^{\mathsf{\ell }\left(\mathsf{\ell }-1\right)}\frac{{\left({\psi }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{{\left(\sigma \left(\xi \right)-\epsilon \right)}^{\mathsf{\ell }}}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(39)

From (25), (26) and (39), we have that

$$\begin{array}{cc}\hfill & \left({\int }_{\epsilon }^{\infty }\frac{Z\left(\xi \right)}{{\left({\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}\right.\hfill \\ \hfill & \times {\left.{\left({\left(Z^{\sigma }\left(\xi \right)\right)}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\psi }^{\sigma }\left(\xi \right)+\frac{1}{{\mathsf{\ell }}^{*}}{\int }_{\epsilon }^{\sigma \left(\xi \right)}{\left(Z^{\sigma }\left(\kappa \right)\right)}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|{\psi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & \le G{\left({\int }_{\epsilon }^{\infty }{\left[\frac{{\Lambda }^{\sigma }\left(\xi \right)}{\Lambda \left(\xi \right)}\right]}^{\mathsf{\ell }\left(\mathsf{\ell }-1\right)}{\left(\frac{{\psi }^{\sigma }\left(\xi \right)}{\sigma \left(\xi \right)-\epsilon }\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}+H{\left({\int }_{\epsilon }^{\infty }{\left(\frac{{\psi }^{\sigma }\left(\xi \right)}{\sigma \left(\xi \right)-\epsilon }\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(40)

Substituting (40) into (24), we observe that

$$\begin{array}{cc}\hfill & {\left({\int }_{\epsilon }^{\infty }Z\left(\xi \right){\left(\frac{{{\rm Y}}^{\sigma }\left(\xi \right)}{{\Lambda }^{\sigma }\left(\xi \right)}\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & \le G{\left({\int }_{\epsilon }^{\infty }{\left[\frac{{\Lambda }^{\sigma }\left(\xi \right)}{\Lambda \left(\xi \right)}\right]}^{\mathsf{\ell }\left(\mathsf{\ell }-1\right)}{\left(\frac{{\psi }^{\sigma }\left(\xi \right)}{\sigma \left(\xi \right)-\epsilon }\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}+H{\left({\int }_{\epsilon }^{\infty }{\left(\frac{{\psi }^{\sigma }\left(\xi \right)}{\sigma \left(\xi \right)-\epsilon }\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}},\hfill \end{array}$$

which is (14). □

Remark 1.

If $\mathbb{T}=\mathbb{R}$ and $\epsilon =0,$ then $\sigma \left(\xi \right)=\xi ,$ $Z^{\sigma }\left(\xi \right)=Z\left(\xi \right),$ $Z^{\Delta }\left(\xi \right)=Z^{\prime }\left(\xi \right),{\Lambda }^{\sigma }\left(\xi \right)=\Lambda \left(\xi \right)$ and $\frac{{\Lambda }^{\sigma }\left(\xi \right)}{\Lambda \left(\xi \right)}=1;$ thus, we obtain (5).

Corollary 1.

If $\mathbb{T}={\mathbb{N}}_0,$ $\epsilon =0,$ $\mathsf{\ell }>1,$ then $Z\left(\xi \right)>0$ is a non-increasing function and there exists $G,H$, which are a positive constants, such that

$$\left(\xi +1\right)\left|\Delta Z\left(\xi \right)\right|\le GZ(\xi +1),and\left(\xi +1\right)Z\left(\xi \right)\le H\Lambda (\xi +1),$$

then $\sigma \left(\xi \right)=\xi +1,$ and the inequality

$$\begin{array}{ccc}& & {\left[\sum _{\xi =0}^{\infty }Z\left(\xi \right){\left(\frac{{\rm Y}(\xi +1)}{\Lambda (\xi +1)}\right)}^{\mathsf{\ell }}\right]}^{\frac{1}{\mathsf{\ell }}}\hfill \\ & \le & \left(2^{\mathsf{\ell }-1}G+H\right){\left(\sum _{\xi =0}^{\infty }{\left(\frac{\psi (\xi +1)}{\xi +1}\right)}^{\mathsf{\ell }}\right)}^{\frac{1}{\mathsf{\ell }}},\hfill \end{array}$$

(41)

is satisfied, where

$${\rm Y}\left(\xi \right)=\sum _{\kappa =0}^{\xi -1}Z\left(\kappa \right)\Xi \left(\kappa \right),\psi \left(\xi \right)=\sum _{j=0}^{\xi -1}{\left[Z\left(j\right)\right]}^{\frac{1}{\mathsf{\ell }}}\Xi \left(j\right),$$

and

$$\Lambda \left(\xi \right)=\sum _{\kappa =0}^{\xi -1}Z\left(\kappa \right).$$

Proof. 

For $\mathbb{T}={\mathbb{N}}_0,$ $\epsilon =0,$ we have $\sigma \left(\xi \right)=\xi +1$ and (14) becomes

$$\begin{array}{ccc}& & {\left[\sum _{\xi =0}^{\infty }Z\left(\xi \right){\left(\frac{{\rm Y}(\xi +1)}{\Lambda (\xi +1)}\right)}^{\mathsf{\ell }}\right]}^{\frac{1}{\mathsf{\ell }}}\hfill \\ & \le & G{\left({\int }_{\epsilon }^{\infty }{\left[\frac{\Lambda (\xi +1)}{\Lambda \left(\xi \right)}\right]}^{\mathsf{\ell }\left(\mathsf{\ell }-1\right)}{\left(\frac{\psi (\xi +1)}{\xi +1}\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ & & +H{\left({\int }_{\epsilon }^{\infty }{\left(\frac{\psi (\xi +1)}{\xi +1}\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(42)

Since $Z\left(\xi \right)$ is a non-increasing function, we obtain $Z\left(\xi \right)\le Z(\xi -1)$, and so

$$\begin{array}{cc}\hfill \Lambda (\xi +1)& =\sum _{\kappa =0}^{\xi }Z\left(\kappa \right)=\sum _{\kappa =0}^{\xi -1}Z\left(\kappa \right)+Z\left(\xi \right)\hfill \\ \hfill & \le \sum _{\kappa =0}^{\xi -1}Z\left(\kappa \right)+Z(\xi -1)\le 2\sum _{\kappa =0}^{\xi -1}Z\left(\kappa \right)=2\Lambda \left(\xi \right).\hfill \end{array}$$

(43)

Substituting (43) into (42), we observe that

$${\left[\sum _{\xi =0}^{\infty }Z\left(\xi \right){\left(\frac{{\rm Y}(\xi +1)}{\Lambda (\xi +1)}\right)}^{\mathsf{\ell }}\right]}^{\frac{1}{\mathsf{\ell }}\prime }\le \left(2^{\mathsf{\ell }-1}G+H\right){\left({\int }_{\epsilon }^{\infty }{\left(\frac{\psi (\xi +1)}{\xi +1}\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}},$$

which is (41). □

Theorem 5.

Assume that $\epsilon \in \mathbb{T},$ $\mathsf{\ell }>1$ with $1/\mathsf{\ell }+1/{\mathsf{\ell }}^{*}=1$ and $\Xi \left(\xi \right)$ is a non-negative function on ${[\epsilon ,\infty )}_{\mathbb{T}}.$ If $Z\left(\xi \right)>0$ is a differentiable and non-increasing function on ${[\epsilon ,\infty )}_{\mathbb{T}}$ and there exists $G,H$, which are positive constants, such that

$$\left(\xi -b\right)\left|Z^{\Delta }\left(\xi \right)\right|\le G{Z}^{\sigma }\left(\xi \right)and\left(\xi -b\right)Z\left(\xi \right)\le B\Lambda \left(\xi \right),$$

(44)

then

$${\left({\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\frac{Z\left(\kappa \right)\Xi \left(\kappa \right)}{\Lambda \left(\kappa \right)}\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\le \left(H+\mathsf{\ell }C\right){\left({\int }_b^{\infty }{\left[\phi \left(\xi \right)\right]}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}},$$

(45)

where $C=1+G+H,$ $\phi \left(\xi \right)={\int }_{\xi }^{\infty }\frac{{\left[Z\left(y\right)\right]}^{\frac{1}{\mathsf{\ell }}}\Xi \left(y\right)}{y-b}\Delta y$ and $\Lambda \left(\xi \right)={\int }_b^{\xi }Z\left(y\right)\Delta y.$

Proof. 

Denote

$$\lambda \left(\kappa \right)=\frac{{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}{\phi }^{\sigma }\left(\kappa \right).$$

(46)

Since

$${\phi }^{\Delta }\left(\xi \right)=\frac{-{\left[Z\left(\xi \right)\right]}^{\frac{1}{\mathsf{\ell }}}\Xi \left(\xi \right)}{\xi -b}\le 0\mathrm{for}\xi >b,$$

then $\phi$ is a non-increasing function and thus

$$\begin{array}{ccc}& & {\int }_{\xi }^{\infty }\frac{Z\left(\kappa \right)\Xi \left(\kappa \right)}{\Lambda \left(\kappa \right)}\Delta \kappa \hfill \\ & =& -{\int }_{\xi }^{\infty }\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}{\phi }^{\Delta }\left(\kappa \right)\Delta \kappa .\hfill \end{array}$$

(47)

Applying (9) to ${\int }_{\xi }^{\infty }\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}{\phi }^{\Delta }\left(\kappa \right)\Delta \kappa ,$ with

$$u\left(\kappa \right)=\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\mathrm{and}v\left(\kappa \right)=\phi \left(\kappa \right),$$

we see that

$$\begin{array}{ccc}& & {\int }_{\xi }^{\infty }\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}{\phi }^{\Delta }\left(\kappa \right)\Delta \kappa \hfill \\ & =& {\left.\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\phi \left(\kappa \right)\right|}_{\xi }^{\infty }\hfill \\ & & -{\int }_{\xi }^{\infty }{\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa .\hfill \end{array}$$

(48)

From Lemma 7, we observe that

$$\underset{\kappa \to \infty }{lim}\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\phi \left(\kappa \right)=0,$$

and then we have from (48) that

$$\begin{array}{ccc}& & {\int }_{\xi }^{\infty }\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}{\phi }^{\Delta }\left(\kappa \right)\Delta \kappa \hfill \\ & =& -\frac{\left(\xi -b\right){\left[Z\left(\xi \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\xi \right)}\phi \left(\xi \right)\hfill \\ & & -{\int }_{\xi }^{\infty }{\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa .\hfill \end{array}$$

(49)

From (47) and (49), we observe that

$$\begin{array}{cc}\hfill & {\int }_{\xi }^{\infty }\frac{Z\left(\kappa \right)\Xi \left(\kappa \right)}{\Lambda \left(\kappa \right)}\Delta \kappa \hfill \\ \hfill & =\frac{\left(\xi -b\right){\left[Z\left(\xi \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\xi \right)}\phi \left(\xi \right)+{\int }_{\xi }^{\infty }{\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa \hfill \\ \hfill & \le \frac{\left(\xi -b\right){\left[Z\left(\xi \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\xi \right)}\phi \left(\xi \right)\hfill \\ \hfill & +{\int }_{\xi }^{\infty }\left|{\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }\right|{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa .\hfill \end{array}$$

(50)

From (50), we see that

$$\begin{array}{ccc}& & {\left({\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\frac{Z\left(\kappa \right)\Xi \left(\kappa \right)}{\Lambda \left(\kappa \right)}\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ & \le & \left({\int }_b^{\infty }Z\left(\xi \right)\left(\frac{\left(\xi -b\right){\left[Z\left(\xi \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\xi \right)}\phi \left(\xi \right)\right.\right.\hfill \\ & & +{\left.{\left.{\int }_{\xi }^{\infty }\left|{\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }\right|{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(51)

Applying (11) onto the right-hand side of (51) with

$$\chi \left(\xi \right)=Z\left(\xi \right),\lambda \left(\xi \right)=\frac{\left(\xi -b\right){\left[Z\left(\xi \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\xi \right)}\phi \left(\xi \right),$$

and

$$\omega \left(\xi \right)={\int }_{\xi }^{\infty }\left|{\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }\right|{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa ,$$

we see that

$$\begin{array}{cc}\hfill & \left({\int }_b^{\infty }Z\left(\xi \right)\left(\frac{\left(\xi -b\right){\left[Z\left(\xi \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\xi \right)}\phi \left(\xi \right)\right.\right.\hfill \\ \hfill & +{\left.{\left.{\int }_{\xi }^{\infty }\left|{\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }\right|{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & \le {\left({\int }_b^{\infty }Z\left(\xi \right){\left(\frac{\left(\xi -b\right){\left[Z\left(\xi \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\xi \right)}\phi \left(\xi \right)\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & +{\left({\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\left|{\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }\right|{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(52)

Substituting (52) into (51), we obtain that

$$\begin{array}{ccc}& & {\left({\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\frac{Z\left(\kappa \right)\Xi \left(\kappa \right)}{\Lambda \left(\kappa \right)}\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ & \le & {\left({\int }_b^{\infty }Z\left(\xi \right){\left(\frac{\left(\xi -b\right){\left[Z\left(\xi \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\xi \right)}\phi \left(\xi \right)\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ & & +{\left({\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\left|{\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }\right|{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(53)

From (44) and (53), we have that

$$\begin{array}{cc}\hfill & {\left({\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\frac{Z\left(\kappa \right)\Xi \left(\kappa \right)}{\Lambda \left(\kappa \right)}\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & \le H{\left({\int }_b^{\infty }{\left[\phi \left(\xi \right)\right]}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & +{\left({\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\left|{\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }\right|{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(54)

Applying the quotient rule of differentiation to $\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right),$ we observe that

$${\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }=\frac{\Lambda \left(\kappa \right){\left[\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}\right]}^{\Delta }-\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{1+\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right){\Lambda }^{\sigma }\left(\kappa \right)}.$$

(55)

Applying the multiple rule of differentiation to $\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}$, we have that

$${\left[\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}\right]}^{\Delta }=\left(\kappa -b\right){\left[{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}\right]}^{\Delta }+{\left[Z^{\sigma }\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}.$$

(56)

Applying (8) to ${\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}},$ ${\mathsf{\ell }}^{*}>1,$ we see that

$${\left[{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}\right]}^{\Delta }=\frac{1}{{\mathsf{\ell }}^{*}}{\left[Z\left(\xi \right)\right]}^{\frac{-1}{\mathsf{\ell }}}{Z}^{\Delta }\left(\kappa \right),\xi \in [\kappa ,\sigma \left(\kappa \right)].$$

(57)

From (55) to (57), we obtain

$$\begin{array}{ccc}& & {\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }\hfill \\ & =& \frac{\frac{1}{{\mathsf{\ell }}^{*}}\left(\kappa -b\right){\left[Z\left(\xi \right)\right]}^{\frac{-1}{\mathsf{\ell }}}{Z}^{\Delta }\left(\kappa \right)}{{\Lambda }^{\sigma }\left(\kappa \right)}+\frac{{\left[Z^{\sigma }\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}-\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{1+\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right){\Lambda }^{\sigma }\left(\kappa \right)},\hfill \end{array}$$

and then

$$\begin{array}{ccc}& & \left|{\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }\right|\hfill \\ & \le & \frac{\frac{1}{{\mathsf{\ell }}^{*}}\left(\kappa -b\right){\left[Z\left(\xi \right)\right]}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|}{{\Lambda }^{\sigma }\left(\kappa \right)}+\frac{{\left[Z^{\sigma }\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}+\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{1+\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right){\Lambda }^{\sigma }\left(\kappa \right)}.\hfill \end{array}$$

(58)

Since $\xi \le \sigma \left(\kappa \right)$ and $Z\left(\kappa \right)$ is a non-increasing function, we have that $Z\left(\xi \right)\ge Z^{\sigma }\left(\kappa \right),$ and then

$${\left[Z\left(\xi \right)\right]}^{\frac{-1}{\mathsf{\ell }}}\le {\left[Z^{\sigma }\left(\kappa \right)\right]}^{\frac{-1}{\mathsf{\ell }}}.$$

(59)

Substituting (59) into (58), we observe that

$$\begin{array}{ccc}& & \left|{\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }\right|\hfill \\ & \le & \frac{\frac{1}{{\mathsf{\ell }}^{*}}\left(\kappa -b\right){\left[Z^{\sigma }\left(\kappa \right)\right]}^{\frac{-1}{\mathsf{\ell }}}\left|Z^{\Delta }\left(\kappa \right)\right|}{{\Lambda }^{\sigma }\left(\kappa \right)}+\frac{{\left[Z^{\sigma }\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}+\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{1+\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right){\Lambda }^{\sigma }\left(\kappa \right)}\hfill \\ & =& \frac{\frac{1}{{\mathsf{\ell }}^{*}}\left(\kappa -b\right){\left[Z^{\sigma }\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}\left|Z^{\Delta }\left(\kappa \right)\right|}{Z^{\sigma }\left(\kappa \right){\Lambda }^{\sigma }\left(\kappa \right)}+\frac{{\left[Z^{\sigma }\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}+\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{1+\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right){\Lambda }^{\sigma }\left(\kappa \right)}.\hfill \end{array}$$

(60)

Using (44), the inequality (60) becomes

$$\begin{array}{ccc}& & \left|{\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }\right|\hfill \\ & \le & \frac{b}{{\mathsf{\ell }}^{*}}\frac{{\left[Z^{\sigma }\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}+\frac{{\left[Z^{\sigma }\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}+H\frac{{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}.\hfill \end{array}$$

(61)

Since $Z\left(\kappa \right)$ is non-increasing and $\sigma \left(\kappa \right)\ge \kappa ,$ we have that $Z^{\sigma }\left(\kappa \right)\le Z\left(\kappa \right),$ and then the inequality (61) is

$$\begin{array}{ccc}& & \left|{\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }\right|\hfill \\ & \le & \frac{b}{{\mathsf{\ell }}^{*}}\frac{{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}+\frac{{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}+H\frac{{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}\hfill \\ & =& \frac{{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}\left(\frac{G}{{\mathsf{\ell }}^{*}}+1+H\right).\hfill \end{array}$$

(62)

Since $1/{\mathsf{\ell }}^{*}<1,$ then the inequality (62) becomes

$$\left|{\left(\frac{\left(\kappa -b\right){\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\Lambda \left(\kappa \right)}\right)}^{\Delta }\right|\le \frac{{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}\left[1+G+H\right]=C\frac{{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}.$$

(63)

Substituting (63) into (54), we have that

$$\begin{array}{cc}\hfill & {\left({\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\frac{Z\left(\kappa \right)\Xi \left(\kappa \right)}{\Lambda \left(\kappa \right)}\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & \le H{\left({\int }_b^{\infty }{\left[\phi \left(\xi \right)\right]}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & +C{\left({\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\frac{{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(64)

Using (46), and by applying Lemma 6, we see that

$$\begin{array}{ccc}& & {\left({\int }_{\xi }^{\infty }\frac{{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\hfill \\ & =& {\left({\int }_{\xi }^{\infty }\lambda \left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\hfill \\ & \le & \mathsf{\ell }{\int }_{\xi }^{\infty }\lambda \left(\kappa \right){\left({\int }_{\kappa }^{\infty }\lambda \left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \kappa ,\hfill \end{array}$$

and then

$$\begin{array}{ccc}& & {\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\frac{{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \hfill \\ & \le & \mathsf{\ell }{\int }_b^{\infty }Z\left(\xi \right)\left[{\int }_{\xi }^{\infty }\lambda \left(\kappa \right){\left({\int }_{\kappa }^{\infty }\lambda \left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \kappa \right]\Delta \xi .\hfill \end{array}$$

(65)

Applying Lemma 5, we observe that

$$\begin{array}{ccc}& & {\int }_b^{\infty }Z\left(\xi \right)\left[{\int }_{\xi }^{\infty }\lambda \left(\kappa \right){\left({\int }_{\kappa }^{\infty }\lambda \left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \kappa \right]\Delta \xi \hfill \\ & =& {\int }_b^{\infty }\lambda \left(\xi \right){\left({\int }_{\xi }^{\infty }\lambda \left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\left({\int }_b^{\sigma \left(\xi \right)}Z\left(\kappa \right)\Delta \kappa \right)\Delta \xi \hfill \\ & =& {\int }_b^{\infty }\lambda \left(\xi \right){\Lambda }^{\sigma }\left(\xi \right){\left({\int }_{\xi }^{\infty }\lambda \left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \xi .\hfill \end{array}$$

(66)

Substituting (66) into (65), we have that

$$\begin{array}{ccc}& & {\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\frac{{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \hfill \\ & \le & \mathsf{\ell }{\int }_b^{\infty }\lambda \left(\xi \right){\Lambda }^{\sigma }\left(\xi \right){\left({\int }_{\xi }^{\infty }\lambda \left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \xi \hfill \\ & =& \mathsf{\ell }{\int }_b^{\infty }\frac{\lambda \left(\xi \right){\Lambda }^{\sigma }\left(\xi \right)}{{\left[Z\left(\xi \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{\left[Z\left(\xi \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\left({\int }_{\xi }^{\infty }\lambda \left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\Delta \xi .\hfill \end{array}$$

(67)

Applying (10), we see that

$$\begin{array}{ccc}& & {\int }_b^{\infty }\frac{\lambda \left(\xi \right){\Lambda }^{\sigma }\left(\xi \right)}{{\left[Z\left(\xi \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}\left[{\left[Z\left(\xi \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}{\left({\int }_{\xi }^{\infty }\lambda \left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }-1}\right]\Delta \xi \hfill \\ & \le & {\left({\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\lambda \left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{\mathsf{\ell }-1}{\mathsf{\ell }}}{\left({\int }_b^{\infty }\frac{{\left(\lambda \left(\xi \right){\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{{\left[Z\left(\xi \right)\right]}^{\mathsf{\ell }-1}}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(68)

Substituting (68) into (67), we see that

$$\begin{array}{ccc}& & {\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\frac{{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \hfill \\ & \le & \mathsf{\ell }{\left({\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\lambda \left(\tau \right)\Delta \tau \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{1-\frac{1}{\mathsf{\ell }}}{\left({\int }_b^{\infty }\frac{{\left(\lambda \left(\xi \right){\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{Z^{\mathsf{\ell }-1}\left(\xi \right)}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

From (46), the last inequality becomes

$$\begin{array}{ccc}& & {\left({\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\frac{{\left[Z\left(\kappa \right)\right]}^{\frac{1}{{\mathsf{\ell }}^{*}}}}{{\Lambda }^{\sigma }\left(\kappa \right)}{\phi }^{\sigma }\left(\kappa \right)\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ & \le & \mathsf{\ell }{\left({\int }_b^{\infty }\frac{{\left(\lambda \left(\xi \right){\Lambda }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}}{{\left[Z\left(\xi \right)\right]}^{\mathsf{\ell }-1}}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ & =& \mathsf{\ell }{\left({\int }_b^{\infty }{\left({\phi }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(69)

Substituting (69) into (64), we see that

$$\begin{array}{cc}\hfill & {\left({\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\frac{Z\left(\kappa \right)\Xi \left(\kappa \right)}{\Lambda \left(\kappa \right)}\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & \le H{\left({\int }_b^{\infty }{\left[\phi \left(\xi \right)\right]}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & +\mathsf{\ell }C{\left({\int }_b^{\infty }{\left({\phi }^{\sigma }\left(\xi \right)\right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}.\hfill \end{array}$$

(70)

Since $\phi$ is a non-increasing function and $\sigma \left(\xi \right)\ge \xi ,$ then ${\phi }^{\sigma }\left(\xi \right)\le \phi \left(\xi \right),$ and then the inequality (70) becomes

$$\begin{array}{cc}\hfill & {\left({\int }_b^{\infty }Z\left(\xi \right){\left({\int }_{\xi }^{\infty }\frac{Z\left(\kappa \right)\Xi \left(\kappa \right)}{\Lambda \left(\kappa \right)}\Delta \kappa \right)}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}}\hfill \\ \hfill & \le \left(H+\mathsf{\ell }C\right){\left({\int }_b^{\infty }{\left[\phi \left(\xi \right)\right]}^{\mathsf{\ell }}\Delta \xi \right)}^{\frac{1}{\mathsf{\ell }}},\hfill \end{array}$$

which is (45). □

Remark 2.

If $\mathbb{T}=\mathbb{R},$ $b=0,$ then $\sigma \left(\xi \right)=\xi ,$ $Z^{\sigma }\left(\xi \right)=Z\left(\xi \right)$ and $Z^{\Delta }\left(\xi \right)=Z^{\prime }\left(\xi \right),$ and so we obtain (6).

Corollary 2.

Assume that $\mathbb{T}={\mathbb{N}}_0$, $b=1,$ $\mathsf{\ell }>1$ and $\Xi \left(\xi \right)$ is a non-negative sequence. Let $Z\left(\xi \right)>0$ be a non-increasing sequence and there exists $G,H$ as positive constants, such that

$$\left(\xi -1\right)\left|\Delta Z\left(\xi \right)\right|\le GZ(\xi +1)and\left(\xi -1\right)Z\left(\xi \right)\le H\Lambda \left(\xi \right).$$

Then

$$\sum _{\xi =1}^{\infty }Z\left(\xi \right){\left(\sum _{\kappa =\xi }^{\infty }\frac{Z\left(\kappa \right)\Xi \left(\kappa \right)}{\Lambda \left(\kappa \right)}\right)}^{\mathsf{\ell }}\le {\left(H+\mathsf{\ell }C\right)}^{\mathsf{\ell }}\sum _{\xi =1}^{\infty }{\left[\phi \left(\xi \right)\right]}^{\mathsf{\ell }},$$

where $C=1+G+H,$

$$\phi \left(\xi \right)=\sum _{\delta =\xi }^{\infty }\frac{{\left[Z\left(\delta \right)\right]}^{\frac{1}{\mathsf{\ell }}}\Xi \left(\delta \right)}{\delta }and\Lambda \left(\xi \right)=\sum _{\kappa =1}^{\xi -1}Z\left(\kappa \right).$$

4. Conclusions

In this manuscript, we have demonstrated novel Copson-type inequalities within the framework of time scales delta calculus for the unification of inequalities in the two (the continuous and the discrete) calculi, focusing on a positive, differentiable, and non-increasing function. Looking ahead, we plan to extend our findings by establishing fresh dynamic inequalities of Copson type using time scales nabla calculus. Furthermore, we aim to generalize these inequalities to diamond alpha calculus, a linear combination of delta calculus and nabla calculus. Additionally, our future work includes the exploration of new dynamic inequalities through conformable delta and nabla fractional calculus.