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Weighted dynamic Hardy-type inequalities involving many functions on arbitrary time scales

Abstract

The objective of this paper is to prove some new dynamic inequalities of Hardy type on time scales which generalize and improve some recent results given in the literature. Further, we derive some new weighted Hardy dynamic inequalities involving many functions on time scales. As special cases, we get continuous and discrete inequalities.

Introduction

In [1], Hardy showed that if \(\alpha >1\) and \(\Psi (\zeta )\geq 0\) over the interval \((0,\infty )\) such that \(\int _{0}^{\infty }\Psi ^{\alpha }(\zeta ) \,d\zeta <\infty \), then

$$ \int _{0}^{\infty } \biggl( \frac{1}{x} \int _{0}^{x}\Psi ( \zeta ) \,d\zeta \biggr) ^{\alpha }\,d{ x\leq } \biggl( \frac{\alpha }{\alpha -1} \biggr) ^{\alpha }{ \int _{0}^{\infty }}\Psi ^{\alpha }{ (x)\,dx,} $$
(1)

where the constant \(( \alpha /(\alpha -1) ) ^{\alpha }\) is sharp. In [2], Hardy obtained that if \(\alpha >1\) and \(m>1\), then

$$ \int _{0}^{\infty }\frac{1}{x^{m}} \biggl( \int _{0}^{x}\Psi ( \zeta ) \,d\zeta \biggr) ^{\alpha }\,d{ x\leq } \biggl( \frac{\alpha }{m-1} \biggr) ^{\alpha } \int _{0}^{\infty } \frac{1}{x^{m-\alpha }}\Psi ^{\alpha }(x){ \,dx.} $$
(2)

In [3], Levinson proved that if \(\alpha >1\), \(\Psi (x)\geq 0\), \(f(x)>0\) is an absolutely continuous function and

$$ \frac{\alpha }{\alpha -1}+\frac{f'(x)}{f(x)}\geq \frac{1}{\beta }>0, \quad \text{for all }x>0, $$

then

$$ \int _{0}^{\infty } \biggl( \frac{1}{xf(x)} \int _{0}^{x}f( \zeta )\Psi (\zeta ) \,d\zeta \biggr) ^{\alpha }\,d{ x}\leq \beta ^{\alpha } \int _{0}^{\infty }\Psi { ^{\alpha }}(x){ \,dx.} $$
(3)

In [4], S. Hussan et al. proved that for any \(\mathfrak{i}=1,2,\dots ,n\), nN, \(f_{\mathfrak{i}}(\zeta )\geq 0\) and integrable function on \((0,\infty )\) and ŵ, \(u_{\mathfrak{i}}\), \(z_{\mathfrak{i}}\) are absolutely continuous functions with \(z_{\mathfrak{i}}'\) essentially bounded and positive, if \(u_{\mathfrak{i}}\) is increasing and

$$\begin{aligned}& 1+ \frac{u_{\mathfrak{i}}(\zeta )\hat{w}'(\zeta )}{(1-2m)u_{\mathfrak{i}}'(\zeta )\hat{w}(\zeta )}\geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0 , \quad \text{for }m> \frac{1}{2},\\& 1+ \frac{u_{\mathfrak{i}}(\zeta )\hat{w}'(\zeta )}{(1-2m)u_{\mathfrak{i}}'(\zeta )\hat{w}(\zeta )}\geq \frac{1}{\delta _{\mathfrak{i}}}>0, \quad \text{for }m< \frac{1}{2}, \end{aligned}$$

then

$$ \sum_{\mathfrak{i}=1}^{n} \int _{0}^{\infty }\hat{w}(\zeta )R_{\mathfrak{i}}(\zeta )R_{\mathfrak{i}+1}(\zeta )\,d \zeta \leq \sum_{\mathfrak{i}=1}^{n} \biggl( \frac{2\beta _{\mathfrak{i}}}{ \vert 2m-1 \vert } \biggr) ^{2} \int _{0}^{\infty }\hat{w}( \zeta )g_{\mathfrak{i}}( \zeta )\,d\zeta , $$
(4)

where

$$\begin{aligned} R_{\mathfrak{i}}(\zeta )&=\textstyle\begin{cases} \frac{\sqrt{u_{\mathfrak{i}}'(\zeta )}}{u_{\mathfrak{i}}^{m}(\zeta )}\int _{0}^{\zeta } \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}'(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x)\,dx, & m>\frac{1}{2}, \\ \frac{\sqrt{u_{\mathfrak{i}}'(\zeta )}}{u_{\mathfrak{i}}^{m}(\zeta )}\int _{\zeta }^{\infty } \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}'(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x)\,dx, & m< \frac{1}{2},\end{cases}\displaystyle \\ g_{\mathfrak{i}}(\zeta )&= \frac{ [ u_{\mathfrak{i}}(\zeta ) ] ^{4-2m} [ z_{\mathfrak{i}}'(\zeta ) ] ^{2}f_{\mathfrak{i}}^{2}(\zeta )}{z_{\mathfrak{i}}^{2}(\zeta )u_{\mathfrak{i}}'(\zeta )}\quad \text{and}\quad \beta _{\mathfrak{i}}=\max_{1\leq \mathfrak{i}\leq n} ( \mathfrak{\lambda }_{\mathfrak{i}},\delta _{\mathfrak{i}} ) . \end{aligned}$$

Also in the same paper [4], the authors proved that for any \(\mathfrak{i}=1,2,\dots ,n\), \(n\geq \kappa -1\), n,κN, if \(\alpha _{\mathfrak{i}}>1\), \(\delta _{\mathfrak{i}}=\alpha _{ \mathfrak{i}}/ ( \kappa \alpha _{\mathfrak{i}}-1 ) \) and

$$\begin{aligned}& 1+ \frac{u_{\mathfrak{i}}(\zeta )\hat{w}'(\zeta )}{(1-\kappa \alpha _{\mathfrak{i}}m)u_{\mathfrak{i}}'(\zeta )\hat{w}(\zeta )}\geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0, \quad \text{for }m> \frac{1}{\kappa \alpha _{\mathfrak{i}}},\\& 1+ \frac{u_{\mathfrak{i}}(\zeta )\hat{w}'(\zeta )}{(1-\kappa \alpha _{\mathfrak{i}}m)u_{\mathfrak{i}}'(\zeta )\hat{w}(\zeta )}\geq \frac{1}{\delta _{\mathfrak{i}}}>0, \quad \text{for }m< \frac{1}{\kappa \alpha _{\mathfrak{i}}}, \end{aligned}$$

then

$$ \sum_{\mathfrak{i}=1}^{n-\kappa +2} \int _{0}^{\infty }\hat{w}( \zeta ) \Biggl[ \prod_{j=\mathfrak{i}}^{\mathfrak{i}+\kappa -1}R_{j}^{ \alpha _{j}}( \zeta ) \Biggr] \,d\zeta \leq \sum_{ \mathfrak{i}=1}^{n} \biggl( \frac{\kappa \alpha _{\mathfrak{i}}\beta _{\mathfrak{i}}}{ \vert \kappa \alpha _{\mathfrak{i}}m-1 \vert } \biggr) ^{ \kappa \alpha _{\mathfrak{i}}} \int _{0}^{\infty }\hat{w}( \zeta )g_{\mathfrak{i}}( \zeta )\,d\zeta , $$
(5)

where

$$\begin{aligned} R_{\mathfrak{i}}(\zeta )&=\textstyle\begin{cases} \frac{\sqrt[\kappa \alpha _{\mathfrak{i}}]{u_{\mathfrak{i}}'(\zeta )}}{u_{\mathfrak{i}}^{m}(\zeta )}\int _{0}^{ \zeta } \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}'(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x)\,dx, & m> \frac{1}{\kappa \alpha _{\mathfrak{i}}}, \\ \frac{\sqrt[\kappa \alpha _{\mathfrak{i}}]{u_{\mathfrak{i}}'(\zeta )}}{u_{\mathfrak{i}}^{m}(\zeta )}\int _{ \zeta }^{\infty } \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}'(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x)\,dx, & m< \frac{1}{\kappa \alpha _{\mathfrak{i}}},\end{cases}\displaystyle \\ g_{\mathfrak{i}}(\zeta )&= \frac{ [ u_{\mathfrak{i}}(\zeta ) ] ^{\kappa \alpha _{\mathfrak{i}} ( 2-m ) } [ z_{\mathfrak{i}}'(\zeta ) ] ^{\kappa \alpha _{\mathfrak{i}}}f_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}}(\zeta )}{z_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}}(\zeta ) [ u_{\mathfrak{i}}'(\zeta ) ] ^{k\alpha _{\mathfrak{i}}-1}}\quad \text{and}\quad \beta _{\mathfrak{i}}=\max_{1\leq \mathfrak{i} \leq n} ( \mathfrak{\lambda }_{\mathfrak{i}},\delta _{ \mathfrak{i}} ) . \end{aligned}$$

The main aim is to establish some dynamic inequalities where the involved functions are defined on the T domain. These results involve the classical discrete and continuous inequalities. For more details, we point the reader to the books [5, 6]. In [7], Řehak found the time scale version of Hardy’s inequality. Especially, Řehak derived that if \(\alpha >1\) and \(\Psi (\zeta )\geq 0\) are such that \(\int _{a}^{\infty }\Psi ^{\alpha }(\zeta )\Delta \zeta < \infty \) then

$$ \int _{a}^{\infty } \biggl( \frac{1}{\sigma (\zeta )-a}\int _{a}^{\sigma (\zeta )}\Psi (x)\Delta x \biggr) ^{ \alpha }\Delta \zeta \leq { \biggl( \frac{\alpha }{\alpha -1} \biggr) ^{\alpha }} \int _{a}^{\infty }\Psi ^{\alpha }( \zeta ) \Delta \zeta . $$

In addition, if \(\mu (\zeta )/\zeta \rightarrow 0\) as \(\zeta \rightarrow \infty \), then the constant \((\alpha /(\alpha -1))^{\alpha }\) is sharp.

In [8], the authors showed that if \(f(\zeta )>0\), \(\Psi (\zeta )\geq 0\) and \(f^{\Delta }(\zeta )\leq 0\) on [0, ) T , \(\alpha >1\) and there exist constants \(\kappa ,\beta >0\) such that \(\zeta /\sigma (\zeta )\geq 1/\kappa \) and

α α 1 + κ α Φ ( ζ ) Φ σ ( ζ ) ζ f Δ ( ζ ) f σ ( ζ ) 1 β ,for ζ[0, ) T ,

then

$$ \int _{0}^{\infty }\frac{1}{\zeta ^{\alpha }} \bigl( \Phi ^{ \sigma }(\zeta ) \bigr) ^{\alpha }\Delta \zeta \leq { \bigl( \beta \kappa ^{\alpha } \bigr) ^{\alpha }}\int _{0}^{\infty } \biggl( \frac{f(\zeta )\Psi (\zeta )}{f^{\sigma }(\zeta )} \biggr) ^{\alpha }\Delta \zeta , $$

where

Φ(ζ)= 1 f ( ζ ) 0 ζ f(x)Ψ(x)Δx,ζ[0, ) T .

The purpose of this manuscript is to establish some new Hardy-type inequalities on time scales T involving many functions which generalize and improve some results in [4]. The following is the format of the paper: In Sect. 2, we begin with some background information about the delta derivative on T. Our main findings are obtained in Sect. 3.

Basic principles

A time scale T is an arbitrary nonempty closed subset of R. We define the forward jump operator σ: T T by σ(ζ)=inf{sT:s>ζ} and define the backward jump operator ρ: T T by ρ(ζ)=sup{sT:s<ζ}, respectively, where sup=infT.

A point ζT is called right-dense if \(\sigma (\zeta )=\zeta \), left-dense if \(\rho (\zeta )=\zeta \), right-scattered if \(\sigma (\zeta )>\zeta \), and left-scattered if \(\rho (\zeta )<\zeta \) If supT is finite and left-scattered, then T k =T{supT}, otherwise, T k =T.

A function f: T R is a right-dense continuous (rd-continuous) if f is continuous at right-dense points and its left-hand limits are finite at left-dense points in T.

Let f: T R be a real-valued function on T. Then for ζ T k , we define \(f^{\Delta }(\zeta )\) to be the number (if it exists) with the property that given any \(\varepsilon >0\) there is a neighborhood u of ζ such that, for all \(s\in u\), we have

$$ \bigl\vert \bigl[ f\bigl(\sigma (\zeta )\bigr)-f(s) \bigr] -f^{ \Delta }( \zeta ) [ \zeta -s ] \bigr\vert \leq \mathfrak{\varepsilon } \bigl\vert \sigma (\zeta )-s \bigr\vert . $$

In this case, we say that f is delta differentiable on T k provided \(f(\zeta )\) exists for all ζ T k . If f,g:TR are delta differentiable at ζT, then

$$ ( fg ) ^{\Delta }=f^{\Delta }g+f^{\sigma }g^{ \Delta }=fg^{\Delta }+f^{\Delta }g^{\sigma }, \quad \text{where }f^{\sigma }(\zeta )= ( f\circ \mathfrak{\sigma } ) (\zeta )=f\bigl(\sigma (\zeta )\bigr). $$
(6)

For a,bT and a delta differentiable function f, the Cauchy integral of \(f^{\Delta }\) is defined by \(\int _{a}^{b}f^{\Delta }(\zeta )\Delta \zeta =f(b)-f(a)\). The integration by parts formula on T is given by

$$ \int _{a}^{b}\Psi (\zeta )\Phi ^{\Delta }( \zeta )\Delta \zeta =(\Psi \Phi ) (b)-(\Psi \Phi ) (a)- \int _{a}^{b} \Psi ^{\Delta }(\zeta )\Phi ^{\sigma }(\zeta )\Delta \zeta . $$
(7)

Lemma 1

(Leibniz rule [9])

If f, \(f^{\Delta }\) are continuous and u,v:TT are delta differentiable functions and \(f^{\Delta }(\zeta ,s)\) mean the delta derivative of \(f(\zeta ,s)\) with respect to ζ, then

$$ \begin{aligned} & \biggl( \int _{u(\zeta )}^{v(\zeta )}f( \zeta ,s)\Delta s \biggr) ^{\Delta } \\ &\quad = \int _{u(\zeta )}^{v(\zeta )}f^{\Delta }(\zeta ,s) \Delta s+v^{ \Delta }(\zeta )f\bigl(\sigma (\zeta ),v( \zeta ) \bigr)-u^{\Delta }(\zeta )f\bigl(\sigma ( \zeta ),u(\zeta )\bigr). \end{aligned} $$
(8)

Lemma 2

(Chain rule [10])

Assume g:RR is continuous, g:TR is delta differentiable on T k , and f:RR is continuously differentiable. Then there exists a point c in the real interval \([\zeta ,\sigma (\zeta )]\) with

$$ ( f\circ g ) ^{\Delta }(\zeta )=f^{ \backprime }\bigl(g(c) \bigr)g^{\Delta }(\zeta ). $$
(9)

Lemma 3

(Hölder’s inequality [10])

Let a,bT. For rd-continuous functions f,g: [ a , b ] T R, we have

$$ \int _{a}^{b} \bigl\vert f(\zeta )g(\zeta ) \bigr\vert \Delta \zeta \leq \biggl( \int _{a}^{b} \bigl\vert f( \zeta ) \bigr\vert ^{\alpha }\Delta \zeta \biggr) ^{ \frac{1}{\alpha }} \biggl( \int _{a}^{b} \bigl\vert g(\zeta ) \bigr\vert ^{ \delta }\Delta \zeta \biggr) ^{\frac{1}{\delta }}, $$
(10)

where \(\alpha >1\) and \(\delta =\alpha /(\alpha -1)\).

Lemma 4

([11])

If \(C_{1},C_{2},\dots ,C_{n}\) are reals and \(C_{n+1}=C_{1}\), then

$$ \sum_{r=1}^{n-\kappa +2}C_{r}C_{r+1} \cdots C_{r+\kappa -1} \leq \sum_{r=1}^{n} ( C_{r} ) ^{\kappa },\quad \textit{where } n\geq \kappa -1. $$
(11)

Lemma 5

([11])

If \(C_{1},C_{2},\dots ,C_{n}\) are reals and \(C_{n+1}=C_{1}\), for \(\kappa \geq 1\), then

$$ \Biggl( \sum_{r=1}^{n}C_{r} \Biggr) ^{\kappa }\leq n^{ \kappa -1}\sum_{r=1}^{n} ( C_{r} ) ^{\kappa }. $$
(12)

Main results

Throughout this section, any time scale T is unbounded above with a,bT. We will make the assumption that the functions ŵ, \(u_{\mathfrak{i}}\), \(z_{\mathfrak{i}}\) in the statements of the theorems are rd-continuous, nonnegative and increasing, and \(f_{\mathfrak{i}}(\zeta )>0\) is an integrable function.

Theorem 6

For any \(1\leq \mathfrak{i}\leq n\), \(n\geq \kappa -1\) and n,κN, if there exist constants \(\mathfrak{\lambda }_{\mathfrak{i}}>0\), \(\delta _{\mathfrak{i}}>0\) such that

$$\begin{aligned} 1- \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{\alpha m}\hat{w}^{\Delta }(\zeta )}{(\alpha m-1) [ u_{\mathfrak{i}}(\zeta ) ] ^{\alpha m-1}u_{\mathfrak{i}}^{\Delta }(\zeta )\hat{w}^{\sigma }(\zeta )}\geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0,\quad \textit{for }m> \frac{1}{2}, \end{aligned}$$
(13)
$$\begin{aligned} 1- \frac{u_{\mathfrak{i}}(\zeta )\hat{w}^{\Delta }(\zeta )}{(\alpha m-1)u_{\mathfrak{i}}^{\Delta }(\zeta )\hat{w}^{\sigma }(\zeta )}\geq \frac{1}{\delta _{\mathfrak{i}}}>0,\quad \textit{for }m< \frac{1}{2}, \end{aligned}$$
(14)

then

$$ \sum_{\mathfrak{i}=1}^{n} \int _{0}^{\infty }\hat{w}^{\sigma }( \zeta )R_{\mathfrak{i}}^{\frac{\alpha }{2}}(\zeta )R_{\mathfrak{i}+1}^{\frac{\alpha }{2}}( \zeta )\Delta \zeta \leq \sum_{\mathfrak{i}=1}^{n} \biggl( \frac{\alpha \beta _{\mathfrak{i}}}{ \vert \alpha m-1 \vert } \biggr) ^{\alpha } \int _{0}^{\infty }\hat{w}^{\sigma }(\zeta )g_{\mathfrak{i}}(\zeta )\Delta \zeta , $$
(15)

where

$$\begin{aligned} &R_{\mathfrak{i}}(\zeta )=\textstyle\begin{cases} \frac{\sqrt[\alpha ]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m}}\int _{0}^{\sigma (\zeta )} \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x)\Delta x \\ \quad \textit{and}\quad\int _{0}^{\infty } \frac{u_{\mathfrak{i}}^{\Delta }(x)}{ ( u_{\mathfrak{i}}^{\sigma }(x) ) ^{\alpha m}} \Delta \varkappa < \infty , & \textit{for } m>\frac{1}{2}, \alpha \geq 2, \\ \frac{\sqrt[\alpha ]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}{u_{\mathfrak{i}}^{m}(\zeta )}\int _{\sigma ( \zeta )}^{\infty } \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x)\Delta x, &\textit{for } m< \frac{1}{2},1\leq \alpha \leq 2,\end{cases}\displaystyle \\ &g_{\mathfrak{i}}(\zeta )=\textstyle\begin{cases} \frac{u_{\mathfrak{i}}^{\alpha ( 2-\alpha m ) }(\zeta ) [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m\alpha ( \alpha -1 ) } [ z_{\mathfrak{i}}^{\Delta }(\zeta ) ] ^{\alpha }f_{\mathfrak{i}}^{\alpha }(\zeta )\hat{w}^{\alpha }(\zeta )}{z_{\mathfrak{i}}^{\alpha }(\zeta ) ( u_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{\alpha -1} ( \hat{w}^{\sigma }(\zeta ) ) ^{\alpha }}, & \textit{for } m>\frac{1}{2},\alpha \geq 2, \\ \frac{u_{\mathfrak{i}}^{\alpha ( 2-m ) }(\zeta ) [ z_{\mathfrak{i}}^{\Delta }(\zeta ) ] ^{\alpha }f_{\mathfrak{i}}^{\alpha }(\zeta )\hat{w}^{\alpha }(\zeta )}{z_{\mathfrak{i}}^{\alpha }(\zeta ) ( u_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{\alpha -1} ( \hat{w}^{\sigma }(\zeta ) ) ^{\alpha }}, & \textit{for } m< \frac{1}{2},1\leq \alpha \leq 2,\end{cases}\displaystyle \end{aligned}$$

and \(\beta _{\mathfrak{i}}=\max_{1\leq \mathfrak{i}\leq n} ( \mathfrak{\lambda }_{\mathfrak{i}},\delta _{\mathfrak{i}} ) \), \(u_{ \mathfrak{i}}(\infty )=\infty \).

Proof

First, let us define for \(m>\frac{1}{2}\), \(\alpha \geq 2\), and \(0< a< b<\infty \),

$$ R_{\mathfrak{i}a}(\zeta )= \frac{\sqrt[\alpha ]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}{ ( u_{\mathfrak{i}}^{\sigma }(\zeta ) ) ^{m}} \int _{a}^{\sigma (\zeta )} \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x) \Delta x, \quad 1\leq \mathfrak{i} \leq n, $$

with \(R_{\mathfrak{i}0}(\zeta )=R_{\mathfrak{i}}(\zeta )\). Using (11) with \(\kappa =2\) for \(C_{\mathfrak{i}}=R_{\mathfrak{i}}^{\frac{\alpha }{2}}(\zeta )\), we get

$$ \sum_{\mathfrak{i}=1}^{n}R_{\mathfrak{i}a}^{\frac{\alpha }{2}}( \zeta )R_{ ( \mathfrak{i}+1 ) a}^{\frac{\alpha }{2}}( \zeta )\leq \sum _{\mathfrak{i}=1}^{n}R_{\mathfrak{i}a}^{\alpha }( \zeta ). $$
(16)

Multiplying (16) by \(\hat{w}^{\sigma }(\zeta )\) and integrating from a to b, we have

$$ \sum_{\mathfrak{i}=1}^{n} \int _{a}^{b}\hat{w}^{\sigma }( \zeta )R_{\mathfrak{i}a}^{\frac{\alpha }{2}}(\zeta )R_{ ( \mathfrak{i}+1 ) a}^{\frac{\alpha }{2}}(\zeta ) \Delta \zeta \leq \sum_{\mathfrak{i}=1}^{n} \int _{a}^{b}\hat{w}^{ \sigma }(\zeta )R_{\mathfrak{i}a}^{\alpha }(\zeta )\Delta \zeta . $$
(17)

Now,

$$ \begin{aligned} J &= \int _{a}^{b}\hat{w}^{\sigma }(\zeta )R_{ \mathfrak{i}a}^{\alpha }(\zeta )\Delta \zeta \\ &= \int _{a}^{b}\hat{w}^{\sigma }(\zeta ) \biggl( \frac{\sqrt[\alpha ]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}{ ( u_{\mathfrak{i}}^{\sigma }(\zeta ) ) ^{m}} \int _{a}^{\sigma ( \zeta )}\frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x) \Delta x \biggr) ^{\alpha } \Delta \zeta \\ &= \int _{a}^{b} \frac{u_{\mathfrak{i}}^{\Delta }(\zeta )}{ ( u_{\mathfrak{i}}^{\sigma }(\zeta ) ) ^{\alpha m}} \biggl( \sqrt[ \alpha ]{\hat{w}^{\sigma }(\zeta )} \int _{a}^{ \sigma (\zeta )} \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x) \Delta x \biggr) ^{ \alpha }\Delta \zeta . \end{aligned} $$
(18)

Integrating (18) by parts using formula (7) with

$$ \Phi ^{\Delta }(\zeta )= \frac{u_{\mathfrak{i}}^{\Delta }(\zeta )}{ ( u_{\mathfrak{i}}^{\sigma }(\zeta ) ) ^{\alpha m}}, \qquad \Psi ^{\sigma }( \zeta )= \biggl( \sqrt[\alpha ]{\hat{w}^{\sigma }(\zeta )}\int _{a}^{\sigma (\zeta )} \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x) \Delta x \biggr) ^{\alpha }, $$

we obtain

$$ \begin{aligned} J &= \bigl[ \Phi (\zeta )\Psi (\zeta ) \bigr] _{a}^{b}+ \int _{a}^{b} \bigl( -\Phi (\zeta ) \bigr) \bigl( \Psi (\zeta ) \bigr) ^{\Delta }\Delta \zeta \\ &= \biggl[ -\Psi (\zeta ) \int _{\zeta }^{\infty } \frac{u_{\mathfrak{i}}^{\Delta }(x)}{ ( u_{\mathfrak{i}}^{\sigma }(x) ) ^{\alpha m}} \Delta x \biggr] _{a}^{b}+ \int _{a}^{b} \bigl( -\Phi ( \zeta ) \bigr) \bigl( \Psi (\zeta ) \bigr) ^{ \Delta }\Delta \zeta \\ &= \biggl[ -\Psi (b) \int _{b}^{\infty } \frac{u_{\mathfrak{i}}^{\Delta }(x)}{ ( u_{\mathfrak{i}}^{\sigma }(x) ) ^{\alpha m}} \Delta x \biggr] + \int _{a}^{b} \bigl( -\Phi (\zeta ) \bigr) \bigl( \Psi (\zeta ) \bigr) ^{\Delta }\Delta \zeta \\ &\leq \int _{a}^{b} \bigl( -\Phi (\zeta ) \bigr) \bigl( \Psi (\zeta ) \bigr) ^{\Delta }\Delta \zeta , \end{aligned} $$
(19)

where \(\Phi (\zeta )=-\int _{\zeta }^{\infty } \frac{u_{\mathfrak{i}}^{\Delta }(x)}{ ( u_{\mathfrak{i}}^{\sigma }(x) ) ^{\alpha m}} \Delta \varkappa \) and \(( \Psi (\zeta ) ) ^{\Delta }>0\). From (9), since \(u_{\mathfrak{i}}^{\Delta }(\zeta )\geq 0\) and \(c\in {}[ \zeta ,\sigma (\zeta )] \), we have

$$ \begin{aligned} \bigl[ u_{\mathfrak{i}}^{1-\alpha m}(\zeta ) \bigr] ^{\Delta } &= ( 1-\alpha m ) u_{ \mathfrak{i}}^{-\alpha m}(c)u_{\mathfrak{i}}^{\Delta }( \zeta ) \\ &= ( 1-\alpha m ) \frac{u_{\mathfrak{i}}^{\Delta }(\zeta )}{u_{\mathfrak{i}}^{\alpha m}(c)} \\ &\leq ( 1-\alpha m ) \frac{u_{\mathfrak{i}}^{\Delta }(\zeta )}{ ( u_{\mathfrak{i}}^{\sigma }(\zeta ) ) ^{\alpha m}}. \end{aligned} $$
(20)

Therefore, integrating (20) from ζ to ∞ with respect to x, we have

$$ -\Phi (\zeta )\leq \frac{1}{\alpha m-1}u_{\mathfrak{i}}^{1- \alpha m}(\zeta ). $$
(21)

Combining (21) and (19), we get

$$ J\leq \frac{1}{\alpha m-1} \int _{a}^{b}u_{\mathfrak{i}}^{1-\alpha m}( \zeta ) \bigl( \Psi (\zeta ) \bigr) ^{\Delta } \Delta \zeta . $$
(22)

Now, by applying (6) to \(\Psi (\zeta )=\hat{w}(\zeta )\hat{Y}^{\alpha }(\zeta )\) and using (9), we obtain

$$ \begin{aligned} \bigl( \Psi (\zeta ) \bigr) ^{\Delta } &= \hat{w}^{\Delta }(\zeta ) \bigl[ \hat{Y}^{\sigma }( \zeta ) \bigr] ^{\alpha }+\hat{w}(\zeta ) \bigl[ \hat{Y}^{\alpha }( \zeta ) \bigr] ^{\Delta } \\ &=\hat{w}^{\Delta }(\zeta ) \bigl[ \hat{Y}^{\sigma }( \zeta ) \bigr] ^{\alpha }+\alpha \hat{w}(\zeta )\hat{Y}^{ \alpha -1}(c) \hat{Y}^{\Delta }(\zeta ) \\ &\leq \hat{w}^{\Delta }(\zeta ) \bigl[ \hat{Y}^{ \sigma }(\zeta ) \bigr] ^{\alpha }+\alpha \hat{w}(\zeta ) \frac{u_{\mathfrak{i}}(\zeta )z_{\mathfrak{i}}^{\Delta }(\zeta )}{z_{\mathfrak{i}}(\zeta )}f_{\mathfrak{i}}( \zeta ) \bigl[ \hat{Y}^{\sigma }(\zeta ) \bigr] ^{\alpha -1}, \end{aligned} $$
(23)

where

$$ \hat{Y}(\zeta )= \int _{a}^{\zeta } \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{ \mathfrak{i}}(x) \Delta x. $$

Substituting (23) into (22), we get

$$\begin{aligned} J &\leq \frac{1}{\alpha m-1} \int _{a}^{b}u_{\mathfrak{i}}^{1- \alpha m}( \zeta )\hat{w}^{\Delta }(\zeta ) \biggl( \int _{a}^{ \sigma (\zeta )} \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{ \mathfrak{i}}(x) \Delta x \biggr) ^{\alpha }\Delta \zeta \\ &\quad {}+\frac{\alpha }{\alpha m-1} \int _{a}^{b}u_{\mathfrak{i}}^{2- \alpha m}( \zeta )\hat{w}(\zeta ) \frac{z_{\mathfrak{i}}^{\Delta }(\zeta )}{z_{\mathfrak{i}}(\zeta )}f_{\mathfrak{i}}( \zeta ) \\ &\quad {}\times \biggl( \int _{a}^{\sigma ( \zeta )} \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{ \mathfrak{i}}(x) \Delta x \biggr) ^{\alpha -1}\Delta \zeta \\ &=\frac{1}{\alpha m-1} \int _{a}^{b} \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{\alpha m}\hat{w}^{\Delta }(\zeta ) }{u_{\mathfrak{i}}^{\alpha m-1}(\zeta )u_{\mathfrak{i}}^{\Delta }(\zeta )}R_{\mathfrak{i}a}^{\alpha }( \zeta )\Delta \zeta \\ &\quad {}+\frac{\alpha }{\alpha m-1} \int _{a}^{b} \frac{u_{\mathfrak{i}}^{2-\alpha m}(\zeta ) [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m ( \alpha -1 ) }z_{\mathfrak{i}}^{\Delta }(\zeta )f_{\mathfrak{i}}(\zeta )\hat{w}(\zeta )}{z_{\mathfrak{i}}(\zeta ) ( u_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{1-\frac{1}{\alpha }}}R_{\mathfrak{i}a}^{\alpha -1}( \zeta )\Delta \zeta . \end{aligned}$$

Hence,

$$ \begin{aligned} & \int _{a}^{b}\hat{w}^{\sigma }(\zeta )R_{ \mathfrak{i}a}^{\alpha }(\zeta ) \biggl( 1- \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{\alpha m}\hat{w}^{\Delta }(\zeta )}{(\alpha m-1)u_{\mathfrak{i}}^{\alpha m-1}(\zeta )u_{\mathfrak{i}}^{\Delta }(\zeta )\hat{w}^{\sigma }(\zeta )} \biggr) \Delta \zeta \\ &\quad \leq \frac{\alpha }{\alpha m-1} \int _{a}^{b} \frac{u_{\mathfrak{i}}^{2-\alpha m}(\zeta ) [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m ( \alpha -1 ) }z_{\mathfrak{i}}^{\Delta }(\zeta )f_{\mathfrak{i}}(\zeta )\hat{w}(\zeta )}{z_{\mathfrak{i}}(\zeta ) ( u_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{1-\frac{1}{\alpha }}}R_{\mathfrak{i}a}^{\alpha -1}( \zeta )\Delta \zeta . \end{aligned} $$
(24)

From (13) and (24), we have

$$ \begin{aligned} & \int _{a}^{b}\hat{w}^{\sigma }(\zeta )R_{ \mathfrak{i}a}^{\alpha }(\zeta )\Delta \zeta \\ &\quad \leq \frac{\alpha \mathfrak{\lambda }_{\mathfrak{i}}}{\alpha m-1}\int _{a}^{b} \frac{u_{\mathfrak{i}}^{2-\alpha m}(\zeta ) [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m ( \alpha -1 ) }z_{\mathfrak{i}}^{\Delta }(\zeta )f_{\mathfrak{i}}(\zeta )\hat{w}(\zeta )}{z_{\mathfrak{i}}(\zeta ) ( u_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{1-\frac{1}{\alpha }}}R_{\mathfrak{i}a}^{\alpha -1}( \zeta ) \Delta \zeta \\ &\quad =\frac{\alpha \mathfrak{\lambda }_{\mathfrak{i}}}{\alpha m-1} \int _{a}^{b} \bigl[ \hat{w}^{\sigma }(\zeta )R_{\mathfrak{i}a}^{\alpha }( \zeta ) \bigr] ^{\frac{\alpha -1}{\alpha }} \frac{u_{\mathfrak{i}}^{2-\alpha m}(\zeta ) [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m ( \alpha -1 ) }z_{\mathfrak{i}}^{\Delta }(\zeta )f_{\mathfrak{i}}(\zeta )\hat{w}(\zeta )}{z_{\mathfrak{i}}(\zeta ) ( u_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{1-\frac{1}{\alpha }} ( \hat{w}^{\sigma }(\zeta ) ) ^{1-\frac{1}{\alpha }}}\Delta \zeta . \end{aligned} $$

Applying Hölder’s inequality with α and \(\alpha / ( \alpha -1 ) \), we have

$$ \begin{aligned} & \int _{a}^{b}\hat{w}^{\sigma }(\zeta )R_{ \mathfrak{i}a}^{\alpha }(\zeta )\Delta \zeta \\ &\quad \leq \biggl( \frac {\alpha \mathfrak{\lambda }_{\mathfrak{i}}}{\alpha m-1} \biggr) ^{\alpha } \int _{a}^{b} \frac {u_{\mathfrak{i}}^{\alpha ( 2-\alpha m ) }(\zeta ) [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m\alpha ( \alpha -1 ) } [ z_{\mathfrak{i}}^{\Delta }(\zeta ) ] ^{\alpha }f_{\mathfrak{i}}^{\alpha }(\zeta )\hat{w}^{\alpha }(\zeta )}{z_{\mathfrak{i}}^{\alpha }(\zeta ) ( u_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{\alpha -1} ( \hat{w}^{\sigma }(\zeta ) ) ^{\alpha -1}}\Delta \zeta \\ &\quad \leq \biggl( \frac {\alpha \mathfrak{\lambda }_{\mathfrak{i}}}{\alpha m-1} \biggr) ^{\alpha } \int _{a}^{\infty } \frac {u_{\mathfrak{i}}^{{\alpha ( 2-\alpha m ) }}(\zeta ) [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m\alpha ( \alpha -1 ) } [ z_{\mathfrak{i}}^{\Delta }(\zeta ) ] ^{\alpha }f_{\mathfrak{i}}^{\alpha }(\zeta )\hat{w}^{\alpha }(\zeta )}{z_{\mathfrak{i}}^{\alpha }(\zeta ) ( u_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{\alpha -1} ( \hat{w}^{\sigma }(\zeta ) ) ^{\alpha -1}}\Delta \zeta . \end{aligned} $$
(25)

By letting \(a\rightarrow 0\), \(b\rightarrow \infty \) and from (25), (17), we have

$$ \sum_{\mathfrak{i}=1}^{n} \int _{0}^{\infty }\hat{w}^{\sigma }( \zeta )R_{\mathfrak{i}a}^{\frac{\alpha }{2}}(\zeta )R_{ ( \mathfrak{i}+1 ) a}^{\frac{\alpha }{2}}( \zeta )\Delta \zeta \leq \sum_{\mathfrak{i}=1}^{n} \biggl( \frac{\alpha \beta _{\mathfrak{i}}}{\alpha m-1} \biggr) ^{\alpha } \int _{0}^{\infty }\hat{w}^{\sigma }( \zeta )g_{\mathfrak{i}}(\zeta )\Delta \zeta . $$
(26)

Second, let us define for \(m<\frac{1}{2}\), \(1\leq \alpha \leq 2\), and \(0< a< b<\infty \),

$$ R_{\mathfrak{i}b}(\zeta )= \frac{\sqrt[\alpha ]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}{u_{\mathfrak{i}}^{m}(\zeta )}\int _{\sigma (\zeta )}^{b} \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x) \Delta x, \quad 1\leq \mathfrak{i}\leq n, $$

with \(R_{\mathfrak{i}\infty }(\zeta )=R_{\mathfrak{i}}(\zeta )\). Following the same steps as in the proof of (26), we obtain

$$ \sum_{\mathfrak{i}=1}^{n} \int _{0}^{\infty }\hat{w}^{\sigma }( \zeta )R_{\mathfrak{i}}^{\frac{\alpha }{2}}(\zeta )R_{\mathfrak{i}+1}^{\frac{\alpha }{2}}( \zeta )\Delta \zeta \leq \sum_{\mathfrak{i}=1}^{n} \biggl( \frac{\alpha \beta _{\mathfrak{i}}}{1-\alpha m} \biggr) ^{\alpha } \int _{0}^{\infty }\hat{w}^{\sigma }( \zeta )g_{\mathfrak{i}}(\zeta )\Delta \zeta . $$
(27)

Inequalities (26) and (27) are equivalent to (15). □

In Theorem 6, if we take T = N , then we have \(\sigma (s)=s+1\) and obtain the next corollary.

Corollary 7

Let \(\{ u(s) \} _{s=1}^{\infty }\), \(\{ \hat{w}(s) \} _{s=1}^{\infty }\), and \(\{ z(s) \} _{s=1}^{\infty }\) be increasing and nonnegative sequences. For any \(1\leq \mathfrak{i}\leq n\), \(n\geq \kappa -1\), n,κ N , and

$$\begin{aligned} &1- \frac{ [ u_{\mathfrak{i}}(s+1) ] ^{\alpha m}\Delta \hat{w}(s)}{(\alpha m-1) [ u_{\mathfrak{i}}(s) ] ^{\alpha m-1}\hat{w}(s+1)\Delta u_{\mathfrak{i}}(s)}\geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0,\quad \textit{for }m>\frac{1}{2}, \\ &1- \frac{u_{\mathfrak{i}}(s)\Delta \hat{w}(s)}{(\alpha m-1)\hat{w}(s+1)\Delta u_{\mathfrak{i}}(s)}\geq \frac{1}{\delta _{\mathfrak{i}}}>0,\quad \textit{for }m< \frac{1}{2}, \end{aligned}$$

we have

$$ \sum_{\mathfrak{i}=1}^{n} \Biggl( \sum _{s=1}^{\mathfrak{\infty }} \hat{w}(s+1)R_{\mathfrak{i}}^{\frac{\alpha }{2}}(s)R_{\mathfrak{i}+1}^{\frac{\alpha }{2}}(s) \Biggr) \leq \sum_{ \mathfrak{i}=1}^{n} \biggl( \frac{\alpha \beta _{\mathfrak{i}}}{ \vert 1-\alpha m \vert } \biggr) ^{\alpha }\sum_{s=1}^{\mathfrak{\infty }} \hat{w}(s+1)g_{\mathfrak{i}}(s), $$

where

$$\begin{aligned} R_{\mathfrak{i}}(s)&=\textstyle\begin{cases} \frac{\sqrt[\alpha ]{ \Delta u_{\mathfrak{i}}(s)}}{ [ u_{\mathfrak{i}}(s+1) ] ^{m}}\sum_{r=1}^{s} \frac{u_{\mathfrak{i}}(r)\Delta z_{\mathfrak{i}}(r)}{z_{\mathfrak{i}}(r)}f_{ \mathfrak{i}}(r), & \textit{for }m>\frac{1}{2},\alpha \geq 2, \\ \frac{\sqrt[\alpha ]{ \Delta u_{\mathfrak{i}}(s)}}{u_{\mathfrak{i}}^{m}(s)}\sum_{r=s+1}^{\mathfrak{\infty }} \frac{u_{\mathfrak{i}}(r)\Delta z_{\mathfrak{i}}(r)}{z_{\mathfrak{i}}(r)}f_{\mathfrak{i}}(r), & \textit{for }m< \frac{1}{2},1\leq \alpha \leq 2,\end{cases}\displaystyle \\ g_{\mathfrak{i}}(s)&=\textstyle\begin{cases} \frac{u_{\mathfrak{i}}^{\alpha ( 2-\alpha m ) }(s) [ u_{\mathfrak{i}}(s+1) ] ^{m\alpha ( \alpha -1 ) } [ \Delta z_{\mathfrak{i}}(s) ] ^{\alpha }f_{\mathfrak{i}}^{\alpha }(s)\hat{w}^{\alpha }(s)}{z_{\mathfrak{i}}^{\alpha }(s) ( \Delta u_{\mathfrak{i}}(s) ) ^{\alpha -1} ( \hat{w}(s+1) ) ^{\alpha }}, & \textit{for }m> \frac{1}{2},\alpha \geq 2, \\ \frac{u_{\mathfrak{i}}^{\alpha ( 2-\alpha m ) }(s) [ \Delta z_{\mathfrak{i}}(s) ] ^{\alpha }f_{\mathfrak{i}}^{\alpha }(s)\hat{w}^{\alpha }(s)}{z_{\mathfrak{i}}^{\alpha }(s) ( \Delta u_{\mathfrak{i}}(s) ) ^{\alpha -1} ( \hat{w}(s+1) ) ^{\alpha }}, & \textit{for }m< \frac{1}{2},1\leq \alpha \leq 2,\end{cases}\displaystyle \end{aligned}$$

with \(\Delta y(s)=y(s+1)-y(s)\), \(\beta _{\mathfrak{i}}=\max_{1\leq \mathfrak{i}\leq n} ( \mathfrak{\lambda }_{\mathfrak{i}},\delta _{\mathfrak{i}} )\), and \(u_{\mathfrak{i}}(\infty )=\infty \).

Remark 8

If we put T = R and \(\alpha =2\), in Theorem 6, then (15) reduces to (4).

The next corollary follows from Theorem 6 by taking \(u_{\mathfrak{i}}(\zeta )=z_{\mathfrak{i}}(\zeta )=\zeta \), \(f_{\mathfrak{i}}(\zeta )=f_{\mathfrak{i}+1}(\zeta )\), \(m=1\), and \(\alpha =2\).

Corollary 9

For any \(1\leq \mathfrak{i}\leq n\), \(n\geq \kappa -1\) and κN, if there exist \(\mathfrak{\lambda }_{\mathfrak{i}}>0 \) such that

$$ 1- \frac{\sigma ^{2}(\zeta )\hat{w}^{\Delta }(\zeta )}{\zeta \hat{w}^{\sigma }(\zeta )}\geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0, $$

then

$$ \sum_{\mathfrak{i}=1}^{n} \int _{0}^{\infty }\hat{w}^{\sigma }( \zeta ) \biggl[ \frac{1}{\sigma (\zeta )} \int _{0}^{\sigma ( \zeta ) }f_{\mathfrak{i}}(x)\Delta x \biggr] ^{2} \Delta \zeta \leq \sum _{\mathfrak{i}=1}^{n} ( 2\mathfrak{\lambda }_{\mathfrak{i}} ) ^{2} \int _{0}^{\infty } \hat{w}^{\sigma }( \zeta )g_{\mathfrak{i}}(\zeta ) \Delta \zeta , $$
(28)

where

$$ g_{\mathfrak{i}}(\zeta )= \frac{\sigma ^{2}(\zeta )f_{\mathfrak{i}}^{2}(\zeta )\hat{w}^{2}(\zeta )}{\zeta ^{2} [ \hat{w}^{\sigma }(\zeta ) ] ^{2}}. $$

Remark 10

Letting T = R in Corollary 9, we have that \(\sigma (\zeta )=\zeta \) and

$$ 1- \frac{\zeta \hat{w}'(\zeta )}{\hat{w}(\zeta )}\geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0. $$

Then

$$ \sum_{\mathfrak{i}=1}^{n} \int _{0}^{\infty }\hat{w}(\zeta ) \biggl[ \frac{1}{\zeta } \int _{0}^{\zeta }f_{ \mathfrak{i}}(x)\,dx \biggr] ^{2}\,d\zeta \leq \sum_{\mathfrak{i}=1}^{n} ( 2 \mathfrak{\lambda }_{\mathfrak{i}} ) ^{2} \int _{0}^{\infty }\hat{w}( \zeta )f_{\mathfrak{i}}^{2}( \zeta )\,d\zeta , $$

which agrees with [4, Corollary 1].

Theorem 11

For any \(1\leq \mathfrak{i}\leq n\), \(n\geq \kappa -1\), and n,κN, if \(\alpha _{\mathfrak{i}}>1\), \(\delta _{\mathfrak{i}}=\alpha _{ \mathfrak{i}}/ ( \kappa \alpha _{\mathfrak{i}}-1 ) \) and there exist \(\mathfrak{\lambda }_{\mathfrak{i}}>0\), \(\delta _{\mathfrak{i}}>0\) such that

$$\begin{aligned} &1- \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{\kappa \alpha _{\mathfrak{i}}m}\hat{w}^{\Delta }(\zeta )}{(\kappa \alpha _{\mathfrak{i}}m-1)u_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}m-1}(\zeta )u_{\mathfrak{i}}^{\Delta }(\zeta )\hat{w}^{\sigma }(\zeta )}\geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0, \quad \textit{for }m> \frac{1}{\kappa \alpha _{\mathfrak{i}}}, \end{aligned}$$
(29)
$$\begin{aligned} &1- \frac{u_{\mathfrak{i}}(\zeta )\hat{w}^{\Delta }(\zeta )}{(\kappa \alpha _{\mathfrak{i}}m-1)u_{\mathfrak{i}}^{\Delta }(\zeta )\hat{w}^{\sigma }(\zeta )} \geq \frac{1}{\delta _{\mathfrak{i}}}>0, \quad \textit{for }m< \frac{1}{\kappa \alpha _{\mathfrak{i}}}, \end{aligned}$$
(30)

then

$$ \begin{aligned} &\sum_{\mathfrak{i}=1}^{n-\kappa +2} \int _{0}^{\infty } \hat{w}^{\sigma }(\zeta ) \Biggl( \prod_{j=\mathfrak{i}}^{\mathfrak{i}+ \kappa -1}R_{j}^{\alpha _{j}}( \zeta ) \Biggr) \Delta \zeta \\ &\quad \leq \sum_{\mathfrak{i}=1}^{n} \biggl( \frac{\kappa \alpha _{\mathfrak{i}}\beta _{\mathfrak{i}}}{ \vert \kappa \alpha _{\mathfrak{i}}m-1 \vert } \biggr) ^{\kappa \alpha _{\mathfrak{i}}} \int _{0}^{ \infty }\hat{w}^{\sigma }( \zeta )g_{\mathfrak{i}}(\zeta )\Delta \zeta , \end{aligned} $$
(31)

where

$$\begin{aligned} R_{\mathfrak{i}}(\zeta )&=\textstyle\begin{cases} \frac{\sqrt[\kappa \alpha _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m}}\int _{0}^{\sigma (\zeta )} \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{ \mathfrak{i}}(x)\Delta x \\ \quad \textit{and}\quad\int _{0}^{\infty } \frac{u_{\mathfrak{i}}^{\Delta }(x)}{ ( u_{\mathfrak{i}}^{\sigma }(x) ) ^{\kappa \alpha _{\mathfrak{i}}m}} \Delta \varkappa < \infty ,& \textit{for } m> \frac{1}{\kappa \alpha _{\mathfrak{i}}}, \\ \frac{\sqrt[\kappa \alpha _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}{u_{\mathfrak{i}}^{m}(\zeta )} \int _{\sigma (\zeta )}^{\infty } \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x)\Delta x, & \textit{for }m< \frac{1}{\kappa \alpha _{\mathfrak{i}}},\end{cases}\displaystyle \\ g_{\mathfrak{i}}(\zeta )&=\textstyle\begin{cases} \frac{u_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}} ( 2-\kappa \alpha _{\mathfrak{i}}m ) }(\zeta ) [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{\kappa \alpha _{\mathfrak{i}}m ( \kappa \alpha _{\mathfrak{i}}-1 ) } ( z_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{\kappa \alpha _{\mathfrak{i}}}f_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}}(\zeta )\hat{w}^{\kappa \alpha _{\mathfrak{i}}}(\zeta )}{z_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}}(\zeta ) ( u_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{\kappa \alpha _{\mathfrak{i}}-1} ( \hat{w}^{\sigma }(\zeta ) ) ^{\kappa \alpha _{\mathfrak{i}}}}, &\textit{for }m>\frac{1}{\kappa \alpha _{\mathfrak{i}}}, \\ \frac{u_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}} ( 2-m ) }(\zeta ) ( z_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{\kappa \alpha _{\mathfrak{i}}}f_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}}(\zeta )\hat{w}^{\kappa \alpha _{\mathfrak{i}}}(\zeta )}{z_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}}(\zeta ) ( u_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{\kappa \alpha _{\mathfrak{i}}-1} ( \hat{w}^{\sigma }(\zeta ) ) ^{\kappa \alpha _{\mathfrak{i}}}},& \textit{for }m< \frac{1}{\kappa \alpha _{\mathfrak{i}}},\end{cases}\displaystyle \end{aligned}$$

and \(\beta _{\mathfrak{i}}=\max_{1\leq \mathfrak{i}\leq n} ( \mathfrak{\lambda }_{\mathfrak{i}},\delta _{\mathfrak{i}} ) \), \(u_{ \mathfrak{i}}(\infty )=\infty \).

Proof

Let us define for \(m>\frac{1}{\kappa \alpha _{\mathfrak{i}}}\) and \(0< a< b<\infty \),

$$ R_{\mathfrak{i}a}(\zeta )= \frac{\sqrt[\kappa \alpha _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m}} \int _{a}^{\sigma ( \zeta )}\frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x) \Delta x,\quad 1\leq \mathfrak{i}\leq n, $$
(32)

with \(R_{\mathfrak{i}0}(\zeta )=R_{\mathfrak{i}}(\zeta )\). Using (11) with \(C_{\mathfrak{i}}=R_{\mathfrak{i}a}^{\alpha _{\mathfrak{i}}}( \zeta )\), we get

$$ \sum_{\mathfrak{i}=1}^{n-\kappa +2}R_{\mathfrak{i}a}^{\alpha _{ \mathfrak{i}}}( \zeta )R_{ ( \mathfrak{i}+1 ) a}^{\alpha _{ \mathfrak{i+1}}}(\zeta )\cdots R_{ ( \mathfrak{i}+\kappa -1 ) a}^{ \alpha _{\mathfrak{i+}\kappa \mathfrak{-1}}}( \zeta )\leq \sum_{ \mathfrak{i}=1}^{n}R_{\mathfrak{i}a}^{\kappa \alpha _{\mathfrak{i}}}( \zeta ). $$
(33)

Multiplying (33) by \(\hat{w}^{\sigma }(\zeta )\) and integrating from a to b, we have

$$ \sum_{\mathfrak{i}=1}^{n-\kappa +2} \int _{a}^{b}\hat{w}^{\sigma }( \zeta ) \Biggl( \prod_{j=\mathfrak{i}}^{\mathfrak{i}+ \kappa -1}R_{ja}^{\alpha _{j}}( \zeta ) \Biggr) \Delta \zeta \leq \sum_{\mathfrak{i}=1}^{n} \int _{a}^{b}\hat{w}^{\sigma }( \zeta )R_{\mathfrak{i}a}^{\kappa \alpha _{\mathfrak{i}}}(\zeta )\Delta \zeta . $$
(34)

Now,

$$ \begin{aligned} \mathfrak{I} &= \int _{a}^{b}\hat{w}^{\sigma }( \zeta )R_{\mathfrak{i}a}^{\kappa \alpha _{\mathfrak{i}}}(\zeta )\Delta \zeta \\ &= \int _{a}^{b}\hat{w}^{\sigma }(\zeta ) \biggl( \frac{\sqrt[\kappa \alpha _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m}}\int _{a}^{\sigma (\zeta )} \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x) \Delta x \biggr) ^{\kappa \alpha _{\mathfrak{i}}} \Delta \zeta \\ &= \int _{a}^{b} \frac{u_{\mathfrak{i}}^{\Delta }(\zeta )}{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{\kappa \alpha _{\mathfrak{i}}m}} \biggl( \sqrt[ \kappa \alpha _{\mathfrak{i}}]{\hat{w}^{\sigma }(\zeta )} \int _{a}^{\sigma (\zeta )} \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{ \mathfrak{i}}(x) \Delta x \biggr) ^{\kappa \alpha _{\mathfrak{i}}} \Delta \zeta . \end{aligned} $$
(35)

Integrating (35) by parts using formula (7) with

$$ \Phi ^{\Delta }(\zeta )= \frac{u_{\mathfrak{i}}^{\Delta }(\zeta )}{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{\kappa \alpha _{\mathfrak{i}}m}}, \qquad \Psi ^{\sigma }( \zeta )= \biggl( \sqrt[\kappa \alpha _{\mathfrak{i}}]{\hat{w}^{\sigma }( \zeta )} \int _{a}^{\sigma (\zeta )} \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x) \Delta x \biggr) ^{\kappa \alpha _{ \mathfrak{i}}}, $$

we obtain

$$ \begin{aligned} J &= \bigl[ \Phi (\zeta )\Psi (\zeta ) \bigr] _{a}^{b}+ \int _{a}^{b} \bigl( -\Phi (\zeta ) \bigr) \bigl( \Psi (\zeta ) \bigr) ^{\Delta }\Delta \zeta \\ &= \biggl[ -\Psi (\zeta ) \int _{\zeta }^{\infty } \frac{u_{\mathfrak{i}}^{\Delta }(x)}{ [ u_{\mathfrak{i}}^{\sigma }(x ] ^{\kappa \alpha _{\mathfrak{i}}m}}\Delta x \biggr] _{a}^{b}+ \int _{a}^{b} \bigl( -\Phi (\zeta ) \bigr) \bigl( \Psi (\zeta ) \bigr) ^{\Delta }\Delta \zeta \\ &= \biggl[ -\Psi (b) \int _{b}^{\infty } \frac{u_{\mathfrak{i}}^{\Delta }(x)}{ [ u_{\mathfrak{i}}^{\sigma }(x) ] ^{\kappa \alpha _{\mathfrak{i}}m}}\Delta x \biggr] + \int _{a}^{b} \bigl( -\Phi ( \zeta ) \bigr) \bigl( \Psi (\zeta ) \bigr) ^{ \Delta }\Delta \zeta \\ &\leq \int _{a}^{b} \bigl( -\Phi (\zeta ) \bigr) \bigl( \Psi (\zeta ) \bigr) ^{\Delta }\Delta \zeta , \end{aligned} $$
(36)

where \(\Phi (\zeta )=-\int _{\zeta }^{\infty } \frac{u_{\mathfrak{i}}^{\Delta }(x)}{ [ u_{\mathfrak{i}}^{\sigma }(x) ] ^{\kappa \alpha _{\mathfrak{i}}m}} \Delta \varkappa \) and \(( \Psi (\zeta ) ) ^{\Delta }>0\). From (9), since \(u_{\mathfrak{i}}^{\Delta }(\zeta )\geq 0\) and \(c\in {}[ \zeta ,\sigma (\zeta )] \), we have

$$ \begin{aligned} \bigl[ u_{\mathfrak{i}}^{1-\kappa \alpha _{\mathfrak{i}}m}( \zeta ) \bigr] ^{\Delta } &= ( 1-\kappa \alpha _{\mathfrak{i}}m ) u_{\mathfrak{i}}^{-\kappa \alpha _{\mathfrak{i}}m}(c)u_{ \mathfrak{i}}^{\Delta }(\zeta ) \\ &= ( 1-\kappa \alpha _{\mathfrak{i}}m ) \frac{u_{\mathfrak{i}}^{\Delta }(\zeta )}{u_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}m}(c)} \\ &\leq ( 1-\kappa \alpha _{\mathfrak{i}}m ) \frac{u_{\mathfrak{i}}^{\Delta }(\zeta )}{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{\kappa \alpha _{\mathfrak{i}}m}}. \end{aligned} $$
(37)

Therefore, integrating (37) from ζ to ∞ with respect to x, we have

$$ -\Phi (\zeta )\leq \frac{1}{\kappa \alpha _{\mathfrak{i}}m-1}u_{\mathfrak{i}}^{1-\kappa \alpha _{\mathfrak{i}}m}(\zeta ). $$
(38)

Combining (38) and (36), we have

$$ J\leq \frac{1}{\kappa \alpha _{\mathfrak{i}}m-1} \int _{a}^{b}u_{ \mathfrak{i}}^{1-\kappa \alpha _{\mathfrak{i}}m}( \zeta ) \bigl( \Psi ( \zeta ) \bigr) ^{\Delta }\Delta \zeta . $$
(39)

Now, by applying (6) to \(\Psi (\zeta )=\hat{w}(\zeta )\hat{Y}^{\kappa \alpha _{\mathfrak{i}}}(\zeta )\) and using (9), we obtain

$$ \begin{aligned} \bigl( \Psi (\zeta ) \bigr) ^{\Delta } &= \hat{w}^{\Delta }(\zeta ) \bigl[ \hat{Y}^{\sigma }( \zeta ) \bigr] ^{\kappa \alpha _{\mathfrak{i}}}+\hat{w}(\zeta ) \bigl[ \hat{Y}^{\kappa \alpha _{\mathfrak{i}}}( \zeta ) \bigr] ^{ \Delta } \\ &=\hat{w}^{\Delta }(\zeta ) \bigl[ \hat{Y}^{\sigma }( \zeta ) \bigr] ^{\kappa \alpha _{\mathfrak{i}}}+\kappa \alpha _{ \mathfrak{i}}\hat{w}(\zeta ) \hat{Y}^{\kappa \alpha _{\mathfrak{i}}-1}(c) \hat{Y}^{\Delta }(\zeta ) \\ &\leq \hat{w}^{\Delta }(\zeta ) \bigl[ \hat{Y}^{ \sigma }(\zeta ) \bigr] ^{\kappa \alpha _{\mathfrak{i}}}+\kappa \alpha _{\mathfrak{i}}\hat{w}(\zeta ) \frac{u_{\mathfrak{i}}(\zeta )z_{\mathfrak{i}}^{\Delta }(\zeta )}{z_{\mathfrak{i}}(\zeta )}f_{\mathfrak{i}}(\zeta ) \bigl[ \hat{Y}^{\sigma }( \zeta ) \bigr] ^{\kappa \alpha _{\mathfrak{i}}-1}, \end{aligned} $$
(40)

where

$$ \hat{Y}(\zeta )= \int _{a}^{\zeta } \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{ \mathfrak{i}}(x) \Delta x. $$

Substituting (40) into (39) and using (32), we get

$$ \begin{aligned} J &\leq \frac{1}{\kappa \alpha _{\mathfrak{i}}m-1} \int _{a}^{b}u_{ \mathfrak{i}}^{1-\kappa \alpha _{\mathfrak{i}}m}( \zeta )\hat{w}^{\Delta }(\zeta ) \biggl( \int _{a}^{\sigma ( \zeta )}\frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x) \Delta x \biggr) ^{ \kappa \alpha _{\mathfrak{i}}}\Delta \zeta \\ &\quad {}+ \frac{\kappa \alpha _{\mathfrak{i}}}{\kappa \alpha _{\mathfrak{i}}m-1}\int _{a}^{b} \frac{u_{\mathfrak{i}}^{2-\kappa \alpha _{\mathfrak{i}}m}(\zeta )\hat{w}(\zeta )z_{\mathfrak{i}}^{\Delta }(\zeta )}{z_{\mathfrak{i}}(\zeta )}f_{\mathfrak{i}}( \zeta ) \\ &\quad {}\times \biggl( \int _{a}^{\sigma ( \zeta )} \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{ \mathfrak{i}}(x) \Delta x \biggr) ^{\kappa \alpha _{\mathfrak{i}}-1} \Delta \zeta \\ &=\frac{1}{\kappa \alpha _{\mathfrak{i}}m-1} \int _{a}^{b} \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{\kappa \alpha _{\mathfrak{i}}m}\hat{w}^{\Delta }(\zeta )}{u_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}m-1}(\zeta )u_{\mathfrak{i}}^{\Delta }(\zeta )}R_{ \mathfrak{i}a}^{\kappa \alpha _{\mathfrak{i}}}( \zeta )\Delta \zeta \\ &\quad {}+ \frac{\kappa \alpha _{\mathfrak{i}}}{\kappa \alpha _{\mathfrak{i}}m-1}\int _{a}^{b} \frac{u_{\mathfrak{i}}^{2-\kappa \alpha _{\mathfrak{i}}m}(\zeta ) [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m ( \kappa \alpha _{\mathfrak{i}}-1 ) }z_{\mathfrak{i}}^{\Delta }(\zeta )f_{\mathfrak{i}}(\zeta )\hat{w}(\zeta )}{z_{\mathfrak{i}}(\zeta ) ( u_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{1-\frac{1}{\kappa \alpha _{\mathfrak{i}}}}}R_{\mathfrak{i}a}^{\kappa \alpha _{\mathfrak{i}}-1}( \zeta )\Delta \zeta . \end{aligned} $$

Hence,

$$ \begin{aligned} & \int _{a}^{b}\hat{w}^{\sigma }(\zeta )R_{ \mathfrak{i}a}^{\kappa \alpha _{\mathfrak{i}}}(\zeta ) \biggl( 1- \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{\kappa \alpha _{\mathfrak{i}}m}\hat{w}^{\Delta }(\zeta )}{(\kappa \alpha _{\mathfrak{i}}m-1)u_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}m-1}(\zeta )u_{\mathfrak{i}}^{\Delta }(\zeta )\hat{w}^{\sigma }(\zeta )} \biggr) \Delta \zeta \\ &\quad \leq \frac{\kappa \alpha _{\mathfrak{i}}}{\kappa \alpha _{\mathfrak{i}}m-1} \int _{a}^{b} \frac{u_{\mathfrak{i}}^{2-\kappa \alpha _{\mathfrak{i}}m}(\zeta ) [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m ( \kappa \alpha _{\mathfrak{i}}-1 ) }z_{\mathfrak{i}}^{\Delta }(\zeta )f_{\mathfrak{i}}(\zeta )\hat{w}(\zeta )}{z_{\mathfrak{i}}(\zeta ) ( u_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{1-\frac{1}{\kappa \alpha _{\mathfrak{i}}}}}R_{\mathfrak{i}a}^{\kappa \alpha _{\mathfrak{i}}-1}( \zeta )\Delta \zeta . \end{aligned} $$
(41)

From (41) and (29), we have

$$ \begin{aligned} & \int _{a}^{b}\hat{w}^{\sigma }(\zeta )R_{ \mathfrak{i}a}^{\kappa \alpha _{\mathfrak{i}}}(\zeta ) \Delta \zeta \\ &\quad \leq \frac{\kappa \alpha _{\mathfrak{i}}\mathfrak{\lambda }_{\mathfrak{i}}}{\kappa \alpha _{\mathfrak{i}}m-1} \int _{a}^{b} \frac{u_{\mathfrak{i}}^{2-\kappa \alpha _{\mathfrak{i}}m}(\zeta ) [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m ( \kappa \alpha _{\mathfrak{i}}-1 ) }z_{\mathfrak{i}}^{\Delta }(\zeta )f_{\mathfrak{i}}(\zeta )\hat{w}(\zeta )}{z_{\mathfrak{i}}(\zeta ) ( u_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{1-\frac{1}{\kappa \alpha _{\mathfrak{i}}}}}R_{\mathfrak{i}a}^{\kappa \alpha _{\mathfrak{i}}-1}( \zeta )\Delta \zeta \\ &\quad = \frac{\kappa \alpha _{\mathfrak{i}}\mathfrak{\lambda }_{\mathfrak{i}}}{\kappa \alpha _{\mathfrak{i}}m-1} \int _{a}^{b} \bigl( \hat{w}^{ \sigma }(\zeta )R_{\mathfrak{i}a}^{\kappa \alpha _{\mathfrak{i}}}( \zeta ) \bigr) ^{ \frac{\kappa \alpha _{\mathfrak{i}}-1}{\kappa \alpha _{\mathfrak{i}}}} \\ &\qquad {}\times \frac{u_{\mathfrak{i}}^{2-\kappa \alpha _{\mathfrak{i}}m}(\zeta ) [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m ( \kappa \alpha _{\mathfrak{i}}-1 ) }z_{\mathfrak{i}}^{\Delta }(\zeta )f_{\mathfrak{i}}(\zeta )\hat{w}(\zeta )}{z_{\mathfrak{i}}(\zeta ) ( u_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{1-\frac{1}{\kappa \alpha _{\mathfrak{i}}}} ( \hat{w}^{\sigma }(\zeta ) ) ^{1-\frac{1}{\kappa \alpha _{\mathfrak{i}}}}}\Delta \zeta . \end{aligned} $$

Applying Hölder’s inequality with \(\kappa \alpha _{\mathfrak{i}}\) and \(\kappa \delta _{\mathfrak{i}}=\kappa \alpha _{\mathfrak{i}}/ ( \kappa \alpha _{\mathfrak{i}}-1 ) \), we have

$$ \begin{aligned} & \int _{a}^{b}\hat{w}^{\sigma }(\zeta )R_{ \mathfrak{i}a}^{\kappa \alpha _{\mathfrak{i}}}(\zeta ) \Delta \zeta \\ &\quad \leq \biggl( \frac{\kappa \alpha _{\mathfrak{i}}\mathfrak{\lambda }_{\mathfrak{i}}}{\kappa \alpha _{\mathfrak{i}}m-1} \biggr) ^{\kappa \alpha _{\mathfrak{i}}} \\ &\qquad{} \times \int _{a}^{b} \frac{u_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}} ( 2-\kappa \alpha _{\mathfrak{i}}m ) }(\zeta ) [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{\kappa \alpha _{\mathfrak{i}}m ( \kappa \alpha _{\mathfrak{i}}-1 ) } ( z_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{\kappa \alpha _{\mathfrak{i}}}f_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}}(\zeta )\hat{w}^{\kappa \alpha _{\mathfrak{i}}}(\zeta )}{z_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}}(\zeta ) ( u_{\mathfrak{i}}^{\Delta }(\zeta ) ) ^{\kappa \alpha _{\mathfrak{i}}-1} ( \hat{w}^{\sigma }(\zeta ) ) ^{\kappa \alpha _{\mathfrak{i}}-1}}\Delta \zeta \\ &\quad = \biggl( \frac{\kappa \alpha _{\mathfrak{i}}\mathfrak{\lambda }_{\mathfrak{i}}}{\kappa \alpha _{\mathfrak{i}}m-1} \biggr) ^{\kappa \alpha _{ \mathfrak{i}}} \int _{a}^{b}\hat{w}^{\sigma }(\zeta )g_{\mathfrak{i}}( \zeta )\Delta \zeta . \end{aligned} $$
(42)

From (42) and (34), we have

$$ \begin{aligned} &\sum_{\mathfrak{i}=1}^{n-\kappa +2} \int _{a}^{b} \hat{w}^{\sigma }(\zeta ) \Biggl( \prod_{j=\mathfrak{i}}^{\mathfrak{i}+ \kappa -1}R_{j}^{\alpha _{j}}( \zeta ) \Biggr) \Delta \zeta \\ &\quad \leq \sum_{\mathfrak{i}=1}^{n} \biggl( \frac{\kappa \alpha _{\mathfrak{i}}\beta _{\mathfrak{i}}}{\kappa \alpha _{\mathfrak{i}}m-1} \biggr) ^{ \kappa \alpha _{\mathfrak{i}}} \int _{a}^{b}\hat{w}^{\sigma }( \zeta )g_{\mathfrak{i}}(\zeta )\Delta \zeta \\ &\quad \leq \sum_{\mathfrak{i}=1}^{n} \biggl( \frac{\kappa \alpha _{\mathfrak{i}}\beta _{\mathfrak{i}}}{\kappa \alpha _{\mathfrak{i}}m-1} \biggr) ^{ \kappa \alpha _{\mathfrak{i}}} \int _{a}^{\infty }\hat{w}^{\sigma }( \zeta )g_{\mathfrak{i}}(\zeta )\Delta \zeta . \end{aligned} $$
(43)

Let us define for \(m<1/\kappa \alpha _{\mathfrak{i}}\) and \(0< a< b<\infty \),

$$ R_{\mathfrak{i}b}(\zeta )= \frac{\sqrt[\kappa \alpha _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}{u_{\mathfrak{i}}^{m}(\zeta )} \int _{\sigma (\zeta )}^{b} \frac{u_{\mathfrak{i}}(x)z_{\mathfrak{i}}^{\Delta }(x)}{z_{\mathfrak{i}}(x)}f_{ \mathfrak{i}}(x) \Delta x, \quad 1\leq \mathfrak{i}\leq n, $$

with \(R_{\mathfrak{i}\infty }(\zeta )=R_{\mathfrak{i}}(\zeta )\). Following the same steps as in the proof of (43), we obtain

$$\begin{aligned} &\sum_{\mathfrak{i}=1}^{n-\kappa +2} \int _{a}^{b} \hat{w}^{\sigma }(\zeta ) \Biggl( \prod_{j=\mathfrak{i}}^{\mathfrak{i}+ \kappa -1}R_{j}^{\alpha _{j}}( \zeta ) \Biggr) \Delta \zeta \\ &\quad \leq \sum_{\mathfrak{i}=1}^{n} \biggl( \frac{\kappa \alpha _{\mathfrak{i}}\beta _{\mathfrak{i}}}{1-\kappa \alpha _{\mathfrak{i}}m} \biggr) ^{ \kappa \alpha _{\mathfrak{i}}} \int _{a}^{b}\hat{w}^{\sigma }( \zeta )g_{\mathfrak{i}}(\zeta )\Delta \zeta \\ &\quad \leq \sum_{\mathfrak{i}=1}^{n} \biggl( \frac{\kappa \alpha _{\mathfrak{i}}\beta _{\mathfrak{i}}}{1-\kappa \alpha _{\mathfrak{i}}m} \biggr) ^{ \kappa \alpha _{\mathfrak{i}}} \int _{a}^{\infty }\hat{w}^{\sigma }( \zeta )g_{\mathfrak{i}}(\zeta )\Delta \zeta . \end{aligned}$$
(44)

By letting \(a\rightarrow 0\) and \(b\rightarrow \infty \) in (43) and (44), we get (31). □

In Theorem 11, if we take T = N , then we have the following corollary.

Corollary 12

Let \(\{ u(s) \} _{s=1}^{\infty }\), \(\{ \hat{w}(s) \} _{s=1}^{\infty }\), and \(\{ z(s) \} _{s=1}^{\infty }\) be increasing and nonnegative sequences. Then for any \(1\leq \mathfrak{i}\leq n\), \(n>\kappa -1\), \(\alpha _{\mathfrak{i}}>1\), κN, \(\delta _{\mathfrak{i}}=\alpha _{\mathfrak{i}}/ ( \kappa \alpha _{\mathfrak{i}}-1 ) \) and

$$\begin{aligned} &1- \frac{ [ u_{\mathfrak{i}}(s+1) ] ^{\kappa \alpha _{\mathfrak{i}}m}\Delta \hat{w}(s)}{(\kappa \alpha _{\mathfrak{i}}m-1) [ u_{\mathfrak{i}}(s) ] ^{\kappa \alpha _{\mathfrak{i}}m-1}\hat{w}(s+1)\Delta u_{\mathfrak{i}}(s)}\geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0,\quad \textit{for }m> \frac{1}{2}, \\ &1- \frac{u_{\mathfrak{i}}(s)\Delta \hat{w}(s)}{(\kappa \alpha _{\mathfrak{i}}m-1)\hat{w}(s+1)\Delta u_{\mathfrak{i}}(s)}\geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0,\quad \textit{for }m>\frac{1}{2}, \end{aligned}$$

we have

$$\begin{aligned}& \begin{aligned} &\sum_{\mathfrak{i}=1}^{n-\kappa +2} \Biggl( \sum_{s=1}^{ \mathfrak{\infty }}\hat{w}(s+1) \Biggl[ \prod_{j=\mathfrak{i}}^{\mathfrak{i}+\kappa -1}R_{j}^{\alpha _{j}}(s) \Biggr] \Biggr) \\ &\quad \leq \sum_{\mathfrak{i}=1}^{n} \biggl( \frac{\kappa \alpha _{\mathfrak{i}}\beta _{\mathfrak{i}}}{ \vert 1-\kappa \alpha _{\mathfrak{i}}m \vert } \biggr) ^{\kappa \alpha _{\mathfrak{i}}}\sum _{s=1}^{ \mathfrak{\infty }}\hat{w}(s+1)g_{\mathfrak{i}}(s), \end{aligned} \\& R_{\mathfrak{i}}(s)=\textstyle\begin{cases} \frac{\sqrt[\kappa \alpha _{\mathfrak{i}}]{ \Delta u_{\mathfrak{i}}(s)}}{ [ u_{\mathfrak{i}}(s+1) ] ^{m}}\sum_{r=1}^{s} \frac{u_{\mathfrak{i}}(r)\Delta z_{\mathfrak{i}}(r)}{z_{\mathfrak{i}}(r)}f_{\mathfrak{i}}(r),& \textit{for } m> \frac{1}{\kappa \alpha _{\mathfrak{i}}}, \\ \frac{\sqrt[\kappa \alpha _{\mathfrak{i}}]{ \Delta u_{\mathfrak{i}}(s)}}{u_{\mathfrak{i}}^{m}(s)}\sum_{r=s+1}^{\mathfrak{\infty }} \frac{u_{\mathfrak{i}}(r)\Delta z_{\mathfrak{i}}(r)}{z_{\mathfrak{i}}(r)}f_{\mathfrak{i}}(r), & \textit{for }m< \frac{1}{\kappa \alpha _{\mathfrak{i}}},\end{cases}\displaystyle \\& g_{\mathfrak{i}}(s)=\textstyle\begin{cases} \frac{u_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}} ( 2-\kappa \alpha _{\mathfrak{i}}m ) }(s) [ u_{\mathfrak{i}}(s+1) ] ^{\kappa \alpha _{\mathfrak{i}}m ( \kappa \alpha _{\mathfrak{i}}-1 ) } [ \Delta z_{\mathfrak{i}}(s) ] ^{\kappa \alpha _{\mathfrak{i}}}f_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}}(s)\hat{w}^{\kappa \alpha _{\mathfrak{i}}}(s)}{z_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}}(s) ( \Delta u_{\mathfrak{i}}(s) ) ^{\kappa \alpha _{\mathfrak{i}}-1} ( \hat{w}(s+1) ) ^{\kappa \alpha _{\mathfrak{i}}}}, & \textit{for }m>\frac{1}{\kappa \alpha _{\mathfrak{i}}}, \\ \frac{u_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}} ( 2-m ) }(s) [ \Delta z_{\mathfrak{i}}(s) ] ^{\kappa \alpha _{\mathfrak{i}}}f_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}}(s)\hat{w}^{\kappa \alpha _{\mathfrak{i}}}(s)}{z_{\mathfrak{i}}^{\kappa \alpha _{\mathfrak{i}}}(s) ( \Delta u_{\mathfrak{i}}(s) ) ^{\kappa \alpha _{\mathfrak{i}}-1} ( \hat{w}(s+1) ) ^{\kappa \alpha _{\mathfrak{i}}}},& \textit{for }m< \frac{1}{\kappa \alpha _{\mathfrak{i}}},\end{cases}\displaystyle \end{aligned}$$

and \(\beta _{\mathfrak{i}}=\max_{1\leq \mathfrak{i}\leq n} ( \mathfrak{\lambda }_{\mathfrak{i}},\delta _{\mathfrak{i}} ) \), \(u_{\mathfrak{i}}(\infty )=\infty \).

Remark 13

If we put T = R in Theorem 11, then (31) reduces to (5).

The next corollary follows from Theorem 11 by taking \(u_{\mathfrak{i}}(\zeta )=z_{\mathfrak{i}}(\zeta )=\zeta \), \(f_{\mathfrak{i}}(\zeta )\rightarrow \zeta ^{m-1}h_{\mathfrak{i}}\), \(h_{ \mathfrak{i}}=h_{\mathfrak{i}+1}\), \(\alpha _{\mathfrak{i}}=\alpha _{\mathfrak{i}+1}\) and \(\kappa =2\).

Corollary 14

For any \(1\leq \mathfrak{i}\leq n\), \(n>\kappa -1\), κN, if \(h_{\mathfrak{i}}\) are rd-continuous functions and there exist \(\mathfrak{\lambda }_{\mathfrak{i}}>0\) such that

$$ 1- \frac{\sigma ^{2\alpha _{\mathfrak{i}}m}(\zeta )\hat{w}^{\Delta }(\zeta )}{ ( 2\alpha _{\mathfrak{i}}m-1 ) \zeta ^{2\alpha _{\mathfrak{i}}m-1}\hat{w}^{\sigma }(\zeta )}\geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0, $$

then

$$ \begin{aligned} &\sum_{\mathfrak{i}=1}^{n} \int _{0}^{\infty }\hat{w}^{ \sigma }(\zeta ) \biggl[ \frac{1}{\sigma ^{m}(\zeta )} \int _{0}^{\sigma ( \zeta )}x^{m-1}h_{\mathfrak{i}} (x)\Delta x \biggr] ^{2 \alpha _{\mathfrak{i}}}\Delta \zeta \\ &\quad \leq \sum_{\mathfrak{i}=1}^{n} \biggl( \frac{2\alpha _{\mathfrak{i}}\mathfrak{\lambda }_{\mathfrak{i}}}{2\alpha _{\mathfrak{i}}m-1} \biggr) ^{2\alpha _{\mathfrak{i}}} \int _{0}^{\infty }\hat{w}^{ \sigma }(\zeta )g_{\mathfrak{i}} (\zeta )\Delta \zeta , \end{aligned} $$

where

$$ g_{\mathfrak{i}}(\zeta )= \frac{\sigma ^{2\alpha _{\mathfrak{i}}m ( 2\alpha _{\mathfrak{i}}-1 ) }(\zeta )\zeta ^{2\alpha _{\mathfrak{i}}m ( 1-\alpha _{\mathfrak{i}} ) }h_{\mathfrak{i}}^{2\alpha _{\mathfrak{i}}}(\zeta )\hat{w}^{2\alpha _{\mathfrak{i}}}(\zeta )}{ [ \hat{w}^{\sigma }(\zeta ) ] ^{2\alpha _{\mathfrak{i}}}},\quad \textit{for } m> \frac{1}{2\alpha _{\mathfrak{i}}}. $$

Remark 15

Letting T = R in Corollary 14, we have that \(\sigma (\zeta )=\zeta \) and

$$ 1- \frac{\zeta \hat{w}'(\zeta )}{(2\alpha _{\mathfrak{i}}m-1)\hat{w}(\zeta )}\geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0. $$

Then

$$ \begin{aligned} &\sum_{\mathfrak{i}=1}^{n} \int _{0}^{\infty }\hat{w}( \zeta ) \biggl[ \frac{1}{\zeta ^{m}} \int _{0}^{ \zeta }x^{m-1}h_{\mathfrak{i}}(x)\,dx \biggr] ^{2\alpha _{\mathfrak{i}}}\,d\zeta \\ &\quad \leq \sum_{\mathfrak{i}=1}^{n} \biggl( \frac{2\alpha _{\mathfrak{i}}\mathfrak{\lambda }_{\mathfrak{i}}}{2\alpha _{\mathfrak{i}}m-1} \biggr) ^{2\alpha _{\mathfrak{i}}} \int _{0}^{\infty }\hat{w}( \zeta )h_{\mathfrak{i}}^{2\alpha _{\mathfrak{i}}}( \zeta )\,d \zeta , \end{aligned} $$

which agrees with [4, Corollary 2].

Theorem 16

For any \(1\leq \mathfrak{i}\leq n\), \(n>\kappa -1\), κN and ζ 2 , σ ( ζ ) 2 T, if \(\alpha _{\mathfrak{i}}>1\), \(\delta _{\mathfrak{i}}=\alpha _{ \mathfrak{i}}/ ( 3\alpha _{\mathfrak{i}}-1 )\), and

$$ 1+ \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{3\alpha _{\mathfrak{i}}m}\hat{w}^{\Delta }(\zeta )}{(1-3\alpha _{\mathfrak{i}}m)u_{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}m-1}(\zeta )u_{\mathfrak{i}}^{\Delta }(\zeta )\hat{w}^{\sigma }(\zeta )}\geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0 , \quad \textit{for }m> \frac{1}{3\alpha _{\mathfrak{i}}}, $$
(45)

then

$$ \begin{aligned} &\sum_{\mathfrak{i}=1}^{n-1} \int _{0}^{x}\hat{w}^{ \sigma }(\zeta ) \bigl[ \Gamma _{\mathfrak{i}}^{\alpha _{\mathfrak{i}}}( \zeta ) \Gamma _{\mathfrak{i}+1}^{\alpha _{\mathfrak{i}}+1}(\zeta ) \Gamma _{\mathfrak{i}+2}^{\alpha _{\mathfrak{i}+2}}( \zeta ) \bigr] \Delta \zeta \\ &\quad \leq \sum_{\mathfrak{i}=1}^{n} \biggl( \frac{3\alpha _{\mathfrak{i}}\mathfrak{\lambda }_{\mathfrak{i}}}{3\alpha _{\mathfrak{i}}m-1} \biggr) ^{3\alpha _{\mathfrak{i}}} \int _{0}^{x}\hat{w}^{\sigma }( \zeta )g_{\mathfrak{i}}(\zeta )\Delta \zeta , \end{aligned} $$
(46)

where

$$\begin{aligned}& \begin{aligned} \Gamma _{\mathfrak{i}}(\zeta )&= \frac{\sqrt[3\alpha _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m}} \int _{ \frac{\sigma (\zeta )}{2}}^{\sigma (\zeta )} \frac{u_{\mathfrak{i}}(\eta )z_{\mathfrak{i}}^{\Delta }(\eta )}{z_{\mathfrak{i}}(\eta )}f_{\mathfrak{i}}( \eta )\Delta \eta \\ &\quad \textit{and}\quad \int _{0}^{\infty } \frac{u_{\mathfrak{i}}^{\Delta }(x)}{ ( u_{\mathfrak{i}}^{\sigma }(x) ) ^{\alpha _{\mathfrak{i}}m}} \Delta \varkappa < \infty , \end{aligned} \\& \begin{aligned} g_{\mathfrak{i}}(\zeta )&= \biggl[ \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{ ( 3\alpha _{\mathfrak{i}}-1 ) m}z_{\mathfrak{i}}^{\Delta }(\zeta )}{u_{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}m-2}(\zeta )\sqrt[3\delta _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}z_{\mathfrak{i}}(\zeta )}f_{\mathfrak{i}}(\zeta ) \\ &\quad {}- \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{ ( 3\alpha _{\mathfrak{i}}-1 ) m}u_{\mathfrak{i}}(\frac{\zeta }{2})z_{\mathfrak{i}}^{\Delta }(\frac{\zeta }{2})}{2u_{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}m-1}(\zeta )z_{\mathfrak{i}}(\frac{\zeta }{2})\sqrt[3\delta _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}f_{ \mathfrak{i}}\biggl(\frac{\zeta }{2} \biggr) \biggr] ^{3\alpha _{ \mathfrak{i}}}, \end{aligned} \\& u_{\mathfrak{i}}(\infty )=\infty . \end{aligned}$$

Proof

Let us define for \(m>\frac{1}{3\alpha _{\mathfrak{i}}}\), \(1\leq \mathfrak{i}\leq n\),

$$ \Gamma _{\mathfrak{i}}(\zeta )= \frac{\sqrt[3\alpha _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m}} \int _{ \frac{\sigma (\zeta )}{2}}^{\sigma (\zeta )} \frac{u_{\mathfrak{i}}(\eta ) z_{\mathfrak{i}}^{\Delta }(\eta )}{z_{\mathfrak{i}}(\eta )}f_{\mathfrak{i}}( \eta )\Delta \eta . $$
(47)

Using (11) with \(\kappa =3\), for \(C_{\mathfrak{i}}=\Gamma _{\mathfrak{i}}^{\alpha _{\mathfrak{i}}}(\zeta )\), we get

$$ \sum_{\mathfrak{i}=1}^{n-1}\Gamma _{\mathfrak{i}}^{\alpha _{ \mathfrak{i}}}(\zeta )\Gamma _{\mathfrak{i}+1}^{\alpha _{\mathfrak{i}+1}}( \zeta )\Gamma _{\mathfrak{i}+2}^{\alpha _{\mathfrak{i}+2}}(\zeta )\leq \sum _{\mathfrak{i}=1}^{n}\Gamma _{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}}( \zeta ). $$
(48)

Multiplying (48) by \(\hat{w}^{\sigma }(\zeta )\) and integrating from 0 to ϰ, we get

$$ \sum_{\mathfrak{i}=1}^{n-1} \int _{0}^{x}\hat{w}^{\sigma }( \zeta ) \bigl[ \Gamma _{\mathfrak{i}}^{\alpha _{\mathfrak{i}}}( \zeta ) \Gamma _{\mathfrak{i}+1}^{\alpha _{\mathfrak{i}+1}}(\zeta ) \Gamma _{\mathfrak{i}+2}^{\alpha _{\mathfrak{i}+2}}( \zeta ) \bigr] \Delta \zeta \leq \sum _{\mathfrak{i}=1}^{n} \int _{0}^{x}\hat{w}^{\sigma }(\zeta )\Gamma _{ \mathfrak{i}}^{3\alpha _{\mathfrak{i}}}(\zeta )\Delta \zeta . $$
(49)

Now,

$$ \begin{aligned} J &= \int _{0}^{x}\hat{w}^{\sigma }(\zeta ) \Gamma _{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}}(\zeta )\Delta \zeta \\ &= \int _{0}^{x}\hat{w}^{\sigma }(\zeta ) \biggl[ \frac{\sqrt[3\alpha _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m}} \int _{ \frac{\sigma (\zeta )}{2}}^{\sigma (\zeta )} \frac{u_{\mathfrak{i}}(\eta )z_{\mathfrak{i}}^{\Delta }(\eta )}{z_{\mathfrak{i}}(\eta )}f_{\mathfrak{i}}( \eta )\Delta \eta \biggr] ^{3\alpha _{ \mathfrak{i}}}\Delta \zeta \\ &= \int _{0}^{x} \frac{u_{\mathfrak{i}}^{\Delta }(\zeta )}{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{3\alpha _{\mathfrak{i}}m}} \biggl[ \sqrt[3 \alpha _{\mathfrak{i}}]{\hat{w}^{\sigma }(\zeta )} \int _{\frac{\sigma (\zeta )}{2}}^{\sigma ( \zeta )} \frac{u_{\mathfrak{i}}(\eta )z_{\mathfrak{i}}^{\Delta }(\eta )}{z_{\mathfrak{i}}(\eta )}f_{ \mathfrak{i}}( \eta )\Delta \eta \biggr] ^{3\alpha _{\mathfrak{i}}}\Delta \zeta . \end{aligned} $$
(50)

Integrating (50) by parts using formula (7) with

$$ \Phi ^{\Delta }(\zeta )= \frac{u_{\mathfrak{i}}^{\Delta }(\zeta )}{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{3\alpha _{\mathfrak{i}}m}}, \qquad \Psi ^{\sigma }( \zeta )= \biggl[ \sqrt[3\alpha _{\mathfrak{i}}]{\hat{w}^{\sigma }( \zeta )} \int _{ \frac{\sigma (\zeta )}{2}}^{\sigma (\zeta )} \frac{u_{\mathfrak{i}}(\varkappa )z_{\mathfrak{i}}^{\Delta }(\varkappa )}{z_{\mathfrak{i}}(\varkappa )}f_{ \mathfrak{i}}( \varkappa )\Delta \varkappa \biggr] ^{3\alpha _{ \mathfrak{i}}}, $$

we obtain

$$ \begin{aligned} J &= \bigl[ \Phi (\zeta )\Psi (\zeta ) \bigr] _{0}^{x}+ \int _{0}^{x} \bigl( -\Phi (\zeta ) \bigr) \bigl( \Psi (\zeta ) \bigr) ^{\Delta }\Delta \zeta \\ &= \biggl[ -\Psi (\zeta ) \int _{\zeta }^{\infty } \frac{u_{\mathfrak{i}}^{\Delta }(\eta )}{ [ u_{\mathfrak{i}}^{\sigma }(\eta ) ] ^{3\alpha _{\mathfrak{i}}m}} \Delta \eta \biggr] _{0}^{x}+ \int _{0}^{x} \bigl( -\Phi ( \zeta ) \bigr) \bigl( \Psi (\zeta ) \bigr) ^{\Delta }\Delta \zeta \\ &= \biggl[ -\Psi (\varkappa ) \int _{x}^{\infty } \frac{u_{\mathfrak{i}}^{\Delta }(\eta )}{ [ u_{\mathfrak{i}}^{\sigma }(\eta ) ] ^{3\alpha _{\mathfrak{i}}m}}\Delta \eta \biggr] + \int _{0}^{x} \bigl( -\Phi (\zeta ) \bigr) \bigl( \Psi (\zeta ) \bigr) ^{\Delta }\Delta \zeta \\ &\leq \int _{0}^{x} \bigl( -\Phi (\zeta ) \bigr) \bigl( \Psi (\zeta ) \bigr) ^{\Delta }\Delta \zeta , \end{aligned} $$
(51)

where \(\Phi (\zeta )=-\int _{\zeta }^{\infty } \frac{u_{\mathfrak{i}}^{\Delta }(\boldsymbol{\eta })}{ [ u_{\mathfrak{i}}^{\sigma }(\boldsymbol{\eta }) ] ^{3\alpha _{\mathfrak{i}}m}} \Delta \eta \) and \(( \Psi (\zeta ) ) ^{\Delta }>0\). From (9), using \(u_{\mathfrak{i}}^{\Delta }(\zeta )\geq 0\) and \(c\in {}[ \zeta ,\sigma (\zeta )]\), we have

$$ \begin{aligned} \bigl[ u_{\mathfrak{i}}^{1-3\alpha _{\mathfrak{i}}m}( \zeta ) \bigr] ^{\Delta } &= ( 1-3\alpha _{ \mathfrak{i}}m ) u_{\mathfrak{i}}^{-3\alpha _{\mathfrak{i}}m}(c)u_{\mathfrak{i}}^{\Delta }( \zeta ) \\ &= ( 1-3\alpha _{\mathfrak{i}}m ) \frac{u_{\mathfrak{i}}^{\Delta }(\zeta )}{u_{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}m}(c)} \\ &\leq ( 1-3\alpha _{\mathfrak{i}}m ) \frac{u_{\mathfrak{i}}^{\Delta }(\zeta )}{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{3\alpha _{\mathfrak{i}}m}}. \end{aligned} $$
(52)

Therefore, integrating (52) from ζ to ∞ with respect to η, we have

$$ -\Phi (\zeta )\leq \frac{1}{3\alpha _{\mathfrak{i}}m-1}u_{ \mathfrak{i}}^{1-3\alpha _{\mathfrak{i}}m}(\zeta ). $$
(53)

Combining (53) and (51), we have

$$ J\leq \frac{1}{3\alpha _{\mathfrak{i}}m-1} \int _{0}^{x}u_{ \mathfrak{i}}^{1-3\alpha _{\mathfrak{i}}m}( \zeta ) \bigl( \Psi ( \zeta ) \bigr) ^{\Delta }\Delta \zeta . $$
(54)

Now, by applying (6) to \(\Psi (\zeta )=\hat{w}(\zeta )\hat{Y}^{3\alpha _{\mathfrak{i}}}(\zeta )\) and using (9), we obtain

$$ \begin{aligned} \bigl( \Psi (\zeta ) \bigr) ^{\Delta } &= \hat{w}^{\Delta }(\zeta ) \bigl[ \hat{Y}^{\sigma }( \zeta ) \bigr] ^{3\alpha _{\mathfrak{i}}}+\hat{w}(\zeta ) \bigl[ \hat{Y}^{3\alpha _{\mathfrak{i}}}(\zeta ) \bigr] ^{\Delta } \\ &=\hat{w}^{\Delta }(\zeta ) \bigl[ \hat{Y}^{\sigma }( \zeta ) \bigr] ^{3\alpha _{\mathfrak{i}}}+3\alpha _{\mathfrak{i}} \hat{w}(\zeta ) \hat{Y}^{ ( 3\alpha _{\mathfrak{i}}-1 ) }(c) \hat{Y}^{\Delta }(\zeta ) \\ &\leq \hat{w}^{\Delta }(\zeta ) \bigl[ \hat{Y}^{ \sigma }(\zeta ) \bigr] ^{3\alpha _{\mathfrak{i}}}+3\alpha _{ \mathfrak{i}}\hat{w}^{\sigma }(\zeta ) \bigl[ \hat{Y}^{\sigma }( \zeta ) \bigr] ^{3\alpha _{\mathfrak{i}}-1}\hat{Y}^{\Delta }( \zeta ). \end{aligned} $$
(55)

From (55) and (54), as well as using (47), we have

$$ \begin{aligned} J &\leq \frac{1}{3\alpha _{\mathfrak{i}}m-1} \int _{0}^{x}u_{ \mathfrak{i}}^{1-3\alpha _{\mathfrak{i}}m}( \zeta )\hat{w}^{ \Delta }(\zeta ) \\ &\quad {}\times \biggl( \int _{ \frac{\sigma (\zeta )}{2}}^{\sigma (\zeta )} \frac{u_{\mathfrak{i}}(\eta )z_{\mathfrak{i}}^{\Delta }(\eta )}{z_{\mathfrak{i}}(\eta )}f_{\mathfrak{i}}( \eta )\Delta \eta \biggr) ^{3\alpha _{\mathfrak{i}}} \Delta \zeta \\ &\quad {}+\frac{3\alpha _{\mathfrak{i}}}{3\alpha _{\mathfrak{i}}m-1} \int _{0}^{x}u_{\mathfrak{i}}^{1-3\alpha _{\mathfrak{i}}m}( \zeta )\hat{w}^{ \sigma }(\zeta ) \\ &\quad {}\times \biggl[ \frac{u_{\mathfrak{i}}(\zeta )z_{\mathfrak{i}}^{\Delta }(\zeta )f_{\mathfrak{i}}(\zeta )}{z_{\mathfrak{i}}(\zeta )}- \frac{u_{\mathfrak{i}}(\frac{\zeta }{2})z_{\mathfrak{i}}^{\Delta }(\frac{\zeta }{2})f_{\mathfrak{i}}(\frac{\zeta }{2})}{2z_{\mathfrak{i}}(\frac{\zeta }{2})} \biggr] \\ &\quad {}\times \biggl( \int _{ \frac{\sigma (\zeta )}{2}}^{\sigma (\zeta )}\frac{u_{\mathfrak{i}}(\eta )z_{\mathfrak{i}}^{\Delta }(\eta )}{z_{\mathfrak{i}}(\eta )}f_{\mathfrak{i}}( \eta )\Delta \eta \biggr) ^{3\alpha _{\mathfrak{i}}-1}\Delta \zeta \\ &=\frac{1}{3\alpha _{\mathfrak{i}}m-1} \int _{0}^{x} \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{3\alpha _{\mathfrak{i}}m}\hat{w}^{\Delta }(\zeta )}{u_{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}m-1}(\zeta )u_{\mathfrak{i}}^{\Delta }(\zeta )}\Gamma _{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}}( \zeta )\Delta \zeta \\ &\quad {}+\frac{3\alpha _{\mathfrak{i}}}{3\alpha _{\mathfrak{i}}m-1} \int _{0}^{x}\frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{ ( 3\alpha _{\mathfrak{i}}-1 ) m}\hat{w}^{\sigma }(\zeta )}{u_{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}m-1}(\zeta ) [ u_{\mathfrak{i}}^{\Delta }(\zeta ) ] ^{1-\frac{1}{3\alpha _{\mathfrak{i}}}}} \Gamma _{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}-1}(\zeta ) \\ &\quad {}\times \biggl[ \frac{u_{\mathfrak{i}}(\zeta )z_{\mathfrak{i}}^{\Delta }(\zeta )f_{\mathfrak{i}}(\zeta )}{z_{\mathfrak{i}}(\zeta )}- \frac{u_{\mathfrak{i}}(\frac{\zeta }{2})z_{\mathfrak{i}}^{\Delta }(\frac{\zeta }{2})f_{\mathfrak{i}}(\frac{\zeta }{2})}{2z_{\mathfrak{i}}(\frac{\zeta }{2})} \biggr] \Delta \zeta . \end{aligned} $$

Hence,

$$ \begin{aligned} & \int _{0}^{x}\hat{w}^{\sigma }(\zeta )\Gamma _{ \mathfrak{i}}^{3\alpha _{\mathfrak{i}}}(\zeta ) \biggl( 1- \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{3\alpha _{\mathfrak{i}}m}\hat{w}^{\Delta }(\zeta )}{ ( 3\alpha _{\mathfrak{i}}m-1 ) u_{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}m-1}(\zeta )u_{\mathfrak{i}}^{\Delta }(\zeta )\hat{w}^{\sigma }(\zeta )} \biggr) \Delta \zeta \\ &\quad \leq \frac{3\alpha _{\mathfrak{i}}}{3\alpha _{\mathfrak{i}}m-1} \int _{0}^{x}\frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{ ( 3\alpha _{\mathfrak{i}}-1 ) m}\hat{w}^{\sigma }(\zeta )}{u_{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}m-1}(\zeta ) [ u_{\mathfrak{i}}^{\Delta }(\zeta ) ] ^{1-\frac{1}{3\alpha _{\mathfrak{i}}}}} \Gamma _{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}-1}(\zeta ) \\ &\qquad{} \times \biggl[ \frac{u_{\mathfrak{i}}(\zeta )z_{\mathfrak{i}}^{\Delta }(\zeta )}{z_{\mathfrak{i}}(\zeta )}f_{\mathfrak{i}}(\zeta )- \frac{u_{\mathfrak{i}}(\frac{\zeta }{2})z_{\mathfrak{i}}^{\Delta }(\frac{\zeta }{2})}{2z_{\mathfrak{i}}(\frac{\zeta }{2})}f_{\mathfrak{i}}\biggl( \frac{\zeta }{2}\biggr) \biggr] \Delta \zeta . \end{aligned} $$
(56)

From (56) and (45), we have

$$ \begin{aligned} & \int _{0}^{x}\hat{w}^{\sigma }(\zeta )\Gamma _{ \mathfrak{i}}^{3\alpha _{\mathfrak{i}}}(\zeta )\Delta \zeta \\ &\quad \leq \frac{3\alpha _{\mathfrak{i}}\mathfrak{\lambda }_{\mathfrak{i}}}{3\alpha _{\mathfrak{i}}m-1} \int _{0}^{x}\hat{w}^{\sigma }( \zeta )\Gamma _{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}-1}(\zeta ) \\ &\qquad{} \times \biggl[ \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{ ( 3\alpha _{\mathfrak{i}}-1 ) m}z_{\mathfrak{i}}^{\Delta }(\zeta )}{u_{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}m-2}(\zeta )\sqrt[3\delta _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}z_{\mathfrak{i}}(\zeta )}f_{\mathfrak{i}}(\zeta )- \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{ ( 3\alpha _{\mathfrak{i}}-1 ) m}u_{\mathfrak{i}}(\frac{\zeta }{2})z_{\mathfrak{i}}^{\Delta }(\frac{\zeta }{2})}{2u_{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}m-1}(\zeta )z_{\mathfrak{i}}(\frac{\zeta }{2})\sqrt[3\delta _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}f_{\mathfrak{i}}\biggl( \frac{\zeta }{2}\biggr) \biggr] \Delta \zeta \\ &\quad = \frac{3\alpha _{\mathfrak{i}}\mathfrak{\lambda }_{\mathfrak{i}}}{3\alpha _{\mathfrak{i}}m-1} \int _{0}^{x} \bigl( \hat{w}^{\sigma }(\zeta )\Gamma _{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}}(\zeta ) \bigr) ^{\frac{3\alpha _{\mathfrak{i}}-1}{3\alpha _{\mathfrak{i}}}} \bigl[ \hat{w}^{\sigma }(\zeta ) \bigr] ^{ \frac{1}{3\alpha _{\mathfrak{i}}}} \\ &\qquad {}\times \biggl[ \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{ ( 3\alpha _{\mathfrak{i}}-1 ) m}z_{\mathfrak{i}}^{\Delta }(\zeta )}{u_{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}m-2}(\zeta )\sqrt[3\delta _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}z_{\mathfrak{i}}(\zeta )}f_{\mathfrak{i}}(\zeta )- \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{ ( 3\alpha _{\mathfrak{i}}-1 ) m}u_{\mathfrak{i}}(\frac{\zeta }{2})z_{\mathfrak{i}}^{\Delta }(\frac{\zeta }{2})}{2u_{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}m-1}(\zeta )z_{\mathfrak{i}}(\frac{\zeta }{2})\sqrt[3\delta _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}}f_{\mathfrak{i}}\biggl( \frac{\zeta }{2}\biggr) \biggr] \Delta \zeta . \end{aligned} $$

Applying Hölder’s inequality with \(3\alpha _{\mathfrak{i}}\) and \(\alpha _{\mathfrak{i}}/ ( 3\alpha _{\mathfrak{i}}-1 ) \), we have

$$ \begin{aligned} & \int _{0}^{x}\hat{w}^{\sigma }(\zeta )\Gamma _{ \mathfrak{i}}^{3\alpha _{\mathfrak{i}}}(\zeta )\Delta \zeta \\ &\quad \leq \biggl( \frac{3\alpha _{\mathfrak{i}}\mathfrak{\lambda }_{\mathfrak{i}}}{3\alpha _{\mathfrak{i}}m-1} \biggr) ^{3\alpha _{\mathfrak{i}}} \int _{0}^{x}\hat{w}^{\sigma }( \zeta ) \\ &\qquad{} \times \biggl[ \frac { [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{ ( 3\alpha _{\mathfrak{i}}-1 ) m}z_{\mathfrak{i}}^{\Delta }(\zeta )f_{\mathfrak{i}}(\zeta )}{u_{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}m-2}(\zeta )\sqrt[3\delta _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}z_{\mathfrak{i}}(\zeta )}- \frac { [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{ ( 3\alpha _{\mathfrak{i}}-1 ) m}u_{\mathfrak{i}}(\frac{\zeta }{2})z_{\mathfrak{i}}^{\Delta }(\frac{\zeta }{2})f_{\mathfrak{i}}(\frac{\zeta }{2})}{2u_{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}m-1}(\zeta )z_{\mathfrak{i}}(\frac{\zeta }{2})\sqrt[3\delta _{\mathfrak{i}}]{u_{\mathfrak{i}}^{\Delta }(\zeta )}} \biggr] ^{3 \alpha _{\mathfrak{i}}}\Delta \zeta \\ &\quad = \biggl( \frac{3\alpha _{\mathfrak{i}}\mathfrak{\lambda }_{\mathfrak{i}}}{3\alpha _{\mathfrak{i}}m-1} \biggr) ^{3\alpha _{\mathfrak{i}}} \int _{0}^{x}\hat{w}^{\sigma }( \zeta )g_{\mathfrak{i}}(\zeta )\Delta \zeta . \end{aligned} $$
(57)

From (57) and (49), we get (46). □

In Theorem 16, if we take T = N , then we obtain the following corollary.

Corollary 17

For any \(\{ u(s) \} _{s=1}^{\infty }\), \(\{ \hat{w}(s) \} _{s=1}^{\infty }\), and \(\{ z(s) \} _{s=1}^{\infty }\) increasing and nonnegative sequences, \(1\leq \mathfrak{i}\leq n\), if \(\alpha _{\mathfrak{i}}>1\), \(\delta _{\mathfrak{i}}=\alpha _{\mathfrak{i}}/ ( 3\alpha _{\mathfrak{i}}-1 )\), and

$$ 1+ \frac{ [ u_{\mathfrak{i}}(s+1) ] ^{3\alpha _{\mathfrak{i}}m}\Delta \hat{w}(s)}{(1-3\alpha _{\mathfrak{i}}m)u_{\mathfrak{i}}^{3\alpha _{\mathfrak{i}}m-1}(s)\hat{w}(s+1)\Delta u(s)}\geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0 , \quad \textit{for }m>\frac{1}{3\alpha _{\mathfrak{i}}}, $$

then

$$ \begin{aligned} &\sum_{\mathfrak{i}=1}^{n\mathfrak{-1}} \Biggl( \sum_{s=1}^{ \mathfrak{q-1}} \hat{w}(s+1) \bigl[ \Gamma _{\mathfrak{i}}^{\alpha _{\mathfrak{i}}}(s)\Gamma _{\mathfrak{i}+1}^{\alpha _{\mathfrak{i}}+1}(s)\Gamma _{\mathfrak{i}+2}^{\alpha _{\mathfrak{i}+2}}(s) \bigr] \Biggr) \\ &\quad \leq \sum_{\mathfrak{i}=1}^{n} \biggl( \frac{3\alpha _{\mathfrak{i}}\mathfrak{\lambda }_{\mathfrak{i}}}{3\alpha _{\mathfrak{i}}m-1} \biggr) ^{3 \alpha _{\mathfrak{i}}}\sum_{s=1}^{\mathfrak{q-1}} \hat{w}(s+1)g_{\mathfrak{i}}(s), \end{aligned} $$

where

Γ i ( s ) = Δ u i ( s ) 3 α i u i m ( s + 1 ) r = s + 1 2 s u i ( r ) Δ z i ( r ) z i ( r ) f i ( r ) , s + 1 2 , s 2 N , g i ( s ) = [ [ u i ( s + 1 ) ] ( 3 α i 1 ) m Δ z i ( s ) u i 3 α i m 2 ( s ) Δ u i ( s ) 3 δ i z i ( s ) f i ( s ) [ u i ( s + 1 ] ( 3 α i 1 ) m u i ( s 2 ) Δ z i ( s 2 ) 2 u i 3 α i m 1 ( s ) z i ( s 2 ) Δ u i ( s ) 3 δ i f i ( s 2 ) ] 3 α i ,

and \(u_{\mathfrak{i}}(\infty )=\infty \).

Remark 18

Clearly, for T = R , Theorem 16 reduces to [4, Theorem 3].

Theorem 19

For any \(1\leq \mathfrak{i}\leq n\), if \(\alpha >1\), \(\kappa \geq 1\), \(\delta =\alpha / ( \kappa \alpha -1 )\), and there exist \(\mathfrak{\lambda }_{\mathfrak{i}}>0\), \(m>0\) such that

$$ 1+ \frac{u_{\mathfrak{i}}^{\sigma }(\zeta )\hat{w}^{\Delta }(\zeta )}{ ( 1+\kappa \alpha m ) u_{\mathfrak{i}}^{\Delta }(\zeta )\hat{w}(\zeta )}\geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0, $$
(58)

then

$$ \int _{a}^{b}\hat{w}^{\sigma }(\zeta ) \Biggl( \sum_{ \mathfrak{i}=1}^{n}\Gamma _{\mathfrak{i}a}(\zeta ) \Biggr) ^{\kappa \alpha }\Delta \zeta \leq \sum _{\mathfrak{i}=1}^{n} \biggl( \frac{-\kappa \alpha \mathfrak{\lambda }_{\mathfrak{i}}\sqrt[\kappa \delta ]{n}}{1+\kappa \alpha m} \biggr) ^{\kappa \alpha } \int _{a}^{b}\hat{w}( \zeta )g_{\mathfrak{i}}( \zeta )\Delta \zeta , $$
(59)

where

Γ i (ζ)= [ u i σ ( ζ ) ] m u i Δ ( ζ ) κ α 0 ζ z i Δ ( x ) u i σ ( x ) z i ( x ) f i (x)Δx,ζ[0, ) T ,

and

$$ g_{\mathfrak{i}}(\zeta )= \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{\kappa \alpha m} [ \hat{w}^{\sigma }(\zeta ) ] ^{\kappa \alpha } [ z_{\mathfrak{i}}^{\Delta }(\zeta ) ] ^{\kappa \alpha }}{ [ u_{\mathfrak{i}}^{\Delta }(\zeta ) ] ^{\frac{\alpha }{\delta }}\hat{w}^{\kappa \alpha }(\zeta )z_{\mathfrak{i}}^{\kappa \alpha }(\zeta )}f_{\mathfrak{i}}^{\kappa \alpha }( \zeta ). $$

Proof

Let us define for \(1\leq \mathfrak{i}\leq n\) and \(0< a< b<\infty \),

$$ \Gamma _{\mathfrak{i}a}(\zeta )= \bigl[ u_{\mathfrak{i}}^{ \sigma }(\zeta ) \bigr] ^{m} \sqrt[\kappa \alpha ]{u_{\mathfrak{i}}^{\Delta }( \zeta )} \int _{a}^{\zeta } \frac{z_{\mathfrak{i}}^{\Delta }(x)}{u_{\mathfrak{i}}^{\sigma }(x)z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x) \Delta x. $$
(60)

Using (12), for \(C_{\mathfrak{i}}=\Gamma _{\mathfrak{i}}(\zeta )\) and \(\kappa \rightarrow \kappa \alpha \), we get

$$ \Biggl( \sum_{\mathfrak{i}=1}^{n}\Gamma _{\mathfrak{i}a}( \zeta ) \Biggr) ^{\kappa \alpha }\leq n^{\alpha \kappa -1}\sum_{ \mathfrak{i}=1}^{n}\Gamma _{\mathfrak{i}a}^{\kappa \alpha }(\zeta ). $$
(61)

Multiplying (61) by \(\hat{w}^{\sigma }(\zeta )\) and integrating from a to b, we get

$$ \int _{a}^{b}\hat{w}(\zeta ) \Biggl( \sum _{\mathfrak{i}=1}^{n} \Gamma _{\mathfrak{i}a}(\zeta ) \Biggr) ^{\kappa \alpha } \Delta \zeta \leq n^{\alpha \kappa -1}\sum _{\mathfrak{i}=1}^{n} \int _{a}^{b} \hat{w}(\zeta )\Gamma _{\mathfrak{i}a}^{\kappa \alpha }( \zeta )\Delta \zeta . $$
(62)

Now,

$$ \begin{aligned} J &= \int _{a}^{b}\hat{w}(\zeta )\Gamma _{ \mathfrak{i}a}^{\kappa \alpha }(\zeta )\Delta \zeta \\ &= \int _{a}^{b}\hat{w}(\zeta ) \biggl[ \bigl[ u_{ \mathfrak{i}}^{\sigma }(\zeta ) \bigr] ^{m} \sqrt[\kappa \alpha ]{u_{\mathfrak{i}}^{\Delta }(\zeta )} \int _{a}^{\zeta } \frac{z_{\mathfrak{i}}^{\Delta }(x)}{u_{\mathfrak{i}}^{\sigma }(x)z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x) \Delta x \biggr] ^{ \kappa \alpha }\Delta \zeta \\ &= \int _{a}^{b} \bigl[ u_{\mathfrak{i}}^{\sigma }( \zeta ) \bigr] ^{\kappa \alpha m}u_{\mathfrak{i}}^{\Delta }( \zeta ) \biggl[ \sqrt[\kappa \alpha ]{\hat{w}(\zeta )} \int _{a}^{ \zeta }\frac{z_{\mathfrak{i}}^{\Delta }(x)}{u_{\mathfrak{i}}^{\sigma }(x)z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x) \Delta x \biggr] ^{ \kappa \alpha }\Delta \zeta . \end{aligned} $$
(63)

Integrating (63) by parts using formula (7) with

$$ \Phi ^{\Delta }(\zeta )= \bigl[ u_{\mathfrak{i}}^{ \sigma }(\zeta ) \bigr] ^{\kappa \alpha m}u_{\mathfrak{i}}^{ \Delta }(\zeta ), \qquad \Psi (\zeta )= \biggl[ \sqrt[\kappa \alpha ]{\hat{w}(\zeta )} \int _{a}^{ \zeta } \frac{z_{\mathfrak{i}}^{\Delta }(x)}{u_{\mathfrak{i}}^{\sigma }(x)z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x) \Delta x \biggr] ^{ \kappa \alpha }, $$

we obtain

$$ \begin{aligned} J &= \bigl[ \Phi (\zeta )\Psi (\zeta ) \bigr] _{a}^{b}+ \int _{a}^{b} \bigl( \Phi ^{\sigma }( \zeta ) \bigr) \bigl( -\Psi (\zeta ) \bigr) ^{ \Delta }\Delta \zeta \\ &= \biggl[ -\Psi (\zeta ) \int _{\zeta }^{b} \bigl[ u_{\mathfrak{i}}^{\sigma }(x) \bigr] ^{\kappa \alpha m}u_{ \mathfrak{i}}^{\Delta }(x)\Delta x \biggr] _{a}^{b}+ \int _{a}^{b} \bigl( \Phi ^{\sigma }(\zeta ) \bigr) \bigl( -\Psi ( \zeta ) \bigr) ^{\Delta }\Delta \zeta \\ &= \int _{a}^{b} \bigl( \Phi ^{\sigma }(\zeta ) \bigr) \bigl( -\Psi (\zeta ) \bigr) ^{\Delta }\Delta \zeta , \end{aligned} $$
(64)

where \(\Phi (\zeta )=-\int _{\zeta }^{b} [ u_{\mathfrak{i}}^{\sigma }( \varkappa ) ] ^{\kappa \alpha m}u_{\mathfrak{i}}^{ \Delta }(\varkappa )\Delta \varkappa \) and \(( \Psi (\zeta ) ) ^{\Delta }>0\). From (9), using \(u_{\mathfrak{i}}^{\Delta }(\zeta )\geq 0\) and \(c\in {}[ \zeta ,\sigma (\zeta )] \), we have

$$ \begin{aligned} \bigl[ u_{\mathfrak{i}}^{1+\kappa \alpha m}( \zeta ) \bigr] ^{\Delta } &= ( 1+\kappa \alpha m ) u_{\mathfrak{i}}^{\kappa \alpha m}(c)u_{\mathfrak{i}}^{\Delta }( \zeta ) \\ &\leq ( 1+\kappa \alpha m ) \bigl[ u_{\mathfrak{i}}^{ \sigma }(\zeta ) \bigr] ^{\kappa \alpha m}u_{\mathfrak{i}}^{ \Delta }(\zeta ). \end{aligned} $$

This implies that

$$ \frac{1}{1+\kappa \alpha m} \int _{\zeta }^{b} \bigl[ u_{ \mathfrak{i}}^{1+\kappa \alpha m}(x) \bigr] ^{\Delta }\Delta x \leq \int _{\zeta }^{b} \bigl[ u_{\mathfrak{i}}^{\sigma }(x) \bigr] ^{\kappa \alpha m}u_{\mathfrak{i}}^{\Delta }(x) \Delta x =-\Phi (\zeta ). $$

Therefore

$$ \begin{aligned} \Phi (\zeta ) &\leq \frac{-1}{1+\kappa \alpha m} \int _{\zeta }^{b} \bigl[ u_{\mathfrak{i}}^{1+\kappa \alpha m}(x) \bigr] ^{\Delta }\Delta x \\ &=\frac{-1}{1+\kappa \alpha m} \bigl[ u_{\mathfrak{i}}^{1+\kappa \alpha m}(b)-u_{\mathfrak{i}}^{1+\kappa \alpha m}( \zeta ) \bigr] \\ &=\frac{1}{1+\kappa \alpha m} \bigl[ u_{\mathfrak{i}}^{1+\kappa \alpha m}(\zeta )-u_{\mathfrak{i}}^{1+\kappa \alpha m}(b) \bigr] \\ &\leq \frac{1}{1+\kappa \alpha m}u_{\mathfrak{i}}^{1+\kappa \alpha m}( \zeta ). \end{aligned} $$
(65)

Substituting (65) into (64), we have

$$ J\leq \frac{1}{1+\kappa \alpha m} \int _{a}^{b} \bigl[ u_{\mathfrak{i}}^{ \sigma }( \zeta ) \bigr] ^{1+\kappa \alpha m} \bigl( -\Psi ( \zeta ) \bigr) ^{\Delta }\Delta \zeta . $$
(66)

Now, by applying (6) to \(\Psi (\zeta )=\hat{w}(\zeta )\hat{Y}^{\kappa \alpha }(\zeta )\) and using (9), we obtain

$$ \begin{aligned} \bigl( \Psi (\zeta ) \bigr) ^{\Delta } &= \hat{w}^{\Delta }(\zeta )\hat{Y}^{\kappa \alpha }( \zeta )+\hat{w}^{\sigma }(\zeta ) \bigl[ \hat{Y}^{\kappa \alpha }( \zeta ) \bigr] ^{\Delta } \\ &=\hat{w}^{\Delta }(\zeta )\hat{Y}^{\kappa \alpha }( \zeta )+\kappa \alpha \hat{w}^{\sigma }(\zeta )\hat{Y}^{ \kappa \alpha -1}(c)\hat{Y}^{\Delta }( \zeta ) \\ &\geq \hat{w}^{\Delta }(\zeta )\hat{Y}^{\kappa \alpha }(\zeta )+ \kappa \alpha \hat{w}^{\sigma }(\zeta ) \biggl[ \frac{z_{\mathfrak{i}}^{\Delta }(\zeta )}{u_{\mathfrak{i}}^{\sigma }(\zeta )z_{\mathfrak{i}}(\zeta )}f_{ \mathfrak{i}}( \zeta ) \biggr] \hat{Y}^{\kappa \alpha -1}(\zeta ). \end{aligned} $$
(67)

From (67) and (66), as well as using (60), we have

$$ \begin{aligned} J &\leq \frac{-1}{1+\kappa \alpha m} \int _{a}^{b} \bigl[ u_{ \mathfrak{i}}^{\sigma }( \zeta ) \bigr] ^{1+\kappa \alpha m}\hat{w}^{\Delta }(\zeta ) \biggl( \int _{a}^{\zeta } \frac{z_{\mathfrak{i}}^{\Delta }(x)}{u_{\mathfrak{i}}^{\sigma }(x)z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x) \Delta x \biggr) ^{ \kappa \alpha }\Delta \zeta \\ &\quad {}+\frac{-\kappa \alpha }{1+\kappa \alpha m} \int _{a}^{b} \bigl[ u_{ \mathfrak{i}}^{\sigma }( \zeta ) \bigr] ^{1+\kappa \alpha m}\hat{w}^{ \sigma }(\zeta ) \biggl( \frac{z_{\mathfrak{i}}^{\Delta }(\zeta )}{u_{\mathfrak{i}}^{\sigma }(\zeta )z_{\mathfrak{i}}(\zeta )}f_{\mathfrak{i}}(\zeta ) \biggr) \\ &\quad {}\times \biggl( \int _{a}^{ \zeta } \frac{z_{\mathfrak{i}}^{\Delta }(x)}{u_{\mathfrak{i}}^{\sigma }(x)z_{\mathfrak{i}}(x)}f_{\mathfrak{i}}(x) \Delta x \biggr) ^{\kappa \alpha -1}\Delta \zeta \\ &=\frac{-1}{1+\kappa \alpha m} \int _{a}^{b} \frac{u_{\mathfrak{i}}^{\sigma }(\zeta )\hat{w}^{\Delta }(\zeta )}{u_{\mathfrak{i}}^{\Delta }(\zeta )}\Gamma _{\mathfrak{i}a}^{\kappa \alpha }(\zeta )\Delta \zeta \\ &\quad {}+\frac{-\kappa \alpha }{1+\kappa \alpha m} \int _{a}^{b} \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{1+m}\hat{w}^{\sigma }(\zeta )}{ [ u_{\mathfrak{i}}^{\Delta }(\zeta ) ] ^{1-\frac{1}{\kappa \alpha }}}\Gamma _{\mathfrak{i}a}^{ \kappa \alpha -1}(\zeta ) \biggl( \frac{z_{\mathfrak{i}}^{\Delta }(\zeta )}{u_{\mathfrak{i}}^{\sigma }(\zeta )z_{\mathfrak{i}}(\zeta )}f_{\mathfrak{i}}( \zeta ) \biggr) \Delta \zeta . \end{aligned} $$

Hence,

$$ \begin{aligned} & \int _{a}^{b}\hat{w}(\zeta )\Gamma _{ \mathfrak{i}}^{\kappa \alpha }(\zeta ) \biggl( 1+ \frac{u_{\mathfrak{i}}^{\sigma }(\zeta )\hat{w}^{\Delta }(\zeta )}{ ( 1+\kappa \alpha m ) u_{\mathfrak{i}}^{\Delta }(\zeta )\hat{w}(\zeta )} \biggr) \Delta \zeta \\ &\quad \leq \frac{-\kappa \alpha }{1+\kappa \alpha m} \int _{a}^{b} \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{1+m}\hat{w}^{\sigma }(\zeta )}{ [ u_{\mathfrak{i}}^{\Delta }(\zeta ) ] ^{1-\frac{1}{\kappa \alpha }}}\Gamma _{\mathfrak{i}a}^{ \kappa \alpha -1}(\zeta ) \biggl[ \frac{z_{\mathfrak{i}}^{\Delta }(\zeta )}{u_{\mathfrak{i}}^{\sigma }(\zeta )z_{\mathfrak{i}}(\zeta )}f_{\mathfrak{i}}( \zeta ) \biggr] \Delta \zeta . \end{aligned} $$
(68)

From (68) and (58), we have

$$ \begin{aligned} & \int _{a}^{b}\hat{w}(\zeta )\Gamma _{ \mathfrak{i}a}^{\kappa \alpha }(\zeta )\Delta \zeta \\ &\quad \leq \frac{-\kappa \alpha \mathfrak{\lambda }_{\mathfrak{i}}}{1+\kappa \alpha m} \int _{a}^{b} \bigl( \hat{w}(\zeta )\Gamma _{\mathfrak{i}a}^{\kappa \alpha }(\zeta ) \bigr) ^{ \frac{\kappa \alpha -1}{\kappa \alpha }} \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{m}\hat{w}^{\sigma }(\zeta )z_{\mathfrak{i}}^{\Delta }(\zeta )}{ [ u_{\mathfrak{i}}^{\Delta }(\zeta ) ] ^{1-\frac{1}{\kappa \alpha }}\hat{w}^{\frac{\kappa \alpha -1}{\kappa \alpha }}(\zeta )z_{\mathfrak{i}}(\zeta )}f_{\mathfrak{i}}( \zeta )\Delta \zeta . \end{aligned} $$

Applying Hölder’s inequality with κα and \(\alpha / ( \kappa \alpha -1 ) \), we have

$$ \begin{aligned} & \int _{a}^{b}\hat{w}(\zeta )\Gamma _{ \mathfrak{i}a}^{\kappa \alpha }(\zeta )\Delta \zeta \\ &\quad \leq \biggl( \frac{-\kappa \alpha \mathfrak{\lambda }_{\mathfrak{i}}}{1+\kappa \alpha m} \biggr) ^{\kappa \alpha } \int _{a}^{b} \frac{ [ u_{\mathfrak{i}}^{\sigma }(\zeta ) ] ^{\kappa \alpha m} [ \hat{w}^{\sigma }(\zeta ) ] ^{\kappa \alpha } [ z_{\mathfrak{i}}^{\Delta }(\zeta ) ] ^{\kappa \alpha }}{ [ u_{\mathfrak{i}}^{\Delta }(\zeta ) ] ^{\kappa \alpha -1}\hat{w}^{\kappa \alpha -1}(\zeta )z_{\mathfrak{i}}^{\kappa \alpha }(\zeta )}\Gamma _{\mathfrak{i}}^{\kappa \alpha }(\zeta )\Delta \zeta \\ &\quad = \biggl( \frac{-\kappa \alpha \mathfrak{\lambda }_{\mathfrak{i}}}{1+\kappa \alpha m} \biggr) ^{\kappa \alpha } \int _{a}^{b}\hat{w}(\zeta )g_{\mathfrak{i}}( \zeta )\Delta \zeta \mathfrak{.} \end{aligned} $$
(69)

From (69) and (62), we get (59). □

In Theorem 19, if we take T = N , then we obtain the following corollary.

Corollary 20

For any \(\{ u(s) \} _{s=1}^{\infty }\), \(\{ \hat{w}(s) \} _{s=1}^{\infty }\), and \(\{ z(s) \} _{s=1}^{\infty }\) increasing and nonnegative sequences, if \(1\leq \mathfrak{i}\leq n\), \(\alpha >1\), \(\kappa \geq 1\), \(\delta =\alpha / ( \kappa \alpha -1 )\), and

$$ 1+ \frac{u_{\mathfrak{i}}(s+1)\Delta \hat{w}(s)}{ ( 1+\kappa \alpha m ) \hat{w}(s)\Delta u_{\mathfrak{i}}(s)} \geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0, $$

then

$$ \sum_{s=1}^{r\mathfrak{-1}}\hat{w}(s+1) \Biggl( \sum _{\mathfrak{i}=1}^{n}\Gamma _{\mathfrak{i}a}(s) \Biggr) ^{\kappa \alpha }\leq \sum_{\mathfrak{i}=1}^{n} \biggl( \frac{-\kappa \alpha \mathfrak{\lambda }_{\mathfrak{i}}\sqrt[\kappa \delta ]{n}}{1+\kappa \alpha m} \biggr) ^{ \kappa \alpha }\sum_{s=1}^{r\mathfrak{-1}} \hat{w}(s)g_{\mathfrak{i}}(s), $$

where

$$ \Gamma _{\mathfrak{i}}(s)= \bigl[ u_{\mathfrak{i}}(s+1) \bigr] ^{m} \sum_{q=1}^{s\mathfrak{-1}} \frac{\Delta z_{\mathfrak{i}}(q)}{u_{\mathfrak{i}}(q+1)z_{\mathfrak{i}}(q)}f_{\mathfrak{i}}(q) $$

and

$$ g_{\mathfrak{i}}(s)= \frac{ [ u_{\mathfrak{i}}(s+1) ] ^{\kappa \alpha m} [ \hat{w}(s+1) ] ^{\kappa \alpha } [ \Delta z_{\mathfrak{i}}(s) ] ^{\kappa \alpha }}{ [ \Delta u_{\mathfrak{i}}(s) ] ^{\frac{\alpha }{\delta }}\hat{w}^{\kappa \alpha }(s)z_{\mathfrak{i}}^{\kappa \alpha }(s)}f_{ \mathfrak{i}}^{\kappa \alpha }(s). $$

Remark 21

Clearly, for T = R , Theorem 16 reduces to [4, Theorem 4].

The next corollary follows from Theorem 19 by taking \(u_{\mathfrak{i}}(\zeta )=z_{\mathfrak{i}}(\zeta )=\zeta \), \(f_{\mathfrak{i}}(\zeta )\rightarrow \zeta ^{1-m}h_{\mathfrak{i}} \).

Corollary 22

For any \(1\leq \mathfrak{i}\leq n\) and \(\alpha >1\), if \(h_{\mathfrak{i}}\) are rd-continuous functions and

$$ 1+ \frac{\sigma (\zeta )\hat{w}^{\Delta }(\zeta )}{ ( \kappa \alpha m+1 ) \hat{w}(\zeta )}\geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0, $$

then

$$ \begin{aligned} & \int _{a}^{b}\hat{w}(\zeta ) \Biggl[ \sum _{ \mathfrak{i}=1}^{n}\sigma ^{m}(\zeta ) \int _{0}^{ \zeta }\frac{x^{2-m}}{\sigma (x)}h_{\mathfrak{i}}(x) \Delta x \Biggr] ^{\kappa \alpha } \Delta \zeta \\ &\quad \leq n^{\alpha \kappa -1}\sum_{\mathfrak{i}=1}^{n} \biggl( \frac{-\kappa \alpha \mathfrak{\lambda }_{\mathfrak{i}}}{1+\kappa \alpha m} \biggr) ^{\kappa \alpha } \int _{a}^{b}\hat{w}(\zeta )g_{ \mathfrak{i}}( \zeta )\Delta \zeta , \end{aligned} $$
(70)

where

$$ g_{\mathfrak{i}}(\zeta )= \frac{ [ \sigma (\zeta ) ] ^{\kappa \alpha m} [ \hat{w}^{\sigma }(\zeta ) ] ^{\kappa \alpha }\zeta ^{\kappa \alpha -\kappa \alpha m}}{\hat{w}^{\kappa \alpha }(\zeta )\zeta ^{\kappa \alpha }}h_{\mathfrak{i}}^{\kappa \alpha }( \zeta ). $$

Remark 23

Letting T=R in Corollary 22, we have that \(\sigma (\zeta )=\zeta \) and

$$ 1+ \frac{\zeta \hat{w}'(\zeta )}{ ( \kappa \alpha m+1 ) \hat{w}(\zeta )} \geq \frac{1}{\mathfrak{\lambda }_{\mathfrak{i}}}>0. $$

Then

$$ \begin{aligned} & \int _{a}^{b}\hat{w}(\zeta ) \Biggl[ \sum _{ \mathfrak{i}=1}^{n}\zeta ^{m} \int _{0}^{\zeta }\frac{1}{\zeta ^{1+m}}h_{\mathfrak{i}}(x)\,dx \Biggr] ^{\kappa \alpha }\,d\zeta \\ &\quad \leq n^{\alpha \kappa -1}\sum_{\mathfrak{i}=1}^{n} \biggl( \frac{-\kappa \alpha \mathfrak{\lambda }_{\mathfrak{i}}}{1+\kappa \alpha m} \biggr) ^{\kappa \alpha } \int _{a}^{b}\hat{w}(\zeta )h_{ \mathfrak{i}}^{\kappa \alpha }( \zeta )\,d\zeta , \end{aligned} $$

which agrees with [4, Corollary 3].

Conclusion

In this work, we explored some new generalized inequalities involving many functions of Hardy type on time scales by using delta calculus. Further, we also applied our inequalities to discrete and continuous calculus to obtain some new Hardy inequalities as special cases. In a future work, we will continue to generalize more dynamic inequalities by conformable delta fractional calculus on time scales.

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Resources and methodology, AAE-D, KAM, DB, HMR; data gathering, AAE-D and KAM; writing-original draft preparation, AAE-D, KAM and HMR; conceptualization, writing-review and editing, AAE-D, KAM, DB and HMR; administration, AAE-D and DB. All authors read and approved the final manuscript.

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El-Deeb, A.A., Mohamed, K.A., Baleanu, D. et al. Weighted dynamic Hardy-type inequalities involving many functions on arbitrary time scales. J Inequal Appl 2022, 120 (2022). https://doi.org/10.1186/s13660-022-02854-5

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  • DOI: https://doi.org/10.1186/s13660-022-02854-5

MSC

  • 26A15
  • 26A16
  • 26D10
  • 26D15
  • 39A13
  • 34A40
  • 34N05

Keywords

  • Delta derivative
  • Hardy’s inequality
  • Hölder’s inequality
  • Time scales