Novel Weighted Dynamic Hardy-Type Inequalities in the Framework of Delta Conformable Calculus on Time Scales

Abstract

This work presents new results concerning weighted Hardy-type inequalities within the framework of delta conformable fractional integrals on arbitrary time scales. The proposed approach unifies the treatment of inequalities across continuous and discrete domains, enabling the derivation of original forms in both settings. The obtained results exhibit symmetry with classical inequalities, and several integral and discrete inequalities arise as special cases. These findings extend and generalize known results and enrich the theory of integral inequalities in fractional and dynamic calculus, providing a versatile platform for further developments in symmetric and weighted inequality analysis.

1. Introduction

In [1], Hardy demonstrated that if $\mathrm{\Psi }\ge 0$ is an integrable function over $\left(0, \eta \right)$ and $\beta >1$, then

$${\int }_0^{\infty }{\displaystyle \frac{1}{{\eta }^{\beta }}}{\left({\int }_0^{\eta }\mathrm{\Psi }\left(\zeta \right)d\zeta \right)}^{\beta }d\eta \le {\left({\displaystyle \frac{\beta }{\beta -1}}\right)}^{\beta }{\int }_0^{\infty }{\mathrm{\Psi }}^{\beta }\left(\eta \right)d\eta ,$$

(1)

where ${(\beta /(\beta -1))}^{\beta }$ is sharp. In [2], Hardy also showed that if $\beta \ge r>1$, then

$${\int }_0^{\infty }{\displaystyle \frac{1}{{\eta }^r}}{\left({\int }_0^{\eta }\mathrm{\Psi }\left(\zeta \right)d\zeta \right)}^{\beta }d\eta \le {\left({\displaystyle \frac{\beta }{r-1}}\right)}^{\beta }{\int }_0^{\infty }{\displaystyle \frac{1}{{\eta }^{r-\beta }}}{\mathrm{\Psi }}^{\beta }\left(\eta \right)d\eta .$$

(2)

In [3], Levinson established that if $\mathrm{\Psi }\left(\eta \right)\ge 0,$ and $\psi \left(\eta \right)>0$ is absolutely continuous, then for $\beta >1,$ if there exist $\delta >0$ such that

$${\displaystyle \frac{\beta }{\beta -1}}+{\displaystyle \frac{{\psi }^{\prime }\left(\eta \right)}{\psi \left(\eta \right)}}\ge {\displaystyle \frac{1}{\delta }}>0,\forall \eta >0,$$

it follows that

$${\int }_0^{\infty }{\left({\displaystyle \frac{1}{\eta \psi \left(\eta \right)}}{\int }_0^{\eta }\psi \left(\zeta \right)\mathrm{\Psi }\left(\zeta \right)d\zeta \right)}^{\beta }d\eta \le {\delta }^{\beta }{\int }_0^{\infty }{\mathrm{\Psi }}^{\beta }\left(\eta \right)d\eta .$$

(3)

In [4], Hussan et al. established the following result: Let ${\psi }_i\left(\zeta \right)\ge 0$ be integrable over $\left(0, \infty \right),$ and $w, u_i, {\phi }_i$ be absolutely continuous functions such that ${\phi }_i^{\prime }$ is essentially bounded and positive, and $u_i$ is increasing. If the following conditions hold:

$$1+{\displaystyle \frac{u_i\left(\zeta \right)w^{\prime }\left(\zeta \right)}{(1-2r)u_i^{\prime }\left(\zeta \right)w\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\lambda }_i}}>0\mathrm{for}r>{\displaystyle \frac{1}{2}},$$

$$1+{\displaystyle \frac{u_{\mathfrak{i}}\left(\zeta \right)w^{\prime }\left(\zeta \right)}{(1-2r)u_{\mathfrak{i}}^{\prime }\left(\zeta \right)w\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\delta }_{\mathfrak{i}}}}>0\mathrm{for}r<{\displaystyle \frac{1}{2}},$$

then

$$\begin{array}{ccc}& & \sum _{i=1}^m{\int }_0^{\infty }w\left(\zeta \right){\mathrm{\Omega }}_i\left(\zeta \right){\mathrm{\Omega }}_{i+1}\left(\zeta \right)d\zeta \hfill \\ & \le & \sum _{i=1}^m{\left({\displaystyle \frac{2{\beta }_{\mathfrak{i}}}{\left|2r-1\right|}}\right)}^2{\int }_0^{\infty }w\left(\zeta \right)g_i\left(\zeta \right)d\zeta \mathrm{for}i=1, 2, \dots m, m\in \mathbb{N},\hfill \end{array}$$

(4)

where

$${\mathrm{\Omega }}_{\mathfrak{i}}\left(\zeta \right)=\left\{\begin{array}{c}{\displaystyle \frac{\sqrt{u_{\mathfrak{i}}^{\prime }\left(\zeta \right)}}{u_{\mathfrak{i}}^r\left(\zeta \right)}}{\int }_0^{\zeta }{\displaystyle \frac{u_{\mathfrak{i}}\left(s\right){\phi }_{\mathfrak{i}}^{\prime }\left(s\right)}{{\phi }_{\mathfrak{i}}\left(s\right)}}{\psi }_{\mathfrak{i}}\left(s\right)ds,r>{\displaystyle \frac{1}{2}},\\ \\ {\displaystyle \frac{\sqrt{u_{\mathfrak{i}}^{\prime }\left(\zeta \right)}}{u_{\mathfrak{i}}^r\left(\zeta \right)}}{\int }_{\zeta }^{\infty }{\displaystyle \frac{u_{\mathfrak{i}}\left(s\right){\phi }_{\mathfrak{i}}^{\prime }\left(s\right)}{{\phi }_{\mathfrak{i}}\left(s\right)}}{\psi }_{\mathfrak{i}}\left(s\right)ds,r<{\displaystyle \frac{1}{2}},\end{array}\right.$$

$$g_{\mathfrak{i}}\left(\zeta \right)={\displaystyle \frac{{\left[u_{\mathfrak{i}}\left(\zeta \right)\right]}^{4-2r}{\left[{\phi }_{\mathfrak{i}}^{\prime }\left(\zeta \right)\right]}^2{\psi }_{\mathfrak{i}}^2\left(\zeta \right)}{{\phi }_{\mathfrak{i}}^2\left(\zeta \right)u_{\mathfrak{i}}^{\prime }\left(\zeta \right)}}\mathrm{and}{\beta }_{\mathfrak{i}}=\underset{1\le \mathfrak{i}\le \tau }{max}\left({\lambda }_{\mathfrak{i}},{\delta }_{\mathfrak{i}}\right).$$

In recent years, many authors have been interested in studying dynamic inequalities on time scales $\mathbb{T}$, which is an arbitrary non-empty closed subset of $\mathbb{R}$ (see [5,6,7,8,9,10,11,12,13,14,15]). In [16], Řehak established a time scale analogue of inequality (1). For $\beta >1$, if $\mathrm{\Psi }\left(\zeta \right)\ge 0$ such that ${\int }_a^{\infty }{\mathrm{\Psi }}^{\beta }\left(\zeta \right)△\zeta <\infty$, then

$${\int }_a^{\infty }{\left({\displaystyle \frac{1}{\sigma \left(\eta \right)-a}}{\int }_a^{\sigma \left(\eta \right)}\mathrm{\Psi }\left(\zeta \right)△\zeta \right)}^{\beta }△\eta \le {\left({\displaystyle \frac{\beta }{\beta -1}}\right)}^{\beta }{\int }_a^{\infty }{\mathrm{\Psi }}^{\beta }\left(\eta \right)△\eta .$$

Furthermore, if $\mu \left(\zeta \right)/\zeta \to 0$ as $\zeta \to \infty$, then the constant ${(\beta /(\beta -1))}^{\beta }$ is sharp.

In [17], Saker et al. extended (3) to general time scales. For $\psi \left(\zeta \right)>0,$ $\mathrm{\Psi }\left(\zeta \right)\ge 0,$ and ${\psi }^{△}\left(\zeta \right)\le 0$ on ${[0, \infty )}_{\mathbb{T}}$, if $p>1$ and there exist constants $\kappa , \beta >0$ such that $\zeta /\sigma \left(\zeta \right)\ge 1/\kappa$ and

$${\displaystyle \frac{p}{p-1}}+{\displaystyle \frac{{\kappa }^p\mathrm{\Phi }\left(\zeta \right)}{{\mathrm{\Phi }}^{\sigma }\left(\zeta \right)}}{\displaystyle \frac{\zeta {\psi }^{△}\left(\zeta \right)}{{\psi }^{\sigma }\left(\zeta \right)}}\ge {\displaystyle \frac{1}{\beta }}\mathrm{for}\zeta \in {[0, \infty )}_{\mathbb{T}},$$

then

$${\int }_0^{\infty }{\displaystyle \frac{1}{{\zeta }^p}}{\left({\mathrm{\Phi }}^{\sigma }\left(\zeta \right)\right)}^p△\zeta \le {\left(\beta {\kappa }^p\right)}^p{\int }_0^{\infty }{\left({\displaystyle \frac{\psi \left(\zeta \right)\mathrm{\Psi }\left(\zeta \right)}{{\psi }^{\sigma }\left(\zeta \right)}}\right)}^p△\zeta ,$$

where

$$\mathrm{\Phi }\left(\zeta \right)={\displaystyle \frac{1}{\psi \left(\zeta \right)}}{\int }_0^{\zeta }\psi \left(t\right)\mathrm{\Psi }\left(t\right)△t\zeta \in {[0, \infty )}_{\mathbb{T}}.$$

In [18], El-Deeb et al. gave the time scale version of (4). For any $1\le \mathfrak{i}\le \tau ,$ where $\tau \ge k-1$ and $\tau ,k\in \mathbb{N}$, if there exist constants ${\lambda }_i,$ ${\delta }_i>0$ such that

$$1-{\displaystyle \frac{{\left[u_{\mathfrak{i}}^{\sigma }\left(\zeta \right)\right]}^{pr}{w}^{△}\left(\zeta \right)}{(pr-1){\left[u_{\mathfrak{i}}\left(\zeta \right)\right]}^{pr-1}{u}_{\mathfrak{i}}^{△}\left(\zeta \right)w^{\sigma }\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\lambda }_{\mathfrak{i}}}}>0\mathrm{for}r>{\displaystyle \frac{1}{2}},$$

$$1-{\displaystyle \frac{u_{\mathfrak{i}}\left(\zeta \right)w^{△}\left(\zeta \right)}{(pr-1)u_{\mathfrak{i}}^{△}\left(\zeta \right)w^{\sigma }\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\delta }_{\mathfrak{i}}}}>0\mathrm{for}r<{\displaystyle \frac{1}{2}},$$

then

$$\sum _{\mathfrak{i}=1}^{\tau }{\int }_0^{\infty }{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{\mathfrak{i}}^{{\displaystyle \frac{p}{2}}}\left(\zeta \right){\mathrm{\Omega }}_{\mathfrak{i}+1}^{{\displaystyle \frac{p}{2}}}\left(\zeta \right)△\zeta \le \sum _{\mathfrak{i}=1}^{\tau }{\left({\displaystyle \frac{p{\beta }_{\mathfrak{i}}}{\left|pr-1\right|}}\right)}^p{\int }_0^{\infty }{w}^{\sigma }\left(\zeta \right)g_{\mathfrak{i}}\left(\zeta \right)△\zeta ,$$

where

$${\mathrm{\Omega }}_{\mathfrak{i}}\left(\zeta \right)=\left\{\begin{array}{c}{\displaystyle \frac{\sqrt[p]{u_{\mathfrak{i}}^{△}\left(\zeta \right)}}{{\left[u_{\mathfrak{i}}^{\sigma }\left(\zeta \right)\right]}^r}}{\int }_0^{\sigma \left(\zeta \right)}{\displaystyle \frac{u_{\mathfrak{i}}\left(t\right){\phi }_{\mathfrak{i}}^{△}\left(t\right)}{{\phi }_{\mathfrak{i}}\left(t\right)}}{\psi }_{\mathfrak{i}}\left(t\right)△t\mathrm{and}{\int }_0^{\infty }{\displaystyle \frac{u_{\mathfrak{i}}^{△}\left(t\right)}{{\left(u_{\mathfrak{i}}^{\sigma }\left(t\right)\right)}^{pr}}}△t<\infty \mathrm{for}r>{\displaystyle \frac{1}{2}},p\ge 2,\\ \\ {\displaystyle \frac{\sqrt[p]{u_{\mathfrak{i}}^{△}\left(\zeta \right)}}{u_{\mathfrak{i}}^r\left(\zeta \right)}}{\int }_{\sigma \left(\zeta \right)}^{\infty }{\displaystyle \frac{u_{\mathfrak{i}}\left(t\right){\phi }_{\mathfrak{i}}^{△}\left(t\right)}{{\phi }_{\mathfrak{i}}\left(t\right)}}{\psi }_{\mathfrak{i}}\left(t\right)△t\mathrm{for}r<{\displaystyle \frac{1}{2}},1\le p\le 2,\end{array}\right.$$

$$g_{\mathfrak{i}}\left(\zeta \right)=\left\{\begin{array}{c}{\displaystyle \frac{u_{\mathfrak{i}}^{p\left(2-pr\right)}\left(\zeta \right){\left[u_{\mathfrak{i}}^{\sigma }\left(\zeta \right)\right]}^{rp\left(p-1\right)}{\left[{\phi }_{\mathfrak{i}}^{△}\left(\zeta \right)\right]}^p{\psi }_{\mathfrak{i}}^p\left(\zeta \right)w^p\left(\zeta \right)}{{\phi }_{\mathfrak{i}}^p\left(\zeta \right){\left(u_{\mathfrak{i}}^{△}\left(\zeta \right)\right)}^{p-1}{\left(w^{\sigma }\left(\zeta \right)\right)}^p}}\mathrm{for}r>{\displaystyle \frac{1}{2}},p\ge 2,\\ \\ {\displaystyle \frac{u_{\mathfrak{i}}^{p\left(2-r\right)}\left(\zeta \right){\left[{\phi }_{\mathfrak{i}}^{△}\left(\zeta \right)\right]}^p{\psi }_{\mathfrak{i}}^p\left(\zeta \right)w^p\left(\zeta \right)}{{\phi }_{\mathfrak{i}}^p\left(\zeta \right){\left(u_{\mathfrak{i}}^{△}\left(\zeta \right)\right)}^{p-1}{\left(w^{\sigma }\left(\zeta \right)\right)}^p}}\mathrm{for}r<{\displaystyle \frac{1}{2}},1\le p\le 2,\end{array}\right.$$

and ${\beta }_i={max}_{1\le i\le \tau }\left({\lambda }_i,{\delta }_i\right),$ $u_i(\infty )=\infty .$

In recent years, many articles have been published on the subject of fractional inequalities, which have been studied by many researchers to prove the inequalities of fractional types by using the definition of Riemann–Liouville and Caputo derivatives [19]. In [20], the authors gave a new definition of conformable fractional calculus on time scales.

Motivated by the above-mentioned work, this research article aims to establish new Hardy-type inequalities by utilizing alpha-conformable calculus on time scales. The proofs of these inequalities rely on employing the properties of differentiation and integration on time scales. They give the generalization of known inequalities as well as a few new ones for nabla-integrals through the choice of some special time scales. The novelty of this research lies in its pioneering application of alpha-conformable calculus on time scales to establish a new class of Hardy-type inequalities. To the best of our knowledge, this is the first work to explore this specific combination. Our study provides not only a direct generalization of several classical inequalities but also introduces several new ones, particularly for nabla-integrals, by judiciously selecting certain time scales. The proofs of our results are built upon the foundational properties of this novel calculus, thereby extending the existing body of knowledge in the field of dynamic inequalities.

The remainder of this article is structured as follows: Section 2 presents some preliminaries about fractional conformable calculus on time scales. Section 3 presents the main results. Section 4 provides an overview of the findings.

2. Fundamental Concepts

In this section, we introduce key concepts related to the conformable fractional derivative and integral of order $\alpha \in \left(0, 1\right]$ on a general time scale $\mathbb{T}$. Let us now proceed with an exploration of time scale calculus. For $\zeta \in \mathbb{T},$ the forward and backward jump operators $\sigma , \rho :\mathbb{T}\to \mathbb{T}$ are defined as follows: $\sigma \left(\zeta \right)=inf\{s\in \mathbb{T}:s>\zeta \}$ and $\rho \left(\zeta \right)=sup\{s\in \mathbb{T}:s<\zeta \}$, respectively. Based on these operators, a point $\zeta \in \mathbb{T}$ is called right-dense if $\sigma \left(\zeta \right)=\zeta ,$ left-dense if $\rho \left(\zeta \right)=\zeta ,$ right-scattered if $\sigma \left(\zeta \right)>\zeta$, and left-scattered if $\rho \left(\zeta \right)<\zeta .$

If $\mathbb{T}$ has left-scattered maximum L, we define ${\mathbb{T}}^k=\mathbb{T}-\left\{L\right\}.$ Otherwise, we let ${\mathbb{T}}^k=\mathbb{T}$.

Definition 1

([20]). Let $0<\alpha \le 1,\mathrm{\Psi }:\mathbb{T}\to \mathbb{R},$ and $\zeta \in {\mathbb{T}}^k.$ For $\zeta >0,$ we define $D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right)$ to be the number with the property that, given any $\epsilon >0$, there is a neighborhood Y of ζ such that for all $\zeta \in Y,$ we get

$$\left|\left[{\mathrm{\Psi }}^{\sigma }\left(\zeta \right)-\mathrm{\Psi }\left(s\right)\right]{\zeta }^{1-\alpha }-D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right)\left[\sigma \left(\zeta \right)-s\right]\right|\le \epsilon \left|\sigma \left(\zeta \right)-s\right|.$$

We say that $D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right)$ is the conformable fractional derivative of Ψ of order α at ζ.

Theorem 1

([20]). Let $\mathrm{\Psi },\mathrm{\Phi }:\mathbb{T}\to \mathbb{R}$ be conformable α-fractional derivative on $\mathbb{T}$ and continuous. Then,

(1) 

$D_{\alpha }^{△}(\mathrm{\Psi }+\mathrm{\Phi })=D_{\alpha }^{△}\left(\mathrm{\Psi }\right)+D_{\alpha }^{△}\left(\mathrm{\Phi }\right).$

(2) 

$D_{\alpha }^{△}\left(\beta \mathrm{\Psi }\right)=\beta D_{\alpha }^{△}\left(\mathrm{\Psi }\right)$ for all $\beta \in \mathbb{R}.$

(3) 

$D_{\alpha }^{△}\left(\mathrm{\Psi }\mathrm{\Phi }\right)={\mathrm{\Phi }}^{\sigma }{D}_{\alpha }^{△}\left(\mathrm{\Psi }\right)+\mathrm{\Psi }{D}_{\alpha }^{△}\left(\mathrm{\Phi }\right)=\mathrm{\Phi }{D}_{\alpha }^{△}\left(\mathrm{\Psi }\right)+{\mathrm{\Psi }}^{\sigma }{D}_{\alpha }^{△}\left(\mathrm{\Phi }\right).$

(4) 

$D_{\alpha }^{△}\left({\displaystyle \frac{\mathrm{\Psi }}{\mathrm{\Phi }}}\right)={\displaystyle \frac{\mathrm{\Phi }{D}_{\alpha }^{△}\left(\mathrm{\Psi }\right)-\mathrm{\Psi }{D}_{\alpha }^{△}\left(\mathrm{\Phi }\right)}{\mathrm{\Phi }{\mathrm{\Phi }}^{\sigma }}}$, provided that $\mathrm{\Phi }{\mathrm{\Phi }}^{\sigma }\ne 0.$

Lemma 1

(Chain rule [20]). For $0<\alpha \le 1.$ Suppose $\mathrm{\Phi }:\mathbb{T}\to \mathbb{R}$ is conformable α-fractional differentiable and $\mathrm{\Psi }:\mathbb{R}\to \mathbb{R}$ is continuously differentiable. Then,

$$D_{\alpha }^{△}(\mathrm{\Psi }\circ \mathrm{\Phi })\left(\zeta \right)={\mathrm{\Psi }}^{\prime }\left(\mathrm{\Phi }\left(c\right)\right)D_{\alpha }^{△}\left(\mathrm{\Phi }\left(\zeta \right)\right),wherec\in {\left[\zeta ,\sigma \left(\zeta \right)\right]}_{\mathbb{R}}.$$

(5)

Definition 2

([20]). For $0<\alpha \le 1$ and for a regulated function $\mathrm{\Psi }:\mathbb{T}\to \mathbb{R}.$ The conformable α-fractional integral of Ψ is defined as:

$$I_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right)=\int \mathrm{\Psi }\left(\zeta \right){△}_{\alpha }\zeta =\int \mathrm{\Psi }\left(\zeta \right){\zeta }^{\alpha -1}△\zeta .$$

The following properties are true for the conformable $\alpha$-fractional integral.

Theorem 2

([20]). For $0<\alpha \le 1,$ ${\eta }_1, {\eta }_2, c\in \mathbb{T}$ and $\beta \in \mathbb{R}$. If $G, H:\mathbb{T}\to \mathbb{R}$ are two rd-continuous functions, then

(1) 

${\int }_{{\eta }_1}^{{\eta }_2}\left[G\left(\zeta \right)+H\left(\zeta \right)\right]{△}_{\alpha }\zeta ={\int }_{{\eta }_1}^{{\eta }_2}G\left(\zeta \right){△}_{\alpha }\zeta +{\int }_{{\eta }_1}^{{\eta }_2}H\left(\zeta \right){△}_{\alpha }\zeta ,$

(2) 

${\int }_{{\eta }_1}^{{\eta }_2}\beta G\left(\zeta \right){△}_{\alpha }\zeta =\beta {\int }_{{\eta }_1}^{{\eta }_2}G\left(\zeta \right){△}_{\alpha }\zeta ,$

(3) 

${\int }_{{\eta }_1}^{{\eta }_2}G\left(\zeta \right){△}_{\alpha }\zeta =-{\int }_{{\eta }_2}^{{\eta }_1}G\left(\zeta \right){△}_{\alpha }\zeta ,$

(4) 

${\int }_{{\eta }_1}^{{\eta }_2}G\left(\zeta \right){△}_{\alpha }\zeta ={\int }_{{\eta }_1}^{c}G\left(\zeta \right){△}_{\alpha }\zeta +{\int }_c^{{\eta }_2}G\left(\zeta \right){△}_{\alpha }\zeta ,$

(5) 

${\int }_{{\eta }_1}^{{\eta }_1}G\left(\zeta \right){△}_{\alpha }\zeta =0.$

The following lemma provides the-conformable integration-by-parts formula on time scales.

Lemma 2

([20]). Let ${\eta }_1, {\eta }_2\in \mathbb{T},$ with ${\eta }_2>{\eta }_1$ and $\alpha \in (0, 1]$. If $G, H:\mathbb{T}\to \mathbb{R}$ are conformable α-fractional derivatives on $\mathbb{T}$, then

$${\int }_{{\eta }_1}^{{\eta }_2}G\left(\zeta \right)D_{\alpha }^{△}\left(H\left(\zeta \right)\right){△}_{\alpha }\zeta ={\left.G\left(\zeta \right)H\left(\zeta \right)\right|}_{{\eta }_1}^{{\eta }_2}-{\int }_{{\eta }_1}^{{\eta }_2}{H}^{\sigma }\left(\zeta \right)D_{\alpha }^{△}\left(G\left(\zeta \right)\right){△}_{\alpha }\zeta .$$

(6)

Finally, a version of Hölder’s inequality adapted to this fractional setting holds.

Lemma 3

([21]). Let ${\eta }_1,{\eta }_2\in \mathbb{T},$ with ${\eta }_2>{\eta }_1,$ $\alpha \in (0, 1],$ and $p>1$. If $G, H:\mathbb{T}\to \mathbb{R}$ are two rd-continuous functions, then

$${\int }_{{\eta }_1}^{{\eta }_2}\left|G\left(\zeta \right)H\left(\zeta \right)\right|{△}_{\alpha }\zeta \le {\left({\int }_{{\eta }_1}^{{\eta }_2}{\left|G\left(\zeta \right)\right|}^p{△}_{\alpha }\zeta \right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{{\eta }_1}^{{\eta }_2}{\left|H\left(\zeta \right)\right|}^l{△}_{\alpha }\zeta \right)}^{{\displaystyle \frac{1}{l}}},$$

where $1/p+1/l=1.$

Additionally, the following discrete inequalities will be useful in subsequent analysis.

Lemma 4

([22]). If $C_1, C_2, \dots , C_{\tau }$ are real numbers and $C_{\tau +1}=C_1$, then for $\tau \ge k-1,$

$$\sum _{r=1}^{\tau -k+2}{C}_r{C}_{r+1}\dots C_{r+k-1}\le \sum _{r=1}^{\tau }{\left(C_r\right)}^k,$$

(7)

and for any $k\ge 1,$

$${\left(\sum _{r=1}^{\tau }{C}_r\right)}^k\le {\tau }^{k-1}\sum _{r=1}^{\tau }{\left(C_r\right)}^k.$$

(8)

3. Results

In this section, we assume that any time scale $\mathbb{T}$ is unbounded above and $a, b\in \mathbb{T}$. The following assumption will be needed throughout the paper: the functions $w, u_i, {\phi }_i$ in the statements of the theorems are rd-continuous functions, increasing, and non-negative. Furthermore, let ${\psi }_i\left(\zeta \right)>0$ be an integrable function.

Theorem 3.

For any $0<\alpha \le 1,$ $i=1, 2, \dots \tau ,$ $\tau \ge k-1,$ $\tau , k\in \mathbb{N}$ and ${\int }_0^{\infty }\frac{D_{\alpha }^{△}\left(u_i\left(s\right)\right)}{{\left[u_i^{\sigma }\left(s\right)\right]}^{p\left(r-\alpha +1\right)}}{△}_{\alpha }s<\infty$. If there exist positive constants ${\lambda }_i$ and ${\delta }_i$ such that

$$1-{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{p\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left(w\left(\zeta \right)\right)}{\left(p\left(r-\alpha +1\right)-1\right){u_i}^{p\left(r-\alpha +1\right)-1}\left(\zeta \right)D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)w^{\sigma }\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\lambda }_i}},forr>{\displaystyle \frac{1}{2}},$$

$$1-{\displaystyle \frac{u_i\left(\zeta \right)D_{\alpha }^{△}\left(w\left(\zeta \right)\right)}{\left(p\left(r+\alpha -1\right)-1\right)D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)w^{\sigma }\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\delta }_i}},forr<{\displaystyle \frac{1}{2}},$$

then

$$\begin{array}{ccc}& & \sum _{i=1}^{\tau }{\int }_0^{\infty }{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_i^{{\displaystyle \frac{p}{2}}}\left(\zeta \right){\mathrm{\Omega }}_{i+1}^{{\displaystyle \frac{p}{2}}}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & \sum _{i=1}^{\tau }{\left({\displaystyle \frac{{\beta }_{i}p}{\left|\eta -1\right|}}\right)}^p{\int }_0^{\infty }{w}^{\sigma }\left(\zeta \right)g_i\left(\zeta \right){△}_{\alpha }\zeta ,\hfill \end{array}$$

(9)

where

$${\mathrm{\Omega }}_i\left(\zeta \right)=\left\{\begin{array}{c}\frac{\sqrt[p]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r-\alpha +1}}{\int }_0^{\sigma \left(\zeta \right)}\frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}{\psi }_i\left(\mu \right){△}_{\alpha }\mu ,forr>{\displaystyle \frac{1}{2}}, p\ge 2,\\ \\ \frac{\sqrt[p]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{u_i^{r+\alpha -1}\left(\zeta \right)}{\int }_{\sigma \left(\zeta \right)}^{\infty }\frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}{\psi }_i\left(\mu \right){△}_{\alpha }\mu ,forr<{\displaystyle \frac{1}{2}}, 1\le p\le 2,\end{array}\right.$$

$$g_i\left(\zeta \right)=\left\{\begin{array}{c}{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(r-\alpha +1\right)\left(p-1\right)p}{w}^p\left(\zeta \right){\left[D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)\right]}^p{\psi }_i^p\left(\zeta \right)}{u_i^{p^2\left(r-\alpha +1\right)-2p}\left(\zeta \right){\left[w^{\sigma }\left(\zeta \right)\right]}^p{\phi }_i^p\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{p-1}}},forr>{\displaystyle \frac{1}{2}}, p\ge 2,\\ \\ \frac{{\left[w\left(\zeta \right)\right]}^p{\left[D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)\right]}^p{\psi }_i^p\left(\zeta \right)}{u_i^{p\left(r-\alpha +1\right)-2p}\left(\zeta \right){\left(w^{\sigma }\left(\zeta \right)\right)}^p{\phi }_i^p\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{p-1}},forr<{\displaystyle \frac{1}{2}}, 1\le p\le 2,\end{array}\right.$$

$$\eta =\left\{\begin{array}{c}p\left(r-\alpha +1\right),forr>{\displaystyle \frac{1}{2}}, p\ge 2,\hfill \\ \\ p\left(r+\alpha -1\right),forr<{\displaystyle \frac{1}{2}}, 1\le p\le 2,\hfill \end{array}\right.$$

and ${\beta }_{\mathfrak{i}}={max}_{1\le \mathfrak{i}\le \tau }\left({\lambda }_{\mathfrak{i}},{\delta }_{\mathfrak{i}}\right),u_{\mathfrak{i}}(\infty )=\infty .$

Proof. 

Firstly, let us define for $r>{\displaystyle \frac{1}{2}}, p\ge 2, 0<a<b<\infty$ and

$${\mathrm{\Omega }}_{ia}\left(\zeta \right)={\displaystyle \frac{\sqrt[p]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r-\alpha +1}}}{\int }_a^{\sigma \left(\zeta \right)}{\displaystyle \frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu ,$$

with ${\mathrm{\Omega }}_{i0}\left(\zeta \right)={\mathrm{\Omega }}_i\left(\zeta \right).$ By using (7) with $k=2$ and $C_i={\mathrm{\Omega }}_{ia}^{{\displaystyle \frac{p}{2}}}\left(\zeta \right),$ we have

$$\sum _{i=1}^{\tau }{\mathrm{\Omega }}_{ia}^{{\displaystyle \frac{p}{2}}}\left(\zeta \right){\mathrm{\Omega }}_{\left(i+1\right)a}^{{\displaystyle \frac{p}{2}}}\left(\zeta \right)\le \sum _{i=1}^{\tau }{\mathrm{\Omega }}_{ia}^p\left(\zeta \right).$$

(10)

Multiplying (10) by $w^{\sigma }\left(\zeta \right)$ and integrating from a to b, we obtain

$$\sum _{i=1}^{\tau }{\int }_a^b{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^{{\displaystyle \frac{p}{2}}}\left(\zeta \right){\mathrm{\Omega }}_{\left(i+1\right)a}^{{\displaystyle \frac{p}{2}}}\left(\zeta \right){△}_{\alpha }\zeta \le \sum _{i=1}^{\tau }{\int }_a^b{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^p\left(\zeta \right){△}_{\alpha }\zeta .$$

(11)

Now,

$$\begin{array}{ccc}\hfill I& =& {\int }_a^b{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^p\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & =& {\int }_a^b{w}^{\sigma }\left(\zeta \right){\left[{\displaystyle \frac{\sqrt[p]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r-\alpha +1}}}{\int }_a^{\sigma \left(\zeta \right)}{\displaystyle \frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu \right]}^p{△}_{\alpha }\zeta \hfill \\ & =& {\int }_a^b{\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{p\left(r-\alpha +1\right)}}}\hfill \\ & & \times {\left[\sqrt[p]{w^{\sigma }\left(\zeta \right)}{\int }_a^{\sigma \left(\zeta \right)}{\displaystyle \frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu \right]}^p{△}_{\alpha }\zeta .\hfill \end{array}$$

(12)

Integrating (12) using (6) with

$$D_{\alpha }^{△}\left(\mathrm{\Phi }\left(\zeta \right)\right)={\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{p\left(r-\alpha +1\right)}}},$$

and

$${\mathrm{\Psi }}^{\sigma }\left(\zeta \right)={\left[\sqrt[p]{w^{\sigma }\left(\zeta \right)}{\int }_a^{\sigma \left(\zeta \right)}{\displaystyle \frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu \right]}^p,$$

we get

$$\begin{array}{ccc}& & {\int }_a^b{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^p\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & =& {\left[\mathrm{\Phi }\left(\zeta \right)\mathrm{\Psi }\left(\zeta \right)\right]}_a^b+{\int }_a^b\left(-\mathrm{\Phi }\left(\zeta \right)\right)D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right){△}_{\alpha }\zeta \hfill \\ & =& {\left[-\mathrm{\Psi }\left(\zeta \right){\int }_{\zeta }^{\infty }{\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(s\right)\right)}{{\left[u_i^{\sigma }\left(s\right)\right]}^{p\left(r-\alpha +1\right)}}}{△}_{\alpha }s\right]}_a^b\hfill \\ & & +{\int }_a^b\left(-\mathrm{\Phi }\left(\zeta \right)\right)D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right){△}_{\alpha }\zeta \hfill \\ & \le & {\int }_a^b\left(-\mathrm{\Phi }\left(\zeta \right)\right)D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right){△}_{\alpha }\zeta ,\hfill \end{array}$$

(13)

where $\mathrm{\Phi }\left(\zeta \right)=-{\int }_{\zeta }^{\infty }{\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(s\right)\right)}{{\left[u_i^{\sigma }\left(s\right)\right]}^{p\left(r-\alpha +1\right)}}}{△}_{\alpha }s.$ From (5), $D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\ge 0$ and $c\in {\left[\zeta , \sigma \left(\zeta \right)\right]}_{\mathbb{T}},$ we see that

$$\begin{array}{ccc}& & D_{\alpha }^{△}\left({u_i}^{1-p\left(r-\alpha +1\right)}\left(\zeta \right)\right)\hfill \\ & =& \left(1-p\left(r-\alpha +1\right)\right)u_i^{-p\left(r-\alpha +1\right)}\left(c\right)D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\hfill \\ & =& \left(1-p\left(r-\alpha +1\right)\right){\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}{u_i^{p\left(r-\alpha +1\right)}\left(c\right)}}\hfill \\ & \le & \left(1-p\left(r-\alpha +1\right)\right){\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{p\left(r-\alpha +1\right)}}}.\hfill \end{array}$$

(14)

Integrating (14) from $\zeta$ to ∞ with respect to $s,$ we obtain

$$-\mathrm{\Phi }\left(\zeta \right)\le {\displaystyle \frac{1}{p\left(r-\alpha +1\right)-1}}{u_i}^{1-p\left(r-\alpha +1\right)}\left(\zeta \right).$$

(15)

Substituting (15) into (13), we have

$$\begin{array}{ccc}& & {\int }_a^b{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^p\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & {\displaystyle \frac{1}{p\left(r-\alpha +1\right)-1}}{\int }_a^b{u_i}^{1-p\left(r-\alpha +1\right)}\left(\zeta \right)D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right){△}_{\alpha }\zeta .\hfill \end{array}$$

(16)

Now, we consider that $\mathrm{\Psi }\left(\zeta \right)=w\left(\zeta \right)L^p\left(\zeta \right)$ and applying Theorem 1, we get

$$\begin{array}{ccc}\hfill D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right)& =& D_{\alpha }^{△}\left(w\left(\zeta \right)\right){\left(L^{\sigma }\left(\zeta \right)\right)}^p+w\left(\zeta \right)D_{\alpha }^{△}\left(L^p\left(\zeta \right)\right)\hfill \\ & =& D_{\alpha }^{△}\left(w\left(\zeta \right)\right){\left(L^{\sigma }\left(\zeta \right)\right)}^p+w\left(\zeta \right)\left[p{L}^{p-1}\left(c\right)D_{\alpha }^{△}\left(L\left(\zeta \right)\right)\right]\hfill \\ & \le & D_{\alpha }^{△}\left(w\left(\zeta \right)\right){\left(L^{\sigma }\left(\zeta \right)\right)}^p\hfill \\ & & +w\left(\zeta \right)\left[p{\left(L^{\sigma }\left(\zeta \right)\right)}^{p-1}{\displaystyle \frac{u_i\left(\zeta \right)D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)}{{\phi }_i\left(\zeta \right)}}{\psi }_i\left(\zeta \right)\right],\hfill \end{array}$$

(17)

where $L\left(\zeta \right)={\int }_a^{\zeta }{\displaystyle \frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu .$ From (16) and (17), we have

$$\begin{array}{ccc}& & {\int }_a^b{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^p\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & {\displaystyle \frac{1}{p\left(r-\alpha +1\right)-1}}{\int }_a^b{u_i}^{1-p\left(r-\alpha +1\right)}\left(\zeta \right)D_{\alpha }^{△}\left(w\left(\zeta \right)\right){\left(L^{\sigma }\left(\zeta \right)\right)}^p{△}_{\alpha }\zeta \hfill \\ & & +{\displaystyle \frac{p}{p\left(r-\alpha +1\right)-1}}{\int }_a^b{u_i}^{1-p\left(r-\alpha +1\right)}\left(\zeta \right)w\left(\zeta \right)\hfill \\ & & \times \left({\left(L^{\sigma }\left(\zeta \right)\right)}^{p-1}{\displaystyle \frac{u_i\left(\zeta \right)D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)}{{\phi }_i\left(\zeta \right)}}{\psi }_i\left(\zeta \right)\right){△}_{\alpha }\zeta \hfill \\ & =& {\displaystyle \frac{1}{p\left(r-\alpha +1\right)-1}}{\int }_a^b{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{p\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left(w\left(\zeta \right)\right)}{{u_i}^{p\left(r-\alpha +1\right)-1}\left(\zeta \right)D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{\mathrm{\Omega }}_{ia}^p\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & & +{\displaystyle \frac{p}{p\left(r-\alpha +1\right)-1}}{\int }_a^b{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(r-\alpha +1\right)\left(p-1\right)}w\left(\zeta \right)D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right){\psi }_i\left(\zeta \right)}{{u_i}^{p\left(r-\alpha +1\right)-2}\left(\zeta \right){\phi }_i\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{p}}}}{\mathrm{\Omega }}_{ia}^{p-1}\left(\zeta \right){△}_{\alpha }\zeta .\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}& & {\int }_a^b{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^p\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & {\displaystyle \frac{{\lambda }_{i}p}{p\left(r-\alpha +1\right)-1}}\hfill \\ & & \times {\int }_a^b{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(r-\alpha +1\right)\left(p-1\right)}w\left(\zeta \right)D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right){\psi }_i\left(\zeta \right)}{{u_i}^{p\left(r-\alpha +1\right)-2}\left(\zeta \right){\phi }_i\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{p}}}}{\mathrm{\Omega }}_{ia}^{p-1}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & =& {\displaystyle \frac{{\lambda }_{i}p}{p\left(r-\alpha +1\right)-1}}\hfill \\ & & \times {\int }_a^b{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(r-\alpha +1\right)\left(p-1\right)}w\left(\zeta \right)D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right){\psi }_i\left(\zeta \right)}{{u_i}^{p\left(r-\alpha +1\right)-2}\left(\zeta \right){\left[w^{\sigma }\left(\zeta \right)\right]}^{\frac{p-1}{p}}{\phi }_i\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{p}}}}{\left[w^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^p\left(\zeta \right)\right]}^{{\displaystyle \frac{p-1}{p}}}{△}_{\alpha }\zeta .\hfill \end{array}$$

By Hölder’s inequality with p and $p/(p-1),$ we get

$$\begin{array}{ccc}& & {\int }_a^b{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^p\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & {\left({\displaystyle \frac{{\lambda }_{i}p}{p\left(r-\alpha +1\right)-1}}\right)}^p\hfill \\ & & \times {\int }_a^b{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(r-\alpha +1\right)\left(p-1\right)p}{w}^p\left(\zeta \right){\left[D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)\right]}^p{\psi }_i^p\left(\zeta \right)}{{u_i}^{p^2\left(r-\alpha +1\right)-2p}\left(\zeta \right){\left[w^{\sigma }\left(\zeta \right)\right]}^{p-1}{\phi }_i^p\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{p-1}}}{△}_{\alpha }\zeta .\hfill \end{array}$$

(18)

By taking $a\to 0, b\to \infty$ and from (18) and (11), we get

$$\begin{array}{ccc}& & \sum _{i=1}^{\tau }{\int }_0^{\infty }{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_i^{{\displaystyle \frac{p}{2}}}\left(\zeta \right){\mathrm{\Omega }}_{i+1}^{{\displaystyle \frac{p}{2}}}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & \sum _{i=1}^{\tau }{\left({\displaystyle \frac{p{\beta }_i}{p\left(r-\alpha +1\right)-1}}\right)}^p{\int }_0^{\infty }{w}^{\sigma }\left(\zeta \right)g_i\left(\zeta \right){△}_{\alpha }\zeta .\hfill \end{array}$$

(19)

Secondly, let us define for $r<{\displaystyle \frac{1}{2}}, 1\le p\le 2, 0<a<b<\infty$ and

$${\mathrm{\Omega }}_{\mathfrak{i}b}\left(\zeta \right)=\frac{\sqrt[p]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r+\alpha -1}}{\int }_{\sigma \left(\zeta \right)}^b\frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}{\psi }_i\left(\mu \right){△}_{\alpha }\mu ,1\le \mathfrak{i}\le \tau ,$$

with ${\mathrm{\Omega }}_{\mathfrak{i}\infty }\left(\zeta \right)={\mathrm{\Omega }}_{\mathfrak{i}}\left(\zeta \right).$ Following the same steps as in the proof of (19), we obtain

$$\begin{array}{ccc}& & \sum _{\mathfrak{i}=1}^{\tau }{\int }_0^{\infty }{\widehat{w}}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{\mathfrak{i}}^{{\displaystyle \frac{\alpha }{2}}}\left(\zeta \right){\mathrm{\Omega }}_{\mathfrak{i}+1}^{{\displaystyle \frac{\alpha }{2}}}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & \sum _{\mathfrak{i}=1}^{\tau }{\left({\displaystyle \frac{\alpha {\beta }_{\mathfrak{i}}}{1-p\left(r+\alpha -1\right)}}\right)}^{\alpha }{\int }_0^{\infty }{\widehat{w}}^{\sigma }\left(\zeta \right)g_{\mathfrak{i}}\left(\zeta \right){△}_{\alpha }\zeta .\hfill \end{array}$$

(20)

Inequalities (19) and (20) are equivalent to (9). □

Remark 1.

If we put $\alpha =1$ in Theorem 3, then we have ([18], Theorem 6).

Corollary 1.

Let $\mathbb{T}=\mathbb{R}$ in Theorem 3. For any $0<\alpha \le 1, i=1, 2, \dots \tau , \tau \ge k-1, \tau , k\in \mathbb{N}$ and ${\int }_0^{\infty }\frac{u_i^{\prime }\left(s\right)}{{\left[u_i\left(s\right)\right]}^{p\left(r-\alpha +1\right)}}ds<\infty$. If there exist positive constants ${\lambda }_i$ and ${\delta }_i$ such that

$$1-{\displaystyle \frac{u_i\left(\zeta \right)w^{\prime }\left(\zeta \right)}{\left(p\left(r-\alpha +1\right)-1\right)u_i^{\prime }\left(\zeta \right)w\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\lambda }_i}},forr>{\displaystyle \frac{1}{2}},$$

$$1-{\displaystyle \frac{u_i\left(\zeta \right)w^{\prime }\left(\zeta \right)}{\left(p\left(r+\alpha -1\right)-1\right)u_i^{\prime }\left(\zeta \right)w\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\delta }_i}},forr<{\displaystyle \frac{1}{2}},$$

then

$$\begin{array}{ccc}& & \sum _{i=1}^{\tau }{\int }_0^{\infty }w\left(\zeta \right){\mathrm{\Omega }}_i^{{\displaystyle \frac{p}{2}}}\left(\zeta \right){\mathrm{\Omega }}_{i+1}^{{\displaystyle \frac{p}{2}}}\left(\zeta \right){\zeta }^{\alpha -1}d\zeta \hfill \\ & \le & \sum _{i=1}^{\tau }{\left({\displaystyle \frac{{\beta }_{i}p}{\left|\eta -1\right|}}\right)}^p{\int }_0^{\infty }w\left(\zeta \right)g_i\left(\zeta \right){\zeta }^{\alpha -1}d\zeta ,\hfill \end{array}$$

(21)

where

$${\mathrm{\Omega }}_i\left(\zeta \right)=\left\{\begin{array}{c}\frac{\sqrt[p]{{\zeta }^{1-\alpha }{u}_i^{\prime }\left(\zeta \right)}}{{\left[u_i\left(\zeta \right)\right]}^{r-\alpha +1}}{\int }_0^{\sigma \left(\zeta \right)}\frac{u_i\left(\mu \right){\phi }_i^{\prime }\left(\mu \right)}{{\phi }_i\left(\mu \right)}{\psi }_i\left(\mu \right)d\mu ,forr>{\displaystyle \frac{1}{2}}, p\ge 2,\\ \\ \frac{\sqrt[p]{{\zeta }^{1-\alpha }{u}_i^{\prime }\left(\zeta \right)}}{{\left[u_i\left(\zeta \right)\right]}^{r+\alpha -1}}{\int }_{\sigma \left(\zeta \right)}^{\infty }\frac{u_i\left(\mu \right){\phi }_i^{\prime }\left(\mu \right)}{{\phi }_i\left(\mu \right)}{\psi }_i\left(\mu \right)d\mu ,forr<{\displaystyle \frac{1}{2}}, 1\le p\le 2,\end{array}\right.$$

$$g_i\left(\zeta \right)=\left\{\begin{array}{c}{\displaystyle \frac{u_i^{\left(-r+\alpha +1\right)p}\left(\zeta \right){\zeta }^{1-\alpha }{\left[{\phi }_i^{\prime }\left(\zeta \right)\right]}^p{\psi }_i^p\left(\zeta \right)}{{\phi }_i^p\left(\zeta \right){\left(u_i^{\prime }\left(\zeta \right)\right)}^{p-1}}},forr>{\displaystyle \frac{1}{2}}, p\ge 2,\\ \\ \frac{{\zeta }^{1-\alpha }{\left[{\phi }_i^{\prime }\left(\zeta \right)\right]}^p{\psi }_i^p\left(\zeta \right)}{u_i^{p\left(r-\alpha +1\right)-2p}\left(\zeta \right){\phi }_i^p\left(\zeta \right){\left[u_i^{\prime }\left(\zeta \right)\right]}^{p-1}},forr<{\displaystyle \frac{1}{2}}, 1\le p\le 2,\end{array}\right.$$

$$\eta =\left\{\begin{array}{c}p\left(r-\alpha +1\right),forr>{\displaystyle \frac{1}{2}}, p\ge 2,\hfill \\ \\ p\left(r+\alpha -1\right),forr<{\displaystyle \frac{1}{2}}, 1\le p\le 2,\hfill \end{array}\right.$$

and ${\beta }_{\mathfrak{i}}={max}_{1\le \mathfrak{i}\le \tau }\left({\lambda }_{\mathfrak{i}},{\delta }_{\mathfrak{i}}\right),u_{\mathfrak{i}}(\infty )=\infty .$

Remark 2.

If we put $p=2$ and $\alpha =1$ in Corollary 1, then (21) reduces to (4).

Corollary 2.

Let $\mathbb{T}=\mathbb{Z}$ in Theorem 3. For any $0<\alpha \le 1, i=1, 2, \dots \tau , \tau \ge k-1$ and $\tau , k\in \mathbb{N}$. If there exist positive constants ${\lambda }_i$ and ${\delta }_i$ such that

$$1-{\displaystyle \frac{{\left[u_i\left(\zeta +1\right)\right]}^{p\left(r-\alpha +1\right)}△w\left(\zeta \right)}{\left(p\left(r-\alpha +1\right)-1\right){u_i}^{p\left(r-\alpha +1\right)-1}\left(\zeta \right)w(\zeta +1)△u_i\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\lambda }_i}},forr>{\displaystyle \frac{1}{2}},$$

$$1-{\displaystyle \frac{u_i\left(\zeta \right)△w\left(\zeta \right)}{\left(p\left(r-\alpha +1\right)-1\right)w(\zeta +1)△u_i\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\delta }_i}},forr<{\displaystyle \frac{1}{2}},$$

then

$$\begin{array}{ccc}& & \sum _{i=1}^{\tau }\sum _{\zeta =0}^{\infty }w\left(\zeta +1\right){\mathrm{\Omega }}_i^{{\displaystyle \frac{p}{2}}}\left(\zeta \right){\mathrm{\Omega }}_{i+1}^{{\displaystyle \frac{p}{2}}}\left(\zeta \right){\zeta }^{\alpha -1}\hfill \\ & \le & \sum _{i=1}^{\tau }{\left({\displaystyle \frac{{\beta }_{i}p}{\left|\eta -1\right|}}\right)}^p\sum _{\zeta =0}^{\infty }w(\zeta +1)g_i\left(\zeta \right){\zeta }^{\alpha -1},\hfill \end{array}$$

where

$${\mathrm{\Omega }}_i\left(\zeta \right)=\left\{\begin{array}{c}{\displaystyle \frac{\sqrt[p]{{\zeta }^{1-\alpha }△u_i\left(\zeta \right)}}{u_i^{r-\alpha +1}\left(\zeta +1\right)}}\sum _{\mu =0}^{\zeta }{\displaystyle \frac{u_i\left(\mu \right)△{\phi }_i\left(\mu \right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right),forr>{\displaystyle \frac{1}{2}}, p\ge 2,\\ \\ {\displaystyle \frac{\sqrt[p]{{\zeta }^{1-\alpha }△u_i\left(\zeta \right)}}{u_i^{r-\alpha +1}\left(\zeta \right)}}\sum _{\mu =\zeta +1}^{\infty }{\displaystyle \frac{u_i\left(\mu \right)△{\phi }_i\left(\mu \right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right),forr<{\displaystyle \frac{1}{2}}, 1\le p\le 2,\end{array}\right.$$

$$g_i\left(\zeta \right)=\left\{\begin{array}{c}{\displaystyle \frac{u_i^{\left(r-\alpha +1\right)\left(p-1\right)p}\left(\zeta +1\right)w^p\left(\zeta \right){\zeta }^{1-\alpha }{\left[△{\phi }_i\left(\zeta \right)\right]}^p{\psi }_i^p\left(\zeta \right)}{{u_i}^{p^2\left(r-\alpha +1\right)-2p}\left(\zeta \right)w^p(\zeta +1){\phi }_i^p\left(\zeta \right){\left[△u_i\left(\zeta \right)\right]}^{p-1}}},forr>{\displaystyle \frac{1}{2}}, p\ge 2,\\ \\ \frac{w^p\left(\zeta \right){\zeta }^{1-\alpha }{\left[△{\phi }_i\left(\zeta \right)\right]}^p{\psi }_i^p\left(\zeta \right)}{u_i^{p\left(r-\alpha +1\right)-2p}\left(\zeta \right)w^p(\zeta +1){\phi }_i^p\left(\zeta \right){\left[△u_i\left(\zeta \right)\right]}^{p-1}},forr<{\displaystyle \frac{1}{2}}, 1\le p\le 2,\end{array}\right.$$

$$\eta =\left\{\begin{array}{c}p\left(r-\alpha +1\right),forr>{\displaystyle \frac{1}{2}}, p\ge 2,\\ \\ p\left(r+\alpha -1\right),forr<{\displaystyle \frac{1}{2}}, 1\le p\le 2,\end{array}\right.$$

and ${\beta }_{\mathfrak{i}}={max}_{1\le \mathfrak{i}\le \tau }\left({\lambda }_{\mathfrak{i}},{\delta }_{\mathfrak{i}}\right),u_{\mathfrak{i}}(\infty )=\infty .$

Theorem 4.

For any $0<\alpha \le 1, i=1, 2, \dots \tau , \tau \ge k-1, \tau \in \mathbb{N}, p_i>1$ and ${\int }_0^{\infty }{\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(s\right)\right)}{{\left[u_i^{\sigma }\left(s\right)\right]}^{k{p}_i\left(r-\alpha +1\right)}}}{△}_{\alpha }s<\infty$. If there exist positive constants ${\lambda }_i$ and ${\delta }_i$ such that

$$1-{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{k{p}_i\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left(w\left(\zeta \right)\right)}{\left(k{p}_i\left(r-\alpha +1\right)-1\right){u_i}^{k{p}_i\left(r-\alpha +1\right)-1}\left(\zeta \right)D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)w^{\sigma }\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\lambda }_i}},forr>{\displaystyle \frac{1}{k{p}_i}}$$

$$1-{\displaystyle \frac{u_i\left(\zeta \right)D_{\alpha }^{△}\left(w\left(\zeta \right)\right)}{\left(k{p}_i\left(r+\alpha -1\right)-1\right)D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)w^{\sigma }\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\delta }_i}},forr<{\displaystyle \frac{1}{k{p}_i}},$$

then

$$\begin{array}{ccc}& & \sum _{i=1}^{\tau -k+2}{\int }_0^{\infty }{w}^{\sigma }\left(\zeta \right)\left(\prod _{j=i}^{i+k-1}{\mathrm{\Omega }}_j^{p_j}\left(\zeta \right)\right){△}_{\alpha }\zeta \hfill \\ & \le & \sum _{i=1}^{\tau }{\left({\displaystyle \frac{{\beta }_{i}k{p}_i}{\left|{\eta }_i-1\right|}}\right)}^{k{p}_i}{\int }_0^{\infty }{w}^{\sigma }\left(\zeta \right)g_i\left(\zeta \right){△}_{\alpha }\zeta ,\hfill \end{array}$$

(22)

where

$${\mathrm{\Omega }}_i\left(\zeta \right)=\left\{\begin{array}{c}\frac{\sqrt[k{p}_i]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r-\alpha +1}}{\int }_0^{\sigma \left(\zeta \right)}\frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}{\psi }_i\left(\mu \right){△}_{\alpha }\mu ,forr>\frac{1}{k{p}_i},\\ \\ {\displaystyle \frac{\sqrt[k{p}_i]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{u_i^{r+\alpha -1}\left(\zeta \right)}}{\int }_{\sigma \left(\zeta \right)}^{\infty }{\displaystyle \frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu ,forr<\frac{1}{k{p}_i},\end{array}\right.$$

$$g_i\left(\zeta \right)=\left\{\begin{array}{c}\frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(r-\alpha +1\right)\left(k{p}_i-1\right)k{p}_i}{w}^{k{p}_i}\left(\zeta \right){\left[D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)\right]}^{k{p}_i}{\psi }_i^{k{p}_i}\left(\zeta \right)}{{u_i}^{{\left(k{p}_i\right)}^2\left(r-\alpha +1\right)-2k{p}_i}\left(\zeta \right){\left[w^{\sigma }\left(\zeta \right)\right]}^{k{p}_i}{\phi }_i^{k{p}_i}\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{k{p}_i-1}},forr>\frac{1}{k{p}_i},\\ \\ {\displaystyle \frac{{\left[u_i\left(\zeta \right)\right]}^{k{p}_i\left(2-\left(r+\alpha -1\right)\right)}{w}^{k{p}_i}\left(\zeta \right){\left[D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)\right]}^{k{p}_i}{\psi }_i^{k{p}_i}\left(\zeta \right)}{{\left[w^{\sigma }\left(\zeta \right)\right]}^{k{p}_i}{\phi }_i^{k{p}_i}\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{k{p}_i-1}}},forr<\frac{1}{k{p}_i},\end{array}\right.$$

$${\eta }_i=\left\{\begin{array}{c}k{p}_i\left(r-\alpha +1\right),forr>{\displaystyle \frac{1}{2}}, p\ge 2,\hfill \\ \\ k{p}_i\left(r+\alpha -1\right),forr<{\displaystyle \frac{1}{2}}, 1\le p\le 2,\hfill \end{array}\right.$$

and ${\beta }_{\mathfrak{i}}={max}_{1\le \mathfrak{i}\le \tau }\left({\lambda }_{\mathfrak{i}},{\delta }_{\mathfrak{i}}\right),u_{\mathfrak{i}}(\infty )=\infty .$

Proof. 

Let $r>{\displaystyle \frac{1}{k{p}_i}}, p_i>1$ and

$${\mathrm{\Omega }}_{ia}\left(\zeta \right)={\displaystyle \frac{\sqrt[k{p}_i]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r-\alpha +1}}}{\int }_a^{\sigma \left(\zeta \right)}{\displaystyle \frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu ,$$

with ${\mathrm{\Omega }}_{i0}\left(\zeta \right)={\mathrm{\Omega }}_i\left(\zeta \right).$ By using (7) with $C_i={\mathrm{\Omega }}_{ia}^{p_i}\left(\zeta \right),$ we have

$$\sum _{i=1}^{\tau -k+2}{\mathrm{\Omega }}_{ia}^{p_i}\left(\zeta \right){\mathrm{\Omega }}_{\left(i+1\right)a}^{p_{i+1}}\left(\zeta \right)\dots {\mathrm{\Omega }}_{\left(i+k-1\right)a}^{p_{i+k-1}}\left(\zeta \right)\le \sum _{i=1}^{\tau }{\mathrm{\Omega }}_{ia}^{k{p}_i}\left(\zeta \right).$$

(23)

Multiplying (23) by $w^{\sigma }\left(\zeta \right)$ and integrating from a to b, we obtain

$$\begin{array}{ccc}& & \sum _{i=1}^{\tau -k+2}{\int }_a^b{w}^{\sigma }\left(\zeta \right)\left(\prod _{j=i}^{i+k-1}{\mathrm{\Omega }}_{ja}^{p_j}\left(\zeta \right)\right){△}_{\alpha }\zeta \hfill \\ & \le & \sum _{i=1}^{\tau }{\int }_a^b{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^{k{p}_i}\left(\zeta \right){△}_{\alpha }\zeta .\hfill \end{array}$$

(24)

Now,

$$\begin{array}{ccc}\hfill I& =& {\int }_a^b{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^{k{p}_i}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & =& {\int }_a^b{w}^{\sigma }\left(\zeta \right){\left[{\displaystyle \frac{\sqrt[k{p}_i]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r-\alpha +1}}}{\int }_a^{\sigma \left(\zeta \right)}{\displaystyle \frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu \right]}^{k{p}_i}{△}_{\alpha }\zeta \hfill \\ & =& {\int }_a^b{\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{k{p}_i\left(r-\alpha +1\right)}}}\hfill \\ & & \times {\left[\sqrt[k{p}_i]{w^{\sigma }\left(\zeta \right)}{\int }_a^{\sigma \left(\zeta \right)}{\displaystyle \frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu \right]}^{k{p}_i}{△}_{\alpha }\zeta .\hfill \end{array}$$

(25)

Integrating (25) by using (6) with

$$D_{\alpha }^{△}\left(\mathrm{\Phi }\left(\zeta \right)\right)={\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{k{p}_i\left(r-\alpha +1\right)}}},$$

and

$${\mathrm{\Psi }}^{\sigma }\left(\zeta \right)={\left[\sqrt[k{p}_i]{w^{\sigma }\left(\zeta \right)}{\int }_a^{\sigma \left(\zeta \right)}\frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}{\psi }_i\left(\mu \right){△}_{\alpha }\mu \right]}^{k{p}_i},$$

we get

$$\begin{array}{ccc}& & {\int }_a^b{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^{k{p}_i}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & =& {\left[\mathrm{\Phi }\left(\zeta \right)\mathrm{\Psi }\left(\zeta \right)\right]}_a^b+{\int }_a^b\left(-\mathrm{\Phi }\left(\zeta \right)\right)D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right){△}_{\alpha }\zeta \hfill \\ & =& {\left[-\mathrm{\Psi }\left(\zeta \right){\int }_{\zeta }^{\infty }{\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(s\right)\right)}{{\left[u_i^{\sigma }\left(s\right)\right]}^{k{p}_i\left(r-\alpha +1\right)}}}{△}_{\alpha }s\right]}_a^b+{\int }_a^b\left(-\mathrm{\Phi }\left(\zeta \right)\right)D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right){△}_{\alpha }\zeta \hfill \\ & \le & {\int }_a^b\left(-\mathrm{\Phi }\left(\zeta \right)\right)D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right){△}_{\alpha }\zeta ,\hfill \end{array}$$

(26)

where $\mathrm{\Phi }\left(\zeta \right)=-{\int }_{\zeta }^{\infty }{\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(s\right)\right)}{{\left[u_i^{\sigma }\left(s\right)\right]}^{k{p}_i\left(r-\alpha +1\right)}}}{△}_{\alpha }s.$ From (5), $D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\ge 0$ and $c\in {\left[\zeta , \sigma \left(\zeta \right)\right]}_{\mathbb{T}},$ we see that

$$\begin{array}{ccc}& & D_{\alpha }^{△}\left({u_i}^{1-k{p}_i\left(r-\alpha +1\right)}\left(\zeta \right)\right)\hfill \\ & =& \left(1-k{p}_i\left(r-\alpha +1\right)\right)u_i^{-k{p}_i\left(r-\alpha +1\right)}\left(c\right)D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\hfill \\ & =& \left(1-k{p}_i\left(r-\alpha +1\right)\right){\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}{u_i^{k{p}_i\left(r-\alpha +1\right)}\left(c\right)}}\hfill \\ & \le & \left(1-k{p}_i\left(r-\alpha +1\right)\right){\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{k{p}_i\left(r-\alpha +1\right)}}}.\hfill \end{array}$$

(27)

Integrating (27) from $\zeta$ to ∞ with respect to $s,$ to obtain

$$-\mathrm{\Phi }\left(\zeta \right)\le {\displaystyle \frac{1}{k{p}_i\left(r-\alpha +1\right)-1}}{u_i}^{1-k{p}_i\left(r-\alpha +1\right)}\left(\zeta \right).$$

(28)

Substituting (28) into (26), we have

$$\begin{array}{ccc}& & {\int }_a^b{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^{k{p}_i}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & {\displaystyle \frac{1}{k{p}_i\left(r-\alpha +1\right)-1}}{\int }_a^b{u_i}^{1-k{p}_i\left(r-\alpha +1\right)}\left(\zeta \right)D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right){△}_{\alpha }\zeta .\hfill \end{array}$$

(29)

Now, we consider that $\mathrm{\Psi }\left(\zeta \right)=w\left(\zeta \right)L^{k{p}_i}\left(\zeta \right)$ and applying Theorem 1, to see that

$$\begin{array}{ccc}\hfill D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right)& =& D_{\alpha }^{△}\left(w\left(\zeta \right)\right){\left(L^{\sigma }\left(\zeta \right)\right)}^{k{p}_i}+w\left(\zeta \right)D_{\alpha }^{△}\left(L^{k{p}_i}\left(\zeta \right)\right)\hfill \\ & =& D_{\alpha }^{△}\left(w\left(\zeta \right)\right){\left(L^{\sigma }\left(\zeta \right)\right)}^{k{p}_i}+w\left(\zeta \right)\left[k{p}_i{L}^{k{p}_i-1}\left(c\right)D_{\alpha }^{△}\left(L\left(\zeta \right)\right)\right]\hfill \\ & \le & D_{\alpha }^{△}\left(w\left(\zeta \right)\right){\left(L^{\sigma }\left(\zeta \right)\right)}^{k{p}_i}\hfill \\ & & +w\left(\zeta \right)\left[k{p}_i{\left(L^{\sigma }\left(\zeta \right)\right)}^{k{p}_i-1}{\displaystyle \frac{u_i\left(\zeta \right)D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)}{{\phi }_i\left(\zeta \right)}}{\psi }_i\left(\zeta \right)\right],\hfill \end{array}$$

(30)

where $L\left(\zeta \right)={\int }_a^{\zeta }{\displaystyle \frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu .$ From (29) and (30), we have

$$\begin{array}{ccc}& & {\int }_a^b{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^{k{p}_i}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & {\displaystyle \frac{1}{k{p}_i\left(r-\alpha +1\right)-1}}{\int }_a^b{u_i}^{1-k{p}_i\left(r-\alpha +1\right)}\left(\zeta \right)D_{\alpha }^{△}\left(w\left(\zeta \right)\right){\left(L^{\sigma }\left(\zeta \right)\right)}^{k{p}_i}{△}_{\alpha }\zeta \hfill \\ & & +{\displaystyle \frac{k{p}_i}{k{p}_i\left(r-\alpha +1\right)-1}}{\int }_a^b{\displaystyle \frac{w\left(\zeta \right)D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right){\psi }_i\left(\zeta \right)}{{u_i}^{k{p}_i\left(r-\alpha +1\right)-2}\left(\zeta \right){\phi }_i\left(\zeta \right)}}{\left(L^{\sigma }\left(\zeta \right)\right)}^{k{p}_i-1}{△}_{\alpha }\zeta \hfill \\ & =& {\displaystyle \frac{1}{k{p}_i\left(r-\alpha +1\right)-1}}{\int }_a^b{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{k{p}_i\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left(w\left(\zeta \right)\right)}{{u_i}^{k{p}_i\left(r-\alpha +1\right)-1}\left(\zeta \right)D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{\mathrm{\Omega }}_{ia}^{k{p}_i}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & & +{\displaystyle \frac{k{p}_i}{k{p}_i\left(r-\alpha +1\right)-1}}\hfill \\ & & \times {\int }_a^b{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(r-\alpha +1\right)\left(k{p}_i-1\right)}w\left(\zeta \right)D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right){\psi }_i\left(\zeta \right)}{{u_i}^{k{p}_i\left(r-\alpha +1\right)-2}\left(\zeta \right){\phi }_i\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{k{p}_i}}}}{\mathrm{\Omega }}_{ia}^{k{p}_i-1}\left(\zeta \right){△}_{\alpha }\zeta .\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}& & {\int }_a^b{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^{k{p}_i}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & {\displaystyle \frac{{\lambda }_{i}k{p}_i}{k{p}_i\left(r-\alpha +1\right)-1}}\hfill \\ & & \times {\int }_a^b{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(r-\alpha +1\right)\left(k{p}_i-1\right)}w\left(\zeta \right)D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right){\psi }_i\left(\zeta \right)}{{u_i}^{k{p}_i\left(r-\alpha +1\right)-2}\left(\zeta \right){\phi }_i\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{k{p}_i}}}}{\mathrm{\Omega }}_{ia}^{k{p}_i-1}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & =& {\displaystyle \frac{{\lambda }_{i}k{p}_i}{k{p}_i\left(r-\alpha +1\right)-1}}\hfill \\ & & \times {\int }_a^b{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(r-\alpha +1\right)\left(k{p}_i-1\right)}w\left(\zeta \right)D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right){\psi }_i\left(\zeta \right)}{{u_i}^{k{p}_i\left(r-\alpha +1\right)-2}\left(\zeta \right){\left[w^{\sigma }\left(\zeta \right)\right]}^{\frac{k{p}_i-1}{k{p}_i}}{\phi }_i\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{k{p}_i}}}}{\left(w^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^{k{p}_i}\left(\zeta \right)\right)}^{{\displaystyle \frac{k{p}_i-1}{k{p}_i}}}{△}_{\alpha }\zeta .\hfill \end{array}$$

By Hölder’s inequality with $k{p}_i$ and $k{p}_i/(k{p}_i-1),$ we get

$$\begin{array}{ccc}& & {\int }_a^b{w}^{\sigma }\left(\zeta \right){\mathrm{\Omega }}_{ia}^{k{p}_i}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & {\left({\displaystyle \frac{k{p}_i}{k{p}_i\left(r-\alpha +1\right)-1}}\right)}^{k{p}_i}\hfill \\ & & \times {\int }_a^b\frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(r-\alpha +1\right)\left(k{p}_i-1\right)k{p}_i}{w}^{k{p}_i}\left(\zeta \right){\left[D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)\right]}^{k{p}_i}{\psi }_i^{k{p}_i}\left(\zeta \right)}{{u_i}^{{\left(k{p}_i\right)}^2\left(r-\alpha +1\right)-2k{p}_i}\left(\zeta \right){\left[w^{\sigma }\left(\zeta \right)\right]}^{k{p}_i-1}{\phi }_i^{k{p}_i}\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{k{p}_i-1}}{△}_{\alpha }\zeta .\hfill \end{array}$$

(31)

By taking $a\to 0, b\to \infty$ and from (24), (31), we get

$$\begin{array}{ccc}& & \sum _{i=1}^{\tau -k+2}{\int }_0^{\infty }{w}^{\sigma }\left(\zeta \right)\left(\prod _{j=i}^{i+k-1}{\mathrm{\Omega }}_j^{p_j}\left(\zeta \right)\right){△}_{\alpha }\zeta \hfill \\ & \le & \sum _{i=1}^{\tau }{\left({\displaystyle \frac{{\lambda }_{i}k{p}_i}{k{p}_i\left(r-\alpha +1\right)-1}}\right)}^{k{p}_i}{\int }_0^{\infty }{w}^{\sigma }\left(\zeta \right)g_i\left(\zeta \right){△}_{\alpha }\zeta .\hfill \end{array}$$

(32)

Let $r<{\displaystyle \frac{1}{k{p}_i}}, p_i>1$ and

$${\mathrm{\Omega }}_{\mathfrak{i}b}\left(\zeta \right)={\displaystyle \frac{\sqrt[k{p}_i]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{u_i^{r+\alpha -1}\left(\zeta \right)}}{\int }_{\sigma \left(\zeta \right)}^b{\displaystyle \frac{u_i\left(\mu \right)D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu ,1\le \mathfrak{i}\le \tau ,$$

with ${\mathrm{\Omega }}_{\mathfrak{i}\infty }\left(\zeta \right)={\mathrm{\Omega }}_{\mathfrak{i}}\left(\zeta \right).$ Following the same steps as in the proof of (32), we obtain

$$\begin{array}{ccc}& & \sum _{i=1}^{\tau -k+2}{\int }_0^{\infty }{w}^{\sigma }\left(\zeta \right)\left(\prod _{j=i}^{i+k-1}{\mathrm{\Omega }}_j^{p_j}\left(\zeta \right)\right){△}_{\alpha }\zeta \hfill \\ & \le & \sum _{i=1}^{\tau }{\left({\displaystyle \frac{{\delta }_{i}k{p}_i}{1-k{p}_i\left(r+\alpha -1\right)}}\right)}^{k{p}_i}{\int }_0^{\infty }{w}^{\sigma }\left(\zeta \right)g_i\left(\zeta \right){△}_{\alpha }\zeta .\hfill \end{array}$$

(33)

Inequalities (33) and (32) are equivalent to (22). □

Remark 3.

If we put $\alpha =1$ in Theorem 4, then we have ([18], Theorem 11).

Corollary 3.

Let $\mathbb{T}=\mathbb{R}$ in Theorem 4. For any $0<\alpha \le 1, i=1, 2, \dots \tau , \tau \ge k-1, \tau \in \mathbb{N}, p_i>1$ and ${\int }_0^{\infty }\frac{u_i^{\prime }\left(s\right)}{{\left[u_i\left(s\right)\right]}^{k{p}_i\left(r-\alpha +1\right)}}ds<\infty$. If there exist positive constants ${\lambda }_i$ and ${\delta }_i$ such that

$$1-{\displaystyle \frac{u_i\left(\zeta \right)w^{\prime }\left(\zeta \right)}{\left(k{p}_i\left(r-\alpha +1\right)-1\right)w\left(\zeta \right)u_i^{\prime }\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\lambda }_i}},forr>{\displaystyle \frac{1}{k{p}_i}},$$

$$1-{\displaystyle \frac{u_i\left(\zeta \right)w^{\prime }\left(\zeta \right)}{\left(k{p}_i\left(r-\alpha +1\right)-1\right)u_i^{\prime }\left(\zeta \right)w\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\delta }_i}},forr<{\displaystyle \frac{1}{k{p}_i}},$$

then

$$\begin{array}{ccc}& & \sum _{i=1}^{\tau -k+2}{\int }_0^{\infty }w\left(\zeta \right)\left(\prod _{j=i}^{i+k-1}{\mathrm{\Omega }}_j^{p_j}\left(\zeta \right)\right){\zeta }^{\alpha -1}d\zeta \hfill \\ & \le & \sum _{i=1}^{\tau }{\left({\displaystyle \frac{{\beta }_{i}k{p}_i}{\left|{\eta }_i-1\right|}}\right)}^{k{p}_i}{\int }_0^{\infty }w\left(\zeta \right)g_i\left(\zeta \right){\zeta }^{\alpha -1}d\zeta ,\hfill \end{array}$$

where

$${\mathrm{\Omega }}_i\left(\zeta \right)=\left\{\begin{array}{c}{\displaystyle \frac{\sqrt[k{p}_i]{{\zeta }^{1-\alpha }{u}_i^{\prime }\left(\zeta \right)}}{u_i^{r-\alpha +1}\left(\zeta \right)}}{\int }_0^{\zeta }{\displaystyle \frac{u_i\left(\mu \right){\phi }_i^{\prime }\left(\mu \right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right)d\mu ,forr>{\displaystyle \frac{1}{k{p}_i}},\\ \\ {\displaystyle \frac{\sqrt[k{p}_i]{{\zeta }^{1-\alpha }{u}_i^{\prime }\left(\zeta \right)}}{u_i^{r+\alpha -1}\left(\zeta \right)}}{\int }_0^{\zeta }{\displaystyle \frac{u_i\left(\mu \right){\phi }_i^{\prime }\left(\mu \right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right)d\mu ,forr<{\displaystyle \frac{1}{k{p}_i}},\end{array}\right.$$

$$g_i\left(\zeta \right)=\left\{\begin{array}{c}{\displaystyle \frac{{\zeta }^{1-\alpha }{u}_i^{\left(1-r+\alpha \right)k{p}_i}\left(\zeta \right){\left[{\phi }_i^{\prime }\left(\zeta \right)\right]}^{k{p}_i}{\psi }_i^{k{p}_i}\left(\zeta \right)}{{\phi }_i^{k{p}_i}\left(\zeta \right){\left[u_i^{\prime }\left(\zeta \right)\right]}^{k{p}_i-1}}},forr>{\displaystyle \frac{1}{k{p}_i}},\\ \\ {\displaystyle \frac{{\zeta }^{1-\alpha }{\left[u_i\left(\zeta \right)\right]}^{k{p}_i\left(2-\left(r+\alpha -1\right)\right)}{\left[{\phi }_i^{\prime }\left(\zeta \right)\right]}^{k{p}_i}{\psi }_i^{k{p}_i}\left(\zeta \right)}{{\phi }_i^{k{p}_i}\left(\zeta \right){\left[u_i^{\prime }\left(\zeta \right)\right]}^{k{p}_i-1}}},forr<{\displaystyle \frac{1}{k{p}_i}},\end{array}\right.$$

$${\eta }_i=\left\{\begin{array}{c}k{p}_i\left(r-\alpha +1\right),forr>{\displaystyle \frac{1}{2}}, p\ge 2,\hfill \\ \\ k{p}_i\left(r+\alpha -1\right),forr<{\displaystyle \frac{1}{2}}, 1\le p\le 2,\hfill \end{array}\right.$$

and ${\beta }_{\mathfrak{i}}={max}_{1\le \mathfrak{i}\le \tau }\left({\lambda }_{\mathfrak{i}},{\delta }_{\mathfrak{i}}\right),u_{\mathfrak{i}}(\infty )=\infty .$

Remark 4.

For $\alpha =1$ in Corollary 3. Then, reduce to ([4], Theorem 2).

Corollary 4.

Let $\mathbb{T}=\mathbb{Z}$ in Theorem 4. For any $0<\alpha \le 1, i=1, 2, \dots \tau , \tau \ge k-1, \tau \in \mathbb{N}, r>{\displaystyle \frac{1}{k{p}_i}}$ and $p_i>1$. If there exist positive constant ${\lambda }_i$ such that

$$1-{\displaystyle \frac{u_i^{k{p}_i\left(r-\alpha +1\right)}\left(\zeta +1\right)△w\left(\zeta \right)}{\left(k{p}_i\left(r-\alpha +1\right)-1\right){u_i}^{k{p}_i\left(r-\alpha +1\right)-1}\left(\zeta \right)w(\zeta +1)△u_i\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\lambda }_i}}>0,forr>{\displaystyle \frac{1}{k{p}_i}}$$

$$1-{\displaystyle \frac{u_i\left(\zeta \right)△w\left(\zeta \right)}{\left(k{p}_i\left(r+\alpha -1\right)-1\right)△u_i\left(\zeta \right)w(\zeta +1)}}\ge {\displaystyle \frac{1}{{\delta }_i}},forr<{\displaystyle \frac{1}{k{p}_i}},$$

then

$$\begin{array}{ccc}& & \sum _{i=1}^{\tau -k+2}\sum _{\zeta =0}^{\infty }w\left(\zeta +1\right)\left(\prod _{j=i}^{i+k-1}{\mathrm{\Omega }}_j^{p_j}\left(\zeta \right)\right){\zeta }^{1-\alpha }\hfill \\ & \le & \sum _{i=1}^{\tau }{\left({\displaystyle \frac{{\beta }_{i}k{p}_i}{\left|{\eta }_i-1\right|}}\right)}^{k{p}_i}\left(\sum _{\zeta =0}^{\infty }w(\zeta +1)g_i\left(\zeta \right){\zeta }^{1-\alpha }\right),\hfill \end{array}$$

where

$${\mathrm{\Omega }}_i\left(\zeta \right)=\left\{\begin{array}{c}{\displaystyle \frac{\sqrt[k{p}_i]{{\zeta }^{1-\alpha }△u_i\left(\zeta \right)}}{u_i^{r-\alpha +1}\left(\zeta +1\right)}}\sum _{\mu =0}^{\zeta }{\displaystyle \frac{u_i\left(\mu \right)△{\phi }_i\left(\mu \right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right),forr>\frac{1}{k{p}_i},\\ \\ {\displaystyle \frac{\sqrt[k{p}_i]{{\zeta }^{1-\alpha }△u_i\left(\zeta \right)}}{u_i^{r+\alpha -1}\left(\zeta \right)}}\sum _{\mu =\zeta }^{\infty }{\displaystyle \frac{u_i\left(\mu \right)△{\phi }_i\left(\mu \right)}{{\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right),forr<\frac{1}{k{p}_i},\end{array}\right.$$

$$g_i\left(\zeta \right)=\left\{\begin{array}{c}{\displaystyle \frac{{\zeta }^{1-\alpha }{u}_i^{\left(r-\alpha +1\right)\left(k{p}_i-1\right)k{p}_i}\left(\zeta +1\right)w^{k{p}_i}\left(\zeta \right){\left[△{\phi }_i\left(\zeta \right)\right]}^{k{p}_i}{\psi }_i^{k{p}_i}\left(\zeta \right)}{{u_i}^{{\left(k{p}_i\right)}^2\left(r-\alpha +1\right)-2k{p}_i}\left(\zeta \right)w^{k{p}_i}(\zeta +1){\phi }_i^{k{p}_i}\left(\zeta \right){\left[△u_i\left(\zeta \right)\right]}^{k{p}_i-1}}},forr>\frac{1}{k{p}_i},\\ \\ {\displaystyle \frac{{\zeta }^{1-\alpha }{\left[u_i\left(\zeta \right)\right]}^{k{p}_i\left(2-\left(r+\alpha -1\right)\right)}{w}^{k{p}_i}\left(\zeta \right){\left[△{\phi }_i\left(\zeta \right)\right]}^{k{p}_i}{\psi }_i^{k{p}_i}\left(\zeta \right)}{{\left[w(\zeta +1)\right]}^{k{p}_i}{\phi }_i^{k{p}_i}\left(\zeta \right){\left[△u_i\left(\zeta \right)\right]}^{k{p}_i-1}}},forr<\frac{1}{k{p}_i},\end{array}\right.$$

$${\eta }_i=\left\{\begin{array}{c}k{p}_i\left(r-\alpha +1\right),forr>{\displaystyle \frac{1}{2}}, p\ge 2,\hfill \\ \\ k{p}_i\left(r+\alpha -1\right),forr<{\displaystyle \frac{1}{2}}, 1\le p\le 2,\hfill \end{array}\right.$$

and ${\beta }_{\mathfrak{i}}={max}_{1\le \mathfrak{i}\le \tau }\left({\lambda }_{\mathfrak{i}},{\delta }_{\mathfrak{i}}\right),u_{\mathfrak{i}}(\infty )=\infty .$

Theorem 5.

For any $0<\alpha \le 1, 1\le i\le \tau , \tau \in \mathbb{N}, r>{\displaystyle \frac{1}{3p_i}}$ and ${\displaystyle \frac{\zeta }{2}}, {\displaystyle \frac{\sigma \left(\zeta \right)}{2}}\in \mathbb{T}$. If there exist positive constant ${\lambda }_i$ such that

$$1-{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{3p_i\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left(w\left(\zeta \right)\right)}{\left(3p_i\left(r-\alpha +1\right)-1\right)u_i^{3p_i\left(r-\alpha +1\right)-1}\left(\zeta \right)D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)w^{\sigma }\left(\zeta \right)}}\ge \frac{1}{{\lambda }_i},$$

(34)

then

$$\begin{array}{ccc}& & \sum _{i=1}^{\tau -1}{\int }_0^{\mu }{w}^{\sigma }\left(\zeta \right)\left[{\mathrm{\Gamma }}_i^{p_i}\left(\zeta \right){\mathrm{\Gamma }}_{i+1}^{p_{i+1}}\left(\zeta \right){\mathrm{\Gamma }}_{i+2}^{p_{i+2}}\left(\zeta \right)\right]{△}_{\alpha }\zeta \hfill \\ & \le & \sum _{i=1}^{\tau }{\left({\displaystyle \frac{3p_i{\lambda }_i}{3p_i\left(r-\alpha +1\right)-1}}\right)}^{3p_i}{\int }_0^{\mu }{w}^{\sigma }\left(\zeta \right)g_i\left(\zeta \right){△}_{\alpha }\zeta ,\hfill \end{array}$$

(35)

where

$${\mathrm{\Gamma }}_i\left(\zeta \right)={\displaystyle \frac{\sqrt[3p_i]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r-\alpha +1}}}{\int }_{{\displaystyle \frac{\sigma \left(\zeta \right)}{2}}}^{\sigma \left(\zeta \right)}{\displaystyle \frac{u_i\left(s\right)D_{\alpha }^{△}\left({\phi }_i\left(s\right)\right)}{{\phi }_i\left(s\right)}}{\psi }_i\left(s\right){△}_{\alpha }s,$$

$$\begin{array}{ccc}\hfill g_i\left(\zeta \right)& =& \left[{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(3p_i-1\right)\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)w\left(\zeta \right)}{u_i^{3p_i\left(r-\alpha +1\right)-2}\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{3p_i}}{\phi }_i\left(\zeta \right)w^{\sigma }\left(\zeta \right)}}{\psi }_i\left(\zeta \right)\right.\hfill \\ & & -{\left.{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(3p_i-1\right)\left(r-\alpha +1\right)}{u}_i\left(\frac{\zeta }{2}\right)D_{\alpha }^{△}\left(\phi \left(\frac{\zeta }{2}\right)\right)w\left(\zeta \right)}{2u_i^{3p_i\left(r-\alpha +1\right)-1}\left(\zeta \right){\phi }_i\left(\frac{\zeta }{2}\right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{3p_i}}{w}^{\sigma }\left(\zeta \right)}}{\psi }_i\left({\displaystyle \frac{\zeta }{2}}\right)\right]}^{3p_i},\hfill \end{array}$$

${\int }_0^{\infty }{\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(s\right)\right)}{{\left[u_i^{\sigma }\left(s\right)\right]}^{3p_i\left(r-\alpha +1\right)}}}{△}_{\alpha }s<\infty$ and $u_i(\infty )=\infty .$

Proof. 

Let us define for $r>{\displaystyle \frac{1}{3p_i}}$ and

$${\mathrm{\Gamma }}_i\left(\zeta \right)={\displaystyle \frac{\sqrt[3p_i]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r-\alpha +1}}}{\int }_{{\displaystyle \frac{\sigma \left(\zeta \right)}{2}}}^{\sigma \left(\zeta \right)}{\displaystyle \frac{u_i\left(s\right)D_{\alpha }^{△}\left({\phi }_i\left(s\right)\right)}{{\phi }_i\left(s\right)}}{\psi }_i\left(s\right){△}_{\alpha }s.$$

(36)

Using (7) with $k=3$ for $C_i={\mathrm{\Gamma }}_i^{p_i}\left(\zeta \right),$ we get

$$\sum _{i=1}^{\tau -1}{\mathrm{\Gamma }}_i^{p_i}\left(\zeta \right){\mathrm{\Gamma }}_{i+1}^{p_{i+1}}\left(\zeta \right){\mathrm{\Gamma }}_{i+2}^{p_{i+2}}\left(\zeta \right)\le \sum _{i=1}^{\tau }{\mathrm{\Gamma }}_i^{3p_i}\left(\zeta \right).$$

(37)

Multiplying (37) by $w^{\sigma }\left(\zeta \right)$ and integrating from 0 to $\mu ,$ we obtain

$$\begin{array}{ccc}& & \sum _{i=1}^{\tau -1}{\int }_0^{\mu }{w}^{\sigma }\left(\zeta \right)\left[{\mathrm{\Gamma }}_i^{p_i}\left(\zeta \right){\mathrm{\Gamma }}_{i+1}^{p_{i+1}}\left(\zeta \right){\mathrm{\Gamma }}_{i+2}^{p_{i+2}}\left(\zeta \right)\right]{△}_{\alpha }\zeta \hfill \\ & \le & \sum _{i=1}^{\tau }{\int }_0^{\mu }{w}^{\sigma }\left(\zeta \right){\mathrm{\Gamma }}_i^{3p_i}\left(\zeta \right){△}_{\alpha }\zeta .\hfill \end{array}$$

(38)

Now,

$$\begin{array}{ccc}\hfill I& =& {\int }_0^{\mu }{w}^{\sigma }\left(\zeta \right){\mathrm{\Gamma }}_i^{3p_i}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & =& {\int }_0^{\mu }{w}^{\sigma }\left(\zeta \right){\left[{\displaystyle \frac{\sqrt[3p_i]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r-\alpha +1}}}{\int }_{{\displaystyle \frac{\sigma \left(\zeta \right)}{2}}}^{\sigma \left(\zeta \right)}{\displaystyle \frac{u_i\left(s\right)D_{\alpha }^{△}\left({\phi }_i\left(s\right)\right)}{{\phi }_i\left(s\right)}}{\psi }_i\left(s\right){△}_{\alpha }s\right]}^{3p_i}{△}_{\alpha }\zeta \hfill \\ & =& {\int }_0^{\mu }{\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{3p_i\left(r-\alpha +1\right)}}}\hfill \\ & & \times {\left[\sqrt[3p_i]{w^{\sigma }\left(\zeta \right)}{\int }_{{\displaystyle \frac{\sigma \left(\zeta \right)}{2}}}^{\sigma \left(\zeta \right)}{\displaystyle \frac{u_i\left(s\right)D_{\alpha }^{△}\left({\phi }_i\left(s\right)\right)}{{\phi }_i\left(s\right)}}{\psi }_i\left(s\right){△}_{\alpha }s\right]}^{3p_i}{△}_{\alpha }\zeta .\hfill \end{array}$$

(39)

Integrating (39) by using (6) with

$$D_{\alpha }^{△}\left(\mathrm{\Phi }\left(\zeta \right)\right)={\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{3p_i\left(r-\alpha +1\right)}}},$$

and

$${\mathrm{\Psi }}^{\sigma }\left(\zeta \right)={\left[\sqrt[3p_i]{w^{\sigma }\left(\zeta \right)}{\int }_{{\displaystyle \frac{\sigma \left(\zeta \right)}{2}}}^{\sigma \left(\zeta \right)}{\displaystyle \frac{u_i\left(s\right)D_{\alpha }^{△}\left({\phi }_i\left(s\right)\right)}{{\phi }_i\left(s\right)}}{\psi }_i\left(s\right){△}_{\alpha }s\right]}^{3p_i},$$

to obtain

$$\begin{array}{ccc}\hfill I& =& {\left[\mathrm{\Phi }\left(\zeta \right)\mathrm{\Psi }\left(\zeta \right)\right]}_0^{\mu }+{\int }_0^{\mu }\left(-\mathrm{\Phi }\left(\zeta \right)\right)D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right){△}_{\alpha }\zeta \hfill \\ & =& {\left[-\mathrm{\Psi }\left(\zeta \right){\int }_{\zeta }^{\infty }{\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(s\right)\right)}{{\left[u_i^{\sigma }\left(s\right)\right]}^{3p_i\left(r-\alpha +1\right)}}}{△}_{\alpha }s\right]}_0^{\mu }+{\int }_0^{\mu }\left(-\mathrm{\Phi }\left(\zeta \right)\right)D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right){△}_{\alpha }\zeta \hfill \\ & =& \left[-\mathrm{\Psi }\left(\mu \right){\int }_{\mu }^{\infty }{\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(s\right)\right)}{{\left[u_i^{\sigma }\left(s\right)\right]}^{3p_i\left(r-\alpha +1\right)}}}{△}_{\alpha }s\right]+{\int }_0^{\mu }\left(-\mathrm{\Phi }\left(\zeta \right)\right)D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right){△}_{\alpha }\zeta \hfill \\ & \le & {\int }_0^{\mu }\left(-\mathrm{\Phi }\left(\zeta \right)\right)D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right){△}_{\alpha }\zeta ,\hfill \end{array}$$

(40)

where $\mathrm{\Phi }\left(\zeta \right)=-{\int }_{\zeta }^{\infty }{\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(s\right)\right)}{{\left[u_i^{\sigma }\left(s\right)\right]}^{3p_i\left(r-\alpha +1\right)}}}{△}_{\alpha }s.$ From (5), $D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\ge 0$ and $c\in {[\zeta , \sigma \left(\zeta \right)]}_{\mathbb{T}},$ we have

$$\begin{array}{ccc}& & D_{\alpha }^{△}\left(u_i^{1-3p_i\left(r-\alpha +1\right)}\left(\zeta \right)\right)\hfill \\ & =& \left(1-3p_i\left(r-\alpha +1\right)\right)u_i^{-3p_i\left(r-\alpha +1\right)}\left(c\right)D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\hfill \\ & =& \left(1-3p_i\left(r-\alpha +1\right)\right){\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}{u_i^{3p_i\left(r-\alpha +1\right)}\left(c\right)}}\hfill \\ & \le & \left(1-3p_i\left(r-\alpha +1\right)\right){\displaystyle \frac{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{3p_i\left(r-\alpha +1\right)}}}.\hfill \end{array}$$

(41)

Therefore, integrating (41) from $\zeta$ to ∞ with respect to s, we have

$$-\mathrm{\Phi }\left(\zeta \right)\le {\displaystyle \frac{1}{3p_i\left(r-\alpha +1\right)-1}}{u}_i^{1-3p_i\left(r-\alpha +1\right)}\left(\zeta \right).$$

(42)

Combining (42) and (40), we have

$$I\le {\displaystyle \frac{1}{3p_i\left(r-\alpha +1\right)-1}}{\int }_0^{\mu }{u}_i^{1-3p_i\left(r-\alpha +1\right)}\left(\zeta \right)D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right){△}_{\alpha }\zeta .$$

(43)

Now, we consider that $\mathrm{\Psi }\left(\zeta \right)=w\left(\zeta \right)L^{3p_i}\left(\zeta \right)$ and applying Theorem 1, to see that

$$\begin{array}{ccc}\hfill D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right)& =& D_{\alpha }^{△}\left(w\left(\zeta \right)\right){\left[L^{\sigma }\left(\zeta \right)\right]}^{3p_i}+w\left(\zeta \right)D_{\alpha }^{△}\left(L^{3p_i}\left(\zeta \right)\right)\hfill \\ & =& D_{\alpha }^{△}\left(w\left(\zeta \right)\right){\left[L^{\sigma }\left(\zeta \right)\right]}^{3p_i}+3p_{i}w\left(\zeta \right)L^{3p_i-1}\left(c\right)D_{\alpha }^{△}\left(L\left(\zeta \right)\right)\hfill \\ & \le & D_{\alpha }^{△}\left(w\left(\zeta \right)\right){\left[L^{\sigma }\left(\zeta \right)\right]}^{3p_i}\hfill \\ & & +3p_{i}w\left(\zeta \right){\left[L^{\sigma }\left(\zeta \right)\right]}^{3p_i-1}\hfill \\ & & \times \left[{\displaystyle \frac{u_i\left(\zeta \right)D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right){\psi }_i\left(\zeta \right)}{{\phi }_i\left(\zeta \right)}}-{\displaystyle \frac{u_i\left(\frac{\zeta }{2}\right)D_{\alpha }^{△}\left({\phi }_i\left(\frac{\zeta }{2}\right)\right){\psi }_i\left(\frac{\zeta }{2}\right)}{2{\phi }_i\left(\frac{\zeta }{2}\right)}}\right],\hfill \end{array}$$

(44)

where $L\left(\zeta \right)={\int }_{{\displaystyle \frac{\zeta }{2}}}^{\zeta }{\displaystyle \frac{u_i\left(s\right)D_{\alpha }^{△}\left({\phi }_i\left(s\right)\right)}{{\phi }_i\left(s\right)}}{\psi }_i\left(s\right){△}_{\alpha }s.$ From (44) and (43), we have

$$\begin{array}{ccc}& & {\int }_0^{\mu }{w}^{\sigma }\left(\zeta \right){\mathrm{\Gamma }}_i^{3p_i}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & {\displaystyle \frac{1}{3p_i\left(r-\alpha +1\right)-1}}{\int }_0^{\mu }{u}_i^{1-3p_i\left(r-\alpha +1\right)}\left(\zeta \right)D_{\alpha }^{△}\left(w\left(\zeta \right)\right){\left[L^{\sigma }\left(\zeta \right)\right]}^{3p_i}{△}_{\alpha }\zeta \hfill \\ & & +{\displaystyle \frac{3p_i}{3p_i\left(r-\alpha +1\right)-1}}{\int }_0^{\mu }{u}_i^{1-3p_i\left(r-\alpha +1\right)}\left(\zeta \right)w\left(\zeta \right)\hfill \\ & & \times \left[{\displaystyle \frac{u_i\left(\zeta \right)D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right){\psi }_i\left(\zeta \right)}{{\phi }_i\left(\zeta \right)}}-{\displaystyle \frac{u_i\left(\frac{\zeta }{2}\right)D_{\alpha }^{△}\left({\phi }_i\left(\frac{\zeta }{2}\right)\right){\psi }_i\left(\frac{\zeta }{2}\right)}{2{\phi }_i\left(\frac{\zeta }{2}\right)}}\right]{\left[L^{\sigma }\left(\zeta \right)\right]}^{3p_i-1}{△}_{\alpha }\zeta \hfill \\ & =& {\displaystyle \frac{1}{3p_i\left(r-\alpha +1\right)-1}}{\int }_0^{\mu }{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{3p_i\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left(w\left(\zeta \right)\right)}{u_i^{3p_i\left(r-\alpha +1\right)-1}\left(\zeta \right)D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{\mathrm{\Gamma }}_i^{3p_i}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & & +{\displaystyle \frac{3p_i}{3p_i\left(r-\alpha +1\right)-1}}{\int }_0^{\mu }{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(3p_i-1\right)\left(r-\alpha +1\right)}w\left(\zeta \right)}{u_i^{3p_i\left(r-\alpha +1\right)-1}\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{3p_i}}}}{\mathrm{\Gamma }}_i^{3p_i-1}\left(\zeta \right)\hfill \\ & & \times \left[{\displaystyle \frac{u_i\left(\zeta \right)D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right){\psi }_i\left(\zeta \right)}{{\phi }_i\left(\zeta \right)}}-{\displaystyle \frac{u_i\left(\frac{\zeta }{2}\right)D_{\alpha }^{△}\left({\phi }_i\left(\frac{\zeta }{2}\right)\right){\psi }_i\left(\frac{\zeta }{2}\right)}{2{\phi }_i\left(\frac{\zeta }{2}\right)}}\right]{△}_{\alpha }\zeta .\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}& & {\int }_0^{\mu }{w}^{\sigma }\left(\zeta \right){\mathrm{\Gamma }}_i^{3p_i}\left(\zeta \right)\left(1-{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{3p_i\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left(w\left(\zeta \right)\right)}{\left(3p_i\left(r-\alpha +1\right)-1\right)u_i^{3p_i\left(r-\alpha +1\right)-1}\left(\zeta \right)D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)w^{\sigma }\left(\zeta \right)}}\right){△}_{\alpha }\zeta \hfill \\ & \le & {\displaystyle \frac{3p_i}{3p_i\left(r-\alpha +1\right)-1}}{\int }_0^{\mu }{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(3p_i-1\right)\left(r-\alpha +1\right)}w\left(\zeta \right){\mathrm{\Gamma }}_i^{3p_i-1}\left(\zeta \right)}{u_i^{3p_i\left(r-\alpha +1\right)-1}\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{3p_i}}}}\hfill \\ & & \times \left[{\displaystyle \frac{u_i\left(\zeta \right)D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right){\psi }_i\left(\zeta \right)}{{\phi }_i\left(\zeta \right)}}-{\displaystyle \frac{u_i\left(\frac{\zeta }{2}\right)D_{\alpha }^{△}\left({\phi }_i\left(\frac{\zeta }{2}\right)\right){\psi }_i\left(\frac{\zeta }{2}\right)}{2{\phi }_i\left(\frac{\zeta }{2}\right)}}\right]{△}_{\alpha }\zeta .\hfill \end{array}$$

(45)

From (45) and (34), we have

$$\begin{array}{ccc}& & {\int }_0^{\mu }{w}^{\sigma }\left(\zeta \right){\mathrm{\Gamma }}_i^{3p_i}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & {\displaystyle \frac{3p_i{\lambda }_i}{3p_i\left(r-\alpha +1\right)-1}}{\int }_0^{\mu }{w}^{\sigma }\left(\zeta \right){\mathrm{\Gamma }}_i^{3p_i-1}\left(\zeta \right)\hfill \\ & & \times \left[{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(3p_i-1\right)\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)w\left(\zeta \right){\psi }_i\left(\zeta \right)}{u_i^{3p_i\left(r-\alpha +1\right)-2}\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{3p_i}}{\phi }_i\left(\zeta \right)w^{\sigma }\left(\zeta \right)}}\right.\hfill \\ & & -\left.{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(3p_i-1\right)\left(r-\alpha +1\right)}{u}_i\left(\frac{\zeta }{2}\right)D_{\alpha }^{△}\left(\phi \left(\frac{\zeta }{2}\right)\right)w\left(\zeta \right){\psi }_i\left(\frac{\zeta }{2}\right)}{2u_i^{3p_i\left(r-\alpha +1\right)-1}\left(\zeta \right){\phi }_i\left(\frac{\zeta }{2}\right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{3p_i}}{w}^{\sigma }\left(\zeta \right)}}\right]{△}_{\alpha }\zeta \hfill \\ & =& {\displaystyle \frac{3p_i{\lambda }_i}{3p_i\left(r-\alpha +1\right)-1}}{\int }_0^{\mu }{\left(w^{\sigma }\left(\zeta \right){\mathrm{\Gamma }}_i^{3p_i}\left(\zeta \right)\right)}^{{\displaystyle \frac{3p_i-1}{3p_i}}}{\left[w^{\sigma }\left(\zeta \right)\right]}^{{\displaystyle \frac{1}{3p_i}}}\hfill \\ & & \times \left[{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(3p_i-1\right)\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)w\left(\zeta \right){\psi }_i\left(\zeta \right)}{u_i^{3p_i\left(r-\alpha +1\right)-2}\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{3p_i}}{\phi }_i\left(\zeta \right)w^{\sigma }\left(\zeta \right)}}\right.\hfill \\ & & -\left.{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(3p_i-1\right)\left(r-\alpha +1\right)}{u}_i\left(\frac{\zeta }{2}\right)D_{\alpha }^{△}\left(\phi \left(\frac{\zeta }{2}\right)\right)w\left(\zeta \right){\psi }_i\left(\frac{\zeta }{2}\right)}{2u_i^{3p_i\left(r-\alpha +1\right)-1}\left(\zeta \right){\phi }_i\left(\frac{\zeta }{2}\right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{3p_i}}{w}^{\sigma }\left(\zeta \right)}}\right]{△}_{\alpha }\zeta .\hfill \end{array}$$

Applying Hölder’s inequality with $3p_i$ and $p_i/\left(3p_i-1\right),$ we have

$$\begin{array}{ccc}& & {\int }_0^{\mu }{w}^{\sigma }\left(\zeta \right){\mathrm{\Gamma }}_i^{3p_i}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & {\left({\displaystyle \frac{3p_i{\lambda }_i}{3p_i\left(r-\alpha +1\right)-1}}\right)}^{3p_i}\hfill \\ & & \times {\int }_0^{\mu }{w}^{\sigma }\left(\zeta \right)\left[{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(3p_i-1\right)\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)w\left(\zeta \right){\psi }_i\left(\zeta \right)}{u_i^{3p_i\left(r-\alpha +1\right)-2}\left(\zeta \right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{3p_i}}{\phi }_i\left(\zeta \right)w^{\sigma }\left(\zeta \right)}}\right.\hfill \\ & & -{\left.{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{\left(3p_i-1\right)\left(r-\alpha +1\right)}{u}_i\left(\frac{\zeta }{2}\right)D_{\alpha }^{△}\left(\phi \left(\frac{\zeta }{2}\right)\right)w\left(\zeta \right){\psi }_i\left(\frac{\zeta }{2}\right)}{2u_i^{3p_i\left(r-\alpha +1\right)-1}\left(\zeta \right){\phi }_i\left(\frac{\zeta }{2}\right){\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{3p_i}}{w}^{\sigma }\left(\zeta \right)}}\right]}^{3p_i}{△}_{\alpha }\zeta \hfill \\ & =& {\left({\displaystyle \frac{3p_i{\lambda }_i}{3p_i\left(r-\alpha +1\right)-1}}\right)}^{3p_i}{\int }_0^{\mu }{w}^{\sigma }\left(\zeta \right)g_i\left(\zeta \right){△}_{\alpha }\zeta .\hfill \end{array}$$

(46)

From (46) and (38), we get (35). □

Remark 5.

If we put $\alpha =1$ in Theorem 5, then we have ([18], Theorem 16).

Corollary 5.

Let $\mathbb{T}=\mathbb{R}$ in Theorem 5. For any $0<\alpha \le 1, 1\le i\le \tau , \tau \in \mathbb{N}, r>{\displaystyle \frac{1}{3p_i}}$ and ${\displaystyle \frac{\zeta }{2}},{\displaystyle \frac{\sigma \left(\zeta \right)}{2}}\in \mathbb{R}$. If there exist positive constant ${\lambda }_i$ such that

$$1-{\displaystyle \frac{u_i\left(\zeta \right)w^{\prime }\left(\zeta \right)}{\left(3p_i\left(r-\alpha +1\right)-1\right)u_i^{\prime }\left(\zeta \right)w\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\lambda }_i}},$$

then

$$\begin{array}{ccc}& & \sum _{i=1}^{\tau -1}{\int }_0^{\mu }w\left(\zeta \right)\left[{\mathrm{\Gamma }}_i^{p_i}\left(\zeta \right){\mathrm{\Gamma }}_{i+1}^{p_{i+1}}\left(\zeta \right){\mathrm{\Gamma }}_{i+2}^{p_{i+2}}\left(\zeta \right)\right]{\zeta }^{\alpha -1}d\zeta \hfill \\ & \le & \sum _{i=1}^{\tau }{\left({\displaystyle \frac{3p_i{\lambda }_i}{3p_i\left(r-\alpha +1\right)-1}}\right)}^{3p_i}{\int }_0^{\mu }w\left(\zeta \right)g_i\left(\zeta \right){\zeta }^{\alpha -1}d\zeta ,\hfill \end{array}$$

where

$${\mathrm{\Gamma }}_i\left(\zeta \right)={\displaystyle \frac{\sqrt[3p_i]{{\zeta }^{1-\alpha }{u}_i^{\prime }\left(\zeta \right)}}{u_i^{r-\alpha +1}\left(\zeta \right)}}{\int }_{{\displaystyle \frac{\zeta }{2}}}^{\zeta }{\displaystyle \frac{u_i\left(s\right){\phi }_i^{\prime }\left(s\right)}{{\phi }_i\left(s\right)}}{\psi }_i\left(s\right)ds,$$

$$g_i\left(\zeta \right)={\left[{\displaystyle \frac{u_i^{\alpha -r+1}\left(\zeta \right){\zeta }^{1-\alpha }{\phi }_i^{\prime }\left(\zeta \right){\psi }_i\left(\zeta \right)}{{\left({\zeta }^{1-\alpha }{u}_i^{\prime }\left(\zeta \right)\right)}^{1-\frac{1}{3p_i}}{\phi }_i\left(\zeta \right)}}-{\displaystyle \frac{u_i^{\alpha -r}\left(\zeta \right)u_i\left(\frac{\zeta }{2}\right){\left(\frac{\zeta }{2}\right)}^{1-\alpha }{\phi }_i^{\prime }\left(\frac{\zeta }{2}\right){\psi }_i\left(\frac{\zeta }{2}\right)}{2{\phi }_i\left(\frac{\zeta }{2}\right){\left({\zeta }^{1-\alpha }{u}_i^{\prime }\left(\zeta \right)\right)}^{1-\frac{1}{3p_i}}}}\right]}^{3p_i},$$

and $u_i(\infty )=\infty .$

Remark 6.

For $\alpha =1$ in Corollary 5. Then, reduce to ([4], Theorem 3).

Corollary 6.

Let $\mathbb{T}=\mathbb{Z}$ in Theorem 5. For any $0<\alpha \le 1, r>{\displaystyle \frac{1}{3p_i}}$ and ${\displaystyle \frac{\zeta +1}{2}}, {\displaystyle \frac{\zeta }{2}}\in \mathbb{Z}$. If there exist positive constant ${\lambda }_i$ such that

$$1-{\displaystyle \frac{u_i^{3p_i\left(r-\alpha +1\right)}\left(\zeta +1\right)△w\left(\zeta \right)}{\left(3p_i\left(r-\alpha +1\right)-1\right)u_i^{3p_i\left(r-\alpha +1\right)-1}\left(\zeta \right)w\left(\zeta +1\right)△u_i\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\lambda }_i}},$$

then

$$\begin{array}{ccc}& & \sum _{i=1}^{\tau -1}\sum _{\zeta =0}^{\mu -1}w\left(\zeta +1\right)\left[{\mathrm{\Gamma }}_i^{p_i}\left(\zeta \right){\mathrm{\Gamma }}_{i+1}^{p_{i+1}}\left(\zeta \right){\mathrm{\Gamma }}_{i+2}^{p_{i+2}}\left(\zeta \right)\right]{\zeta }^{\alpha -1}\hfill \\ & \le & \sum _{i=1}^{\tau }{\left({\displaystyle \frac{3p_i{\lambda }_i}{3p_i\left(r-\alpha +1\right)-1}}\right)}^{3p_i}\left(\sum _{\zeta =0}^{\mu -1}w\left(\zeta +1\right)g_i\left(\zeta \right){\zeta }^{\alpha -1}\right),for1\le i\le \tau ,\tau \in \mathbb{N}\hfill \end{array}$$

where

$${\mathrm{\Gamma }}_i\left(\zeta \right)={\displaystyle \frac{\sqrt[3p_i]{{\zeta }^{1-\alpha }△u_i\left(\zeta \right)}}{u_i^{r-\alpha +1}\left(\zeta +1\right)}}\sum _{s={\displaystyle \frac{\zeta +1}{2}}}^{\zeta }{\displaystyle \frac{u_i\left(s\right)△{\phi }_i\left(s\right)}{{\phi }_i\left(s\right)}}{\psi }_i\left(s\right),$$

$$\begin{array}{ccc}\hfill g_i\left(\zeta \right)& =& \left[{\displaystyle \frac{u_i^{\left(3p_i-1\right)\left(r-\alpha +1\right)}\left(\zeta +1\right){\zeta }^{1-\alpha }△{\phi }_i\left(\zeta \right){\psi }_i\left(\zeta \right)}{u_i^{3p_i\left(r-\alpha +1\right)-2}(\zeta ){\left({\zeta }^{1-\alpha }△u_i\left(\zeta \right)\right)}^{1-\frac{1}{3p_i}}{\phi }_i\left(\zeta \right)}}\right.\hfill \\ & & -{\left.{\displaystyle \frac{u_i^{\left(3p_i-1\right)\left(r-\alpha +1\right)}\left(\zeta +1\right)u_i\left(\frac{\zeta }{2}\right){\left(\frac{\zeta }{2}\right)}^{1-\alpha }{\phi }_i\left(\frac{\zeta }{2}\right){\psi }_i\left(\frac{\zeta }{2}\right)}{2u_i^{3p_i\left(r-\alpha +1\right)-1}\left(\zeta \right){\phi }_i\left(\frac{\zeta }{2}\right){\left({\zeta }^{1-\alpha }△u_i\left(\zeta \right)\right)}^{1-\frac{1}{3p_i}}}}\right]}^{3p_i},\hfill \end{array}$$

and $u_i(\infty )=\infty .$

Theorem 6.

For any $0<\alpha \le 1, k\ge 1, p>1$ and $r>0$. If there exist positive constant ${\lambda }_i$ such that

$$1+{\displaystyle \frac{u_i^{\sigma }\left(\zeta \right)D_{\alpha }^{△}\left(w\left(\zeta \right)\right)}{\left(1+kp\left(r-\alpha +1\right)\right)D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)w\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\lambda }_i}},$$

(47)

then

$$\begin{array}{ccc}& & {\int }_a^{b}w\left(\zeta \right){\left(\sum _{i=1}^{\tau }{\mathrm{\Gamma }}_{ia}\left(\zeta \right)\right)}^{kp}{△}_{\alpha }\zeta \hfill \\ & \le & \sum _{i=1}^{\tau }{\left({\displaystyle \frac{-{\tau }^{pk-1}kp{\lambda }_i}{1+kp\left(r-\alpha +1\right)}}\right)}^{kp}{\int }_a^{b}w\left(\zeta \right)g_i\left(\zeta \right){△}_{\alpha }\zeta ,\hfill \end{array}$$

(48)

where

$${\mathrm{\Gamma }}_i\left(\zeta \right)={\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r-\alpha +1}\sqrt[kp]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}{\int }_0^{\zeta }{\displaystyle \frac{D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{u_i^{\sigma }\left(\mu \right){\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu ,$$

and

$$g_i\left(\zeta \right)={\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{kp\left(r-\alpha +1\right)}{\left[w^{\sigma }\left(\zeta \right)\right]}^{kp}{\left[D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)\right]}^{kp}}{{\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{kp-1}{w}^{kp}\left(\zeta \right){\phi }_i^{kp}\left(\zeta \right)}}{\psi }_i^{kp}\left(\zeta \right).$$

Proof. 

Let us define

$${\mathrm{\Gamma }}_{ia}\left(\zeta \right)={\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r-\alpha +1}\sqrt[kp]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}{\int }_a^{\zeta }{\displaystyle \frac{D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{u_i^{\sigma }\left(\mu \right){\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu ,$$

(49)

with ${\mathrm{\Gamma }}_{i0}\left(\zeta \right)={\mathrm{\Gamma }}_i\left(\zeta \right).$ Using (8) for $C_i={\mathrm{\Gamma }}_i\left(\zeta \right)$ and $k\to kp$, we get

$${\left(\sum _{i=1}^{\tau }{\mathrm{\Gamma }}_{ia}\left(\zeta \right)\right)}^{kp}\le {\tau }^{pk-1}\sum _{i=1}^{\tau }{\mathrm{\Gamma }}_{ia}^{kp}\left(\zeta \right).$$

(50)

Multiplying (50) by $w\left(\zeta \right)$ and integrating from a to b, we get

$${\int }_a^{b}w\left(\zeta \right){\left(\sum _{i=1}^{\tau }{\mathrm{\Gamma }}_{ia}\left(\zeta \right)\right)}^{kp}{△}_{\alpha }\zeta \le {\tau }^{pk-1}\sum _{i=1}^{\tau }{\int }_a^{b}w\left(\zeta \right){\mathrm{\Gamma }}_{ia}^{kp}\left(\zeta \right){△}_{\alpha }\zeta .$$

(51)

Now,

$$\begin{array}{ccc}\hfill I& =& {\int }_a^{b}w\left(\zeta \right){\mathrm{\Gamma }}_{ia}^{kp}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & =& {\int }_a^{b}w\left(\zeta \right){\left[{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r-\alpha +1}\sqrt[kp]{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}{\int }_a^{\zeta }{\displaystyle \frac{D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{u_i^{\sigma }\left(\mu \right){\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu \right]}^{kp}{△}_{\alpha }\zeta \hfill \\ & =& {\int }_a^b{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{kp\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\hfill \\ & & {\left[\sqrt[kp]{w\left(\zeta \right)}{\int }_a^{\zeta }{\displaystyle \frac{D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{u_i^{\sigma }\left(\mu \right){\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu \right]}^{kp}{△}_{\alpha }\zeta .\hfill \end{array}$$

(52)

Integrating (52) by the parts Formula (6) with

$$D_{\alpha }^{△}\left(\mathrm{\Phi }\left(\zeta \right)\right)={\left[u_i^{\sigma }\left(\zeta \right)\right]}^{kp\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left(u_i\left(\zeta \right)\right),$$

and

$$B\left(\zeta \right)={\left[\sqrt[kp]{w\left(\zeta \right)}{\int }_a^{\zeta }{\displaystyle \frac{D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{u_i^{\sigma }\left(\mu \right){\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu \right]}^{kp},$$

to obtain

$$\begin{array}{ccc}\hfill I& =& {\left[\mathrm{\Phi }\left(\zeta \right)\mathrm{\Psi }\left(\zeta \right)\right]}_a^b+{\int }_a^b{\mathrm{\Phi }}^{\sigma }\left(\zeta \right){\left(-\mathrm{\Psi }\left(\zeta \right)\right)}^{△}{△}_{\alpha }\zeta \hfill \\ & =& {\left[-\mathrm{\Psi }\left(\zeta \right){\int }_{\zeta }^b{\left[u_i^{\sigma }\left(\mu \right)\right]}^{kp\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left(u_i\left(\mu \right)\right){△}_{\alpha }\mu \right]}_a^b+{\int }_a^b{\mathrm{\Phi }}^{\sigma }\left(\zeta \right)\left(-D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right)\right){△}_{\alpha }\zeta \hfill \\ & =& {\int }_a^b{\mathrm{\Phi }}^{\sigma }\left(\zeta \right)\left(-D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right)\right){△}_{\alpha }\zeta ,\hfill \end{array}$$

(53)

where $\mathrm{\Phi }\left(\zeta \right)=-{\int }_{\zeta }^b{\left[u_i^{\sigma }\left(\mu \right)\right]}^{kp\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left(u_i\left(\mu \right)\right){△}_{\alpha }\mu .$ From (5), $D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\ge 0$ and $c\in {[\zeta , \sigma \left(\zeta \right)]}_{\mathbb{T}},$ we have

$$\begin{array}{ccc}& & D_{\alpha }^{△}\left(u_i^{1+kp\left(r-\alpha +1\right)}\left(\zeta \right)\right)\hfill \\ & =& \left(1+kp\left(r-\alpha +1\right)\right)u_i^{kp\left(r-\alpha +1\right)}\left(c\right)D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\hfill \\ & \le & \left(1+kp\left(r-\alpha +1\right)\right){\left[u_i^{\sigma }\left(\zeta \right)\right]}^{kp\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left(u_i\left(\zeta \right)\right).\hfill \end{array}$$

This implies that

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{1+kp\left(r-\alpha +1\right)}}{\int }_{\zeta }^b{D}_{\alpha }^{△}\left(u_i^{1+kp\left(r-\alpha +1\right)}\left(\mu \right)\right){△}_{\alpha }\mu \hfill \\ & \le & {\int }_{\zeta }^b{\left[u_i^{\sigma }\left(\mu \right)\right]}^{kp\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left(u_i\left(\mu \right)\right){△}_{\alpha }\mu =-\mathrm{\Phi }\left(\zeta \right),\hfill \end{array}$$

therefore,

$$\begin{array}{ccc}\hfill \mathrm{\Phi }\left(\zeta \right)& \le & {\displaystyle \frac{1}{1+kp\left(r-\alpha +1\right)}}\left[u_i^{1+kp\left(r-\alpha +1\right)}\left(\zeta \right)-u_i^{1+kp\left(r-\alpha +1\right)}\left(b\right)\right]\hfill \\ & \le & {\displaystyle \frac{1}{1+kp\left(r-\alpha +1\right)}}{u}_i^{1+kp\left(r-\alpha +1\right)}\left(\zeta \right).\hfill \end{array}$$

(54)

Substituting (54) into (53), we have

$$I\le {\displaystyle \frac{1}{1+kp\left(r-\alpha +1\right)}}{\int }_a^b{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{1+kp\left(r-\alpha +1\right)}\left(-D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right)\right){△}_{\alpha }\zeta .$$

(55)

Now, we consider that $\mathrm{\Psi }\left(\zeta \right)=w\left(\zeta \right)L^{kp}\left(\zeta \right)$ and applying Theorem 1, to see that

$$\begin{array}{ccc}\hfill D_{\alpha }^{△}\left(\mathrm{\Psi }\left(\zeta \right)\right)& =& D_{\alpha }^{△}\left(w\left(\zeta \right)\right)L^{kp}\left(\zeta \right)+w^{\sigma }\left(\zeta \right)D_{\alpha }^{△}\left(L^{kp}\left(\zeta \right)\right)\hfill \\ & =& D_{\alpha }^{△}\left(w\left(\zeta \right)\right)L^{kp}\left(\zeta \right)+kp{w}^{\sigma }\left(\zeta \right)L^{kp-1}\left(c\right)D_{\alpha }^{△}\left(L\left(\zeta \right)\right)\hfill \\ & \ge & D_{\alpha }^{△}\left(w\left(\zeta \right)\right)L^{kp}\left(\zeta \right)\hfill \\ & & +kp{w}^{\sigma }\left(\zeta \right)\left[{\displaystyle \frac{D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)}{u_i^{\sigma }\left(\zeta \right){\phi }_i\left(\zeta \right)}}{\psi }_i\left(\zeta \right)\right]L^{kp-1}\left(\zeta \right),\hfill \end{array}$$

(56)

where $L\left(\zeta \right)={\int }_a^{\zeta }{\displaystyle \frac{D_{\alpha }^{△}\left({\phi }_i\left(\mu \right)\right)}{u_i^{\sigma }\left(\mu \right){\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right){△}_{\alpha }\mu .$ From (56) and (55), we have

$$\begin{array}{ccc}& & {\int }_a^{b}w\left(\zeta \right){\mathrm{\Gamma }}_{ia}^{kp}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & {\displaystyle \frac{-1}{1+kp\left(r-\alpha +1\right)}}{\int }_a^b{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{1+kp\left(r-\alpha +1\right)}{D}_{\alpha }^{△}\left(w\left(\zeta \right)\right)L^{kp}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & & +{\displaystyle \frac{-kp}{1+kp\left(r-\alpha +1\right)}}{\int }_a^b{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{1+kp\left(r-\alpha +1\right)}{w}^{\sigma }\left(\zeta \right)\left[{\displaystyle \frac{D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)}{u_i^{\sigma }\left(\zeta \right){\phi }_i\left(\zeta \right)}}{\psi }_i\left(\zeta \right)\right]L^{kp-1}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & =& {\displaystyle \frac{-1}{1+kp\left(r-\alpha +1\right)}}{\int }_a^b{\displaystyle \frac{u_i^{\sigma }\left(\zeta \right)D_{\alpha }^{△}\left(w\left(\zeta \right)\right)}{D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)}}{\mathrm{\Gamma }}_{ia}^{kp}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & & +{\displaystyle \frac{-kp}{1+kp\left(r-\alpha +1\right)}}{\int }_a^b{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r-\alpha +2}{w}^{\sigma }\left(\zeta \right)}{{\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{kp}}}}{\mathrm{\Gamma }}_{ia}^{kp-1}\left(\zeta \right)\left({\displaystyle \frac{D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)}{u_i^{\sigma }\left(\zeta \right){\phi }_i\left(\zeta \right)}}{\psi }_i\left(\zeta \right)\right){△}_{\alpha }\zeta .\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}& & {\int }_a^{b}w\left(\zeta \right){\mathrm{\Gamma }}_i^{kp}\left(\zeta \right)\left(1+{\displaystyle \frac{u_i^{\sigma }\left(\zeta \right)D_{\alpha }^{△}\left(w\left(\zeta \right)\right)}{\left(1+kp\left(r-\alpha +1\right)\right)D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)w\left(\zeta \right)}}\right){△}_{\alpha }\zeta \hfill \\ & \le & {\displaystyle \frac{-kp}{1+kp\left(r-\alpha +1\right)}}\hfill \\ & & \times {\int }_a^b{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r-\alpha +2}{w}^{\sigma }\left(\zeta \right)}{{\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{kp}}}}{\mathrm{\Gamma }}_{ia}^{kp-1}\left(\zeta \right)\left[{\displaystyle \frac{D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)}{u_i^{\sigma }\left(\zeta \right){\phi }_i\left(\zeta \right)}}{\psi }_i\left(\zeta \right)\right]{△}_{\alpha }\zeta .\hfill \end{array}$$

(57)

From (57) and (47), we have

$$\begin{array}{ccc}& & {\int }_a^{b}w\left(\zeta \right){\mathrm{\Gamma }}_{ia}^{kp}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & {\displaystyle \frac{-kp{\lambda }_i}{1+kp\left(r-\alpha +1\right)}}\hfill \\ & & \times {\int }_a^b{\left(w\left(\zeta \right){\mathrm{\Gamma }}_{ia}^{kp}\left(\zeta \right)\right)}^{{\displaystyle \frac{kp-1}{kp}}}{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{r-\alpha +1}{w}^{\sigma }\left(\zeta \right)D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)}{{\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{1-\frac{1}{kp}}{w}^{\frac{kp-1}{kp}}\left(\zeta \right){\phi }_i\left(\zeta \right)}}{\psi }_i\left(\zeta \right){△}_{\alpha }\zeta .\hfill \end{array}$$

By Hölder’s inequality with $kp$ and $kp/\left(kp-1\right),$ we have

$$\begin{array}{ccc}& & {\int }_a^{b}w\left(\zeta \right){\mathrm{\Gamma }}_{ia}^{kp}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & \le & {\left({\displaystyle \frac{-kp{\lambda }_i}{1+kp\left(r-\alpha +1\right)}}\right)}^{kp}{\int }_a^b{\displaystyle \frac{{\left[u_i^{\sigma }\left(\zeta \right)\right]}^{kp\left(r-\alpha +1\right)}{\left[w^{\sigma }\left(\zeta \right)\right]}^{kp}{\left[D_{\alpha }^{△}\left({\phi }_i\left(\zeta \right)\right)\right]}^{kp}}{{\left[D_{\alpha }^{△}\left(u_i\left(\zeta \right)\right)\right]}^{kp-1}{w}^{kp-1}\left(\zeta \right){\phi }_i^{kp}\left(\zeta \right)}}{\psi }_i^{kp}\left(\zeta \right){△}_{\alpha }\zeta \hfill \\ & =& {\left({\displaystyle \frac{-kp{\lambda }_i}{1+kp\left(r-\alpha +1\right)}}\right)}^{kp}{\int }_a^{b}w\left(\zeta \right)g_i\left(\zeta \right){△}_{\alpha }\zeta .\hfill \end{array}$$

(58)

From (58) and (51), we get (48). □

Remark 7.

If we put $\alpha =1$ in Theorem 6, then we have ([18], Theorem 19).

Corollary 7.

Let $\mathbb{T}=\mathbb{R}$ in Theorem 6. For any $0<\alpha \le 1, k\ge 1, p>1$ and $r>0$. If there exist positive constant ${\lambda }_i$ such that

$$1+{\displaystyle \frac{u_i\left(\zeta \right)w^{\prime }\left(\zeta \right)}{\left(1+kp\left(r-\alpha +1\right)\right)u_i^{\prime }\left(\zeta \right)w\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\lambda }_i}},$$

then

$${\int }_a^{b}w\left(\zeta \right){\left(\sum _{i=1}^{\tau }{\mathrm{\Gamma }}_{ia}\left(\zeta \right)\right)}^{kp}{\zeta }^{\alpha -1}d\zeta \le \sum _{i=1}^{\tau }{\left({\displaystyle \frac{-{\tau }^{pk-1}kp{\lambda }_i}{1+kp\left(r-\alpha +1\right)}}\right)}^{kp}{\int }_a^{b}w\left(\zeta \right)g_i\left(\zeta \right){\zeta }^{\alpha -1}d\zeta ,$$

where

$${\mathrm{\Gamma }}_i\left(\zeta \right)=u_i^{r-\alpha +1}\left(\zeta \right)\sqrt[kp]{{\zeta }^{1-\alpha }{u}_i^{\prime }\left(\zeta \right)}{\int }_0^{\zeta }{\displaystyle \frac{{\phi }_i^{\prime }\left(\mu \right)}{u_i\left(\mu \right){\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right)d\mu ,$$

and

$$g_i\left(\zeta \right)={\displaystyle \frac{u_i^{kp\left(r-\alpha +1\right)}\left(\zeta \right){\zeta }^{1-\alpha }{\left({\phi }_i^{\prime }\left(\zeta \right)\right)}^{kp}}{{\left(u_i^{\prime }\left(\zeta \right)\right)}^{kp-1}{\phi }_i^{kp}\left(\zeta \right)}}{\psi }_i^{kp}\left(\zeta \right).$$

Remark 8.

For $\alpha =1$ in Corollary 7, then reduce to ([4], Theorem 4).

Corollary 8.

Let $\mathbb{T}=\mathbb{Z}$ in Theorem 3. For any $0<\alpha \le 1, k\ge 1, p>1$ and $r>0$. If there exist positive constant ${\lambda }_i$ such that

$$1+{\displaystyle \frac{u_i\left(\zeta +1\right)△w\left(\zeta \right)}{\left(1+kp\left(r-\alpha +1\right)\right)w\left(\zeta \right)△u_i\left(\zeta \right)}}\ge {\displaystyle \frac{1}{{\lambda }_i}},$$

then

$$\begin{array}{ccc}& & \sum _{a=\zeta }^{b-1}w\left(\zeta \right){\left(\sum _{i=1}^{\tau }{\mathrm{\Gamma }}_{ia}\left(\zeta \right)\right)}^{kp}{\zeta }^{\alpha -1}\hfill \\ & \le & \sum _{i=1}^{\tau }{\left({\displaystyle \frac{-{\tau }^{pk-1}kp{\lambda }_i}{1+kp\left(r-\alpha +1\right)}}\right)}^{kp}\left(\sum _{a=\zeta }^{b-1}w\left(\zeta \right)g_i\left(\zeta \right){\zeta }^{\alpha -1}\right),\hfill \end{array}$$

where

$${\mathrm{\Gamma }}_i\left(\zeta \right)=u_i^{r-\alpha +1}\left(\zeta +1\right)\sqrt[kp]{{\zeta }^{1-\alpha }△u_i\left(\zeta \right)}\sum _{\mu =0}^{\zeta -1}{\displaystyle \frac{△{\phi }_i\left(\mu \right)}{u_i(\mu +1){\phi }_i\left(\mu \right)}}{\psi }_i\left(\mu \right),$$

and

$$g_i\left(\zeta \right)={\displaystyle \frac{u_i^{kp\left(r-\alpha +1\right)}\left(\zeta +1\right)w^{kp}\left(\zeta +1\right){\zeta }^{1-\alpha }{\left(△{\phi }_i\left(\zeta \right)\right)}^{kp}}{{\left(△u_i\left(\zeta \right)\right)}^{kp-1}{w}^{kp}\left(\zeta \right){\phi }_i^{kp}\left(\zeta \right)}}{\psi }_i^{kp}\left(\zeta \right).$$

4. Conclusions

In this article, we used delta conformable calculus on time scales to explain some new weighted dynamic Hardy-type inequalities. Our proposed results show the possibility of producing some original discrete and continuous inequalities. Furthermore, we presented some inequalities as special cases: integral inequalities and discrete inequalities.