Bennett-Leindler nabla type inequalities via conformable fractional derivatives on time scales

Ahmed A. El-Deeb; Samer D. Makharesh; Sameh S. Askar; Dumitru Baleanu; Ahmed A. El-Deeb; Samer D. Makharesh; Sameh S. Askar; Dumitru Baleanu

doi:10.3934/math.2022777

AIMS Mathematics

2022, Volume 7, Issue 8: 14099-14116. doi: 10.3934/math.2022777

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Research article Special Issues

Bennett-Leindler nabla type inequalities via conformable fractional derivatives on time scales

1.
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City (11884), Cairo, Egypt
2.
Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
3.
Institute of Space Science, Magurele-Bucharest, Romania
4.
Department of Mathematics, Cankaya University, Ankara 06530, Turkey
5.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

Received: 06 January 2022 Revised: 27 March 2022 Accepted: 07 April 2022 Published: 27 May 2022
MSC : 26D10, 26D15, 34N05, 26E70

In this work, we prove several new $(\gamma, a)$ -nabla Bennett and Leindler dynamic inequalities on time scales. The results proved here generalize some known dynamic inequalities on time scales, unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proved by using integration by parts, chain rule and Hölder inequality for the $(\gamma, a)$ -nabla-fractional derivative on time scales.

Keywords:

Citation: Ahmed A. El-Deeb, Samer D. Makharesh, Sameh S. Askar, Dumitru Baleanu. Bennett-Leindler nabla type inequalities via conformable fractional derivatives on time scales[J]. AIMS Mathematics, 2022, 7(8): 14099-14116. doi: 10.3934/math.2022777

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Abstract

1. Introduction

In ^[13], Hardy presented the discrete version.

Theorem 1.1. If $\{r(n)\}_{n = 0}^\infty$ is a nonnegative real sequence and $l > 1$ , then

$\begin{equation} \sum\limits_{n = 1}^{\infty }\bigg(\frac{1}{n}\sum\limits_{m = 1}^{n}r(m)\bigg)^{l}\leq \Big( \frac{l}{l-1}\Big)^{l}\sum\limits_{n = 1}^{\infty }r^l(n). \end{equation}$

(1.1)

Also, Hardy ^[14] gave the continuous version of (1.1).

Theorem 1.2. Let $r\geqslant 0$ be a continuous over $[0, \infty)$ and $l > 1$ . Then

$\begin{equation} \int_{0}^{\infty }\bigg(\frac{1}{x}\int_{0}^{x}r(s)ds\bigg)^{l}dx\leq \Big( \frac{l}{l-1}\Big)^{l}\int_{0}^{\infty }r^{l}(x)dx. \end{equation}$

(1.2)

Copson ^[6] obtained a new version of inequality (1.1) by replacing the arithmetic mean of a sequence by a weighted arithmetic mean in the following manner: Let $b(m)\geqslant 0,$ $\eta(m)\geqslant 0$ for all $m.$ If $l > 1,$ $c > 1,$ then

$\begin{equation} \sum\limits_{m = 1}^{\infty} \frac{\eta(m)}{[\overline{\xi}(m)]^c} \bigg(\sum\limits_{j = 1}^{m} b(j) \eta(j) \bigg)^l \leqslant \bigg(\frac{l}{c-1} \bigg)^l \sum\limits_{m = 1}^{\infty} \eta(m)[\overline{\xi}(m)]^{l-c} b^l(m), \end{equation}$

(1.3)

where $\overline{\xi}(m) = \sum_{j = 1}^m \eta(j),$ and if $0\leqslant c < 1 < l,$ then

$\begin{equation} \sum\limits_{m = 1}^{\infty} \frac{\eta(m)}{[\overline{\xi}(m)]^c} \bigg(\sum\limits_{j = 1}^{\infty} b(j) \eta(j) \bigg)^l \leqslant \bigg(\frac{l}{1-c} \bigg)^l \sum\limits_{m = 1}^{\infty} \eta(m)[\overline{\xi}(m)]^{l-c} b^l(m). \end{equation}$

(1.4)

The reverse versions of the inequalities (1.3) and (1.4), which have been derived by Bennett and Leindler ^[4,17], can be deduced for $\overline{\xi}(m)\rightarrow \infty$ , $b(m)\geqslant 0$ and $\eta(m)\geqslant 0$ for all $m$ that if $0 < l < 1 < c,$ then

$\begin{equation} \sum\limits_{m = 1}^{\infty} \frac{\eta(m)}{[\overline{\xi}(m)]^c} \bigg(\sum\limits_{j = 1}^{m} b(j) \eta(j) \bigg)^l \geqslant \bigg(\frac{lL}{c-1} \bigg)^l \sum\limits_{m = 1}^{\infty} \eta(m)[\overline{\xi}(m)]^{l-c} b^l(m), \end{equation}$

(1.5)

where $L = \inf \frac{\overline{\xi}(m)}{\overline{\xi}(m+1)},$ and if $c \leqslant 0 < l < 1,$ then

$\begin{equation} \sum\limits_{m = 1}^{\infty} \frac{\eta(m)}{[\overline{\xi}(m)]^c} \bigg(\sum\limits_{j = 1}^{\infty} b(j) \eta(j) \bigg)^l \geqslant \bigg(\frac{l}{1-c} \bigg)^l \sum\limits_{m = 1}^{\infty} \eta(m)[\overline{\xi}(m)]^{l-c} b^l(m), \end{equation}$

(1.6)

respectively. Copson ^[7] gave the continuous version of the inequalities (1.5) and (1.6), respectively, as follows: Let $\eta$ and $f$ be nonnegative functions and $\overline{\xi}(\wp) = \int_0^\wp \eta(\varrho) d \varrho,$ $B(\wp) = \int_0^\wp \eta(\varrho)f(\varrho) d \varrho$ , $\overline{B}(\wp) = \int_\wp^\infty \eta(\varrho)f(\varrho) d \varrho.$ If $0 < l \leqslant 1 < c,$ $a > 0$ then

$\begin{equation} \int_a^\infty \frac{\eta(\wp)}{[\overline{\xi}(\wp)]^c} [{B}(\wp)]^l dt \geqslant \bigg( \frac{p}{c-1}\bigg)^l \int_a^\infty \eta(\wp) [\overline{\xi}(\wp)]^{l-c} f^l(\wp) dt, \end{equation}$

(1.7)

and if $0 < l < 1,$ $c < 1$ then

$\begin{equation} \int_a^\infty \frac{\eta(\wp)}{[\overline{\xi}(\wp)]^c} [\overline{B}(\wp)]^l dt \geqslant \bigg( \frac{l}{1-c}\bigg)^l \int_a^\infty \eta(\wp) [\overline{\xi}(\wp)]^{l-c} f^l(\wp) dt. \end{equation}$

(1.8)

For further results on Hardy inequalities and other types see ^{[1,2,3,5,8,9,10,12,15,18,19,20,21,23,24,25,26,27]}. In ^[20] the author proved the time scales version of (1.1) and (1.2).

$\begin{equation} \int_{a}^{\infty}\bigg(\frac{\int_{a}^{\sigma(\wp)}\eta(\varrho)\Delta \varrho}{\sigma(\wp)-a}\bigg)^l\Delta \wp < \Big(\frac{l}{l-1}\Big)^l\int_{a}^{\infty}\eta^l(\wp)\Delta \wp, \end{equation}$

(1.9)

unless $\eta\equiv0$ .

In ^[11] El-Deeb et al. extended (1.9)

$\begin{equation} \int_{a}^{\infty}\frac{\tilde{\lambda} (\varsigma)\breve{\Psi}^p(\varsigma)}{\tilde {\Lambda} ^{\hat{\gamma}} (\varsigma)}\Delta \varsigma\geq \frac{p}{\hat{\gamma} -1}\int_{a}^{\infty}\tilde{\lambda} (\varsigma)\tilde {\Lambda} ^{p-\hat{\gamma} }(\varsigma)\upsilon^p(\varsigma)\Delta \varsigma, \end{equation}$

(1.10)

where

$\begin{equation*} \breve{\Psi}(\varsigma) = \int_{a}^{\varsigma}\tilde{\lambda} (\upsilon)\upsilon(\varrho)\Delta \upsilon, \qquad {\rm{and}} \qquad \tilde {\Lambda} (\varsigma) = \int_{a}^{\varsigma}\tilde{\lambda} (\upsilon)\Delta \upsilon. \end{equation*}$

In 2021 Kayar et al. ^[16], established the time scale version unification of discrete and continuous Bennett-Leindler inequalities (1.5) and (1.7) as following theorem.

Theorem 1.3. Let $\lambda,$ $f$ be nonnegative, ld-continuous, $\nabla$ -differentiable and $\nabla$ -integrable functions on $[a, \infty)_{\mathbb{T}}$ where $a \in [0, \infty)_{\mathbb{T}}$ . Define

$\begin{equation*} \overline{\xi}(\wp) = \int_{a}^{\wp} \lambda(\varrho) \nabla \varrho \quad \quad B(\wp) = \int_{a}^{\wp} \lambda(\varrho)f(\varrho)\nabla \varrho. \end{equation*}$

If $L = \inf_{\wp \in \mathbb{T}} \frac{\overline{\xi}^\rho(\wp)}{\overline{\xi}(\wp)} > 0,$ $0 < p < 1$ and $c \geqslant 1$ , then

$\begin{eqnarray} &&\int_{a}^{\infty} \frac{\lambda(\wp)}{[\overline{\xi}^c(\wp)]} [B(\wp)]^{p} \nabla \wp \geqslant \bigg(\frac{pL^{c}}{c-1} \bigg)^p \int_a^\infty \frac{\lambda(\wp)f^p(\wp)}{\big(\overline{\xi}(\wp)\big)^{c-p}} \nabla \wp.\\ \end{eqnarray}$

(1.11)

Lately, Zakarya et al. proved an $\alpha$ -conformable version of Hardy inequalities ^[22].

Theorem 1.4. Assume that $\mathbb{T}$ is a time scale with $\omega \in (0, \infty)_{\mathbb{T}}.$ Let $\lambda$ and $\xi$ be rd-continuous and $\alpha$ -fractional differentiable functions on $[\omega, \infty)_\mathbb{T}.$ Define

$\begin{eqnarray*} \chi(\wp) = \int_t^\infty \lambda(\varrho) \Delta_\alpha \varrho \ \ and \ \ \Theta(\wp) = \int_{\omega}^{\wp} \lambda(\varrho)\xi(\varrho) \Delta_\alpha \varrho. \end{eqnarray*}$

Then, for $k\leqslant 0 < h < 1,$ and $\alpha \in (0, 1],$ we have that

$\begin{eqnarray*} \int_{\omega}^\infty \dfrac{\lambda(\wp)}{\chi^{k-\alpha+1}(\wp)} (\Theta^{\sigma}(\wp))^h \Delta_\alpha \wp \geqslant \bigg(\frac{h}{\alpha-k}\bigg)^h \int_{\omega}^{\infty} \lambda(\wp) \xi^h(\wp) \chi^{h-k+\alpha-1}(\wp) \Delta_\alpha \wp. \end{eqnarray*}$

We will need the following chain rule for $\gamma$ -nabla derivative, integration by parts for $\gamma$ -nabla derivative ^[29] and generalized $\gamma$ -nabla Hölder fractional inequality on timescales ^[28] respectively

$\begin{equation} \nabla_a^\gamma (\varpi\circ \xi)(\wp) = \varpi'(\xi(c)) \nabla_a^\gamma (\xi(\wp)). \end{equation}$

(1.12)

$\begin{equation} \int_{d}^{b} \varpi(\wp)\big[\nabla_a^\gamma \xi(\wp) \big] \nabla_a^\gamma \wp = \Big[\varpi(\wp)\xi(\wp)\Big]_d^b-\int_{d}^{b} \big[\nabla_a^\gamma \varpi(\wp)\big] \xi^\rho(\wp) \nabla_a^\gamma \wp. \end{equation}$

(1.13)

$\begin{equation} \int_{d}^{b}|\varpi(\wp)\xi(\wp)|\nabla_a^\gamma \wp\leq \bigg(\int_{d}^{b}|\varpi(\wp)|^p\nabla_a^\gamma \wp\bigg)^{1/p}\bigg(\int_{d}^{b} |\xi(\wp)|^q \nabla_a^\gamma \wp\bigg)^{1/q}, \end{equation}$

(1.14)

where $\frac{1}{p}+\frac{1}{q} = 1$ and $0 < \gamma\leq1$ . Now, we start to state our main results.

2. Main results

We focus in this section, on investigating corresponding results for $\gamma$ -nabla conformable time scales.

Theorem 2.1. Let $\mathbb{T}$ be a time scale with $r\in[0, \infty)_\mathbb{T}$ , $\gamma \in (0, 1]$ and $\wp\geqslant a$ . In addition, let $\gimel$ and $\lambda$ be nonnegative ld-continuous and $(\gamma, a)$ -nabla fractional differentiable functions on $[r, \infty)_\mathbb{T}$ where

$\begin{equation*} \Omega(\wp) = \int_{\wp}^{\infty} \lambda(\varrho) \nabla_a^\gamma \varrho \quad and \quad \Psi(\wp) = \int_{r}^{\wp} \lambda(\varrho)\gimel(\varrho)\nabla_a^\gamma \varrho, \quad \wp\in[r, \infty)_\mathbb{T}. \end{equation*}$

If $0 < p < \gamma$ and $c\leqslant \gamma-1$ , then

$\begin{eqnarray} &&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\Psi(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \geqslant \bigg(\frac{p-\gamma+1)}{\gamma-c} \bigg)^p \int_r^\infty \frac{\lambda(\wp)\gimel^p(\wp) [\Psi(\wp)]^{1-\gamma}}{\big(\Omega^\rho(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp. \end{eqnarray}$

(2.1)

Proof. Using (1.13), with

$\begin{equation*} \nabla^{\gamma}_{a}\eta (\wp) = \lambda(\wp)/ [\Omega^\rho(\wp)]^{c-\gamma+1}, \qquad \qquad \xi(\wp) = [\Psi(\wp)]^{p-\gamma+1}, \end{equation*}$

we have

$\begin{eqnarray*} \nonumber&&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\Psi(\wp)]^{p-\alpha+1} \nabla_a^\gamma \wp = \big[\eta(\wp)\Psi^{p-\alpha+1}(\wp)\big]_r^\infty+\int_{r}^{\infty}\big(-\eta^\rho(\wp)\big) \nabla_a^\gamma \big( \Psi^{p-\gamma+1}(\wp)\big) \nabla_a^\gamma \wp, \end{eqnarray*}$

where we assumed that

$\begin{equation} \eta(\wp) = -\int_{\wp}^{\infty} \lambda(\wp)/ \Omega^{c-\gamma+1}(\wp) \nabla_a^\gamma \varrho. \end{equation}$

(2.2)

Using $\Psi(r) = 0$ and $\eta(\infty) = 0,$ we get

$\begin{eqnarray} &&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\Psi(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp = \int_{r}^{\infty} -\eta^\rho(\wp) \nabla_a^\gamma \big( \Psi^{p-\gamma+1}(\wp)\big) \nabla_a^\gamma \wp. \end{eqnarray}$

(2.3)

Applying (1.12), then there exists $d\in[\wp, \rho(\wp)]$ such that

$\begin{eqnarray} \nabla_a^\gamma \big( \Psi^{p-\gamma+1}(\wp)\big)& = & \frac{p-\gamma+1}{\Psi^{\gamma-p}(d)} \nabla_a^\gamma \Psi(\wp) \geqslant \frac{(p-\gamma+1)\lambda(\wp)\gimel(\wp)}{\Psi^{\gamma-p}(\wp)}. \end{eqnarray}$

(2.4)

Next note $\nabla_a^\gamma \Omega(\wp) = -\lambda(\wp)\leqslant 0.$ By chain rule, we see that

$\begin{eqnarray*} \nonumber \nabla_a^\gamma \big(\Omega(\wp)\big)^{\gamma-c}& = &(\gamma-c) \int_0^1 \frac{\nabla_a^\gamma \Omega(\wp) dh}{\big[h\Omega(\wp)+(1-h)\Omega^\rho(\wp)\big]^{c-\alpha+1}} \\ \nonumber & = & -(\gamma-c) \int_0^1 \frac{\lambda(\wp)dh}{\big[h\Omega(\wp)+(1-h)\Omega^\rho(\wp)\big]^{c-\gamma+1}} \\ \nonumber &\geqslant & -(\gamma-c) \int_0^1 \frac{\lambda(\wp)dh}{\big[h\Omega^\rho(\wp)+(1-h)\Omega^\rho(\wp)\big]^{c-\gamma+1}} \\ & = & -(\gamma-c) \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}}. \end{eqnarray*}$

This implies that

$\begin{equation} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} \geqslant \frac{-1}{\gamma-c} \nabla_a^\gamma \big(\Omega(\wp)\big)^{\gamma-c}, \end{equation}$

(2.5)

and then, we have that

$\begin{eqnarray} -\eta^\rho(\wp) = \int_{\rho(\wp)}^{\infty} \frac{\lambda(\varrho)}{[\Omega^\rho(\varrho)]^{c-\gamma+1}} \nabla_a^\gamma \varrho &\geqslant&\frac{-1}{\gamma-c}\int_{\rho(\wp)}^{\infty} \nabla_a^\gamma \big(\Omega(\varrho)\big)^{\gamma-c} \nabla_a^\gamma \varrho\\ & = &\frac{1}{(\gamma-c) \big(\Omega^\rho(\wp)\big)^{c-\gamma}}. \end{eqnarray}$

(2.6)

Using (2.4) and (2.6) in (2.3) yields

$\begin{eqnarray} &&\bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\Psi(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^p \geqslant \bigg(\frac{p-\gamma+1)}{\gamma-c} \bigg)^p\bigg(\int_{r}^{\infty} \bigg[ \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big(\Psi(\wp)\big)^{p(\gamma-p)} \big(\Omega^\rho(\wp)\big)^{p(c-\gamma)}}\bigg]^\frac{1}{p} \nabla_a^\gamma \wp \bigg)^p.\\ \end{eqnarray}$

(2.7)

Applying Hölder inequality on the term

$\bigg(\int_{r}^{\infty} \bigg[ \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big(\Psi(\wp)\big)^{p(\gamma-p)} \big(\Omega^\rho(\wp)\big)^{p(\gamma-c)}}\bigg]^\frac{1}{p} \nabla_a^\gamma \wp \bigg)^p,$

with indices $1/p$ and $1/1-p$ and

$F(\wp) = \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big(\Psi(\wp)\big)^{p(\gamma-p)} \big(\Omega^\rho(\wp)\big)^{p(c-\gamma)}} \quad and \quad \gimel(\wp) = \bigg(\frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}}\bigg)^{1-p} [\Psi(\wp)]^{(1-p)(p-\gamma+1)}$

$\begin{eqnarray} &&\bigg(\int_{r}^{\infty}F^{1/p}(\wp) \nabla_a^\gamma \wp \bigg)^p = \bigg(\int_{r}^{\infty} \bigg[ \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big(\Psi(\wp)\big)^{p(\gamma-p)} \big(\Omega^\rho(\wp)\big)^{p(c-\gamma)}}\bigg]^\frac{1}{p} \nabla_a^\gamma \wp \bigg)^p\\ &&\geqslant \frac{ \int_{r}^{\infty} F(\wp)\gimel(\wp) \nabla_a^\gamma \wp}{\big(\int_{r}^{\infty} \gimel^{\frac{1}{1-p}}(\wp) \nabla_a^\gamma \wp \big)^{p-1}} = \bigg[\int_r^\infty \frac{\lambda^p(\wp) \big(\lambda(\wp) [\Omega^\rho(\wp)]^{\gamma-c-1}\big)^{1-p} \gimel^p(\wp)[\Psi(\wp)]^{(1-p)(p-\gamma+1)} }{\big(\Psi(\wp)\big)^{p(\alpha-p)} \big(\Omega^\rho(\wp)\big)^{p(c-\alpha)}} \nabla_a^\gamma \wp \bigg]\\ && \quad \times \bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\alpha+1}} [\Psi(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^{p-1}\\ && = \bigg(\int_a^\infty \frac{\lambda(\wp)\gimel^p(\wp) [\Psi(\wp)]^{1-\gamma}}{\big(\Omega^\rho(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp\bigg) \bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\Psi(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^{p-1}. \end{eqnarray}$

(2.8)

From (2.8) and (2.7) yields

$\begin{eqnarray} &&\bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\Psi(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^p\\ &&\geqslant \bigg(\frac{p-\gamma+1)}{\gamma-c} \bigg)^p \bigg(\int_r^\infty \frac{\lambda(\wp)\gimel^p(\wp) [\Psi(\wp)]^{1-\gamma}}{\big(\Omega^\rho(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp\bigg) \bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\Psi(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^{p-1}. \end{eqnarray}$

(2.9)

Therefore,

$\begin{eqnarray*} &&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\Psi(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \geqslant \bigg(\frac{p-\gamma+1)}{\gamma-c} \bigg)^p \int_r^\infty \frac{\lambda(\wp)\gimel^p(\wp) [\Psi(\wp)]^{1-\gamma}}{\big(\Omega^\rho(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp. \end{eqnarray*}$

Remark 2.1. In Theorem 2, if we take $\gamma = 1$ then inequality (2.1) reduces to

$\begin{equation*} \int_{r}^{\infty}\frac{\lambda(\wp)\Psi^{p}(\wp)}{\big(\Omega^\rho(\wp)\big)^{c}} \nabla \wp \geqslant \bigg(\frac{p}{1-c} \bigg)^p \int_{r}^{\infty}\frac{\lambda(\wp)\gimel^{p}(\wp)}{\big(\Omega^\rho(\wp)\big)^{c}} \nabla \wp, \end{equation*}$

where

$\begin{equation*} \Psi(\wp) = \int_{r}^{\wp}\lambda(\varrho)\gimel(\varrho) \nabla \varrho \qquad and \qquad \Omega(\wp) = \int_{\wp}^{\infty}\lambda(\varrho)\nabla \varrho, \end{equation*}$

which is Theorem 3.1 in ^[16].

Corollary 2.1. From Theorem 2, assume $\mathbb{T} = \mathbb{R}$ , then (2.1) obtains

$\begin{equation*} \int_{r}^{\infty}\frac{\lambda(\wp)\Psi^{p-\gamma+1}(\wp)}{\big(\Omega(\wp)\big)^{c-\gamma+1}}(\wp-a)^{\gamma-1} d \wp \geqslant \bigg(\frac{p-\gamma+1}{\gamma-c} \bigg)^p \int_{r}^{\infty}\frac{\lambda(\wp)\gimel^{p}(\wp)\Psi^{1-\gamma}(\wp)}{\big(\Omega(\wp)\big)^{c-\gamma+1}} (\wp-a)^{\gamma-1} dt, \end{equation*}$

where

$\begin{equation*} \Psi(\wp) = \int_{r}^{\wp}\lambda(\varrho)\gimel(\varrho) (\wp-a)^{\gamma-1} d \varrho \qquad and \qquad \Omega(\wp) = \int_{\wp}^{\infty}\lambda(\varrho)(\wp-a)^{\gamma-1} ds. \end{equation*}$

Corollary 2.2. From Theorem 2, assume $\mathbb{T} = h\mathbb{Z}$ , then (2.1) get

$\begin{equation*} \sum\limits_{\wp = \frac{r}{h}}^{\infty} \frac{\lambda(h\wp)\Psi^{p-\gamma+1}(h\wp)}{\Omega^{c-\gamma+1}(h\wp-h)} (\rho^{\gamma-1}(h\wp)-a)_{h}^{(\gamma-1)} \geqslant \bigg(\frac{p-\gamma+1}{\gamma-c} \bigg)^p \sum\limits_{\wp = \frac{r}{h}}^{\infty}\frac{\lambda(h\wp)\gimel^{p}(h\wp)\Psi^{1-\gamma}(\wp)}{\Omega^{c-\gamma+1}(h\wp-h)} (\rho^{\gamma-1}(h\wp)-a)_{h}^{(\gamma-1)}, \end{equation*}$

where

$\begin{equation*} \Psi(\wp) = h\sum\limits_{\varrho = \frac{\wp}{h}}^{\wp}\lambda(h\varrho)\gimel(h\varrho)(\rho^{\gamma-1}(h\varrho)-a)_{h}^{(\gamma-1)} \qquad and \qquad \Omega(\wp) = h\sum\limits_{\varrho = \frac{\wp}{h}}^{\infty}\lambda(h\varrho) (\rho^{\gamma-1}(h\varrho)-a)_{h}^{(\gamma-1)}. \end{equation*}$

Corollary 2.3. From Corollary 2, assumer $\mathbb{T} = \mathbb{Z}$ and $h = 1$ , then (2.1) obtains

$\begin{equation*} \sum\limits_{\wp = r}^{\infty} \frac{\lambda(\wp)\Psi^{p-\gamma+1}(h\wp)}{\Omega^{c-\gamma+1}(\wp-1)} (\rho^{\gamma-1}(\wp)-a)^{(\gamma-1)} \geqslant \bigg(\frac{p-\gamma+1}{\gamma-c} \bigg)^p \sum\limits_{\wp = r}^{\infty}\frac{\lambda(\wp)\gimel^{p}(\wp)\Psi^{1-\gamma}(\wp)}{\Omega^{c-\gamma+1}(\wp-1)} (\rho^{\gamma-1}(\wp)-a)^{(\gamma-1)}, \end{equation*}$

where

$\begin{equation*} \Psi(\wp) = \sum\limits_{\varrho = r}^{\wp}\lambda(\varrho)\gimel(\varrho)(\rho^{\gamma-1}(h\varrho)-a)^{(\gamma-1)} \qquad and \qquad \Omega(\wp) = h\sum\limits_{\varrho = \wp}^{\infty}\lambda(h\varrho) (\rho^{\gamma-1}(h\varrho)-a)^{(\gamma-1)}. \end{equation*}$

Corollary 2.4. From Theorem 2, assume $\mathbb{T} = q^{\mathbb{N}_0}$ , then (2.10) obtains

$\begin{equation*} \sum\limits_{\wp\in (r, \infty)}\frac{\wp(\rho^{\gamma-1}(\wp)-a)_{\tilde{q}}^{(\gamma-1)}\lambda(\wp)\Psi^{p-\gamma+1}(\wp)}{\Omega^{c-\gamma+1}(\rho(\wp))} \geqslant \bigg(\frac{p-\gamma+1}{\gamma-c} \bigg)^p \sum\limits_{\wp \in (r, \infty)}\frac{\wp(\rho^{\gamma-1}(\wp)-a)_{\tilde{q}}^{(\gamma-1)}\lambda(\wp)\gimel^{p}(\wp) \Psi^{1-\gamma}(\wp)}{\Omega^{c-\gamma+1}(\rho(\wp))}, \end{equation*}$

where

$\begin{equation*} \Psi(\wp) = (\tilde{q}-1)\sum\limits_{\varrho\in (r, \wp)}\varrho\lambda(\varrho)\gimel(\varrho) (\rho^{\gamma-1}(\varrho)-a)_{\tilde{q}}^{(\gamma-1)}\qquad and \qquad \Omega(\wp) = (\tilde{q}-1)\sum\limits_{\varrho \in (\wp, \infty)}\varrho \lambda(\varrho) (\rho^{\gamma-1}(\varrho)-a)_{\tilde{q}}^{(\gamma-1)}. \end{equation*}$

Theorem 2.2. Let $\mathbb{T}$ be a time scale with $r\in[0, \infty)_\mathbb{T}$ , $\gamma \in (0, 1]$ and $\wp\geqslant a$ . In addition, let $\gimel$ and $\lambda$ be nonnegative ld-continuous and $(\gamma, a)$ -nabla fractional differentiable functions on $[r, \infty)_\mathbb{T}$ where

$\begin{equation*} \Omega(\wp) = \int_{\wp}^{\infty} \lambda(\varrho) \nabla_a^\gamma \varrho \quad and \quad \overline{\Psi}(\wp) = \int_{\wp}^{\infty} \lambda(\varrho)\gimel(\varrho)\nabla_a^\gamma \varrho, \quad \wp\in[r, \infty)_\mathbb{T}. \end{equation*}$

If $0 < p < \gamma$ and $c \geqslant \gamma$ , then

$\begin{eqnarray} &&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \geqslant \bigg(\frac{p-\gamma+1)}{\gamma-c} \bigg)^p \int_r^\infty \frac{\lambda(\wp)\gimel^p(\wp) [\overline{\Psi^\rho}(\wp)]^{1-\gamma}}{\big(\Omega^\rho(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp.\\ \end{eqnarray}$

(2.10)

Proof. Using (1.13), with

$\begin{equation*} \nabla^{\gamma}_{a}\eta (\wp) = \lambda(\wp)/ [\Omega^\rho(\wp)]^{c-\gamma+1}, \qquad \qquad \xi(\wp) = [ \overline{\Psi}(\wp)]^{p-\gamma+1}, \end{equation*}$

we have

$\begin{eqnarray} \label{101} &&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\overline{\Psi}^\rho(\wp)]^{p-\alpha+1} \nabla_a^\gamma \wp = \big[\eta(\wp) \overline{\Psi}^{p-\alpha+1}(\wp)\big]_r^\infty+\int_{r}^{\infty}\big(\eta(\wp)\big) \nabla_a^\gamma \big(-\overline{\Psi}^{p-\gamma+1}(\wp)\big) \nabla_a^\gamma \wp, \end{eqnarray}$

where we assumed that

$\begin{equation*} \eta(\wp) = \int_{r}^{\wp} \lambda(\wp)/ [\Omega^\rho(\wp)]^{c-\gamma+1} \nabla_a^\gamma \varrho. \end{equation*}$

Using $\overline{\Psi}(\infty) = 0$ and $\eta(r) = 0,$ we get

$\begin{eqnarray} &&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp = \int_{r}^{\infty} \eta(\wp) \nabla_a^\gamma \big(- \overline{\Psi}^{p-\gamma+1}(\wp)\big) \nabla_a^\gamma \wp. \end{eqnarray}$

(2.11)

Applying (1.12), then there exists $d\in[\wp, \rho(\wp)]$ such that

$\begin{eqnarray} \nabla_a^\gamma \big(- \overline{\Psi}^{p-\gamma+1}(\wp)\big)& = & -\frac{p-\gamma+1}{\Psi^{\gamma-p}(d)} \nabla_a^\gamma \Psi(\wp) \geqslant \frac{(p-\gamma+1)\lambda(\wp)\gimel(\wp)}{\overline{\Psi^\rho}^{\gamma-p}(\wp)}. \end{eqnarray}$

(2.12)

Next note $\nabla_a^\gamma \Omega(\wp) = -\lambda(\wp)\leqslant 0.$ By chain rule, we see that

$\begin{eqnarray} \nabla_a^\gamma \big(\Omega(\wp)\big)^{\gamma-c}& = &(\gamma-c) \int_0^1 \frac{\nabla_a^\gamma \Omega(\wp) dh}{\big[h\Omega(\wp)+(1-h)\Omega^\rho(\wp)\big]^{c-\alpha+1}} \\ & = & (c-\gamma) \int_0^1 \frac{\lambda(\wp)dh}{\big[h\Omega(\wp)+(1-h)\Omega^\rho(\wp)\big]^{c-\gamma+1}} \\ &\geqslant & (c-\gamma) \int_0^1 \frac{\lambda(\wp)dh}{\big[h\Omega^\rho(\wp)+(1-h)\Omega^\rho(\wp)\big]^{c-\gamma+1}} \\ & = & (c-\gamma) \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}}. \end{eqnarray}$

This implies that

$\begin{equation} \frac{-\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} \geqslant \frac{-1}{c-\gamma} \nabla_a^\gamma \big(\Omega(\wp)\big)^{\gamma-c}, \end{equation}$

(2.13)

and thus, we get

$\begin{eqnarray} \eta(\wp) = \int_{r}^{\wp} \frac{\lambda(\varrho)}{[\Omega^\rho(\varrho)]^{c-\gamma+1}} \nabla_a^\gamma \varrho & = &\int_{r}^{\infty} \frac{\lambda(\varrho)}{[\Omega^\rho(\varrho)]^{c-\gamma+1}} \nabla_a^\gamma \varrho - \int_{\wp}^{\infty} \frac{\lambda(\varrho)}{[\Omega^\rho(\varrho)]^{c-\gamma+1}} \nabla_a^\gamma \varrho \geqslant -\int_{\wp}^{\infty} \frac{\lambda(\varrho)}{[\Omega^\rho(\varrho)]^{c-\gamma+1}} \nabla_a^\gamma \varrho \\ & = & \frac{-1}{c-\gamma}\int_{\wp}^{\infty} \nabla_a^\gamma \big(\Omega(\varrho)\big)^{\gamma-c} \nabla_a^\gamma \varrho = \frac{1}{(c-\gamma) \big(\Omega^\rho(\wp)\big)^{\gamma-c}}. \end{eqnarray}$

(2.14)

Substituting (2.13), (2.14) into (2.11) obtains

$\begin{eqnarray} &&\bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^p \geqslant \bigg(\frac{p-\gamma+1)}{c-\gamma} \bigg)^p\bigg(\int_{r}^{\infty} \bigg[ \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big(\overline{\Psi^\rho}(\wp)\big)^{p(\gamma-p)} \big(\Omega^\rho(\wp)\big)^{p(c-\gamma)}}\bigg]^\frac{1}{p} \nabla_a^\gamma \wp \bigg)^p.\\ \end{eqnarray}$

(2.15)

Applying Hölder inequality on the term

$\bigg(\int_{r}^{\infty} \bigg[ \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big(\overline{\Psi^\rho}(\wp)\big)^{p(\gamma-p)} \big(\Omega^\rho(\wp)\big)^{p(\gamma-c)}}\bigg]^\frac{1}{p} \nabla_a^\gamma \wp \bigg)^p,$

with indices $1/p$ and $1/1-p$ and

$F(\wp) = \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big(\overline{\Psi^\rho}(\wp)\big)^{p(\gamma-p)} \big(\Omega^\rho(\wp)\big)^{p(c-\gamma)}} \quad and \quad \gimel(\wp) = \bigg(\frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}}\bigg)^{1-p} [\overline{\Psi^\rho}(\wp)]^{(1-p)(p-\gamma+1)}$

$\begin{eqnarray} &&\bigg(\int_{r}^{\infty}F^{1/p}(\wp) \nabla_a^\gamma \wp \bigg)^p = \bigg(\int_{r}^{\infty} \bigg[ \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big(\Psi(\wp)\big)^{p(\gamma-p)} \big(\Omega^\rho(\wp)\big)^{p(c-\gamma)}}\bigg]^\frac{1}{p} \nabla_a^\gamma \wp \bigg)^p\\ &&\geqslant \frac{ \int_{r}^{\infty} F(\wp)\gimel(\wp) \nabla_a^\gamma \wp}{\big(\int_{r}^{\infty} \gimel^{\frac{1}{1-p}}(\wp) \nabla_a^\gamma \wp \big)^{p-1}} = \bigg[\int_r^\infty \frac{\lambda^p(\wp) \big(\lambda(\wp) [\Omega^\rho(\wp)]^{\gamma-c-1}\big)^{1-p} \gimel^p(\wp)[\overline{\Psi^\rho}(\wp)]^{(1-p)(p-\gamma+1)} }{\big(\overline{\Psi^\rho}(\wp)\big)^{p(\alpha-p)} \big(\Omega^\rho(\wp)\big)^{p(c-\alpha)}} \nabla_a^\gamma \wp \bigg]\\ && \quad \times \bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\alpha+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^{p-1}\\ && = \bigg(\int_r^\infty \frac{\lambda(\wp)\gimel^p(\wp) [\overline{\Psi^\rho}(\wp)]^{1-\gamma}}{\big(\Omega^\rho(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp\bigg) \bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^{p-1}. \end{eqnarray}$

(2.16)

Substituting (2.16) into (2.15) yields

$\begin{eqnarray} &&\bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^p\\ &&\geqslant \bigg(\frac{p-\gamma+1)}{c-\gamma} \bigg)^p \bigg(\int_r^\infty \frac{\lambda(\wp)\gimel^p(\wp) [\overline{\Psi^\rho}(\wp)]^{1-\gamma}}{\big(\Omega^\rho(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp\bigg) \bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^{p-1}. \end{eqnarray}$

(2.17)

Therefore,

$\begin{eqnarray*} &&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\Omega^\rho(\wp)]^{c-\gamma+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \geqslant \bigg(\frac{p-\gamma+1)}{c-\gamma} \bigg)^p \int_r^\infty \frac{\lambda(\wp)\gimel^p(\wp) [\overline{\Psi^\rho}(\wp)]^{1-\gamma}}{\big(\Omega^\rho(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp. \end{eqnarray*}$

Remark 2.2. In Theorem 2, if we take $\gamma = 1$ then inequality (2.10) reduces to

$\begin{equation*} \int_{r}^{\infty}\frac{\lambda(\wp) \overline{\Psi}^{p}(\wp)}{\big(\Omega^\rho(\wp)\big)^{c}} \nabla \wp \geqslant \bigg(\frac{p}{c-1} \bigg)^p \int_{r}^{\infty}\frac{\lambda(\wp)\gimel^{p}(\wp)}{\big(\Omega^\rho(\wp)\big)^{c-p}} \nabla \wp, \end{equation*}$

where

$\begin{equation*} \overline{\Psi}(\wp) = \int_{\wp}^{\infty}\lambda(\varrho)\gimel(\varrho) \nabla \varrho \qquad and \qquad \Omega(\wp) = \int_{\wp}^{\infty}\lambda(\varrho)\nabla \varrho, \end{equation*}$

which is Theorem 3.4 in ^[16].

Corollary 2.5. From Theorem 2, assume $\mathbb{T} = \mathbb{R}$ , then (2.10) gets

$\begin{equation*} \int_{r}^{\infty}\frac{\lambda(\wp)\Psi^{p-\gamma+1}(\wp)}{\big(\Omega(\wp)\big)^{c-\gamma+1}}(\wp-a)^{\gamma-1} d \wp \geqslant \bigg(\frac{p-\gamma+1}{c-\gamma} \bigg)^p \int_{r}^{\infty}\frac{\lambda(\wp)\gimel^{p}(\wp)\Psi^{1-\gamma}(\wp)}{\big(\Omega(\wp)\big)^{c-p-\gamma+1}} (\wp-a)^{\gamma-1} dt, \end{equation*}$

where

$\begin{equation*} \overline{\Psi}(\wp) = \int_{\wp}^{\infty}\lambda(\varrho)\gimel(\varrho) (\wp-a)^{\gamma-1} d \varrho \qquad and \qquad \Omega(\wp) = \int_{\wp}^{\infty}\lambda(\varrho)(\wp-a)^{\gamma-1} ds. \end{equation*}$

Corollary 2.6. From Theorem 2, assume $\mathbb{T} = h\mathbb{Z}$ , then (2.10) gets

$\begin{equation*} \sum\limits_{\wp = \frac{r}{h}}^{\infty} \frac{\lambda(h\wp)\overline{\Psi}^{p-\gamma+1}(h\wp)}{\Omega^{c-\gamma+1}(h\wp-h)} (\rho^{\gamma-1}(h\wp)-a)_{h}^{(\gamma-1)} \geqslant \bigg(\frac{p-\gamma+1}{c-\gamma} \bigg)^p \sum\limits_{\wp = \frac{r}{h}}^{\infty}\frac{\lambda(h\wp)\gimel^{p}(h\wp) \overline{\Psi}^{1-\gamma}(\wp)}{\Omega^{c-p-\gamma+1}(h\wp-h)} (\rho^{\gamma-1}(h\wp)-a)_{h}^{(\gamma-1)} , \end{equation*}$

where

$\begin{equation*} \overline{\Psi}(\wp) = h\sum\limits_{\varrho = \frac{\wp}{h}}^{\infty}\lambda(h\varrho)\gimel(h\varrho)(\rho^{\gamma-1}(h\varrho)-a)_{h}^{(\gamma-1)} \qquad and \qquad \Omega(\wp) = h\sum\limits_{\varrho = \frac{\wp}{h}}^{\infty}\lambda(h\varrho) (\rho^{\gamma-1}(h\varrho)-a)_{h}^{(\gamma-1)}. \end{equation*}$

Corollary 2.7. From Corollary 2, assume $\mathbb{T} = \mathbb{Z}$ , and $h = 1$ , then (2.10) obtains

$\begin{equation*} \sum\limits_{\wp = r}^{\infty} \frac{\lambda(\wp)\overline{\Psi}^{p-\gamma+1}(h\wp)}{\Omega^{c-p-\gamma+1}(\wp-1)} (\rho^{\gamma-1}(\wp)-a)^{(\gamma-1)} \geqslant \bigg(\frac{p-\gamma+1}{c-\gamma} \bigg)^p \sum\limits_{\wp = r}^{\infty}\frac{\lambda(\wp)\gimel^{p}(\wp)\overline{\Psi}^{1-\gamma}(\wp)}{\Omega^{c-p-\gamma+1}(\wp-1)} (\rho^{\gamma-1}(\wp)-a)^{(\gamma-1)} , \end{equation*}$

where

$\begin{equation*} \overline{\Psi}(\wp) = \sum\limits_{\varrho = \wp}^{\infty}\lambda(\varrho)\gimel(\varrho)(\rho^{\gamma-1}(h\varrho)-a)^{(\gamma-1)} \qquad and \qquad \Omega(\wp) = h\sum\limits_{\varrho = \wp}^{\infty}\lambda(h\varrho) (\rho^{\gamma-1}(h\varrho)-a)^{(\gamma-1)}. \end{equation*}$

Corollary 2.8. From Theorem 2, assume $\mathbb{T} = q^{\mathbb{N}_0}$ , then (2.10) obtains

$\begin{equation*} \sum\limits_{\wp\in (r, \infty)}\frac{\wp(\rho^{\gamma-1}(\wp)-a)_{\tilde{q}}^{(\gamma-1)}\lambda(\wp)\overline{\Psi}^{p-\gamma+1}(\wp)}{\Omega^{c-\gamma+1}(\rho(\wp))} \geqslant \bigg(\frac{p-\gamma+1}{c-\gamma} \bigg)^p \sum\limits_{\wp \in (r, \infty)}\frac{\wp(\rho^{\gamma-1}(\wp)-a)_{\tilde{q}}^{(\gamma-1)}\lambda(\wp)\gimel^{p}(\wp) \overline{\Psi}^{1-\gamma}(\wp)}{\Omega^{c-p-\gamma+1}(\rho(\wp))}, \end{equation*}$

where

$\begin{equation*} \overline{\Psi}(\wp) = (\tilde{q}-1)\sum\limits_{\varrho\in (\wp, \infty)}\varrho\lambda(\varrho)\gimel(\varrho) (\rho^{\gamma-1}(\varrho)-a)_{\tilde{q}}^{(\gamma-1)}\qquad and \qquad \Omega(\wp) = (\tilde{q}-1)\sum\limits_{\varrho \in (\wp, \infty)}\varrho \lambda(\varrho) (\rho^{\gamma-1}(\varrho)-a)_{\tilde{q}}^{(\gamma-1)}. \end{equation*}$

Theorem 2.3. Let $\mathbb{T}$ be a time scale with $r\in[0, \infty)_\mathbb{T}$ , $\gamma \in (0, 1]$ and $\wp\geqslant a$ . In addition, let $\gimel$ and $\lambda$ be nonnegative ld-continuous and $(\gamma, a)$ -nabla fractional differentiable functions on $[r, \infty)_\mathbb{T}$ where

$\begin{equation*} \overline{\Omega}(\wp) = \int_{r}^{\wp} \lambda(\varrho) \nabla_a^\gamma \varrho \quad and \quad \overline{\Psi}(\wp) = \int_{\wp}^{\infty} \lambda(\varrho)\gimel(\varrho)\nabla_a^\gamma \varrho, \quad \wp\in[r, \infty)_\mathbb{T}. \end{equation*}$

If $0 < p < \gamma$ and $c \leqslant \gamma-1$ , then

$\begin{eqnarray} &&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \geqslant \bigg(\frac{p-\gamma+1)}{\gamma-c} \bigg)^p \int_r^\infty \frac{\lambda(\wp)\gimel^p(\wp) [\overline{\Psi^\rho}(\wp)]^{1-\gamma}}{\big(\overline{\Omega}(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp. \end{eqnarray}$

(2.18)

Proof. Using (1.13), with

$\begin{equation*} \nabla^{\gamma}_{a}\eta (\wp) = \lambda(\wp)/ [\overline{\Omega}(\wp)]^{c-\gamma+1}, \qquad \qquad \xi(\wp) = [ \overline{\Psi}(\wp)]^{p-\gamma+1}, \end{equation*}$

we have

$\begin{eqnarray} \label{1015} &&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [\overline{\Psi}^\rho(\wp)]^{p-\alpha+1} \nabla_a^\gamma \wp = \big[\eta(\wp) \overline{\Psi}^{p-\alpha+1}(\wp)\big]_r^\infty+\int_{r}^{\infty}\big(-\eta(\wp)\big) \nabla_a^\gamma \big(\overline{\Psi}^{p-\gamma+1}(\wp)\big) \nabla_a^\gamma \wp, \end{eqnarray}$

where we assumed that

$\begin{equation*} \eta(\wp) = \int_{r}^{\wp} \lambda(\wp)/ [\overline{\Omega}(\wp)]^{c-\gamma+1} \nabla_a^\gamma \varrho. \end{equation*}$

Using $\overline{\Psi}(\infty) = 0$ and $\eta(r) = 0,$ we have that

$\begin{eqnarray} &&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp = \int_{r}^{\infty} -\eta(\wp) \nabla_a^\gamma \big( \overline{\Psi}^{p-\gamma+1}(\wp)\big) \nabla_a^\gamma \wp. \end{eqnarray}$

(2.19)

Applying (1.12), then there exists $d\in[\wp, \rho(\wp)]$ such that

$\begin{eqnarray} -\nabla_a^\gamma \big(\overline{\Psi}^{p-\gamma+1}(\wp)\big)& = & -\frac{p-\gamma+1}{\overline{\Psi}^{\gamma-p}(d)} \nabla_a^\gamma \overline{\Psi}(\wp) \geqslant \frac{(p-\gamma+1)\lambda(\wp)\gimel(\wp)}{\overline{\Psi^\rho}^{\gamma-p}(\wp)}. \end{eqnarray}$

(2.20)

Next note $\nabla_a^\gamma \Omega(\wp) = \lambda(\wp)\leqslant 0.$ By chain rule, we see that

$\begin{eqnarray} \label{n126} \nabla_a^\gamma \big(\Omega(\wp)\big)^{\gamma-c}& = &(\gamma-c) \int_0^1 \frac{\nabla_a^\gamma \overline{\Omega}(\wp) dh}{\big[h\overline{\Omega}(\wp)+(1-h)\overline{\Omega}^\rho(\wp)\big]^{c-\gamma+1}} \\ & = & (\gamma-c) \int_0^1 \frac{\lambda(\wp)dh}{\big[h\Omega(\wp)+(1-h)\Omega^\rho(\wp)\big]^{c-\gamma+1}} \\ &\leqslant & (\gamma-c) \int_0^1 \frac{\lambda(\wp)dh}{\big[h\overline{\Omega}(\wp)+(1-h)\overline{\Omega}(\wp)\big]^{c-\gamma+1}} \\ & = & (\gamma-c) \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}}. \end{eqnarray}$

This implies that

$\begin{equation} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} \geqslant \frac{-1}{\gamma-c} \nabla_a^\gamma \big(\overline{\Omega}(\wp)\big)^{\gamma-c}, \end{equation}$

(2.21)

and then, we have that

$\begin{eqnarray} \eta(\wp) = \int_{r}^{\wp} \frac{\lambda(\varrho)}{[\overline{\Omega}(\varrho)]^{c-\gamma+1}} \nabla_a^\gamma \varrho & = &\int_{r}^{\infty} \frac{\lambda(\varrho)}{[\Omega^\rho(\varrho)]^{c-\gamma+1}} \nabla_a^\gamma \varrho - \int_{\wp}^{\infty} \frac{\lambda(\varrho)}{[\overline{\Omega}(\varrho)]^{c-\gamma+1}} \nabla_a^\gamma \varrho \geqslant -\int_{\wp}^{\infty} \frac{\lambda(\varrho)}{[\overline{\Omega}(\varrho)]^{c-\gamma+1}} \nabla_a^\gamma \varrho \\ & = & \frac{1}{\gamma-c}\int_{\wp}^{\infty} \nabla_a^\gamma \big(\Omega(\varrho)\big)^{\gamma-c} \nabla_a^\gamma \varrho = \frac{1}{(\gamma-c) \big(\overline{\Omega}(\wp)\big)^{c-\gamma}}. \end{eqnarray}$

(2.22)

Substituting (2.21), (2.22) into (2.19) yields

$\begin{eqnarray} &&\bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^p \geqslant \bigg(\frac{p-\gamma+1)}{c-\gamma} \bigg)^p\bigg(\int_{r}^{\infty} \bigg[ \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big(\overline{\Psi^\rho}(\wp)\big)^{p(\gamma-p)} \big(\overline{\Omega}(\wp)\big)^{p(c-\gamma)}}\bigg]^\frac{1}{p} \nabla_a^\gamma \wp \bigg)^p.\\ \end{eqnarray}$

(2.23)

Applying Hölder inequality on the term

$\bigg(\int_{r}^{\infty} \bigg[ \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big(\overline{\Psi^\rho}(\wp)\big)^{p(\gamma-p)} \big(\overline{\Omega}(\wp)\big)^{p(\gamma-c)}}\bigg]^\frac{1}{p} \nabla_a^\gamma \wp \bigg)^p,$

with indices $1/p$ and $1/1-p$ and

$F(\wp) = \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big(\overline{\Psi^\rho}(\wp)\big)^{p(\gamma-p)} \big(\overline{\Omega}(\wp)\big)^{p(c-\gamma)}} \quad and \quad \gimel(\wp) = \bigg(\frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}}\bigg)^{1-p} [\overline{\Psi^\rho}(\wp)]^{(1-p)(p-\gamma+1)}$

$\begin{eqnarray} &&\bigg(\int_{r}^{\infty}F^{1/p}(\wp) \nabla_a^\gamma \wp \bigg)^p = \bigg(\int_{r}^{\infty} \bigg[ \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big(\overline{\Psi}^\rho(\wp)\big)^{p(\gamma-p)} \big(\overline{\Omega}(\wp)\big)^{p(c-\gamma)}}\bigg]^\frac{1}{p} \nabla_a^\gamma \wp \bigg)^p\\ &&\geqslant \frac{ \int_{r}^{\infty} F(\wp)\gimel(\wp) \nabla_a^\gamma \wp}{\big(\int_{r}^{\infty} \gimel^{\frac{1}{1-p}}(\wp) \nabla_a^\gamma \wp \big)^{p-1}} = \bigg[\int_r^\infty \frac{\lambda^p(\wp) \big(\lambda(\wp) [\overline{\Omega}(\wp)]^{\gamma-c-1}\big)^{1-p} \gimel^p(\wp)[\overline{\Psi^\rho}(\wp)]^{(1-p)(p-\gamma+1)} }{\big(\overline{\Psi^\rho}(\wp)\big)^{p(\alpha-p)} \big(\overline{\Omega}(\wp)\big)^{p(c-\alpha)}} \nabla_a^\gamma \wp \bigg]\\ && \quad \times \bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\alpha+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^{p-1}\\ && = \bigg(\int_r^\infty \frac{\lambda(\wp)\gimel^p(\wp) [\overline{\Psi^\rho}(\wp)]^{1-\gamma}}{\big(\overline{\Omega}(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp\bigg) \bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^{p-1}. \end{eqnarray}$

(2.24)

From (2.24) and (2.23) gets

$\begin{eqnarray} &&\bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^p\\ &&\geqslant \bigg(\frac{p-\gamma+1)}{c-\gamma} \bigg)^p \bigg(\int_r^\infty \frac{\lambda(\wp)\gimel^p(\wp) [\overline{\Psi^\rho}(\wp)]^{1-\gamma}}{\big(\overline{\Omega}(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp\bigg) \bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^{p-1}. \end{eqnarray}$

(2.25)

Therefore,

$\begin{eqnarray*} &&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [\overline{\Psi^\rho}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \geqslant \bigg(\frac{p-\gamma+1)}{\gamma-c} \bigg)^p \int_r^\infty \frac{\lambda(\wp)\gimel^p(\wp) [\overline{\Psi^\rho}(\wp)]^{1-\gamma}}{\big(\overline{\Omega}(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp. \end{eqnarray*}$

Remark 2.3. In Theorem 2, if we take $\gamma = 1$ then inequality (2.18) reduces to

$\begin{equation*} \int_{r}^{\infty}\frac{\lambda(\wp)[\overline{\Psi^\rho}(\wp)]^{p}}{\big(\overline{\Omega}(\wp)\big)^{c}} \nabla \wp \geqslant \bigg(\frac{p}{1-c} \bigg)^p \int_{r}^{\infty}\frac{\lambda(\wp)\gimel^{p}(\wp)}{\big(\overline{\Omega}(\wp)\big)^{c-p}} \nabla \wp, \end{equation*}$

where

$\begin{equation*} \overline{\Psi}(\wp) = \int_{\wp}^{\infty}\lambda(\varrho)\gimel(\varrho) \nabla \varrho \qquad and \qquad \overline{\Omega}(\wp) = \int_{r}^{\wp}\lambda(\varrho)\nabla \varrho, \end{equation*}$

which is Theorem 3.9 in ^[16].

Corollary 2.9. From Theorem 2, assume $\mathbb{T} = \mathbb{R}$ , then (2.18) gets

$\begin{equation*} \int_{r}^{\infty}\frac{\lambda(\wp) \overline{\Psi}^{p-\gamma+1}(\wp)}{\big(\overline{\Omega}(\wp)\big)^{c-\gamma+1}}(\wp-a)^{\gamma-1} d \wp \geqslant \bigg(\frac{p-\gamma+1}{\gamma-c} \bigg)^p \int_{r}^{\infty}\frac{\lambda(\wp)\gimel^{p}(\wp)\overline{\Psi}^{1-\gamma}(\wp)}{\big(\overline{\Omega}(\wp)\big)^{c-p-\gamma+1}} (\wp-a)^{\gamma-1} dt, \end{equation*}$

where

$\begin{equation*} \overline{\Psi}(\wp) = \int_{\wp}^{\infty}\lambda(\varrho)\gimel(\varrho) (\wp-a)^{\gamma-1} d \varrho \qquad and \qquad \overline{\Omega}(\wp) = \int_{r}^{\wp}\lambda(\varrho)(\wp-a)^{\gamma-1} ds. \end{equation*}$

Remark 2.4. In Corollary 2, if we take $\gamma = 1$ yields discrete Bennett-Leindler type inequality (1.8).

Corollary 2.10. From Theorem 2, assume $\mathbb{T} = h\mathbb{Z}$ , then (2.18) gets

$\begin{equation*} \sum\limits_{\wp = \frac{r}{h}}^{\infty} \frac{\lambda(h\wp)\overline{\Psi}^{p-\gamma+1}(h\wp-h)}{\overline{\Omega}^{c-\gamma+1}(h\wp)} (\rho^{\gamma-1}(h\wp)-a)_{h}^{(\gamma-1)} \geqslant \bigg(\frac{p-\gamma+1}{\gamma-c} \bigg)^p \sum\limits_{\wp = \frac{r}{h}}^{\infty}\frac{\lambda(h\wp)\gimel^{p}(h\wp)\overline{\Psi}^{1-\gamma}(\wp)}{\overline{\Omega}^{c-p-\gamma+1}(h\wp)} (\rho^{\gamma-1}(h\wp)-a)_{h}^{(\gamma-1)} , \end{equation*}$

where

$\begin{equation*} \overline{\Psi}(\wp) = h\sum\limits_{\varrho = \frac{\wp}{h}}^{\infty}\lambda(h\varrho)\gimel(h\varrho)(\rho^{\gamma-1}(h\varrho)-a)_{h}^{(\gamma-1)} \qquad and \qquad \overline{\Omega}(\wp) = h\sum\limits_{\varrho = \frac{r}{h}}^{\wp}\lambda(h\varrho) (\rho^{\gamma-1}(h\varrho)-a)_{h}^{(\gamma-1)}. \end{equation*}$

Corollary 2.11. For $\mathbb{T} = \mathbb{Z}$ , we take $h = 1$ in Corollary 2. In this case, inequality (2.18) reduces to

$\begin{equation*} \sum\limits_{\wp = r}^{\infty} \frac{\lambda(\wp) \overline{\Psi}^{p-\gamma+1}(\wp-1)}{\overline{\Omega}^{c-\gamma+1}(\wp)} (\rho^{\gamma-1}(\wp)-a)^{(\gamma-1)} \geqslant \bigg(\frac{p-\gamma+1}{\gamma-c} \bigg)^p \sum\limits_{\wp = r}^{\infty}\frac{\lambda(\wp)\gimel^{p}(\wp)\overline{\Psi}^{1-\gamma}(\wp)}{\overline{\Omega}^{c-p-\gamma+1}(\wp)} (\rho^{\gamma-1}(\wp)-a)^{(\gamma-1)} , \end{equation*}$

where

$\begin{equation*} \overline{\Psi}(\wp) = \sum\limits_{\varrho = \wp}^{\infty}\lambda(\varrho)\gimel(\varrho)(\rho^{\gamma-1}(h\varrho)-a)^{(\gamma-1)} \qquad and \qquad \overline{\Omega}(\wp) = h\sum\limits_{\varrho = r}^{\wp}\lambda(h\varrho) (\rho^{\gamma-1}(h\varrho)-a)^{(\gamma-1)}. \end{equation*}$

Remark 2.5. In Corollary 2, if we take $\gamma = 1$ and $r = 1$ , yields discrete Bennett-Leindler type inequality (1.6), which is the converse of Copson inequality (1.4).

Corollary 2.12. From Theorem 2, assume $\mathbb{T} = q^{\mathbb{N}_0}$ , then (2.18) gets

$\begin{equation*} \sum\limits_{\wp\in (r, \infty)}\frac{\wp(\rho^{\gamma-1}(\wp)-a)_{\tilde{q}}^{(\gamma-1)}\lambda(\wp)\overline{\Psi}^{p-\gamma+1}(\rho(\wp))}{\overline{\Omega}^{c-\gamma+1}(\wp)} \geqslant \bigg(\frac{p-\gamma+1}{\gamma-c} \bigg)^p \sum\limits_{\wp \in (r, \infty)}\frac{\wp(\rho^{\gamma-1}(\wp)-a)_{\tilde{q}}^{(\gamma-1)}\lambda(\wp)\gimel^{p}(\wp) \overline{\Psi}^{1-\gamma}(\wp)}{\overline{\Omega}^{c-p-\gamma+1}(\wp)}, \end{equation*}$

where

$\begin{equation*} \overline{\Psi}(\wp) = (\tilde{q}-1)\sum\limits_{\varrho\in (\wp, \infty)}\varrho\lambda(\varrho)\gimel(\varrho) (\rho^{\gamma-1}(\varrho)-a)_{\tilde{q}}^{(\gamma-1)}\qquad and \qquad \overline{\Omega}(\wp) = (\tilde{q}-1)\sum\limits_{\varrho \in (r, \wp)}\varrho \lambda(\varrho) (\rho^{\gamma-1}(\varrho)-a)_{\tilde{q}}^{(\gamma-1)}. \end{equation*}$

Theorem 2.4. Let $\mathbb{T}$ be a time scale with $r\in[0, \infty)_\mathbb{T}$ , $\gamma \in (0, 1]$ and $\wp\geqslant a$ . In addition, let $\gimel$ and $\lambda$ be nonnegative ld-continuous and $(\gamma, a)$ -nabla fractional differentiable functions on $[r, \infty)_\mathbb{T}$ where

$\begin{equation*} \overline{\Omega}(\wp) = \int_{r}^{\wp} \lambda(\varrho) \nabla_a^\gamma \varrho, \quad \overline{\Omega}(\infty) = \infty, \quad \Psi(\wp) = \int_{r}^{\wp} \lambda(\varrho)\gimel(\varrho)\nabla_a^\gamma \varrho, \quad \wp\in[r, \infty)_\mathbb{T}. \end{equation*}$

If $L = \inf_{\wp \in \mathbb{T}} \frac{\overline{\Omega}^\rho(\wp)}{\overline{\Omega}(\wp)} > 0,$ $0 < p < \gamma$ and $c \geqslant \gamma$ , then

$\begin{eqnarray} &&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [{\Psi}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \geqslant \bigg(\frac{(p-\gamma+1)L^{c-\gamma+1}}{c-\gamma} \bigg)^p \int_r^\infty \frac{\lambda(\wp)\gimel^p(\wp) [{\Psi}(\wp)]^{1-\gamma}}{\big(\overline{\Omega}(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp.\\ \end{eqnarray}$

(2.26)

Proof. Using (1.13), with

$\begin{equation*} \nabla^{\gamma}_{a}\eta (\wp) = \lambda(\wp)/ [\overline{\Omega}(\wp)]^{c-\gamma+1}, \qquad \qquad \xi(\wp) = [ \Psi(\wp)]^{p-\gamma+1}, \end{equation*}$

we have

$\begin{eqnarray} &&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [\Psi(\wp)]^{p-\alpha+1} \nabla_a^\gamma \wp = \big[\eta(\wp) \Psi^{p-\alpha+1}(\wp)\big]_r^\infty+\int_{r}^{\infty}\big(-\eta^\rho(\wp)\big) \nabla_a^\gamma \big(\Psi^{p-\gamma+1}(\wp)\big) \nabla_a^\gamma \wp, \end{eqnarray}$

where we assumed that

$\begin{equation*} \eta(\wp) = -\int_{\wp}^{\infty} \lambda(\wp)/ [\overline{\Omega}(\wp)]^{c-\gamma+1} \nabla_a^\gamma \varrho. \end{equation*}$

Using $\Psi(r) = 0$ and $\eta(\infty) = 0,$ we have that

$\begin{eqnarray} &&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [\Psi(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp = \int_{r}^{\infty} -\eta^\rho(\wp) \nabla_a^\gamma \big( \Psi^{p-\gamma+1}(\wp)\big) \nabla_a^\gamma \wp. \end{eqnarray}$

(2.27)

Applying (1.12), then there exists $d\in[\wp, \rho(\wp)]$ such that

$\begin{eqnarray} \nabla_a^\gamma \big(\Psi^{p-\gamma+1}(\wp)\big)& = & \frac{p-\gamma+1}{\Psi^{\gamma-p}(d)} \nabla_a^\gamma \Psi(\wp) \geqslant \frac{(p-\gamma+1)\lambda(\wp)\gimel(\wp)}{\Psi^{\gamma-p}(\wp)}. \end{eqnarray}$

(2.28)

Next note $\nabla_a^\gamma \overline{\Omega}(\wp) = \lambda(\wp)\leqslant 0.$ By chain rule, we see that

$\begin{eqnarray} \nabla_a^\gamma \big(\overline{\Omega}(\wp)\big)^{\gamma-c}& = &(\gamma-c) \int_0^1 \frac{\nabla_a^\gamma \overline{\Omega}(\wp) dh}{\big[h\overline{\Omega}(\wp)+(1-h)\overline{\Omega}^\rho(\wp)\big]^{c-\gamma+1}} \\ & = & -(c-\gamma) \int_0^1 \frac{\lambda(\wp)dh}{\big[h\overline{\Omega}(\wp)+(1-h)\overline{\Omega}^\rho(\wp)\big]^{c-\gamma+1}} \\ &\geqslant & -(c-\gamma) \int_0^1 \frac{\lambda(\wp)dh}{\big[h\overline{\Omega}^\rho(\wp)+(1-h)\overline{\Omega}^\rho(\wp)\big]^{c-\gamma+1}} \\ & = & -(c-\gamma) \frac{\lambda(\wp)}{[\overline{\Omega}^\rho(\wp)]^{c-\gamma+1}} = -(\gamma-c) \frac{\lambda(\wp)}{[\overline{\Omega}^\rho(\wp)]^{c-\gamma+1}}\frac{[\overline{\Omega}(\wp)]^{c-\gamma+1}}{[\overline{\Omega}(\wp)]^{c-\gamma+1}}\\ & \geqslant & -(c-\gamma) \frac{\lambda(\wp)}{L^{c-\gamma+1}[\overline{\Omega}(\wp)]^{c-\gamma+1}}. \end{eqnarray}$

This implies that

$\begin{equation} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} \geqslant \frac{-L^{c-\gamma+1}}{c-\gamma} \nabla_a^\gamma \big(\overline{\Omega}(\wp)\big)^{\gamma-c} \end{equation}$

(2.29)

and then, we have that

$\begin{eqnarray} -\eta^\rho(\wp) = \int_{\rho(\wp)}^{\infty} \frac{\lambda(\varrho)}{[\overline{\Omega}(\varrho)]^{c-\gamma+1}} \nabla_a^\gamma \varrho &\geqslant& -\int_{\rho(\wp)}^{\infty} \frac{L^{c-\gamma+1}}{c-\gamma} \nabla_a^\gamma \big(\overline{\Omega}(\wp)\big)^{\gamma-c} \nabla_a^\gamma \varrho = \frac{L^{c-\gamma+1}}{c-\gamma}\bigg\{\big(\overline{\Omega}^\rho(\wp)\big)^{\gamma-c} -\big(\overline{\Omega}(\infty)\big)^{\gamma-c}\bigg \}\\ & = & \frac{L^{c-\gamma+1}}{c-\gamma}(\overline{\Omega}^\rho(\wp)\big)^{\gamma-c}\geqslant \frac{L^{c-\gamma+1}}{c-\gamma}(\overline{\Omega}(\wp)\big)^{\gamma-c}. \end{eqnarray}$

(2.30)

Substituting (2.30), (2.28) into (2.27) yields

$\begin{eqnarray} &&\bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [{\Psi}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^p \geqslant \bigg(\frac{p-\gamma+1)L^{c-\gamma+1}}{c-\gamma} \bigg)^p\bigg(\int_{r}^{\infty} \bigg[ \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big({\Psi}(\wp)\big)^{p(\gamma-p)} \big(\overline{\Omega}(\wp)\big)^{p(c-\gamma)}}\bigg]^\frac{1}{p} \nabla_a^\gamma \wp \bigg)^p.\\ \end{eqnarray}$

(2.31)

Applying Hölder inequality on the term

$\bigg(\int_{r}^{\infty} \bigg[ \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big({\Psi}(\wp)\big)^{p(\gamma-p)} \big(\overline{\Omega}(\wp)\big)^{p(\gamma-c)}}\bigg]^\frac{1}{p} \nabla_a^\gamma \wp \bigg)^p,$

with indices $1/p$ and $1/1-p$ and

$F(\wp) = \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big({\Psi}(\wp)\big)^{p(\gamma-p)} \big(\overline{\Omega}(\wp)\big)^{p(c-\gamma)}} \quad and \quad \gimel(\wp) = \bigg(\frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}}\bigg)^{1-p} [{\Psi}(\wp)]^{(1-p)(p-\gamma+1)}$

$\begin{eqnarray} &&\bigg(\int_{r}^{\infty}F^{1/p}(\wp) \nabla_a^\gamma \wp \bigg)^p = \bigg(\int_{r}^{\infty} \bigg[ \frac{\lambda^p(\wp)\gimel^p(\wp)}{\big({\Psi}(\wp)\big)^{p(\gamma-p)} \big(\overline{\Omega}(\wp)\big)^{p(c-\gamma)}}\bigg]^\frac{1}{p} \nabla_a^\gamma \wp \bigg)^p\\ &&\geqslant \frac{ \int_{r}^{\infty} F(\wp)\gimel(\wp) \nabla_a^\gamma \wp}{\big(\int_{r}^{\infty} \gimel^{\frac{1}{1-p}}(\wp) \nabla_a^\gamma \wp \big)^{p-1}} = \bigg[\int_r^\infty \frac{\lambda^p(\wp) \big(\lambda(\wp) [\overline{\Omega}(\wp)]^{\gamma-c-1}\big)^{1-p} \gimel^p(\wp)[{\Psi}(\wp)]^{(1-p)(p-\gamma+1)} }{\big({\Psi}(\wp)\big)^{p(\alpha-p)} \big(\overline{\Omega}(\wp)\big)^{p(c-\alpha)}} \nabla_a^\gamma \wp \bigg]\\ && \quad \times \bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\alpha+1}} [{\Psi}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^{p-1}\\ && = \bigg(\int_r^\infty \frac{\lambda(\wp)\gimel^p(\wp) [\overline{\Psi^\rho}(\wp)]^{1-\gamma}}{\big(\overline{\Omega}(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp\bigg) \bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [{\Psi}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^{p-1}. \end{eqnarray}$

(2.32)

From (2.32) and (2.31) gets

$\begin{eqnarray} &&\bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [{\Psi}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^p\\ &&\geqslant \bigg(\frac{p-\gamma+1)}{c-\gamma} \bigg)^p \bigg(\int_r^\infty \frac{\lambda(\wp)\gimel^p(\wp) [\overline{\Psi^\rho}(\wp)]^{1-\gamma}}{\big(\overline{\Omega}(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp\bigg) \bigg(\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [{\Psi}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \bigg)^{p-1}. \end{eqnarray}$

(2.33)

Therefore,

$\begin{eqnarray*} &&\int_{r}^{\infty} \frac{\lambda(\wp)}{[\overline{\Omega}(\wp)]^{c-\gamma+1}} [{\Psi}(\wp)]^{p-\gamma+1} \nabla_a^\gamma \wp \geqslant \bigg(\frac{(p-\gamma+1)L^{c-\gamma+1}}{c-\gamma} \bigg)^p \int_r^\infty \frac{\lambda(\wp)\gimel^p(\wp) [{\Psi}(\wp)]^{1-\gamma}}{\big(\overline{\Omega}(\wp)\big)^{c-p-\gamma+1}} \nabla_a^\gamma \wp. \end{eqnarray*}$

Remark 2.6. In Theorem 2, if we take $\gamma = 1$ then we get Theorem 1.

Corollary 2.13. From Theorem 2, assume $\mathbb{T} = \mathbb{R}$ , then (2.26) gets

$\begin{equation*} \int_{r}^{\infty}\frac{\lambda(\wp)\Psi^{p-\gamma+1}(\wp)}{\big(\overline{\Omega}(\wp)\big)^{c-\gamma+1}}(\wp-a)^{\gamma-1} d \wp \geqslant \bigg(\frac{(p-\gamma+1)L^{c-\gamma+1}}{c-\gamma} \bigg)^p \int_{r}^{\infty}\frac{\lambda(\wp)\gimel^{p}(\wp)\Psi^{1-\gamma}(\wp)}{\big(\overline{\Omega}(\wp)\big)^{c-p-\gamma+1}} (\wp-a)^{\gamma-1} dt, \end{equation*}$

where

$\begin{equation*} {\Psi}(\wp) = \int_{r}^{\wp}\lambda(\varrho)\gimel(\varrho) (\wp-a)^{\gamma-1} d \varrho \qquad and \qquad \overline{\Omega}(\wp) = \int_{r}^{\wp}\lambda(\varrho)(\wp-a)^{\gamma-1} ds. \end{equation*}$

Remark 2.7. In Corollary 2, if we take $L = \gamma = 1$ yields continuous variant of Bennett-Leindler type inequality (1.7).

Corollary 2.14. From Theorem 2, assume $\mathbb{T} = h\mathbb{Z}$ , then (2.26) gets

$\begin{equation*} \sum\limits_{\wp = \frac{r}{h}}^{\infty} \frac{\lambda(h\wp){\Psi}^{p-\gamma+1}(h\wp)}{\Omega^{c-\gamma+1}(h\wp)} (\rho^{\gamma-1}(h\wp)-a)_{h}^{(\gamma-1)} \geqslant \bigg(\frac{(p-\gamma+1)L^{c-\gamma+1}}{c-\gamma} \bigg)^p \sum\limits_{\wp = \frac{r}{h}}^{\infty}\frac{\lambda(h\wp)\gimel^{p}(h\wp) {\Psi}^{1-\gamma}(\wp)}{\overline{\Omega}^{c-p-\gamma+1}(h\wp)} (\rho^{\gamma-1}(h\wp)-a)_{h}^{(\gamma-1)} , \end{equation*}$

where

$\begin{equation*} {\Psi}(\wp) = h\sum\limits_{\varrho = \frac{r}{h}}^{\wp}\lambda(h\varrho)\gimel(h\varrho)(\rho^{\gamma-1}(h\varrho)-a)_{h}^{(\gamma-1)} \qquad and \qquad \overline{\Omega}(\wp) = h\sum\limits_{\varrho = \frac{r}{h}}^{\wp}\lambda(h\varrho) (\rho^{\gamma-1}(h\varrho)-a)_{h}^{(\gamma-1)}. \end{equation*}$

Corollary 2.15. For $\mathbb{T} = \mathbb{Z}$ , we take $h = 1$ in Corollary 2. In this case, inequality (2.26) reduces to

$\begin{equation*} \sum\limits_{\wp = r}^{\infty} \frac{\lambda(\wp){\Psi}^{p-\gamma+1}(h\wp)}{\overline{\Omega}^{c-p-\gamma+1}(\wp)} (\rho^{\gamma-1}(\wp)-a)^{(\gamma-1)} \geqslant \bigg(\frac{(p-\gamma+1)L^{c-\gamma+1}}{c-\gamma} \bigg)^p \sum\limits_{\wp = r}^{\infty}\frac{\lambda(\wp)\gimel^{p}(\wp){\Psi}^{1-\gamma}(\wp)}{\overline{\Omega}^{c-p-\gamma+1}(\wp)} (\rho^{\gamma-1}(\wp)-a)^{(\gamma-1)} , \end{equation*}$

where

$\begin{equation*} {\Psi}(\wp) = \sum\limits_{\varrho = r}^{\wp}\lambda(\varrho)\gimel(\varrho)(\rho^{\gamma-1}(h\varrho)-a)^{(\gamma-1)} \qquad and \qquad \overline{\Omega}(\wp) = h\sum\limits_{\varrho = r}^{\wp}\lambda(h\varrho) (\rho^{\gamma-1}(h\varrho)-a)^{(\gamma-1)}. \end{equation*}$

Remark 2.8. In Corollary 2, if we take $\gamma = 1$ and $r = 1$ , yields discrete Bennett-Leindler type inequality (1.5), which is the converse of Copson inequality (1.3).

Corollary 2.16. From Theorem 2, assume $\mathbb{T} = q^{\mathbb{N}_0}$ , then (2.26) gets

$\begin{equation*} \sum\limits_{\wp\in (r, \infty)}\frac{\wp(\rho^{\gamma-1}(\wp)-a)_{\tilde{q}}^{(\gamma-1)}\lambda(\wp){\Psi}^{p-\gamma+1}(\wp)}{\overline{\Omega}^{c-\gamma+1}(\rho(\wp))} \geqslant \bigg(\frac{(p-\gamma+1)L^{c-\gamma+1}}{c-\gamma} \bigg)^p \sum\limits_{\wp \in (r, \infty)}\frac{\wp(\rho^{\gamma-1}(\wp)-a)_{\tilde{q}}^{(\gamma-1)}\lambda(\wp)\gimel^{p}(\wp) {\Psi}^{1-\gamma}(\wp)}{\overline{\Omega}^{c-p-\gamma+1}(\wp)}, \end{equation*}$

where

$\begin{equation*} {\Psi}(\wp) = (\tilde{q}-1)\sum\limits_{\varrho\in (r, \wp)}\varrho\lambda(\varrho)\gimel(\varrho) (\rho^{\gamma-1}(\varrho)-a)_{\tilde{q}}^{(\gamma-1)}\qquad and \qquad \overline{\Omega}(\wp) = (\tilde{q}-1)\sum\limits_{\varrho \in (r, \wp)}\varrho \lambda(\varrho) (\rho^{\gamma-1}(\varrho)-a)_{\tilde{q}}^{(\gamma-1)}. \end{equation*}$

3. Conclusions

In this paper, with the help of a simple consequence of Keller's chain rule and Hölder inequality for the $(\gamma, a)$ -nabla-fractional derivative on time scales, we generalized a number of Bennett and Leindler Hardy-type inequalities to a general time scale. Besides that, in order to obtain some new inequalities as special cases, we also extended our inequalities to discrete and continuous calculus. In order to illustrate the theorems for each type of inequality applied to various time scales such as $\mathbb{R}$ , $h \mathbb{Z}$ , $\overline{q^\mathbb{Z}}$ and $\mathbb{Z}$ as a sub case of $h \mathbb{Z}$ . For future studies researchers may obtain some different generalizations for dynamic Hardy inequality and its companion inequalities by using the results presented in this paper.

Acknowledgments

The authors extend their appreciation to the Research Supporting Project number (RSP-2022/167), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare that there is no competing interest.

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AIMS Mathematics

Bennett-Leindler nabla type inequalities via conformable fractional derivatives on time scales

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Abstract

1. Introduction

2. Main results

3. Conclusions

Acknowledgments

Conflict of interest

References

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AIMS Mathematics

Bennett-Leindler nabla type inequalities via conformable fractional derivatives on time scales

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Abstract

1. Introduction

2. Main results

3. Conclusions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

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