A Variety of Nabla Hardy’s Type Inequality on Time Scales

Abstract

The primary goal of this research is to prove some new Hardy-type ∇-conformable dynamic inequalities by employing product rule, integration by parts, chain rule and $(\gamma ,a)$-nabla Hölder inequality on time scales. The inequalities proved here extend and generalize existing results in the literature. Further, in the case when $\gamma =1$, we obtain some well-known time scale inequalities due to Hardy inequalities. Many special cases of the proposed results are obtained and analyzed such as new conformable fractional h-sum inequalities, new conformable fractional q-sum inequalities and new classical conformable fractional integral inequalities.

1. Introduction

Over several decades Hardy-type inequalities have been attracted many researchers and several refinements and extensions have been done to the previous results, we refer the reader to the works [1,2,3,4] and also the book [5].

Hardy [6] proved that.

Theorem 1.

If a sequence${\left\{\varrho \left(\varsigma \right)\right\}}_{\varsigma =1}^{\infty }\ge 0$is real valued. If$p>1$, we get

$$\sum _{\varsigma =1}^{\infty }\frac{1}{{\varsigma }^p}{\left(\sum _{\iota =1}^{\varsigma }\varrho \left(\iota \right)\right)}^p\le {\left(\frac{p}{p-1}\right)}^p\sum _{\varsigma =1}^{\infty }{\varrho }^p\left(\varsigma \right).$$

(1)

Hardy [7] proved the continuous version of (1).

Theorem 2.

Assume$\eta \ge 0$is continuous function on$[0,\infty )$. If$p>1$, we get

$${\int }_0^{\infty }\frac{1}{{\pi }^p}{\left({\int }_0^{\pi }\eta \left(\varpi \right)ds\right)}^{p}d\pi \le {\left(\frac{p}{p-1}\right)}^p{\int }_0^{\infty }{\eta }^p\left(\pi \right)d\pi .$$

(2)

where ${\left(\frac{p}{p-1}\right)}^p$ in (2) is sharp.

Copson [8] proved variant discrete version of Hardy inequality.

Theorem 3.

Let a sequence${\left\{\varrho \left(n\right)\right\}}_{n=1}^{\infty }\ge 0$be real valued nonnegative. For$1<p$, we have

$$\sum _{n=1}^{\infty }{\left(\sum _{j=n}^{\infty }\zeta \left(j\right)\right)}^p\le p^p\sum _{n=1}^{\infty }{\left(n\zeta \left(n\right)\right)}^p.$$

(3)

Renaud [9] proved the reverse versions of inequality (3).

Theorem 4.

Assume${\left\{\zeta \left(n\right)\right\}}_{n=1}^{\infty }$is a nonnegative and nonincreasing sequence of real numbers. If$p>1$, we get

$$\sum _{\varsigma =1}^{\infty }{\left(\sum _{\iota =\varsigma }^{\infty }\zeta \left(\iota \right)\right)}^p\ge \sum _{\varsigma =1}^{\infty }{\varsigma }^p{\zeta }^p\left(\varsigma \right).$$

(4)

Furthermore, he proved the following inequalities:

$${\int }_0^{\infty }{\left({\int }_{\pi }^{\infty }\zeta \left(\varpi \right)d\varpi \right)}^{p}d\pi \ge {\int }_0^{\infty }{\pi }^p{\zeta }^p\left(\pi \right)d\pi .$$

(5)

$${\int }_0^{\infty }\frac{1}{{\pi }^p}{\left({\int }_0^{\pi }\zeta \left(\varpi \right)d\varpi \right)}^{p}d\pi \ge \frac{p}{p-1}{\int }_0^{\infty }{\zeta }^p\left(\varpi \right)d\varpi .$$

(6)

In [10,11,12,13,14] many authors have studied many new dynamic inequalities. Řehák [14] is the first author proved the version of Hardy inequality on time scales that unifies (1) and (2).

Theorem 5.

Suppose the time scale$\mathbb{T}$, with$\mho \in C_{rd}\left({[a,\infty )}_{\mathbb{T}},[0,\infty )\right)$. If$p>1$, then

$${\int }_a^{\infty }{\left(\frac{{\int }_a^{\sigma \left(\varrho \right)}\eta \left(\varpi \right)\Delta \varpi }{\sigma \left(\varrho \right)-a}\right)}^p\Delta \varrho <{\left(\frac{p}{p-1}\right)}^p{\int }_a^{\infty }{\eta }^p\left(\varrho \right)\Delta \varrho ,$$

(7)

unless$\eta \equiv 0$.

Furthermore, if$\mu \left(\varrho \right)/\varrho \to 0$as$\varrho \to \infty$, then we get the sharp (7).

In [15] the authors extended (6) on genera time scales as follows: If $p>1$,

$${\int }_0^{\infty }\frac{1}{{\wp }^p}{\left({\int }_0^{\wp }\eta \left(\varpi \right)\Delta \varpi \right)}^p\Delta \wp \ge \frac{p}{p-1}{\int }_0^{\infty }{\eta }^p(\wp )\Delta \wp .$$

(8)

El-Deeb et al. [16] generalized (8) that unify (4) and (5). If $p\ge 1$ and $\gamma >1$, we get

$${\int }_a^{\infty }\frac{\tilde{\lambda }\left(\zeta \right){\breve{\Psi }}^p\left(\zeta \right)}{\tilde{\Lambda }\widehat{\gamma }\left(\zeta \right)}\Delta \zeta \ge \frac{p}{\widehat{\gamma }-1}{\int }_a^{\infty }\tilde{\lambda }\left(\zeta \right){\tilde{\Lambda }}^{p-\widehat{\gamma }}\left(\zeta \right){\eta }^p\left(\zeta \right)\Delta \zeta ,$$

(9)

where

$$\breve{\Psi }\left(\zeta \right)={\int }_a^{\zeta }\tilde{\lambda }\left(\varpi \right)\varpi \left(\varpi \right)\Delta \varpi ,\mathrm{and}\tilde{\Lambda }\left(\zeta \right)={\int }_a^{\zeta }\tilde{\lambda }\left(\varpi \right)\Delta \varpi .$$

Furthermore, in 2020, El-Deeb et al. [17] proved that if $\vartheta ,{\daleth }_2\ge 0$ such that ${\displaystyle \frac{w^{\Delta }(\wp )}{w(\wp )}\le \theta \left(\frac{G^{\Delta }(\wp )}{G^{\sigma }(\wp )}\right)}$ and ${\displaystyle \frac{v^{\Delta }(\wp )}{v^{\sigma }(\wp )}\le {\daleth }_2\left(\frac{K^{\Delta }(\wp )}{K(\wp )}\right)}$, where

$$G(\wp )={\int }_a^{\wp }g\left(\varpi \right)\Delta \varpi \mathrm{with}G(\infty )=\infty \mathrm{and}K(\wp )={\int }_a^{\wp }r\left(\varpi \right)\mho \left(\varpi \right)\Delta \varpi ,\wp \in {[a,\infty )}_{\mathbb{T}},$$

with $\alpha >\theta +1$, and $p\ge 1$, then

$$\begin{array}{ccc}\hfill & & {\int }_a^{\infty }{k}^{\sigma }(\wp )v^{\sigma }(\wp )w(\wp )g(\wp ){\left(G^{\sigma }(\wp )\right)}^{-\alpha }{\left(K^{\sigma }(\wp )\right)}^p\Delta \wp \hfill \\ & & \le {\left(\frac{p+{\daleth }_2}{\alpha -\theta -1}\right)}^p{\int }_a^{\infty }\frac{k^{\sigma }(\wp )v^{\sigma }(\wp )w(\wp )r^p(\wp ){\mho }^p(\wp ){\left(G^{\sigma }(\wp )\right)}^{\alpha (p-1)}}{g^{p-1}(\wp )G^{p(\alpha -1)}(\wp )}\Delta \wp .\hfill \end{array}$$

(10)

In [1,3,4,5], several authors proved many new inequalities. See also [18,19].

In [19], the authors proved:

$$\begin{array}{c}\hfill {\int }_{\omega }^{\infty }{\displaystyle \frac{\lambda (\wp )}{{\chi }^{k-\alpha +1}(\wp )}}{\left({\Theta }^{\sigma }(\wp )\right)}^h{\Delta }_{\alpha }\wp ⩾{\left(\frac{h}{\alpha -m}\right)}^h{\int }_{\omega }^{\infty }\lambda (\wp ){\xi }^h(\wp ){\chi }^{h-m+\alpha -1}(\wp ){\Delta }_{\alpha }\wp .\end{array}$$

Lemma 1

([20], $(\gamma ,a)$-nabla Hölder inequality on timescales). Let $d,b\in \mathbb{T}$ where $b>d$. If $\gamma \in (0,1]$ and $\eta ,\xi :\mathbb{T}⟶\mathbb{R}$, then

$${\int }_d^b\left|\eta (\wp )\xi (\wp )\right|{\nabla }_a^{\gamma }\wp \le {\left({\int }_d^b{\left|\eta (\wp )\right|}^p{\nabla }_a^{\gamma }\wp \right)}^{1/p}{\left({\int }_d^b{\left|\xi (\wp )\right|}^q{\nabla }_a^{\gamma }\wp \right)}^{1/q},$$

(11)

where $p,q>1$ and $1/p+1/q=1$. This inequality is reversed if $0<p<1$ and if $p<0,$ or $q<0$.

In this work, we prove and extend some new Hardy’s inequality obtained in [15,16,17] to a standard time scale and set up numerous new sharpened types of fractional conformable ∇-integral of order $\gamma \in (0,1]$ on time scales. As a new work, we generalize the inequalities presented in those papers. New discrete Hardy’s inequalities maybe proved by our results via conformable fractional on time scales.

2. Preliminaries

In PhD thesis of S. Hilger initiated the theory of time scale. This subject has since gained tremendous attention among many mathematician. It combines the continuous and discrete branch of mathematics into one theory [21]. In [11,22], the authors stated all basic rules and definitions regarding to the calculus of time scales. Most of the fractional analysis concepts maybe found in [11,22,23,24]. Suppose $\sigma :\mathbb{T}\to \mathbb{T}$, is forward jump operator

$$\sigma (\wp ):=inf\{\varpi \in \mathbb{T}:\varpi >\wp \},\wp \in \mathbb{T},$$

$\mu (\wp ):=\sigma (\wp )-\wp$ and is forward graininess function.

In [25], Bendouma et al. presenred a new version of the nabla fractional derivative on timescales. Given $ϵ>0,$ there is a $\delta$- neighborhood $U_{\wp }\subset \mathbb{T}$ of $\wp ,$ $\delta >0,$ such that

$$|[\eta \left(\rho (\wp )\right)-\eta \left(\varpi \right)]{\wp }^{1-\alpha }-{\wp }_{\nabla ,\alpha }\left(\eta \right)(\wp )[\rho (\wp )-\varpi ]|\le \epsilon |\rho (\wp )-\varpi |$$

for all $\varpi \in U_{\wp }.$ We define the nabla fractional integral as

$$\int \eta (\wp ){\nabla }_{\alpha }\wp =\int \eta (\wp ){\wp }^{\alpha -1}\nabla \wp .$$

By using ${\widehat{G}}_n(\wp ,\varpi )$ for $\varpi$, $\wp \in \mathbb{T}$, Rahmat et al. [23] presented nabla derivative conformable which generalizes the claim in [26].

Definition 1.

Suppose$[\varpi ,\wp ]\subset \mathbb{T}$and$\varpi <\wp .$${\widehat{G}}_n:\mathbb{T}\times \mathbb{T}⟶{\mathbb{R}}^{+}$, $n\in {\mathbb{N}}_0$is presented as the generalized time scale power function by

$${\widehat{G}}_n(\wp ,\varpi )=\left\{\begin{array}{cc}{(\wp -\varpi )}^n,\hfill & \mathit{if}[\wp ,\varpi ]\mathit{dense};\hfill \\ \prod _{j=0}^{n-1}(\wp -{\rho }^j\left(\varpi \right)),\hfill & \mathit{if}[\wp ,\varpi ]\mathit{isolated};\hfill \end{array}\right.$$

(12)

with inverse${\widehat{G}}_{-n}:\mathbb{T}\times \mathbb{T}⟶{\mathbb{R}}^{+}$

$${\widehat{G}}_{-n}(\wp ,\varpi )=\left\{\begin{array}{cc}{(\wp -\varpi )}^{-n},\hfill & \mathit{if}[\wp ,\varpi ]\mathit{dense};\hfill \\ \frac{1}{{\prod }_{j=0}^{n-1}({\rho }^n(\wp )-{\rho }^j\left(\varpi \right))},\hfill & \mathit{if}[\wp ,\varpi ]\mathit{isolated}.\hfill \end{array}\right.$$

(13)

We use the convention${\widehat{G}}_0(\wp ,\varpi )=1$for all$\varpi ,\wp \in \mathbb{T}.$

Definition 2

(Conformable nabla derivative). Suppose $a\in \mathbb{T},$℧ is ∇ differentiable at $\wp >a,$$\mho :\mathbb{T}⟶\mathbb{R}$ and let it be nable differentiable at ℘, and its ∇ derivative is given by

$${\nabla }_a^{\gamma }\mho (\wp )={\widehat{G}}_{1-\gamma }(\wp ,a){\mho }^{\nabla }(\wp )\wp >a,$$

since ${\widehat{G}}_{1-\gamma }(\wp ,a)$ as defined in (12). Suppose ${\nabla }_a^{\gamma }[\mho (\wp )]$ exists in ${(a,a+ϵ)}_{\mathbb{T}},ϵ>0,$ then

$${\nabla }_a^{\gamma }[\mho \left(a\right)]=\underset{\wp ⟶a^{+}}{lim}{\nabla }_a^{\gamma }[\mho (\wp )]$$

suppose the existence of ${lim}_{\wp ⟶a^{+}}{\nabla }_a^{\gamma }[\mho (\wp )]$.

We have $\zeta \xi :\mathbb{T}⟶\mathbb{R}$ is differentiable with

$${\nabla }_a^{\gamma }\left(\zeta \xi \right)=\left[{\nabla }_a^{\gamma }\zeta (\wp )\right]\xi (\wp )+{\zeta }^{\rho }(\wp )\left[{\nabla }_a^{\gamma }\xi (\wp )\right].$$

(14)

Lemma 2

(Integration by parts). Suppose that $d,$ $b\in \mathbb{T}$ where $b>d.$ If $\eta ,$ ξ are conformable $(\gamma ,a)$- nabla fractional differentiable and $\gamma \in (0,1],$ then:

$${\int }_d^b\eta (\wp )\left[{\nabla }_a^{\gamma }\xi (\wp )\right]{\nabla }_a^{\gamma }\wp ={\left[\eta (\wp )\xi (\wp )\right]}_d^b-{\int }_d^b\left[{\nabla }_a^{\gamma }\eta (\wp )\right]{\xi }^{\rho }(\wp ){\nabla }_a^{\gamma }\wp .$$

(15)

If $c\in {[\rho \left(\tau \right),\tau ]}_{\mathbb{R}}$, we have

$${\nabla }_a^{\gamma }({\rm Y}\circ \zeta )\left(t\right)={{\rm Y}}^{\prime }\left(\zeta \left(c\right)\right){\nabla }_a^{\gamma }\left(\zeta \left(t\right)\right).$$

(16)

Furthermore, we have

$$\begin{array}{ccc}\hfill {\nabla }_a^{-\gamma }\mho (\wp )& =& {\int }_{{\wp }_1}^{{\wp }_2}\mho \left(\tau \right){\nabla }_a^{\gamma }\tau \hfill \\ & =& {\int }_{{\wp }_1}^{{\wp }_2}\mho \left(\tau \right){\widehat{G}}_{\gamma -1}({\sigma }^{\gamma -1}\left(\tau \right),a)\nabla \tau ,\hfill \end{array}$$

(17)

and for $\mathbb{T}=\mathbb{R}$

$${\int }_a^{\wp }\mho \left(\pi \right){\nabla }_a^{\gamma }\pi ={\int }_a^{\wp }\mho \left(\pi \right){(\pi -a)}^{\gamma -1}d\pi .$$

(18)

With $h>0,$, if $\mathbb{T}=h\mathbb{Z}$, we have h-sum given by

$${\int }_a^{\wp }\mho \left(\pi \right){\nabla }_a^{\gamma }\pi =\sum _{\pi \in (a,\wp ]}h\mho \left(\pi \right){\left({\rho }^{\gamma -1}\left(\pi \right)-a\right)}_h^{(\gamma -1)}.$$

(19)

For $\mathbb{T}=q^{{\mathbb{N}}_0},$ we get

$${\int }_a^{\wp }\mho \left(\pi \right){\nabla }_a^{\gamma }\pi =\sum _{\pi \in (a,\wp ]}\pi (1-\tilde{q})\mho \left(\pi \right){\left({\rho }^{\gamma -1}\left(\pi \right)-a\right)}_{\tilde{q}}^{(\gamma -1)}.$$

(20)

3. Main Results

In this section, we assume that $\mathbb{T}$ is unbounded above.

Theorem 6.

Let$\mathbb{T}$be time scale for$m\in {[0,\infty )}_{\mathbb{T}}$, $\gamma \in (0,1]$and$a>\wp$. Further, assume the rd-continuous functions v, r, g, ℧, k,$w\ge 0$, on${[m,\infty )}_{\mathbb{T}}$with k is nondecreasing. Further, suppose we have$\vartheta ,{\daleth }_2\ge 0$: ${\displaystyle \frac{{\nabla }_a^{\gamma }w(\wp )}{w(\wp )}⩾\vartheta \left(\frac{{\nabla }_a^{\gamma }G(\wp )}{G(\wp )}\right)}$and${\displaystyle \frac{{\nabla }_a^{\gamma }v(\wp )}{v^{\rho }(\wp )}⩾{\daleth }_2\left(\frac{{\nabla }_a^{\gamma }K(\wp )}{K(\wp )}\right)}$, where

$$G(\wp )={\int }_m^{\wp }g\left(\varpi \right){\nabla }_a^{\gamma }\varpi withG(\infty )=\infty andK(\wp )={\int }_m^{\wp }r\left(\varpi \right)\mho \left(\varpi \right){\nabla }_a^{\gamma }\varpi ,\wp \in {[m,\infty )}_{\mathbb{T}}.$$

If$0<p<\gamma$and${\daleth }_1>\vartheta +1$, then

$$\begin{array}{ccc}\hfill & & {\int }_m^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1-1}{\left(K^{\rho }(\wp )\right)}^{p-\gamma +1}{\nabla }_a^{\gamma }\wp \hfill \\ \hfill & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }\right)}^p{\int }_m^{\infty }\frac{k^{\rho }(\wp )v^{\rho }(\wp )w(\wp )r^p(\wp ){\mho }^p(\wp ){\left(G^{\rho }(\wp )\right)}^{(1-\gamma +{\daleth }_1)(p-1)}{K}^{1-\gamma }(\wp )}{g^{p-1}(\wp )G^{p({\daleth }_1-\gamma )}(\wp )}{\nabla }_a^{\gamma }\wp .\hfill \end{array}$$

Proof. 

Using (15) with

$${\nabla }_a^{\gamma }u(\wp )=w(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1-1}{z}^{\rho }(\wp )=k^{\rho }(\wp )v^{\rho }(\wp ){\left(K^{\rho }(\wp )\right)}^{p-\gamma +1},$$

we have

$$\begin{array}{ccc}\hfill & & {\int }_m^{\infty }{k}^{\rho }(\wp )v\rho (\wp )w(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1-1}{\left(K^{\rho }(\wp )\right)}^{p-\gamma +1}{\nabla }_a^{\gamma }\wp \hfill \\ & & ={\left[u(\wp )k(\wp )v(\wp )K^{p-\gamma +1}(\wp )\right]}_m^{\infty }+{\int }_m^{\infty }\left(-u(\wp )\right){\nabla }_a^{\gamma }\left(k(\wp )v(\wp )K^{p-\gamma +1}(\wp )\right){\nabla }_a^{\gamma }\wp ,\hfill \end{array}$$

(21)

where we assumed that

$$u(\wp )=-{\int }_{\wp }^{\infty }w\left(\varpi \right)g\left(\varpi \right){\left(G^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1-1}{\nabla }_a^{\gamma }\varpi .$$

Applying (16), (14), and the hypothesis ${\displaystyle \frac{{\nabla }_a^{\gamma }w(\wp )}{w(\wp )}⩾\vartheta \left(\frac{{\nabla }_a^{\gamma }G(\wp )}{G^{\rho }(\wp )}\right)}$, we have $c\in \left[\rho \right(\varpi ),\varpi ]$ with

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left(w\left(\varpi \right)G^{\gamma -{\daleth }_1}\left(\varpi \right)\right)& =& {\nabla }_a^{\gamma }w\left(\varpi \right){\left(G^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1}+w\left(\varpi \right){\nabla }_a^{\gamma }(G^{\gamma -{\daleth }_1}\left(\varpi \right))\hfill \\ & ⩾& \vartheta w\left(\varpi \right){\nabla }_a^{\gamma }G\left(\varpi \right){\left(G^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1-1}+(\gamma -{\daleth }_1)w\left(\varpi \right)G^{\gamma -{\daleth }_1-1}\left(c\right){\nabla }_a^{\gamma }G\left(\varpi \right).\hfill \end{array}$$

Since ${\nabla }_a^{\gamma }G\left(\varpi \right)=g\left(\varpi \right)\ge 0$, $c⩾\rho (\wp )$ and ${\daleth }_1>\vartheta +\gamma$, we get

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left(w\left(\varpi \right)G^{\gamma -{\daleth }_1}\left(\varpi \right)\right)& ⩾& \vartheta w\left(\varpi \right)g\left(\varpi \right){\left(G^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1-1}+(\gamma -{\daleth }_1)w\left(\varpi \right)g\left(\varpi \right){\left(G^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1-1}\hfill \\ & =& (\gamma -{\daleth }_1+\vartheta )w\left(\varpi \right)g\left(\varpi \right){\left(G^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1-1}.\hfill \end{array}$$

This gives us that

$$w\left(\varpi \right)g\left(\varpi \right){\left(G^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1-1}⩾\frac{1}{\gamma -{\daleth }_1+\vartheta }{\nabla }_a^{\gamma }\left(w\left(\varpi \right)G^{\gamma -{\daleth }_1}\left(\varpi \right)\right).$$

Hence

$$\begin{array}{ccc}\hfill -u(\wp )={\int }_{\wp }^{\infty }w\left(\varpi \right)g\left(\varpi \right){\left(G^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1-1}{\nabla }_a^{\gamma }\varpi & ⩾& \frac{1}{\gamma -{\daleth }_1+\vartheta }{\int }_{\wp }^{\infty }{\nabla }_a^{\gamma }\left(w\left(\varpi \right)G^{\gamma -{\daleth }_1}\left(\varpi \right)\right){\nabla }_a^{\gamma }\varpi \hfill \\ & =& \frac{1}{{\daleth }_1-\vartheta -\gamma }w(\wp )G^{\gamma -{\daleth }_1}(\wp ).\hfill \end{array}$$

(22)

Applying (14) and (16), we have $c\in \left[\rho \right(\wp ),\wp ]$ with

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left(k(\wp )v(\wp )K^{p-\gamma +1}(\wp )\right)& =& {\nabla }_a^{\gamma }\left(k(\wp )v(\wp )\right)K^{p-\gamma +1}(\wp )+k^{\rho }(\wp )v^{\rho }(\wp ){\nabla }_a^{\gamma }\left(K^{p-\gamma +1}(\wp )\right)\hfill \\ & =& {\nabla }_a^{\gamma }k(\wp )v(\wp )K^{p-\gamma +1}(\wp )+k^{\rho }(\wp ){\nabla }_a^{\gamma }v(\wp )K^{p-\gamma +1}(\wp )\hfill \\ & & +(p-\gamma +1)k^{\rho }(\wp )v^{\rho }(\wp )K^{p-\gamma }\left(c\right){\nabla }_a^{\gamma }K(\wp ).\hfill \end{array}$$

Since ${\nabla }_a^{\gamma }k(\wp )⩾0$, ${\nabla }_a^{\gamma }K(\wp )=r(\wp )\mho (\wp )\ge 0$, $c\le \wp$, $0<p<\gamma$ and ${\displaystyle \frac{{\nabla }_a^{\gamma }v(\wp )}{v^{\rho }(\wp )}⩾{\daleth }_2\left(\frac{{\nabla }_a^{\gamma }K(\wp )}{K(\wp )}\right)}$, we have

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left(k(\wp )v(\wp )K^{p-\gamma +1}(\wp )\right)& ⩾& {\daleth }_2{k}^{\rho }(\wp )v^{\rho }(\wp )r(\wp )\mho (\wp )K^{p-\gamma }(\wp )+(p-\gamma +1)k^{\rho }(\wp )v^{\rho }(\wp )r(\wp )\mho (\wp )K^{p-\gamma }(\wp )\hfill \\ & ⩾& (p+{\daleth }_2-\gamma +1)k^{\rho }(\wp )v^{\rho }(\wp )r(\wp )\mho (\wp )K^{p-\gamma }(\wp ).\hfill \end{array}$$

(23)

From (21)–(23), with $K\left(m\right)=0$ and $u(\infty )=0$ we have

$$\begin{array}{ccc}& & {\int }_m^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1-1}{\left(K^{\rho }(\wp )\right)}^{p-\gamma +1}{\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }{\int }_m^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )r(\wp )\mho (\wp )G^{\gamma -{\daleth }_1}(\wp )K^{p-\gamma }(\wp ){\nabla }_a^{\gamma }\wp ,\hfill \end{array}$$

or equivalently,

$$\begin{array}{ccc}& & {\int }_m^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1-1}{\left(K^{\rho }(\wp )\right)}^{p-\gamma +1}{\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }{\int }_m^{\infty }\left({\left(k^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp )\right)}^{(p-1)/p}{\left(G^{\rho }(\wp )\right)}^{(\gamma -{\daleth }_1-1)(p-1)/p}{K}^{(p-1)(p-\gamma +1)/p}(\wp )\right)\hfill \\ & & \times \left(\frac{{\left(k^{\rho }(\wp )v^{\rho }(\wp )w(\wp )\right)}^{1/p}r(\wp )\mho (\wp ){\left(G^{\rho }(\wp )\right)}^{(1-\gamma +{\daleth }_1)(p-1)/p}{K}^{(1-\gamma )/p}(\wp )}{g^{(p-1)/p}(\wp )G^{{\daleth }_1-\gamma }(\wp )}\right){\nabla }_a^{\gamma }\wp .\hfill \end{array}$$

Applying (11) with index p and index $p/(p-1)$, we obtain

$$\begin{array}{ccc}& & {\int }_a^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{(\gamma -{\daleth }_1-1)}{\left(K^{\rho }(\wp )\right)}^{(p-\gamma +1)}{\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }{\left({\int }_a^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{(\gamma -{\daleth }_1-1)}{K}^{(p-\gamma +1)}(\wp ){\nabla }_a^{\gamma }\wp \right)}^{(p-1)/p}\hfill \\ & & \times {\left({\int }_a^{\infty }\frac{k^{\rho }(\wp )v^{\rho }(\wp )w(\wp )r^p(\wp ){\mho }^p(\wp ){\left(G^{\rho }(\wp )\right)}^{(1+{\daleth }_1-\gamma )(p-1)}{K}^{1-\gamma }(\wp )}{g^{p-1}(\wp )G^{p({\daleth }_1-\gamma )}(\wp )}{\nabla }_a^{\gamma }\wp \right)}^{1/p},\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill & & {\int }_m^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1-1}{\left(K^{\rho }(\wp )\right)}^{p-\gamma +1}{\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }\right)}^p{\int }_m^{\infty }\frac{k^{\rho }(\wp )v^{\rho }(\wp )w(\wp )r^p(\wp ){\mho }^p(\wp ){\left(G^{\rho }(\wp )\right)}^{(1-\gamma +{\daleth }_1)(p-1)}{K}^{1-\gamma }(\wp )}{g^{p-1}(\wp )G^{p({\daleth }_1-\gamma )}(\wp )}{\nabla }_a^{\gamma }\wp .\hfill \end{array}$$

We get our claim. □

Remark 1.

If one put$\gamma =1$in Theorem 6, then we get the following inequality

$$\begin{array}{ccc}\hfill & & {\int }_m^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{-{\daleth }_1}{\left(K^{\rho }(\wp )\right)}^p\nabla \wp \hfill \\ & & ⩾{\left(\frac{p+{\daleth }_2}{{\daleth }_1-\vartheta -1}\right)}^p{\int }_m^{\infty }\frac{k^{\rho }(\wp )v^{\rho }(\wp )w(\wp )r^p(\wp ){\mho }^p(\wp ){\left(G^{\rho }(\wp )\right)}^{{\daleth }_1(p-1)}}{g^{p-1}(\wp )G^{p({\daleth }_1-1)}(\wp )}\nabla \wp .\hfill \end{array}$$

Corollary 1.

When one put$\mathbb{T}=\mathbb{R}$in (21) we get

$$\begin{array}{ccc}& & {\int }_m^{\infty }k(\wp )v(\wp )w(\wp )g(\wp )G^{\gamma -{\daleth }_1-1}(\wp )K^{p-\gamma +1}(\wp ){(\wp -a)}^{\gamma -1}dt\hfill \\ \hfill & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }\right)}^p{\int }_m^{\infty }\frac{k(\wp )v(\wp )w(\wp )r^p(\wp ){\mho }^p(\wp )G^{p-{\daleth }_1+\gamma -1}(\wp )K^{1-\gamma }(\wp )}{g^{p-1}(\wp )}{(\wp -a)}^{\gamma -1}dt.\hfill \end{array}$$

where

$$G(\wp )={\int }_m^{\wp }g\left(\varpi \right){(\varpi -a)}^{\gamma -1}d\varpi withG(\infty )=\infty andK(\wp )={\int }_m^{\wp }r\left(\varpi \right)\mho \left(\varpi \right){(\varpi -a)}^{\gamma -1}d\varpi .$$

Corollary 2.

For$\mathbb{T}=h\mathbb{Z},$$h>0$in (21) we obtain

$$\begin{array}{ccc}& & \sum _{\wp =m}^{\infty }k(\wp -h)v(\wp -h)w(\wp )g(\wp )G^{\gamma -{\daleth }_1-1}(\wp -h)K^{p-\gamma +1}(\wp -h)({\rho }^{\gamma -1}{(\wp )-a)}_h^{(\gamma -1)}\hfill \\ \hfill & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }\right)}^p\sum _{\wp =m}^{\infty }\frac{k(\wp -h)v(\wp -h)w(\wp )r^p(\wp ){\mho }^p(\wp ){\left(G(\wp -h)\right)}^{(1-\gamma +{\daleth }_1)(p-1)}{K}^{1-\gamma }(\wp )({\rho }^{\gamma -1}{(\wp )-a)}_h^{(\gamma -1)}}{g^{p-1}(\wp )G^{p({\daleth }_1-\gamma )}(\wp )}.\hfill \end{array}$$

where

$$G(\wp )=h\sum _{\varpi =m}^{\wp }g\left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}_h^{(\gamma -1)}withG(\infty )=\infty andK(\wp )=h\sum _{\varpi =m}^{\wp }r\left(\varpi \right)\mho \left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}_h^{(\gamma -1)}.$$

Corollary 3.

If$\mathbb{T}=\mathbb{Z},$with$h=1$in our Corollary 16 we have from (21) that

$$\begin{array}{ccc}& & \sum _{\wp =m}^{\infty }k(\wp -1)v(\wp -1)w(\wp )g(\wp )G^{\gamma -{\daleth }_1-1}(\wp -1)K^{p-\gamma +1}(\wp -1)({\rho }^{\gamma -1}{(\wp )-a)}^{(\gamma -1)}\hfill \\ \hfill & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }\right)}^p\sum _{\wp =m}^{\infty }\frac{k(\wp -1)v(\wp -1)w(\wp )r^p(\wp ){\mho }^p(\wp ){\left(G(\wp -1)\right)}^{(1-\gamma +{\daleth }_1)(p-1)}{K}^{1-\gamma }(\wp )({\rho }^{\gamma -1}{(\wp )-a)}^{(\gamma -1)}}{g^{p-1}(\wp )G^{p({\daleth }_1-\gamma )}(\wp )}.\hfill \end{array}$$

where

$$G(\wp )=h\sum _{\varpi =m}^{\wp }g\left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}^{(\gamma -1)}withG(\infty )=\infty andK(\wp )=h\sum _{\varpi =m}^{\wp }r\left(\varpi \right)\mho \left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}^{(\gamma -1)}.$$

Corollary 4.

For$\mathbb{T}=q^{{\mathbb{N}}_0},$in (21) we obtain

$$\begin{array}{ccc}& & \sum _{\wp =m}^{\infty }\wp ({\rho }^{\gamma -1}{(\wp )-a)}_{\tilde{q}}^{(\gamma -1)}{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1-1}{\left(K^{\rho }(\wp )\right)}^{p-\gamma +1}\hfill \\ \hfill & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }\right)}^p\sum _{\wp =m}^{\infty }\frac{\wp ({\rho }^{\gamma -1}{(\wp )-a)}_{\tilde{q}}^{(\gamma -1)}{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )r^p(\wp ){\mho }^p(\wp ){\left(G^{\rho }(\wp )\right)}^{(1-\gamma +{\daleth }_1)(p-1)}{K}^{1-\gamma }(\wp )}{g^{p-1}(\wp )G^{p({\daleth }_1-\gamma )}(\wp )}.\hfill \end{array}$$

where

$$G(\wp )=(\tilde{q}-1)\sum _{\varpi =m}^{\wp }\varpi ({\rho }^{\gamma -1}\left(\varpi \right)-a{)}_{\tilde{q}}^{(\gamma -1)}g\left(\varpi \right)withG(\infty )=\infty andK(\wp )=(\tilde{q}-1)\sum _{\varpi =m}^{\wp }\varpi ({\rho }^{\gamma -1}\left(\varpi \right)-a{)}_{\tilde{q}}^{(\gamma -1)}r\left(\varpi \right)\mho \left(\varpi \right).$$

Theorem 7.

Suppose$\mathbb{T}$is a time scale with$m\in {[0,\infty )}_{\mathbb{T}},$$\gamma \in (0,1]$and$a>\wp$. Furthermore, suppose the rd-continuous functions w, k, ℧, g, r,$v\ge 0$on${[m,\infty )}_{\mathbb{T}}$with k nonincreasing. Furthermore, assume we have$\vartheta ,{\daleth }_2\ge 0$with${\displaystyle \frac{{\nabla }_a^{\gamma }w(\wp )}{w^{\rho }(\wp )}⩽\vartheta \left(\frac{{\nabla }_a^{\gamma }G(\wp )}{G(\wp )}\right)}$and${\displaystyle \frac{{\nabla }_a^{\gamma }v(\wp )}{v(\wp )}⩽{\daleth }_2\left(\frac{{\nabla }_a^{\gamma }\mho (\wp )}{{\mho }^{\rho }(\wp )}\right)}$, where

$$G(\wp )={\int }_m^{\wp }g\left(\varpi \right){\nabla }_a^{\gamma }\varpi withG(\infty )=\infty ,and\mho (\wp )={\int }_{\wp }^{\infty }r\left(\varpi \right)\mho \left(\varpi \right){\nabla }_a^{\gamma }\varpi ,\wp \in {[m,\infty )}_{\mathbb{T}}.$$

If $0<p<\gamma$ and $0\le {\daleth }_1<\gamma$, then

$$\begin{array}{ccc}\hfill & & {\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1+1}{\mho }^{p-\gamma +1}(\wp ){\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{\gamma -{\daleth }_1+\vartheta }\right)}^p{\int }_m^{\infty }\frac{k(\wp )v(\wp )w^{\rho }(\wp )r^p(\wp ){\mho }^p(\wp ){\left(G^{\rho }(\wp )\right)}^{p-{\daleth }_1+\gamma -1}{\left({\mho }^{\rho }(\wp )\right)}^{1-\gamma }}{g^{p-1}(\wp )}{\nabla }_a^{\gamma }\wp .\hfill \end{array}$$

(24)

Proof. 

Applying (15), we get

$$\begin{array}{ccc}\hfill & & {\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1-1}{\mho }^{p-\gamma +1}(\wp ){\nabla }_a^{\gamma }\wp \hfill \\ & & ={\left[u(\wp )k(\wp )v(\wp ){\mho }^{p-\gamma +1}(\wp )\right]}_a^{\infty }+{\int }_m^{\infty }{u}^{\rho }(\wp ){\nabla }_a^{\gamma }\left(-k(\wp )v(\wp ){\mho }^{p-\gamma +1}(\wp )\right){\nabla }_a^{\gamma }\wp ,\hfill \end{array}$$

(25)

where

$$u(\wp )={\int }_m^{\wp }{w}^{\rho }\left(\varpi \right)g\left(\varpi \right){\left(G^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1-1}{\nabla }_a^{\gamma }\varpi .$$

Applying (16), (14), and the assumption ${\displaystyle \frac{{\nabla }_a^{\gamma }w(\wp )}{w^{\rho }(\wp )}⩽\vartheta \left(\frac{{\nabla }_a^{\gamma }G(\wp )}{G(\wp )}\right)}$, we have $c\in \left[\rho \right(\varpi ),\varpi ]$ with

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left(w\left(\varpi \right)G^{\gamma -{\daleth }_1}\left(\varpi \right)\right)& =& {\nabla }_a^{\gamma }w\left(\varpi \right)G^{\gamma -{\daleth }_1}\left(\varpi \right)+w^{\rho }\left(\varpi \right){\nabla }_a^{\gamma }(G^{\gamma -{\daleth }_1}\left(\varpi \right))\hfill \\ & ⩽& \vartheta w^{\rho }\left(\varpi \right)G^{\gamma -{\daleth }_1-1}\left(\varpi \right){\nabla }_a^{\gamma }G\left(\varpi \right)+(\gamma -{\daleth }_1)w^{\rho }\left(\varpi \right)G^{\gamma -{\daleth }_1-1}\left(c\right){\nabla }_a^{\gamma }G\left(\varpi \right).\hfill \end{array}$$

As ${\nabla }_a^{\gamma }G\left(\varpi \right)=g\left(\varpi \right)\ge 0$, $c⩾\rho \left(\varpi \right)$ and $0\le {\daleth }_1<\gamma$, we get

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left(w\left(\varpi \right)G^{\gamma -{\daleth }_1}\left(\varpi \right)\right)& ⩽& \vartheta w^{\rho }\left(\varpi \right)g\left(\varpi \right){\left(G^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1-1}+(\gamma -{\daleth }_1)w^{\rho }\left(\varpi \right)g\left(\varpi \right){\left(G^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1-1}\hfill \\ & =& (\gamma -{\daleth }_1+\vartheta )w^{\rho }\left(\varpi \right)g\left(\varpi \right){\left(G^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1-1},\hfill \end{array}$$

which implies

$$w^{\rho }\left(\varpi \right)g\left(\varpi \right){\left(G^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1-1}⩾\frac{1}{\gamma -{\daleth }_1+\vartheta }{\nabla }_a^{\gamma }\left(w\left(\varpi \right)G^{\gamma -{\daleth }_1}\left(\varpi \right)\right).$$

Therefore

$$\begin{array}{ccc}\hfill u^{\rho }(\wp )={\int }_m^{\rho (\wp )}{w}^{\rho }\left(\varpi \right)g\left(\varpi \right){\left(G^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1-1}{\nabla }_a^{\gamma }\varpi & ⩾& \frac{1}{\gamma -{\daleth }_1+\vartheta }{\int }_m^{\rho (\wp )}{\nabla }_a^{\gamma }\left(w\left(\varpi \right)G^{\gamma -{\daleth }_1}\left(\varpi \right)\right){\nabla }_a^{\gamma }\varpi \hfill \\ & =& \frac{1}{\gamma -{\daleth }_1+\vartheta }{w}^{\rho }(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1}.\hfill \end{array}$$

(26)

Utilizing (14) and (16), we have $c\in \left[\rho \right(\wp ),\wp ]$ with

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left(-k(\wp )v(\wp ){\mho }^{p-{\daleth }_1+1}(\wp )\right)& =& -\left({\nabla }_a^{\gamma }\left(k(\wp )v(\wp )\right){\left({\mho }^{\rho }(\wp )\right)}^{p-{\daleth }_1+1}+k(\wp )v(\wp ){\nabla }_a^{\gamma }\left({\mho }^{p-\gamma +1}(\wp )\right)\right)\hfill \\ & =& -({\nabla }_a^{\gamma }k(\wp )v^{\rho }(\wp ){\left({\mho }^{\rho }(\wp )\right)}^{p-\gamma +1}+k(\wp ){\nabla }_a^{\gamma }v(\wp ){\left({\mho }^{\rho }(\wp )\right)}^{p-\gamma +1}\hfill \\ & & +(p-\gamma +1)k(\wp )v(\wp ){\mho }^{p-\gamma }\left(c\right){\nabla }_a^{\gamma }\mho (\wp )).\hfill \end{array}$$

Since ${\nabla }_a^{\gamma }k(\wp )⩽0$, ${\nabla }_a^{\gamma }\mho (\wp )=-r(\wp )\mho (\wp )\le 0$, $c⩾\rho (\wp )$, $0<p<\gamma$ and ${\displaystyle \frac{{\nabla }_a^{\gamma }v(\wp )}{v(\wp )}⩽{\daleth }_2\left(\frac{{\nabla }_a^{\gamma }\mho (\wp )}{{\mho }^{\rho }(\wp )}\right)}$, we get

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left(-k(\wp )v(\wp ){\mho }^{p-\gamma +1}(\wp )\right)& ⩾& {\daleth }_{2}k(\wp )v(\wp )r(\wp )\mho (\wp ){\left({\mho }^{\rho }(\wp )\right)}^{p-\gamma }+(p-\gamma +1)k(\wp )v(\wp )r(\wp )\mho (\wp ){\mho }^{p-\gamma }(\wp )\hfill \\ & ⩾& (p-\gamma +{\daleth }_2+1)k(\wp )v(\wp )r(\wp )\mho (\wp ){\left({\mho }^{\rho }(\wp )\right)}^{p-\gamma }.\hfill \end{array}$$

(27)

From (25)–(27), with $\mho (\infty )=0$ and $u\left(m\right)=0$ we obtain

$$\begin{array}{ccc}& & {\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1-1}{\mho }^{p-\gamma +1}(\wp ){\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾\frac{p-\gamma +{\daleth }_2+1}{\gamma -{\daleth }_1+\vartheta }{\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )r(\wp )\mho (\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1}{\left({\mho }^{\rho }(\wp )\right)}^{p-\gamma }{\nabla }_a^{\gamma }\wp ,\hfill \end{array}$$

which is equivalent to

$$\begin{array}{ccc}& & {\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1-1}{\mho }^{p-\gamma +1}(\wp ){\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾\frac{p+{\daleth }_2-\gamma +1}{\gamma -{\daleth }_1+\vartheta }{\int }_m^{\infty }\left({\left(k(\wp )v(\wp )w^{\rho }(\wp )g(\wp )\right)}^{(p-1)/p}{\left(G^{\rho }(\wp )\right)}^{(\gamma -{\daleth }_1-1)(p-1)/p}{\left({\mho }^{\rho }(\wp )\right)}^{(p-1)(p-\gamma +1)/p}\right)\hfill \\ & & \times \left(\frac{{\left(k(\wp )v(\wp )w^{\rho }(\wp )\right)}^{1/p}r(\wp )\mho (\wp ){\left(G^{\rho }(\wp )\right)}^{(p-{\daleth }_1+\gamma -1)/p}{\left({\mho }^{\rho }(\wp )\right)}^{(1-\gamma )/p}}{g^{(p-1)/p}(\wp )}\right){\nabla }_a^{\gamma }\wp .\hfill \end{array}$$

Applying (11) with the index p and the index $p/(p-1)$, obtains

$$\begin{array}{ccc}& & {\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1+1}{\mho }^{p-\gamma +1}(\wp ){\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾\frac{p+{\daleth }_2-\gamma +1}{\gamma -{\daleth }_1+\vartheta }{\left({\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1-1}{\left({\mho }^{\rho }(\wp )\right)}^{p-\gamma +1}{\nabla }_a^{\gamma }\wp \right)}^{(p-1)/p}\hfill \\ & & \times {\left({\int }_m^{\infty }\frac{k(\wp )v(\wp )w^{\rho }(\wp )r^p(\wp ){\mho }^p(\wp ){\left(G^{\rho }(\wp )\right)}^{p-{\daleth }_1+\gamma -1}{\left({\mho }^{\rho }(\wp )\right)}^{1-\gamma }}{g^{p-1}(\wp )}{\nabla }_a^{\gamma }\wp \right)}^{1/p}.\hfill \end{array}$$

This leads to

$$\begin{array}{ccc}& & {\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1+1}{\mho }^{p-\gamma +1}(\wp ){\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{\gamma -{\daleth }_1+\vartheta }\right)}^p{\int }_m^{\infty }\frac{k(\wp )v(\wp )w^{\rho }(\wp )r^p(\wp ){\mho }^p(\wp ){\left(G^{\rho }(\wp )\right)}^{p-{\daleth }_1+\gamma -1}{\left({\mho }^{\rho }(\wp )\right)}^{1-\gamma }}{g^{p-1}(\wp )}{\nabla }_a^{\gamma }\wp ,\hfill \end{array}$$

which is our claim. □

Remark 2.

In Theorem 7, if we take$\gamma =1$then we get the following inequality

$$\begin{array}{ccc}& & {\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{-{\daleth }_1}{\mho }^p(\wp )\nabla \wp \hfill \\ & & ⩾{\left(\frac{p+{\daleth }_2}{1-{\daleth }_1+\vartheta }\right)}^p{\int }_m^{\infty }\frac{k(\wp )v(\wp )w^{\rho }(\wp )r^p(\wp ){\mho }^p(\wp ){\left(G^{\rho }(\wp )\right)}^{p-{\daleth }_1}}{g^{p-1}(\wp )}\nabla \wp .\hfill \end{array}$$

Corollary 5.

Putting$\mathbb{T}=\mathbb{R}$in (24) we obtain

$$\begin{array}{ccc}\hfill & & {\int }_m^{\infty }k(\wp )v(\wp )w(\wp )g(\wp ){\left(G(\wp )\right)}^{\gamma -{\daleth }_1+1}{\mho }^{p-\gamma +1}(\wp ){(\wp -a)}^{\gamma -1}dt\hfill \\ & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{\gamma -{\daleth }_1+\vartheta }\right)}^p{\int }_m^{\infty }\frac{k(\wp )v(\wp )w(\wp )r^p(\wp ){\mho }^p(\wp ){\left(G(\wp )\right)}^{p-{\daleth }_1+\gamma -1}{\mho }^{1-\gamma }(\wp ){(\wp -a)}^{\gamma -1}dt}{g^{p-1}(\wp )}dt,\hfill \end{array}$$

where

$$G(\wp )={\int }_m^{\wp }g\left(\varpi \right){(\varpi -a)}^{\gamma -1}d\varpi withG(\infty )=\infty ,and\mho (\wp )={\int }_{\wp }^{\infty }r\left(\varpi \right)\mho \left(\varpi \right){(\varpi -a)}^{\gamma -1}d\varpi .$$

Corollary 6.

Putting$\mathbb{T}=h\mathbb{Z},$$h>0$in (24) we obtain

$$\begin{array}{ccc}& & \sum _{\wp =m}^{\infty }k(\wp )v(\wp )w(\wp -h)g(\wp )G^{\gamma -{\daleth }_1-1}(\wp -h){\mho }^{p-\gamma +1}(\wp )({\rho }^{\gamma -1}{(\wp )-a)}_h^{(\gamma -1)}\hfill \\ \hfill & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{\gamma -{\daleth }_1+\vartheta }\right)}^p\sum _{\wp =m}^{\infty }\frac{k(\wp )v(\wp )w(\wp -h)r^p(\wp ){\mho }^p(\wp ){\left(G(\wp -h)\right)}^{p-{\daleth }_1+\gamma -1}{\mho }^{1-\gamma }(\wp -h)({\rho }^{\gamma -1}{(\wp )-a)}_h^{(\gamma -1)}}{g^{p-1}(\wp )}.\hfill \end{array}$$

where

$$G(\wp )=h\sum _{\varpi =m}^{\wp }g\left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}_h^{(\gamma -1)}withG(\infty )=\infty and\mho (\wp )=h\sum _{\varpi =\wp }^{\infty }r\left(\varpi \right)\mho \left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}_h^{(\gamma -1)}.$$

Corollary 7.

If$\mathbb{T}=\mathbb{Z},$with$h=1$in our Corollary 6 we have from (24) that

$$\begin{array}{ccc}& & \sum _{\wp =m}^{\infty }k(\wp )v(\wp )w(\wp -1)g(\wp )G^{\gamma -{\daleth }_1-1}(\wp -1){\mho }^{p-\gamma +1}(\wp )({\rho }^{\gamma -1}{(\wp )-a)}^{(\gamma -1)}\hfill \\ \hfill & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{\gamma -{\daleth }_1+\vartheta }\right)}^p\sum _{\wp =m}^{\infty }\frac{k(\wp )v(\wp )w(\wp -1)r^p(\wp ){\mho }^p(\wp ){\left(G(\wp -1)\right)}^{p-{\daleth }_1+\gamma -1}{\mho }^{1-\gamma }(\wp -1)({\rho }^{\gamma -1}{(\wp )-a)}^{(\gamma -1)}}{g^{p-1}(\wp )}.\hfill \end{array}$$

where

$$G(\wp )=\sum _{\varpi =m}^{\wp }g\left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}^{(\gamma -1)}withG(\infty )=\infty and\mho (\wp )=\sum _{\varpi =\wp }^{\infty }r\left(\varpi \right)\mho \left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}^{(\gamma -1)}.$$

Corollary 8.

For$\mathbb{T}=q^{{\mathbb{N}}_0},$in (24) we get

$$\begin{array}{ccc}& & \sum _{\wp =m}^{\infty }\wp ({\rho }^{\gamma -1}{(\wp )-a)}_{\tilde{q}}^{(\gamma -1)}k(\wp )v(\wp )w^{\rho }(\wp )g(\wp ){\left(G^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1-1}{\mho }^{p-\gamma +1}(\wp )\hfill \\ \hfill & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{\gamma -{\daleth }_1+\vartheta }\right)}^p\sum _{\wp =m}^{\infty }\frac{\wp ({\rho }^{\gamma -1}{(\wp )-a)}_{\tilde{q}}^{(\gamma -1)}k(\wp )v(\wp )w^{\rho }(\wp )r^p(\wp ){\mho }^p(\wp ){\left(G^{\rho }(\wp )\right)}^{p-{\daleth }_1+\gamma -1}{\mho }^{1-\gamma }\left(\rho (\wp )\right)}{g^{p-1}(\wp )}.\hfill \end{array}$$

where

$$G(\wp )=(\tilde{q}-1)\sum _{\varpi =m}^{\wp }\varpi ({\rho }^{\gamma -1}\left(\varpi \right){-a)}_{\tilde{q}}^{(\gamma -1)}g\left(\varpi \right)withG(\infty )=\infty .\mho (\wp )=,$$

and

$$\tilde{q}-1)\sum _{\varpi =\wp }^{\infty }\varpi ({\rho }^{\gamma -1}\left(\varpi \right)-a{)}_{\tilde{q}}^{(\gamma -1)}r\left(\varpi \right)\mho \left(\varpi \right).$$

Theorem 8.

Let$\mathbb{T}$be a time scale with$m\in {[0,\infty )}_{\mathbb{T}}$, $\gamma \in (0,1]$and$a>\wp$. Furthermore, assume the rd-continuous functions v, g, r, ℧, k,$w\ge 0$, on ${[m,\infty )}_{\mathbb{T}}$with k nondecreasing. Further, suppose we have$\vartheta ,{\daleth }_2\ge 0$with${\displaystyle \frac{{\nabla }_a^{\gamma }w(\wp )}{w(\wp )}⩾\vartheta \left(\frac{{\nabla }_a^{\gamma }H(\wp )}{H^{\rho }(\wp )}\right)}$and${\displaystyle \frac{{\nabla }_a^{\gamma }v(\wp )}{v^{\rho }(\wp )}⩾{\daleth }_2\left(\frac{{\nabla }_a^{\gamma }K(\wp )}{K(\wp )}\right)}$, where

$$H(\wp )={\int }_{\wp }^{\infty }g\left(\varpi \right){\nabla }_a^{\gamma }\varpi andK(\wp )={\int }_m^{\wp }r\left(\varpi \right)\mho \left(\varpi \right){\nabla }_a^{\gamma }\varpi ,\wp \in {[m,\infty )}_{\mathbb{T}}.$$

If $0<p<\gamma$, $0\le {\daleth }_1<\gamma$, then

$$\begin{array}{ccc}\hfill & & {\int }_m^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\left(K^{\rho }(\wp )\right)}^{p-\gamma +1}{\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾{\left(\frac{p-\gamma +{\daleth }_2+1}{\gamma -{\daleth }_1+\vartheta }\right)}^p{\int }_a^{\infty }\frac{k^{\rho }(\wp )v^{\rho }(\wp )w(\wp )r^p(\wp ){\mho }^p(\wp )H^{p-{\daleth }_1+\gamma -1}(\wp )(K^{\rho }(\wp {\left)\right)}^{(1-\gamma )}}{g^{p-1}(\wp )}{\nabla }_a^{\gamma }\wp .\hfill \end{array}$$

(28)

Proof. 

Applying (15) with

$${\nabla }_a^{\gamma }u(\wp )=w(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp )\mathrm{and}{z}^{\rho }(\wp )=k^{\sigma }(\wp )v^{\rho }(\wp ){\left(K^{\rho }(\wp )\right)}^{p-\gamma +1},$$

we have

$$\begin{array}{ccc}\hfill & & {\int }_m^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\left(K^{\rho }(\wp )\right)}^{p-\gamma +1}{\nabla }_a^{\gamma }\wp \hfill \\ & & ={\left[u(\wp )k(\wp )v(\wp )K^{p-{\daleth }_1+1}(\wp )\right]}_m^{\infty }+{\int }_m^{\infty }\left(-u(\wp )\right){\nabla }_a^{\gamma }\left(k(\wp )v(\wp )K^{p-\gamma +1}(\wp )\right){\nabla }_a^{\gamma }\wp ,\hfill \end{array}$$

(29)

where

$$u(\wp )=-{\int }_{\wp }^{\infty }w\left(\varpi \right)g\left(\varpi \right)H^{\gamma -{\daleth }_1-1}\left(\varpi \right){\nabla }_a^{\gamma }\varpi .$$

From (16), (14), and the assumption ${\displaystyle \frac{{\nabla }_a^{\gamma }w(\wp )}{w(\wp )}⩾\vartheta \left(\frac{{\nabla }_a^{\gamma }H(\wp )}{H^{\rho }(\wp )}\right)}$, we have

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left(-w\left(\varpi \right)H^{\gamma -{\daleth }_1}\left(\varpi \right)\right)& =& -\left({\nabla }_a^{\gamma }w\left(\varpi \right){\left(H^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1}+w\left(\varpi \right){\nabla }_a^{\gamma }(H^{\gamma -{\daleth }_1}\left(\varpi \right))\right)\hfill \\ & ⩽& -\left(\vartheta w\left(\varpi \right){\left(H^{\rho }\left(\varpi \right)\right)}^{\gamma -{\daleth }_1-1}{\nabla }_a^{\gamma }H\left(\varpi \right)+(\gamma -{\daleth }_1)w\left(\varpi \right)H^{\gamma -{\daleth }_1-1}\left(c\right){\nabla }_a^{\gamma }H\left(\varpi \right)\right).\hfill \end{array}$$

Because ${\nabla }_a^{\gamma }H\left(\varpi \right)=-g\left(\varpi \right)\le 0$, $c⩽\varpi$ and $0\le {\daleth }_1<\gamma$, we get

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left(-w\left(\varpi \right)H^{\gamma -{\daleth }_1}\left(\varpi \right)\right)& ⩽& \vartheta w\left(\varpi \right)g\left(\varpi \right)H^{\gamma -{\daleth }_1-1}\left(\varpi \right)+(\gamma -{\daleth }_1)w\left(\varpi \right)g\left(\varpi \right)H^{\gamma -{\daleth }_1-1}\left(\varpi \right)\hfill \\ & =& (\gamma -{\daleth }_1+\vartheta )w\left(\varpi \right)g\left(\varpi \right)H^{\gamma -{\daleth }_1-1}\left(\varpi \right).\hfill \end{array}$$

Thus

$$w\left(\varpi \right)g\left(\varpi \right)H^{\gamma -{\daleth }_1-1}\left(\varpi \right)⩾\frac{1}{\gamma -{\daleth }_1+\vartheta }{\nabla }_a^{\gamma }\left(-w\left(\varpi \right)H^{\gamma -{\daleth }_1}\left(\varpi \right)\right).$$

Hence

$$\begin{array}{ccc}\hfill -u(\wp )={\int }_{\wp }^{\infty }w\left(\varpi \right)g\left(\varpi \right)H^{\gamma -{\daleth }_1-1}\left(\varpi \right){\nabla }_a^{\gamma }\varpi & ⩾& \frac{1}{\gamma -{\daleth }_1+\vartheta }{\int }_{\wp }^{\infty }{\nabla }_a^{\gamma }\left(-w\left(\varpi \right)H^{\gamma -{\daleth }_1}\left(\varpi \right)\right){\nabla }_a^{\gamma }\varpi \hfill \\ & =& \frac{1}{\gamma -{\daleth }_1+\vartheta }w(\wp )H^{\gamma -{\daleth }_1}(\wp ).\hfill \end{array}$$

(30)

Applying (14) and (16), we have $c\in \left[\rho \right(\wp ),\wp ]$ with

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left(k(\wp )v(\wp )K^{p-\gamma +1}(\wp )\right)& =& {\nabla }_a^{\gamma }\left(k(\wp )v(\wp )\right)K^{p-\gamma +1}(\wp )+k^{\rho }(\wp )v^{\rho }(\wp ){\nabla }_a^{\gamma }\left(K^{p-\gamma +1}(\wp )\right)\hfill \\ & =& {\nabla }_a^{\gamma }k(\wp )v(\wp )K^{p-\gamma +1}(\wp )+k^{\rho }(\wp ){\nabla }_a^{\gamma }v(\wp )K^{p-\gamma +1}(\wp )\hfill \\ & & +(p-\gamma +1)k^{\rho }(\wp )v^{\rho }(\wp )K^{p-\gamma }\left(c\right){\nabla }_a^{\gamma }K(\wp ).\hfill \end{array}$$

Considering ${\nabla }_a^{\gamma }k(\wp )⩾0$, ${\nabla }_a^{\gamma }K(\wp )=r(\wp )\mho (\wp )\ge 0$, $c⩽\wp$, $0<p<\gamma$ and ${\displaystyle \frac{{\nabla }_a^{\gamma }v(\wp )}{v^{\rho }(\wp )}⩾{\daleth }_2\left(\frac{K(\wp )}{K(\wp )}\right)}$, we get

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left(k(\wp )v(\wp )K^{p-\gamma +1}(\wp )\right)& ⩾& {\daleth }_2{k}^{\rho }(\wp )v^{\rho }(\wp )r(\wp )\mho (\wp )K^{p-\gamma }(\wp )+(p-\gamma +1)k^{\rho }(\wp )v^{\rho }(\wp )r(\wp )\mho (\wp )K^{p-\gamma }(\wp )\hfill \\ & ⩾& (p-\gamma +{\daleth }_2+1)k^{\rho }(\wp )v^{\rho }(\wp )r(\wp )\mho (\wp )K^{p-\gamma }(\wp ).\hfill \end{array}$$

(31)

Combining (29)–(31), with $K\left(m\right)=0$ and $u(\infty )=0$ obtains

$$\begin{array}{ccc}& & {\int }_m^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\left(K^{\rho }(\wp )\right)}^{p-\gamma +1}{\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾\frac{p-\gamma +{\daleth }_2+1}{\gamma -{\daleth }_1+\vartheta }{\int }_m^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )r(\wp )\mho (\wp )H^{\gamma -{\daleth }_1}(\wp )K^{p-\gamma }(\wp ){\nabla }_a^{\gamma }\wp .\hfill \end{array}$$

The last inequality can be rewritten as

$$\begin{array}{ccc}& & {\int }_m^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp )H^{\gamma -{\daleth }_1+1}(\wp ){\left(K^{\rho }(\wp )\right)}^{p-\gamma +1}{\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾\frac{p-\gamma +{\daleth }_2+1}{\gamma -{\daleth }_1+\vartheta }{\int }_m^{\infty }\left({\left(k^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp )\right)}^{(p-1)/p}{H}^{(\gamma -{\daleth }_1-1)(p-1)/p}(\wp )K^{(p-1)(p-\gamma +1)/p}(\wp )\right)\hfill \\ & & \times \left(\frac{{\left(k^{\rho }(\wp )v^{\rho }(\wp )w(\wp )\right)}^{1/p}r(\wp )\mho (\wp )H^{(p-{\daleth }_1+\gamma -1)/p}(\wp )K^{(1-\gamma )/p}(\wp )}{g^{(p-1)/p}(\wp )}\right){\nabla }_a^{\gamma }\wp .\hfill \end{array}$$

Applying (11) with the index p and the index $p/(p-1)$, gets

$$\begin{array}{ccc}& & {\int }_a^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\left(K^{\rho }(\wp )\right)}^{p-\rho +1}{\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾\frac{p-\gamma +{\daleth }_2+1}{\gamma -{\daleth }_1+\vartheta }{\left({\int }_m^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp )K^{p-\gamma +1}(\wp ){\nabla }_a^{\gamma }\wp \right)}^{(p-1)/p}\hfill \\ & & \times {\left({\int }_m^{\infty }\frac{k^{\rho }(\wp )v^{\rho }(\wp )w(\wp )r^p(\wp ){\mho }^p(\wp )H^{p-{\daleth }_1+\gamma -1}(\wp )K^{1-\gamma }(\wp )}{g^{p-1}(\wp )}{\nabla }_a^{\gamma }\wp \right)}^{1/p}.\hfill \end{array}$$

This implies that

$$\begin{array}{ccc}& & {\int }_m^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\left(K^{\rho }(\wp )\right)}^{p-\gamma +1}{\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾{\left(\frac{p-\gamma +{\daleth }_2+1}{\gamma -{\daleth }_1+\vartheta }\right)}^p{\int }_a^{\infty }\frac{k^{\rho }(\wp )v^{\rho }(\wp )w(\wp )r^p(\wp ){\mho }^p(\wp )H^{p-{\daleth }_1+\gamma -1}(\wp )K^{1-\gamma }(\wp )}{g^{p-1}(\wp )}{\nabla }_a^{\gamma }\wp .\hfill \end{array}$$

This concludes our result. □

Remark 3.

Putting $\gamma =1$ in our result Theorem 8, then we get the following inequality

$$\begin{array}{ccc}& & {\int }_m^{\infty }{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp ){\left(H(\wp )\right)}^{-{\daleth }_1}{\left(K^{\rho }(\wp )\right)}^p\nabla \wp \hfill \\ & & ⩾{\left(\frac{p+{\daleth }_2}{1-{\daleth }_1+\vartheta }\right)}^p{\int }_m^{\infty }\frac{k^{\rho }(\wp )v^{\rho }(\wp )w(\wp )r^p(\wp ){\mho }^p(\wp )H^{p-{\daleth }_1}(\wp )}{g^{p-1}(\wp )}\nabla \wp .\hfill \end{array}$$

Corollary 9.

Putting$\mathbb{T}=\mathbb{R}$in (28) gets

$$\begin{array}{ccc}\hfill & & {\int }_m^{\infty }k(\wp )v(\wp )w(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}{\left(K^{\rho }(\wp )\right)}^{p-\gamma +1}(\wp ){(\wp -a)}^{\gamma -1}dt\hfill \\ & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{\gamma -{\daleth }_1+\vartheta }\right)}^p{\int }_m^{\infty }\frac{k(\wp )v(\wp )w(\wp )r^p(\wp ){\mho }^p(\wp )H^{p-{\daleth }_1+\gamma -1}(\wp )K^{1-\gamma }(\wp ){(\wp -a)}^{\gamma -1}dt}{g^{p-1}(\wp )}dt,\hfill \end{array}$$

where

$$H(\wp )={\int }_m^{\infty }g\left(\varpi \right){(\varpi -a)}^{\gamma -1}d\varpi andK(\wp )={\int }_m^{\wp }r\left(\varpi \right)\mho \left(\varpi \right){(\varpi -a)}^{\gamma -1}d\varpi .$$

Corollary 10.

Putting$\mathbb{T}=h\mathbb{Z},$$h>0$in (28) gets

$$\begin{array}{ccc}& & \sum _{\wp =m}^{\infty }k(\wp -h)v(\wp -h)w(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp )K^{p-\gamma +1}(\wp -h)({\rho }^{\gamma -1}{(\wp )-a)}_h^{(\gamma -1)}\hfill \\ \hfill & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{\gamma -{\daleth }_1+\vartheta }\right)}^p\sum _{\wp =m}^{\infty }\frac{k(\wp -h)v(\wp -h)w(\wp )r^p(\wp ){\mho }^p(\wp )H^{p-{\daleth }_1+\gamma -1}(\wp )K^{1-\gamma }(\wp )({\rho }^{\gamma -1}{(\wp )-a)}_h^{(\gamma -1)}}{g^{p-1}(\wp )}.\hfill \end{array}$$

where

$$H(\wp )=h\sum _{\varpi =\wp }^{\infty }g\left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}_h^{(\gamma -1)}withandK(\wp )=h\sum _{\varpi =m}^{\wp }r\left(\varpi \right)\mho \left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}_h^{(\gamma -1)}.$$

Corollary 11.

If$\mathbb{T}=\mathbb{Z},$with$h=1$in our result Theorem 10 then, (28) obtains to

$$\begin{array}{ccc}& & \sum _{\wp =m}^{\infty }k(\wp -1)v(\wp -1)w(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp )K^{p-\gamma +1}(\wp -1)({\rho }^{\gamma -1}{(\wp )-a)}^{(\gamma -1)}\hfill \\ \hfill & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{\gamma -{\daleth }_1+\vartheta }\right)}^p\sum _{\wp =m}^{\infty }\frac{k(\wp -1)v(\wp -1)w(\wp )r^p(\wp ){\mho }^p(\wp )H^{p-{\daleth }_1+\gamma -1}(\wp )K^{1-\gamma }(\wp )({\rho }^{\gamma -1}{(\wp )-a)}^{(\gamma -1)}}{g^{p-1}(\wp )}.\hfill \end{array}$$

where

$$H(\wp )=\sum _{\varpi =\wp }^{\infty }g\left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}^{(\gamma -1)}withandK(\wp )=h\sum _{\varpi =m}^{\wp }r\left(\varpi \right)\mho \left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}^{(\gamma -1)}.$$

Corollary 12.

Putting$\mathbb{T}=q^{{\mathbb{N}}_0},$in (28) we obtains

$$\begin{array}{ccc}& & \sum _{\wp =m}^{\infty }\wp ({\rho }^{\gamma -1}{(\wp )-a)}_{\tilde{q}}^{(\gamma -1)}{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}{\left(K^{\rho }(\wp )\right)}^{p-\gamma +1}\hfill \\ \hfill & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{\gamma -{\daleth }_1+\vartheta }\right)}^p\sum _{\wp =m}^{\infty }\frac{\wp ({\rho }^{\gamma -1}{(\wp )-a)}_{\tilde{q}}^{(\gamma -1)}{k}^{\rho }(\wp )v^{\rho }(\wp )w(\wp )r^p(\wp ){\mho }^p(\wp )H^{p-{\daleth }_1+\gamma -1}(\wp )K^{1-\gamma }(\wp )}{g^{p-1}(\wp )}.\hfill \end{array}$$

where

$$H(\wp )=(\tilde{q}-1)\sum _{\varpi =\wp }^{\infty }\varpi ({\rho }^{\gamma -1}\left(\varpi \right)-a{)}_{\tilde{q}}^{(\gamma -1)}g\left(\varpi \right)andK(\wp )=(\tilde{q}-1)\sum _{\varpi =m}^{\wp }\varpi ({\rho }^{\gamma -1}\left(\varpi \right)-a{)}_{\tilde{q}}^{(\gamma -1)}r\left(\varpi \right)\mho \left(\varpi \right).$$

Theorem 9.

Assume $\mathbb{T}$ is a time scale with $m\in {[0,\infty )}_{\mathbb{T}}$, $\gamma \in (0,1]$ and $\wp >a$. In addition, let the rd-continuous functions g, v, r, ℧, k, $w\ge 0$, on ${[m,\infty )}_{\mathbb{T}}$ with k nonincreasing. Moreover, suppose there exist $\vartheta ,{\daleth }_2\ge 0$ such that ${\displaystyle \frac{{\nabla }_a^{\gamma }w(\wp )}{w^{\rho }(\wp )}⩽\vartheta \left(\frac{H(\wp )}{H(\wp )}\right)}$ and ${\displaystyle \frac{{\nabla }_a^{\gamma }v(\wp )}{v(\wp )}⩽{\daleth }_2\left(\frac{{\nabla }_a^{\gamma }\mho (\wp )}{{\mho }^{\rho }(\wp )}\right)}$, where

$$w\left(m\right)=0,H(\wp )={\int }_{\wp }^{\infty }g\left(\varpi \right){\nabla }_a^{\gamma }\varpi and\mho (\wp )={\int }_{\wp }^{\infty }r\left(\varpi \right)\mho \left(\varpi \right){\nabla }_a^{\gamma }\varpi ,\wp \in {[m,\infty )}_{\mathbb{T}}.$$

If $0<p<\gamma$ and ${\daleth }_1>\vartheta +1$, then

$$\begin{array}{ccc}\hfill & & {\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\mho }^{p-\gamma +1}(\wp ){\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }\right)}^p{\int }_m^{\infty }\frac{k(\wp )v(\wp )w^{\rho }(\wp )r^p(\wp ){\mho }^p(\wp )H^{(1-\gamma +{\daleth }_1)(p-1)}(\wp ){\left({\mho }^{\rho }(\wp )\right)}^{(1-\gamma )}}{g^{p-1}(\wp ){\left(H^{\rho }(\wp )\right)}^{p({\daleth }_1-\gamma )}}{\nabla }_a^{\gamma }\wp .\hfill \end{array}$$

(32)

Proof. 

Applying (15), obtains

$$\begin{array}{ccc}\hfill & & {\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\mho }^{p-\gamma +1}(\wp ){\nabla }_a^{\gamma }\wp \hfill \\ & & ={\left[u(\wp )k(\wp )v(\wp ){\mho }^{p-\gamma +1}(\wp )\right]}_m^{\infty }+{\int }_m^{\infty }{u}^{\rho }(\wp ){\nabla }_a^{\gamma }(-k(\wp )v(\wp ){\mho }^{p-\gamma +1}(\wp )){\nabla }_a^{\gamma }\wp ,\hfill \end{array}$$

(33)

where

$$u(\wp )={\int }_m^{\wp }{w}^{\rho }\left(\varpi \right)g\left(\varpi \right)H^{\gamma -{\daleth }_1-1}\left(\varpi \right){\nabla }_a^{\gamma }\varpi .$$

Using (16), (14), and the hypothesis ${\displaystyle \frac{{\nabla }_a^{\gamma }w(\wp )}{w^{\rho }(\wp )}⩽\vartheta \left(\frac{{\nabla }_a^{\gamma }H(\wp )}{H(\wp )}\right)}$, we obtain

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left(w\left(\varpi \right)H^{\gamma -{\daleth }_1}\left(\varpi \right)\right)& =& {\nabla }_a^{\gamma }w\left(\varpi \right)H^{\gamma -{\daleth }_1}\left(\varpi \right)+w^{\rho }\left(\varpi \right){\nabla }_a^{\gamma }\left(H^{\gamma -{\daleth }_1}\left(\varpi \right)\right)\hfill \\ & ⩽& \vartheta w^{\rho }\left(\varpi \right){\nabla }_a^{\gamma }H\left(\varpi \right)H^{\gamma -{\daleth }_1-1}\left(\varpi \right)+(\gamma -{\daleth }_1)w^{\rho }\left(\varpi \right)H^{\gamma -{\daleth }_1-1}\left(c\right){\nabla }_a^{\gamma }H\left(\varpi \right).\hfill \end{array}$$

Since ${\nabla }_a^{\gamma }H\left(\varpi \right)=-g\left(\varpi \right)\le 0$, $c⩽\varpi$ and ${\daleth }_1>1$, we get

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left(w\left(\varpi \right)H^{\gamma -{\daleth }_1}\left(\varpi \right)\right)& ⩽& -\vartheta w^{\rho }\left(\varpi \right)g\left(\varpi \right)H^{\gamma -{\daleth }_1-1}\left(\varpi \right)+({\daleth }_1-\gamma )w^{\rho }\left(\varpi \right)g\left(\varpi \right)H^{\gamma -{\daleth }_1-1}\left(\varpi \right)\hfill \\ & =& ({\daleth }_1-\vartheta -\gamma )w^{\rho }\left(\varpi \right)g\left(\varpi \right)H^{\gamma -{\daleth }_1-1}\left(\varpi \right).\hfill \end{array}$$

This gives us that

$$w^{\rho }\left(\varpi \right)g\left(\varpi \right)H^{\gamma -{\daleth }_1-1}\left(\varpi \right)⩾\frac{1}{{\daleth }_1-\vartheta -\gamma }{\nabla }_a^{\gamma }\left(w\left(\varpi \right)H^{\gamma -{\daleth }_1}\left(\varpi \right)\right).$$

Therefore

$$\begin{array}{ccc}\hfill u^{\rho }(\wp )={\int }_m^{\rho (\wp )}{w}^{\rho }\left(\varpi \right)g\left(\varpi \right)H^{\gamma -{\daleth }_1-1}\left(\varpi \right){\nabla }_a^{\gamma }\varpi & ⩾& \frac{1}{{\daleth }_1-\vartheta -\gamma }{\int }_m^{\rho (\wp )}{\nabla }_a^{\gamma }\left(w\left(\varpi \right)H^{\gamma -{\daleth }_1}\left(\varpi \right)\right){\nabla }_a^{\gamma }\varpi \hfill \\ \hfill & =& \frac{1}{{\daleth }_1-\vartheta -\gamma }\left(w^{\rho }(\wp ){\left(H^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1}-w\left(m\right)H^{\gamma -{\daleth }_1}\left(m\right)\right)\hfill \\ & ⩾& \frac{1}{{\daleth }_1-\vartheta -\gamma }{w}^{\rho }(\wp ){\left(H^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1}.\hfill \end{array}$$

(34)

Employing (14) and (16), one have $c\in \left[\rho \right(\wp ),\wp ]$ with

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }(-k(\wp )v(\wp ){\mho }^{p-\gamma +1}(\wp ))& =& -\left({\nabla }_a^{\gamma }\left(k(\wp )v(\wp )\right){\left({\mho }^{\rho }(\wp )\right)}^{p-\rho +1}+k(\wp )v(\wp ){\nabla }_a^{\gamma }\left({\mho }^{p-\rho +1}(\wp )\right)\right)\hfill \\ & =& -({\nabla }_a^{\gamma }k(\wp )v^{\rho }(\wp ){\left({\mho }^{\rho }(\wp )\right)}^{p-\gamma +1}+k(\wp ){\nabla }_a^{\gamma }v(\wp ){\left({\mho }^{\rho }(\wp )\right)}^{p-\gamma +1}\hfill \\ & & +(p-\gamma +1)k(\wp )v(\wp ){\mho }^{p-\gamma }\left(c\right){\nabla }_a^{\gamma }\mho (\wp )).\hfill \end{array}$$

As ${\nabla }_a^{\gamma }k(\wp )⩽0$, ${\nabla }_a^{\gamma }\mho (\wp )=-r(\wp )\mho (\wp )\le 0$, $c⩾\rho (\wp )$, $0<p<\gamma$ and ${\displaystyle \frac{{\nabla }_a^{\gamma }v(\wp )}{v(\wp )}⩽{\daleth }_2\left(\frac{{\nabla }_a^{\gamma }\mho (\wp )}{{\mho }^{\rho }(\wp )}\right)}$, we have

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left(-k(\wp )v(\wp ){\mho }^{p-{\daleth }_1+1}(\wp )\right)& ⩾& {\daleth }_{2}k(\wp )v(\wp )r(\wp )\mho (\wp ){\left({\mho }^{\rho }(\wp )\right)}^{p-\gamma }+(p-\gamma +1)k(\wp )v(\wp )r(\wp )\mho (\wp ){\left({\mho }^{\rho }(\wp )\right)}^{p-\gamma }\hfill \\ & ⩾& (p+{\daleth }_2+\gamma -1)k(\wp )v(\wp )r(\wp )\mho (\wp ){\left({\mho }^{\rho }(\wp )\right)}^{p-\gamma }.\hfill \end{array}$$

(35)

Combining (33), (34) and (35), we obtain (note that $u\left(m\right)=0$ and $\mho (\infty )=0$ )

$$\begin{array}{ccc}& & {\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\mho }^{p-\gamma +1}(\wp ){\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾\frac{p+{\daleth }_2+\gamma -1}{{\daleth }_1-\vartheta -\gamma }{\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )r(\wp )\mho (\wp ){\left(H^{\rho }(\wp )\right)}^{\gamma -{\daleth }_1}{\left({\mho }^{\rho }(\wp )\right)}^{p-\gamma }{\nabla }_a^{\gamma }\wp ,\hfill \end{array}$$

or equivalently,

$$\begin{array}{ccc}& & {\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\mho }^{p-\gamma +1}(\wp ){\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾\frac{p+{\daleth }_2+\gamma -1}{{\daleth }_1-\vartheta -\gamma }{\int }_m^{\infty }\left({\left(k(\wp )v(\wp )w^{\rho }(\wp )g(\wp )\right)}^{(p-1)/p}{H}^{(\gamma -{\daleth }_1-1)(p-1)/p}(\wp ){\left({\mho }^{\rho }(\wp )\right)}^{(p-\gamma +1)(p-1)/p}\right)\hfill \\ & & \times \left(\frac{{\left(k(\wp )v(\wp )w^{\rho }(\wp )\right)}^{1/p}r(\wp )\mho (\wp )H^{(1-\gamma +{\daleth }_1)(p-1)/p}(\wp ){\left({\mho }^{\rho }(\wp )\right)}^{(1-\gamma )/p}}{g^{(p-1)/p}(\wp ){\left(H^{\rho }(\wp )\right)}^{{\daleth }_1-\gamma }}\right){\nabla }_a^{\gamma }\wp .\hfill \end{array}$$

Applying the dynamic Hölder inequality (11) with the index p and the index $p/(p-1)$, obtains

$$\begin{array}{ccc}& & {\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\mho }^{p-\gamma +1}(\wp ){\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }{\left({\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\left({\mho }^{\rho }(\wp )\right)}^{p-\gamma +1}{\nabla }_a^{\gamma }\wp \right)}^{(p-1)/p}\hfill \\ & & \times {\left({\int }_m^{\infty }\frac{k(\wp )v(\wp )w^{\rho }(\wp )r^p(\wp ){\mho }^p(\wp )H^{(1-\gamma +{\daleth }_1)(p-1)}(\wp ){\left({\mho }^{\rho }(\wp )\right)}^{(1-\gamma )}}{g^{p-1}(\wp ){\left(H^{\rho }(\wp )\right)}^{p({\daleth }_1-\gamma )}}{\nabla }_a^{\gamma }\wp \right)}^{1/p},\hfill \end{array}$$

which implies

$$\begin{array}{ccc}& & {\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\mho }^{p-\gamma +1}(\wp ){\nabla }_a^{\gamma }\wp \hfill \\ & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }\right)}^p{\int }_m^{\infty }\frac{k(\wp )v(\wp )w^{\rho }(\wp )r^p(\wp ){\mho }^p(\wp )H^{(1-\gamma +{\daleth }_1)(p-1)}(\wp ){\left({\mho }^{\rho }(\wp )\right)}^{(1-\gamma )}}{g^{p-1}(\wp ){\left(H^{\rho }(\wp )\right)}^{p({\daleth }_1-\gamma )}}{\nabla }_a^{\gamma }\wp ,\hfill \end{array}$$

that is our required inequality. □

Remark 4.

In our result Theorem 9, putting$\gamma =1$then we get the following inequality

$$\begin{array}{ccc}\hfill & & {\int }_m^{\infty }k(\wp )v(\wp )w^{\rho }(\wp )g(\wp )H^{-{\daleth }_1}(\wp ){\mho }^p(\wp )\nabla \wp \hfill \\ & & ⩾{\left(\frac{p+{\daleth }_2}{{\daleth }_1-\vartheta -1}\right)}^p{\int }_m^{\infty }\frac{k(\wp )v(\wp )w^{\rho }(\wp )r^p(\wp ){\mho }^p(\wp )H^{{\daleth }_1(p-1)}}{g^{p-1}(\wp ){\left(H^{\rho }(\wp )\right)}^{p({\daleth }_1-1)}}\nabla \wp .\hfill \end{array}$$

Corollary 13.

Putting$\mathbb{T}=\mathbb{R}$in the result 9, inequality (32) reduces to

$$\begin{array}{ccc}& & {\int }_m^{\infty }k(\wp )v(\wp )w(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\mho }^{p-\gamma +1}(\wp ){(\wp -a)}^{\gamma -1}dt\hfill \\ \hfill & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }\right)}^p{\int }_m^{\infty }\frac{k(\wp )v(\wp )w(\wp )r^p(\wp ){\mho }^p(\wp )H^{p-{\daleth }_1+\gamma -1}(\wp ){\mho }^{1-\gamma }(\wp )}{g^{p-1}(\wp )}{(\wp -a)}^{\gamma -1}dt.\hfill \end{array}$$

where

$$w\left(m\right)=0,H(\wp )={\int }_{\wp }^{\infty }g\left(\varpi \right){(\varpi -a)}^{\gamma -1}d\varpi and\mho (\wp )={\int }_{\wp }^{\infty }r\left(\varpi \right)\mho \left(\varpi \right){(\varpi -a)}^{\gamma -1}d\varpi .$$

Corollary 14.

For$\mathbb{T}=h\mathbb{Z},$$h>0$in the result 9, inequality (32) reduces to

$$\begin{array}{ccc}& & \sum _{\wp =m}^{\infty }k(\wp )v(\wp )w(\wp -h)g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\mho }^{p-\gamma +1}(\wp -h)({\rho }^{\gamma -1}{(\wp )-a)}_h^{(\gamma -1)}\hfill \\ \hfill & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }\right)}^p\sum _{\wp =m}^{\infty }\frac{k(\wp )v(\wp )w(\wp -h)r^p(\wp ){\mho }^p(\wp )H^{(1-\gamma +{\daleth }_1)(p-1)}(\wp ){\mho }^{1-\gamma }(\wp -h)({\rho }^{\gamma -1}(\wp )-a{)}_h^{(\gamma -1)}}{g^{p-1}(\wp ){\left(H^{\rho }(\wp )\right)}^{p({\daleth }_1-\gamma )}}.\hfill \end{array}$$

where

$$w\left(m\right)=0,H(\wp )=h\sum _{\varpi =\wp }^{\infty }g\left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}_h^{(\gamma -1)}and\mho (\wp )=h\sum _{\varpi =\wp }^{\infty }r\left(\varpi \right)\mho \left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}_h^{(\gamma -1)}.$$

Corollary 15.

If$\mathbb{T}=\mathbb{Z},$with$h=1$in the result 14 then, (32) reduces to

$$\begin{array}{ccc}& & \sum _{\wp =m}^{\infty }k(\wp )v(\wp )w(\wp -1)g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\mho }^{p-\gamma +1}(\wp -1)({\rho }^{\gamma -1}{(\wp )-a)}^{(\gamma -1)}\hfill \\ \hfill & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }\right)}^p\sum _{\wp =m}^{\infty }\frac{k(\wp )v(\wp )w(\wp -1)r^p(\wp ){\mho }^p(\wp )H^{(1-\gamma +{\daleth }_1)(p-1)}(\wp ){\mho }^{1-\gamma }(\wp -1)({\rho }^{\gamma -1}{(\wp )-a)}^{(\gamma -1)}}{g^{p-1}(\wp ){\left(H^{\rho }(\wp )\right)}^{p({\daleth }_1-\gamma )}}.\hfill \end{array}$$

where

$$w\left(m\right)=0,H(\wp )=\sum _{\varpi =\wp }^{\infty }g\left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}^{(\gamma -1)}and\mho (\wp )=\sum _{\varpi =\wp }^{\infty }r\left(\varpi \right)\mho \left(\varpi \right)({\rho }^{\gamma -1}\left(\varpi \right)-a{)}^{(\gamma -1)}.$$

Corollary 16.

Putting$\mathbb{T}=q^{{\mathbb{N}}_0},$in (32) we get

$$\begin{array}{ccc}& & \sum _{\wp =m}^{\infty }\wp ({\rho }^{\gamma -1}{(\wp )-a)}_{\tilde{q}}^{(\gamma -1)}k(\wp )v(\wp )w^{\rho }(\wp )g(\wp )H^{\gamma -{\daleth }_1-1}(\wp ){\mho }^{p-\gamma +1}(\wp )\hfill \\ \hfill & & ⩾{\left(\frac{p+{\daleth }_2-\gamma +1}{{\daleth }_1-\vartheta -\gamma }\right)}^p\sum _{\wp =m}^{\infty }\frac{\wp ({\rho }^{\gamma -1}{(\wp )-a)}_{\tilde{q}}^{(\gamma -1)}k(\wp )v(\wp )w^{\rho }(\wp )r^p(\wp ){\mho }^p(\wp )H^{(1-\gamma +{\daleth }_1)(p-1)}(\wp ){\left({\mho }^{\rho }(\wp )\right)}^{p-\gamma }}{g^{p-1}(\wp )H^{p({\daleth }_1-\gamma )}(\wp )}.\hfill \end{array}$$

where

$$w\left(m\right)=0,H(\wp )=(\tilde{q}-1)\sum _{\varpi =\wp }^{\infty }\varpi ({\rho }^{\gamma -1}\left(\varpi \right)-a{)}_{\tilde{q}}^{(\gamma -1)}g\left(\varpi \right)and\mho (\wp )=(\tilde{q}-1)\sum _{\varpi =\wp }^{\infty }\varpi ({\rho }^{\gamma -1}\left(\varpi \right)-a{)}_{\tilde{q}}^{(\gamma -1)}r\left(\varpi \right)\mho \left(\varpi \right).$$

4. Conclusions

In this manuscript, by employing ∇-integral fractional of order $\gamma \in (0,1]$, many ∇ inequalities Hardy-type are introduced. For the sake of completeness, we applied the main results to some nonuniform time scales.