Some New Bennett–Leindler Type Inequalities via Conformable Fractional Nabla Calculus

Abstract

In this article, we prove several new fractional nabla Bennett–Leindler dynamic inequalities with the help of a simple consequence of Keller’s chain rule, integration by parts, mean inequalities and Hölder’s inequality for the nabla fractional derivative on time scales. As a result of this, some new classical inequalities are obtained as special cases, and we extended our inequalities to discrete and continuous calculus. In addition, when $\alpha =1$, we shall obtain some well-known dynamic inequalities as special instances from our results. Symmetrical properties are critical in determining the best ways to solve inequalities.

1. Introduction

In [1], Hardy presented the discrete version

$${\displaystyle \sum _{n=1}^{\infty }}{\left(\frac{1}{n}{\displaystyle \sum _{j=1}^n}z(j)\right)}^r\le {\left(\frac{r}{r-1}\right)}^r{\displaystyle \sum _{n=1}^{\infty }}{z}^r(n),r>1,$$

(1)

where ${\displaystyle {\{z(n)\}}_{n=0}^{\infty }}$ is a nonnegative real sequence.

In [2], Hardy gave the continuous form of (1) in the following

$${\displaystyle {\int }_0^{\infty }}{\left(\frac{1}{s}{\displaystyle {\int }_0^s}g(t)dt\right)}^{r}ds\le {\left(\frac{r}{r-1}\right)}^r{\displaystyle {\int }_0^{\infty }}{g}^r(s)ds,r>1,$$

(2)

where $g\ge 0$ is continuous over $[0,\infty )$, and ${\left(r/\left(r-1\right)\right)}^r$ in (1) and (2) is optimal.

In [3], Copson replaced the arithmetic mean of a sequence with a weighted arithmetic mean to create a new version of (1) as follows: Let $p\ge r>1,$ $z(j)\ge 0$ and $\varkappa (j)>0$ for all j. Then,

$${\displaystyle \sum _{i=1}^{\infty }}\frac{\varkappa (i)}{{\Phi }^r(i)}{\left({\displaystyle \sum _{j=1}^i}z(j)\varkappa (j)\right)}^p\le {\left(\frac{p}{r-1}\right)}^p{\displaystyle \sum _{i=1}^{\infty }}\varkappa (i){\Phi }^{p-r}(i)z^p(i),$$

(3)

where $\Phi (i)={\displaystyle {\sum }_{j=1}^i}\varkappa (j)$, and if $0\le r<1<p$, then

$${\displaystyle \sum _{i=1}^{\infty }}\frac{\varkappa (i)}{{\Phi }^r(i)}{\left({\displaystyle \sum _{j=i}^{\infty }}z(j)\varkappa (j)\right)}^p\le {\left(\frac{p}{1-r}\right)}^p{\displaystyle \sum _{i=1}^{\infty }}\varkappa (i){\Phi }^{p-r}(i)z^p(i).$$

(4)

In [4], Copson gave the continuous versions of (3) and (4), respectively, in the following: Let $\varkappa$ and g be nonnegative functions and $p\ge 1,$ $r>1.$ Then,

$${\displaystyle {\int }_0^a}\frac{\varkappa (t)}{{\Phi }^r(t)}{\Psi }^p(t)dt\le {\left(\frac{p}{r-1}\right)}^p{\displaystyle {\int }_0^a}\varkappa (t){\Phi }^{p-r}(t)g^p(t)dt,$$

(5)

and if $p\ge 1,$ $0\le r<1,$ then

$${\displaystyle {\int }_a^{\infty }}\frac{\varkappa (t)}{{\Phi }^r(t)}{\overline{\Psi }}^p(t)dt\le {\left(\frac{p}{1-r}\right)}^p{\displaystyle {\int }_a^{\infty }}\varkappa (t){\Phi }^{p-r}(t)g^p(t)dt,$$

(6)

where

$$\Phi (t)={\displaystyle {\int }_0^t}\varkappa (s)ds,\Psi (t)={\displaystyle {\int }_0^t}\varkappa (s)g(s)ds\mathrm{and}\overline{\Psi }(t)={\displaystyle {\int }_t^{\infty }}\varkappa (s)g(s)ds.$$

In [5], Leindler demonstrated that if $\Phi (i)={\displaystyle {\sum }_{j=i}^{\infty }}\varkappa (j)<\infty ,$ $p>1$ and $0\le r<1,$ then

$${\displaystyle \sum _{i=1}^{\infty }}\frac{\varkappa (i)}{{\Phi }^r(i)}{\left({\displaystyle \sum _{j=1}^i}z(j)\varkappa (j)\right)}^p\le {\left(\frac{p}{1-r}\right)}^p{\displaystyle \sum _{i=1}^{\infty }}\varkappa (i){\Phi }^{p-r}(i)z^p(i).$$

(7)

In [6], Bennett explicated that if $\Phi (i)={\displaystyle {\sum }_{j=i}^{\infty }}\varkappa (j)<\infty$ and $1<r\le p,$ then

$${\displaystyle \sum _{i=1}^{\infty }}\frac{\varkappa (i)}{{\Phi }^r(i)}{\left({\displaystyle \sum _{j=i}^{\infty }}z(j)\varkappa (j)\right)}^p\le {\left(\frac{p}{r-1}\right)}^p{\displaystyle \sum _{i=1}^{\infty }}\varkappa (i){\Phi }^{p-r}(i)z^p(i).$$

(8)

In [7,8], Bennett and Leindler established the reverse versions of (3) and (4), respectively. In particular, they exemplified that if $\Phi (i)\to \infty ,$ $z(j)\ge 0,$ $\varkappa (j)\ge 0$ for all j and $0<p<1<r,$ then

$${\displaystyle \sum _{i=1}^{\infty }}\frac{\varkappa (i)}{{\Phi }^r(i)}{\left({\displaystyle \sum _{j=1}^{\infty }}z(j)\varkappa (j)\right)}^p\ge {\left(\frac{pL}{r-1}\right)}^p{\displaystyle \sum _{i=1}^{\infty }}\varkappa (i){\Phi }^{p-r}(i)z^p(i),$$

(9)

where $L={inf}_{i\in \mathbb{N}}\frac{\varkappa (i)}{\varkappa (i+1)},$ and if $r\le 0<p<1,$ then

$${\displaystyle \sum _{i=1}^{\infty }}\frac{\varkappa (i)}{{\Phi }^r(i)}{\left({\displaystyle \sum _{j=1}^{\infty }}z(j)\varkappa (j)\right)}^p\ge {\left(\frac{p}{1-r}\right)}^p{\displaystyle \sum _{i=1}^{\infty }}\varkappa (i){\Phi }^{p-r}(i)z^p(i).$$

(10)

Moreover, in [4], Copson established the continuous versions of (9) and (10), respectively, as follows: Let $\varkappa$ and g be nonnegative functions, $0<p\le 1<r,$ $a>0,$ $\Phi (t)={\displaystyle {\int }_0^t}\varkappa (s)ds,$ $\Psi (t)={\displaystyle {\int }_0^t}\varkappa (s)g(s)ds$ and $\overline{\Psi }(t)={\displaystyle {\int }_t^{\infty }}\varkappa (s)g(s)ds.$ Then

$${\displaystyle {\int }_a^{\infty }}\frac{\varkappa (t)}{{\left[\overline{\Psi }(t)\right]}^r}{\Psi }^p(t)dt\ge {\left(\frac{pL}{r-1}\right)}^p{\displaystyle {\int }_a^{\infty }}\varkappa (t){\Phi }^{p-r}(t)g^p(t)dt,$$

(11)

where $L={inf}_{i\in \mathbb{R}}\frac{\varkappa (i)}{\varkappa (i)}=1,$ and if $0<p<1,$ $r<1,$ then

$${\displaystyle {\int }_a^{\infty }}\frac{\varkappa (t)}{{\left[\overline{\Psi }(t)\right]}^r}{\left[\overline{\Psi }(t)\right]}^{p}dt\ge {\left(\frac{p}{1-r}\right)}^p{\displaystyle {\int }_a^{\infty }}\varkappa (t){\Phi }^{p-r}(t)g^p(t)dt.$$

(12)

Newly, time scale delta calculus was introduced for the first time in 1988 by Hilger [9] to unify the theory of difference equations and the theory of differential equations. Since the appearance of this calculus, many mathematicians have used it to unify the discrete and continuous forms of numerous inequalities.

For example, in [10], the author proved the time scale versions of (1) and (2) in the following: Let $r>1$ and $g\ge 0$ be a right-dense continuous function such that the delta integral ${\displaystyle {\int }_a^{\infty }}{g}^r(s)\Delta s$ exists as a finite number as follows

$${\displaystyle {\int }_a^{\infty }}{\left(\frac{{\displaystyle {\int }_a^{\sigma (t)}}g(s)\Delta s}{\sigma (t)-a}\right)}^r\Delta t<{\left(\frac{r}{r-1}\right)}^r{\displaystyle {\int }_a^{\infty }}{g}^r(t)dt.$$

(13)

The authors of [11] constructed the following time scale versions of (5) and (6), respectively: Let $p\ge r>1.$ Then,

$${\displaystyle {\int }_a^{\infty }}\frac{\varkappa (t)}{{\left[{\Phi }^{\sigma }(t)\right]}^r}{\left[{\Psi }^{\sigma }(t)\right]}^p\Delta t\le {\left(\frac{p}{r-1}\right)}^p{\displaystyle {\int }_a^{\infty }}\frac{{\left[{\Phi }^{\sigma }(t)\right]}^{r(p-1)}}{{\left[\Phi (t)\right]}^{(r-1)p}}\varkappa (t)g^p(t)\Delta t,$$

(14)

and if $p>1,$ $0\le r<1,$ then

$${\displaystyle {\int }_a^{\infty }}\frac{\varkappa (t)}{{\left[{\Phi }^{\sigma }(t)\right]}^r}{\overline{\Psi }}^p(t)\Delta t\le {\left(\frac{p}{1-r}\right)}^p{\displaystyle {\int }_a^{\infty }}\frac{\varkappa (t)}{{\left[{\Phi }^{\sigma }(t)\right]}^{r-p}}{g}^p(t)\Delta t,$$

(15)

where

$$\Phi (t)={\displaystyle {\int }_a^t}\varkappa (s)\Delta s,\Psi (t)={\displaystyle {\int }_a^t}\varkappa (s)g(s)\Delta s\mathrm{and}\overline{\Psi }(t)={\displaystyle {\int }_t^{\infty }}\varkappa (s)g(s)\Delta s,\mathrm{for}\mathrm{any}t\in {[a,\infty )}_{\mathbb{T}}.$$

Additionally, in [11], the authors constructed a time scale version of Leindler’s inequality (7) as follows: Let $p>1$ and $0\le r<1.$ Then,

$${\displaystyle {\int }_a^{\infty }}\frac{\varkappa (t)}{{\left[\Omega (t)\right]}^r}{\left[{\Phi }^{\sigma }(t)\right]}^p\Delta t\le {\left(\frac{p}{1-r}\right)}^p{\displaystyle {\int }_a^{\infty }}\frac{\varkappa (t)}{{\left[\Omega (t)\right]}^{r-p}}{g}^p(t)\Delta t,$$

(16)

where

$$\Phi (t)={\displaystyle {\int }_a^t}\varkappa (s)g(s)\Delta s\mathrm{and}\Omega (t)={\displaystyle {\int }_t^{\infty }}\varkappa (s)\Delta s,\mathrm{for}\mathrm{any}t\in {[a,\infty )}_{\mathbb{T}}.$$

In 2001, the time scale nabla calculus (or simply nabla calculus) was introduced by Atici and Guseinov. Any field that requires the study of both continuous and discrete data can benefit from nabla calculus. Recently, a lot of authors have contributed to the development of many inequalities on time scales. For example, in [12], Kayar et al. showed that if $\chi ,\eta$ is nonnegative, ld-continuous and ▽-differentiable and if the ▽-integrable functions on ${[a,\infty )}_{\mathbb{T}}$ where $a\in {[0,\infty )}_{\mathbb{T}},$ $r\le 0<p<1,$ $\Phi (s)={\displaystyle {\int }_s^{\infty }}\chi (t)▽t$ and $\Omega (s)={\displaystyle {\int }_a^s}\chi (t)\eta (t)▽t$, then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{\left({\Phi }^{\rho }(s)\right)}^r}{\Omega }^p(s)▽s\ge {\left(\frac{p}{1-r}\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\left({\Phi }^{\rho }(s)\right)}^{{}^{p-r}}{\eta }^p(s)▽s.$$

(17)

If $0<p<1<r,\Phi (s)={\displaystyle {\int }_s^{\infty }}\varkappa (t)▽t$ and $\overline{\Omega }(s)={\displaystyle {\int }_s^{\infty }}\varkappa (t)\eta (t)▽t,$ then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{\left({\Phi }^{\rho }(s)\right)}^r}{({\overline{\Omega }}^{\rho }(s))}^p▽s\ge {\left(\frac{p{M}^r}{r-1}\right)}^p{\displaystyle {\int }_p^{\infty }}\chi (s){\left({\Phi }^{\rho }(s)\right)}^{{}^{p-r}}{\eta }^p(s)▽s,$$

(18)

where

$$M=\underset{s\in \mathbb{T}}{inf}\frac{\Phi (s)}{{\Phi }^{\rho }(s)}>0.$$

Furthermore, they proved that if $r\le 0<p<1$ and $\overline{\Phi }(s)={\displaystyle {\int }_a^s}\chi (t)▽t$, then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{(\overline{\Phi }(s))}^{{}^r}}{\left({\overline{\Omega }}^{\rho }(s)\right)}^p▽s\ge {\left(\frac{p}{1-r}\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\eta }^p(s){(\overline{\Phi }(s))}^{p-r}▽s.$$

(19)

If $0<p\le 1<r$ and $\Omega (s)={\displaystyle {\int }_a^s}\varkappa (\tau )\eta (\tau )▽\tau$, such that

$$L=\underset{s\in \mathbb{T}}{inf}\frac{{\overline{\Phi }}^{\rho }(s)}{\overline{\Phi }(s)}>0,$$

then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{(\overline{\Phi }(s))}^{{}^r}}{\Omega }^p(s)▽s\ge {\left(\frac{p{L}^{1-r}}{r-1}\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\eta }^p(s){(\overline{\Phi }(s))}^{p-r}▽s.$$

(20)

Fractional calculus is a useful tool for both explaining physical phenomena and solving practical issues. A novel idea in both applied sciences and mathematics is the notion of fractional order derivatives and integrals, which will clarify some aspects of differential equations that are still unknown as well as some solutions to fractional order differential equations that have proven useless for solving other problems. In addition to fractional order derivatives and integrals, new derivatives and integrals aid in the solution of differential equations that are presented and resolved in classical analysis. Its applications in fields including engineering, biostatistics, and mathematical biology have also enhanced its contribution to the literature. Not only do fractional derivative and integral operators differ from one another in terms of singularity, locality, and kernels, but they also introduced new concepts to fractional analysis in terms of the utilizations regions and spaces they could cover. Fractional calculus and the theory of inequality have become the cornerstone of the literature today. Fractional calculus was the answer to the question of whether fractional derivatives and fractional integrals can be taken. Therefore, it has offered solutions to many problems in many disciplines. The most famous of the fractional approaches that are developing day by day are the Riemann–Liouville, Caputo, and Conformable fractional approaches. The theory of inequalities is one of the most important topics of recent research. In particular, its use in analysis, applied mathematics, and pure mathematics is very wide. One of the most studied of these inequalities is the Bennett–Leindler type inequality. The topic of fractional inequalities has received a great deal of attention in recent years, and many writers are now interested in proving the inequalities of this sort using a new fractional calculus called the conformable calculus. For example, in [13], Zakarya et al. proved an $\alpha$-conformable version of Hardy-type inequalities. One of those results is

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{\Phi }^{k-\alpha +1}(s)}{({\Omega }^{\sigma }(s))}^p{\Delta }_{\alpha }s\ge {\left(\frac{p}{\alpha -k}\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\Phi }^{p-k+\alpha -1}(s){\eta }^p(s){\Delta }_{\alpha }s,$$

(21)

wherer $\chi ,\eta$ are rd-continuous and the $\alpha$-fractional differentiable functions on ${[a,\infty )}_{\mathbb{T}}$ where $a\in {[0,\infty )}_{\mathbb{T}},$ $k\le 0<p<1,$ $\alpha \in (0,1],$ $\Phi (s)={\displaystyle {\int }_s^{\infty }}\chi (t){\Delta }_{\alpha }t$ and $\Omega (s)={\displaystyle {\int }_a^s}\chi (t)\eta (t){\Delta }_{\alpha }t$.

The symmetry properties of functions used to define an equation or inequality can be studied in order to determine solutions with particular properties. As far as inequalities are concerned, the study of special functions such as hypergeometric functions and special polynomials considering their symmetry properties may provide some interesting outcomes. Studies on symmetry properties for different types of operators and inequalities associated with the concept of time scales calculus may also be investigated. These properties and results are symmetrical to the properties of the differential superobligation that form the inequalities theorems. For generalizations and implementations of dynamic inequalities, we refer to the papers [14,15,16,17,18,19,20,21,22,23].

In this study, we generalized a number of Bennett and Leindler Hardy-type inequalities to a general time scale with the use of a straightforward Keller’s chain rule and Hölder’s inequality result for the $\alpha$-nabla-fractional derivative on time scales. We also extended our inequalities to discrete and continuous calculus in order to derive some more inequalities as specific cases. A variety of time scales, including $\mathbb{R}$ and $\mathbb{Z}$, were used to show the theorems for each kind of inequality.

The following is the format of the paper: In Section 2, we go over the fundamentals of the conformable fractional on time scales, which will be needed to verify our main points. The primary ramifications will be discussed in Section 3. Our results (when $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{Z}$) are essentially new.

2. Fundamental Concepts

In this section, we will cover the fundamentals of conformable fractional integrals and derivatives of order $\alpha \in \left[0,1\right]$ on the time scales that will be used in this paper (see [24,25]). A time scale $\mathbb{T}$ is defined as any arbitrary non-empty closed subset of $\mathbb{R}$. A backward jump operator on time scales is defined by: $\rho (\tau ):=sup\{r\in \mathbb{T}:r<\tau \}.$ For any function $\psi :\mathbb{T}\to \mathbb{R}$, the notation ${\psi }^{\rho }(\tau )$ denotes $\psi (\rho (\tau )).$ Additionally, we define the function $\nu :\mathbb{T}\to [0,\infty )$ by

$$\nu (\theta )=\theta -\rho (\theta ).$$

Definition 1

(Conformable$\mathit{\alpha }$-fractional derivative [26]).Suppose that $\psi :\mathbb{T}\to \mathbb{R}$ is a function and $\alpha \in (0,1].$ Then, for $\eta >0$, we define ${\displaystyle D_{\alpha }^{▽}}(\psi )(\eta )$ as the number that has the property that for each $\eta >0,$ ∃ a neighborhood U of η s.t. $\forall \eta \in U,$ we have

$$\left|\left[{\psi }^{\rho }(\eta )-\psi (s)\right]{\eta }^{1-\alpha }-{\displaystyle D_{\alpha }^{▽}}(\psi )(\eta )(\rho (\eta )-s)\right|\le ϵ\left|\rho (\eta )-s)\right|.$$

We say that ${\displaystyle D_{\alpha }^{▽}}(\psi )(\eta )$ is the conformable nabla fractional derivative of ψ of order α at $\eta \in \mathbb{T}.$ Moreover, we say ψ is a conformable nabla fractional differentiable of order α on $\eta \in \mathbb{T}$ provided that ${\displaystyle D_{\alpha }^{▽}}(\psi )(\eta )$ is $\forall \eta \in \mathbb{T}.$

Theorem 1

([26]).Let $z,\psi :\mathbb{T}\to \mathbb{R}$ be the nabla conformable $\alpha$-fractional differentiable and $\alpha \in (0,1].$ Then,

(i) 

$z+\psi$ is a nabla conformable $\alpha$-fractional differentiable, and

$${\displaystyle D_{\alpha }^{▽}}(z+\psi )={\displaystyle D_{\alpha }^{▽}}(z)+{\displaystyle D_{\alpha }^{▽}}(\psi ).$$

(ii) 

$z\psi$ is a nabla conformable $\alpha$-fractional differentiable, and

$${\displaystyle D_{\alpha }^{▽}}(z\psi )={\displaystyle D_{\alpha }^{▽}}(z)\psi +(z\circ \rho ){\displaystyle D_{\alpha }^{▽}}(\psi )={\displaystyle D_{\alpha }^{▽}}(z)(\psi \circ \rho )+z{\displaystyle D_{\alpha }^{▽}}(\psi ).$$

(iii) 

$z/\psi$ is a nabla conformable $\alpha$-fractional differentiable, and

$${\displaystyle D_{\alpha }^{▽}}\left(\frac{z}{\psi }\right)=\frac{\psi {\displaystyle D_{\alpha }^{▽}}(z)-z{\displaystyle D_{\alpha }^{▽}}(\psi )}{\psi (\psi \circ \rho )},$$

such that $\psi (\psi \circ \rho )\ne 0.$

Lemma 1

([26]).Assume $\psi :\mathbb{T}\to \mathbb{R}$ is a continuous and nabla conformable $\alpha$-fractional differentiable at $\eta \in \mathbb{T}$ for $\alpha \in (0,1]$ and $z:\mathbb{R}\to \mathbb{R}$ is continuously differentiable. Then, $z\circ \psi :\mathbb{T}\to \mathbb{R}$ is a nabla conformable $\alpha$-fractional differentiable and

$${\displaystyle D_{\alpha }^{▽}}(z\circ \psi )(\eta )=z^{{}^{\prime }}(\psi (d)){\displaystyle D_{\alpha }^{▽}}(\psi )(\eta )\mathit{where}d\in [\eta ,\sigma (\eta )].$$

(22)

Definition 2

(Conformable$\mathit{\alpha }$-fractional integral [26]).For $0<\alpha \le 1,$ the nabla conformable $\alpha$-fractional integral of$\varphi$ is

$${\displaystyle I_{\alpha }^{▽}}(\varphi )(\eta )=\int \varphi (\eta ){▽}_{\alpha }\eta =\int \varphi (\eta ){\eta }^{\alpha -1}▽\eta .$$

Theorem 2

([26]).Let $c,d,e\in \mathbb{T}$ and $\lambda \in \mathbb{R}$. If $z,\psi :\mathbb{T}\to \mathbb{R}$ are ld-continuous functions, then

(i) ${\displaystyle {\int }_c^d}[z(\eta )+\psi (\eta )]{▽}_{\alpha }\eta ={\displaystyle {\int }_c^d}z(\eta ){▽}_{\alpha }\eta +{\displaystyle {\int }_c^d}\psi (\eta ){▽}_{\alpha }\eta .$

(ii) ${\displaystyle {\int }_c^d}\lambda z(\eta ){▽}_{\alpha }\eta =\lambda {\displaystyle {\int }_c^d}z(\eta ){▽}_{\alpha }\eta .$

(iii) ${\displaystyle {\int }_c^d}z(\eta ){▽}_{\alpha }\eta =-{\displaystyle {\int }_d^c}z(\eta ){▽}_{\alpha }\eta .$

(iv) ${\displaystyle {\int }_c^d}z(\eta ){▽}_{\alpha }\eta ={\displaystyle {\int }_c^e}z(\eta ){▽}_{\alpha }\eta +{\displaystyle {\int }_e^d}z(\eta ){▽}_{\alpha }\eta .$

(v) ${\displaystyle {\int }_c^c}z(\eta ){▽}_{\alpha }\eta =0.$

Lemma 2

([26]).Assume that $c,d\in \mathbb{T}$ and $\nu ,\xi$ are nabla conformable $\alpha$-fractional differentiables and $\alpha \in (0,1]$. Then,

$${\displaystyle {\int }_c^d}\xi (s){\displaystyle D_{\alpha }^{▽}}(v)(s){▽}_{\alpha }s={\displaystyle {\left[v(s)\xi (s)\right]}_c^d}-{\displaystyle {\int }_c^d}v(\rho (s)){\displaystyle D_{\alpha }^{▽}}(\xi )(s){▽}_{\alpha }s.$$

(23)

$${\displaystyle {\int }_c^d}\xi (\rho (s)){\displaystyle D_{\alpha }^{▽}}(v)(s){▽}_{\alpha }s={\displaystyle {\left[v(s)\xi (s)\right]}_c^d}-{\displaystyle {\int }_c^d}v(s){\displaystyle D_{\alpha }^{▽}}(\xi )(s){▽}_{\alpha }s.$$

(24)

Lemma 3

(Hölder’s inequality [26]).Let $c,d\in \mathbb{T}$ such that $c>d.$ If $0<\alpha \le 1$ and $z,\psi :\mathbb{T}\to \mathbb{R}$, then

$${\displaystyle {\int }_c^d}\left|z(\eta )\psi (\eta )\right|{▽}_{\alpha }\eta \le {\left({\displaystyle {\int }_c^d}{\left|z(\eta )\right|}^s{▽}_{\alpha }\eta \right)}^{\frac{1}{s}}{\left({\displaystyle {\int }_c^d}{\left|\psi (\eta )\right|}^t{▽}_{\alpha }\eta \right)}^{\frac{1}{t}},$$

(25)

where $s>1$ and $1/s+1/t=1.$

3. Main Results and Some Corollaries

In this section, we will demonstrate the key findings of this article. Throughout this article, we will consider the integrals given to exist (i.e., to be convergent).

Theorem 3.

Suppose that $\mathbb{T}$ is a time scale with $a\in {[0,\infty )}_{\mathbb{T}}$ and $\alpha \in \left(0,1\right].$ Furthermore, let χ and η be nonnegative ld-continuous and α-nabla fractional differentiable functions on $a\in {[0,\infty )}_{\mathbb{T}}$. Define

$$\Phi (s)={\displaystyle {\int }_s^{\infty }}\chi (\theta ){▽}_{\alpha }\theta \mathrm{and}\Omega (s)={\displaystyle {\int }_a^s}\chi (\theta )\eta (\theta ){▽}_{\alpha }\theta ,s\in {[a,\infty )}_{\mathbb{T}}.$$

If $0<p<\alpha$ and $r\le \alpha -1,$ then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{\Omega }^p(s){▽}_{\alpha }s\ge {\left(\frac{p}{\alpha -r}\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\left[{\Phi }^{\rho }(s)\right]}^{{}^{p-r+\alpha -1}}{\eta }^p(s){▽}_{\alpha }s.$$

(26)

Proof. 

By applying the formula (23) to

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{\Omega }^p(s){▽}_{\alpha }s,$$

with $\xi (s)={\Omega }^p(s)$ and ${\displaystyle D_{\alpha }^{▽}}v(s)=\chi (s)/{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1},$ we have

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{▽}_{\alpha }s={\displaystyle {\left.v(s){\Omega }^p(s)\right|}_a^{\infty }}+{\displaystyle {\int }_a^{\infty }}(-v^{\rho }(s)){\displaystyle D_{\alpha }^{▽}}\left({\Omega }^p(s)\right){▽}_{\alpha }s,$$

(27)

where

$$v(s)=-{\displaystyle {\int }_s^{\infty }}\frac{\chi (t)}{{\left[{\Phi }^{\rho }(t)\right]}^{r-\alpha +1}}{▽}_{\alpha }t.$$

Since $\Omega (a)=0$ and $\nu (\infty )=0$ in (27), then we see that

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{▽}_{\alpha }s=-{\displaystyle {\int }_a^{\infty }}{v}^{\rho }(s){\displaystyle D_{\alpha }^{▽}}\left({\Omega }^p(s)\right){▽}_{\alpha }s.$$

(28)

By (22) and ${\displaystyle D_{\alpha }^{▽}}\Omega (s)=\chi (s)\eta (s)$, we obtain

$$\begin{array}{ccc}\hfill {\displaystyle D_{\alpha }^{▽}}\left({\Omega }^p(s)\right)& =& p{\Omega }^{p-1}(d){\displaystyle D_{\alpha }^{▽}}\Omega (s),\mathrm{where}d\in [\rho (s),s]\hfill \\ & =& \frac{p{\displaystyle D_{\alpha }^{▽}}\Omega (s)}{{\Omega }^{1-p}(d)}\ge \frac{p{\displaystyle D_{\alpha }^{▽}}\Omega (s)}{{\Omega }^{1-p}(s)}=\frac{p\chi (s)\eta (s)}{{\Omega }^{1-p}(s)}.\hfill \end{array}$$

(29)

Again using (22) and ${\displaystyle D_{\alpha }^{▽}}\Phi (s)=-\chi (s)\le 0$, we have

$$\begin{array}{ccc}\hfill {\displaystyle D_{\alpha }^{▽}}\left({\Phi }^{\alpha -r}(s)\right)& =& (\alpha -r){\Phi }^{\alpha -r-1}(d){\displaystyle D_{\alpha }^{▽}}\Phi (s)=\frac{\alpha -r}{{\Phi }^{r+1-\alpha }(d)}{\displaystyle D_{\alpha }^{▽}}\Phi (s)\hfill \\ & \ge & \frac{\alpha -r}{{\Phi }^{r+1-\alpha }(s)}{\displaystyle D_{\alpha }^{▽}}\Phi (s)\ge \frac{-(\alpha -r)\chi (s)}{{\Phi }^{r+1-\alpha }(\rho (s))}.\hfill \end{array}$$

This leads to

$$\frac{\chi (s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}\ge \frac{-1}{\alpha -r}{\displaystyle D_{\alpha }^{▽}}\left({\Phi }^{\alpha -r}(s)\right),$$

and then we obtain

$$\begin{array}{ccc}\hfill -v^{\rho }(s)& =& {\displaystyle {\int }_{\rho (s)}^{\infty }}\frac{\chi (t)}{{\left[{\Phi }^{\rho }(t\right]}^{r-\alpha +1}}{▽}_{\alpha }t\hfill \\ & \ge & \frac{1}{\alpha -r}{\displaystyle {\int }_{\rho (s)}^{\infty }}-{\displaystyle D_{\alpha }^{▽}}\left({\Phi }^{\alpha -r}(t)\right){▽}_{\alpha }t\hfill \\ & =& \frac{1}{(\alpha -r){\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha }}.\hfill \end{array}$$

(30)

Substituting (29) and (30) into (28) yields

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{▽}_{\alpha }s\ge \left(\frac{p}{\alpha -r}\right){\displaystyle {\int }_a^{\infty }}\frac{\chi (s)\eta (s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha }{\Omega }^{1-p}(s)}{▽}_{\alpha }s.$$

(31)

Raising (31) to the factor p, we have

$${\left({\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{▽}_{\alpha }s\right)}^p\ge {\left(\frac{p}{\alpha -r}\right)}^p{\left({\displaystyle {\int }_a^{\infty }}{\left(\frac{{\chi }^p(s){\eta }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{p(r-\alpha )}{\Omega }^{{}^{p(1-p)}}(s)}\right)}^{\frac{1}{p}}{▽}_{\alpha }s\right)}^p.$$

(32)

By applying (25) to

$${\left({\displaystyle {\int }_a^{\infty }}{\left(\frac{{\chi }^p(s){\eta }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{p(r-\alpha )}{\Omega }^{{}^{p(1-p)}}(s)}\right)}^{\frac{1}{p}}{▽}_{\alpha }s\right)}^p,$$

with indices $\lambda =1/p>1,$ $\mu =1/(1-p)$ (note that $1/\lambda +1/\mu =1,$ where $\lambda >1$) and

$$F(s)=\frac{{\chi }^p(s){\eta }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{p(r-\alpha )}{\Omega }^{{}^{p(1-p)}}(s)}\mathrm{and}G(s)={\left(\frac{\chi (s){\Omega }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}\right)}^{1-p},$$

we see that

$$\begin{array}{ccc}\hfill {\left[{\displaystyle {\int }_a^{\infty }}{\left(F(s)\right)}^{\frac{1}{p}}{▽}_{\alpha }s\right]}^p& =& {\left({\displaystyle {\int }_a^{\infty }}{\left(\frac{{\chi }^p(s){\eta }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{p(r-\alpha )}{\Omega }^{{}^{p(1-p)}}(s)}\right)}^{\frac{1}{p}}{▽}_{\alpha }s\right)}^p\hfill \\ & \ge & \frac{{\displaystyle {\int }_a^{\infty }}F(s)G(s){▽}_{\alpha }s}{{\left({\displaystyle {\int }_a^{\infty }}{G}^{\frac{1}{1-p}}(s){▽}_{\alpha }s\right)}^{1-p}}\hfill \\ & =& \left({\displaystyle {\int }_a^{\infty }}\left(\frac{{\chi }^p(s){\eta }^p(s){\chi }^{1-p}(s){\Omega }^{p(1-p)}(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{p(r-\alpha )}{\Omega }^{{}^{p(1-p)}}(s){\left[{\Phi }^{\rho }(s)\right]}^{(1-p)(r-\alpha +1)}}\right){▽}_{\alpha }s\right)\hfill \\ & & \times {\left({\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{▽}_{\alpha }s\right)}^{p-1}\hfill \end{array}$$

$$=\left({\displaystyle {\int }_a^{\infty }}\left(\frac{\chi (s){\eta }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-p-\alpha +1}}\right){▽}_{\alpha }s\right){\left({\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{▽}_{\alpha }s\right)}^{p-1}.$$

This implies that

$${\left({\displaystyle {\int }_a^{\infty }}{\left(\frac{{\chi }^p(s){\eta }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{p(r-\alpha )}{\Omega }^{{}^{p(1-p)}}(s)}\right)}^{\frac{1}{p}}{▽}_{\alpha }s\right)}^p\ge \frac{{\displaystyle {\int }_a^{\infty }}\chi (s){\eta }^p(s){\left[{\Phi }^{\rho }(s)\right]}^{p-r+\alpha -1}{▽}_{\alpha }s}{{\left({\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{▽}_{\alpha }s\right)}^{1-p}},$$

(33)

By substituting (33) into (32), we obtain

$${\left({\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{▽}_{\alpha }s\right)}^p\ge {\left(\frac{p}{\alpha -r}\right)}^p\frac{{\displaystyle {\int }_a^{\infty }}\chi (s){\eta }^p(s){\left[{\Phi }^{\rho }(s)\right]}^{p-r+\alpha -1}{▽}_{\alpha }s}{{\left({\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{▽}_{\alpha }s\right)}^{1-p}}.$$

This means that

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{▽}_{\alpha }s\ge {\left(\frac{p}{\alpha -r}\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\left[{\Phi }^{\rho }(s)\right]}^{{}^{p-r+\alpha -1}}{\eta }^p(s){▽}_{\alpha }s,$$

which is (26). □

Remark 1.

If we use $\alpha =1$ in Theorem 3, then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{\left[{\Phi }^{\rho }(s)\right]}^r}▽s\ge {\left(\frac{p}{1-r}\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\left[{\Phi }^{\rho }(s)\right]}^{p-r}{\eta }^p(s)▽s,$$

(34)

where $a\in {[0,\infty )}_{\mathbb{T}},$ $r\le 0<p<1,$

$$\Phi (s)={\displaystyle {\int }_s^{\infty }}\chi (\theta )▽\theta \mathit{and}\Omega (s)={\displaystyle {\int }_a^s}\chi (\theta )\eta (\theta )▽\theta ,$$

which is (17).

Corollary 1.

If $\mathbb{T}=\mathbb{R}$ $(i.e.$ $\rho (s)=s)$ in Theorem 3, then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{\left[\Phi (s)\right]}^{r-\alpha +1}}{s}^{\alpha -1}{d}_{\alpha }s\ge {\left(\frac{p}{\alpha -r}\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\Phi }^{{}^{{}^{p-r+\alpha -1}}}(s){\eta }^p(s)s^{\alpha -1}{d}_{\alpha }s.$$

(35)

where $a\in [0,\infty ),$ $\alpha \in \left(0,1\right],$ $0<p<\alpha ,$ $r\le \alpha -1,$

$$\Phi (s)={\displaystyle {\int }_s^{\infty }}\chi (\theta ){\theta }^{\alpha -1}{d}_{\alpha }\theta \mathit{and}\Omega (s)={\displaystyle {\int }_a^s}\chi (\theta )\eta (\theta ){\theta }^{\alpha -1}{d}_{\alpha }\theta .$$

This is a new result for a conformable continuous calculus.

Remark 2.

Clearly, for $\alpha =1$ and $a=1,$ Corollary 1 corresponds to [12, Remark 3.2].

Corollary 2.

If $\mathbb{T}=\mathbb{Z}$ $(i.e.$ $\rho (s)=s-1)$ in (26), $0<p<\alpha ,$ $r\le \alpha -1$ and $a=1,$ then

$${\displaystyle \sum _{i=2}^{\infty }}\frac{\chi (i){\Omega }^p(i)}{{\Phi }^{r-\alpha +1}(i-1)}{i}^{\alpha -1}\ge {\left(\frac{p}{\alpha -r}\right)}^p{\displaystyle \sum _{i=2}^{\infty }}\chi (i){\Phi }^{{}^{p-r+\alpha -1}}(i-1){\eta }^p(i)i^{\alpha -1}.$$

(36)

where $\alpha \in \left(0,1\right],$

$$\Phi (i)={\displaystyle \sum _{i=2}^{\infty }}\chi (i)i^{\alpha -1}\mathit{and}\Omega (i)={\displaystyle \sum _{j=2}^i}\chi (j)\eta (j)j^{\alpha -1}.$$

This is a new result for a discrete calculus.

Remark 3.

If $\alpha =1,$ then (36) becomes

$${\displaystyle \sum _{i=2}^{\infty }}\frac{\chi (i)}{{\Phi }^r(i-1)}{\Omega }^p(i)\ge {\left(\frac{p}{1-r}\right)}^p{\displaystyle \sum _{i=2}^{\infty }}\chi (i){\Phi }^{{}^{p-r}}(i-1){\eta }^p(i).$$

(37)

which is Remark 3.3 in [12].

Theorem 4.

Suppose that $\mathbb{T}$ is a time scale with $a\in {[0,\infty )}_{\mathbb{T}}$ and $\alpha \in \left(0,1\right].$ Furthermore, let χ and η be nonnegative ld-continuous and α-nabla fractional differentiable functions on $a\in {[0,\infty )}_{\mathbb{T}}.$ Assume $\Phi (s)$ is defined as in Theorem 3 such that

$$M=\underset{s\in \mathbb{T}}{inf}\frac{\Phi (s)}{{\Phi }^{\rho }(s)}>0,$$

(38)

and define $\overline{\Omega }(s)={\displaystyle {\int }_s^{\infty }}\chi (t)\eta (t){▽}_{\alpha }t.$ If $0<p<\alpha$ and $r\ge \alpha ,$ then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){({\overline{\Omega }}^{\rho }(s))}^p}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{▽}_{\alpha }s\ge {\left(\frac{p{M}^{r-\alpha +1}}{r-\alpha }\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\left[{\Phi }^{\rho }(s)\right]}^{{}^{p-r+\alpha -1}}{\eta }^p(s){▽}_{\alpha }s.$$

(39)

Proof. 

By applying (24) to

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{({\overline{\Omega }}^{\rho }(s))}^p{▽}_{\alpha }s,$$

with ${\displaystyle D_{\alpha }^{▽}}v(s)=\chi (s)/{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}$ and $\xi (s)={({\overline{\Omega }}^{\rho }(s))}^p,$ we have

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{({\overline{\Omega }}^{\rho }(s))}^p{▽}_{\alpha }s={\displaystyle {\left.v(s){(\overline{\Omega }(s))}^p\right|}_a^{\infty }}+{\displaystyle {\int }_a^{\infty }}v(s)(-{\displaystyle D_{\alpha }^{▽}}{(\overline{\Omega }(s))}^p){▽}_{\alpha }s,$$

where $v(s)={\displaystyle {\int }_a^s}(\chi (s)/{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}){▽}_{\alpha }t.$ This with $\overline{\Omega }(\infty )=0$ and $v(a)=0$ imply that

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{({\overline{\Omega }}^{\rho }(s))}^p{▽}_{\alpha }s={\displaystyle {\int }_a^{\infty }}v(s)(-{\displaystyle D_{\alpha }^{▽}}{(\overline{\Omega }(s))}^p){\Delta }_{\alpha }s.$$

(40)

By utilizing (22) and ${\displaystyle D_{\alpha }^{▽}}\overline{\Omega }(s)=-\chi (s)\eta (s)\le 0$, we obtain

$$\begin{array}{ccc}\hfill -{\displaystyle D_{\alpha }^{▽}}{(\overline{\Omega }(s))}^p& =& -p{(\overline{\Omega }(d))}^{p-1}{\displaystyle D_{\alpha }^{▽}}\overline{\Omega }(s),\mathrm{where}d\in [\rho (s),s]\hfill \\ & =& \frac{p\chi (s)\eta (s)}{{(\overline{\Omega }(d))}^{1-p}}\ge \frac{p\chi (s)\eta (s)}{{({\overline{\Omega }}^{\rho }(s))}^{1-p}}.\hfill \end{array}$$

(41)

By substituting (41) into (40) and using ${\displaystyle D_{\alpha }^{▽}}\xi (s)\ge 0$, we obtain

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{({\overline{\Omega }}^{\rho }(s))}^p{▽}_{\alpha }s\ge p{\displaystyle {\int }_a^{\infty }}v(s)\frac{\chi (s)\eta (s)}{{({\overline{\Omega }}^{\rho }(s))}^{1-p}}{▽}_{\alpha }s.$$

(42)

Again using (22) and ${\displaystyle D_{\alpha }^{▽}}\Phi (s)=-\chi (s)\le 0$, we have

$$\begin{array}{ccc}\hfill {\displaystyle D_{\alpha }^{▽}}\left({\Phi }^{\alpha -r}(s)\right)& =& (\alpha -r){\Phi }^{\alpha -r-1}(d){\displaystyle D_{\alpha }^{▽}}\Phi (s)=\frac{\alpha -r}{{\Phi }^{r+1-\alpha }(d)}{\displaystyle D_{\alpha }^{▽}}\Phi (s)\hfill \\ & \le & \frac{\alpha -r}{{\Phi }^{r+1-\alpha }(s)}{\displaystyle D_{\alpha }^{▽}}\Phi (s)\le \frac{\alpha -r}{{\Phi }^{r+1-\alpha }(s)}(-\chi (s))\hfill \\ & =& \frac{(r-\alpha )\chi (s)}{{\Phi }^{r+1-\alpha }(\rho (s))}\frac{{\Phi }^{r+1-\alpha }(\rho (s))}{{\Phi }^{r+1-\alpha }(s)}\le \frac{(r-\alpha )}{M^{r-\alpha +1}}\frac{\chi (s)}{{\Phi }^{r+1-\alpha }(\rho (s))}.\hfill \end{array}$$

This implies that

$$\begin{array}{ccc}\hfill v(s)& =& {\displaystyle {\int }_a^s}\frac{\chi (\theta )}{{\left[{\Phi }^{\rho }(\theta )\right]}^{r-\alpha +1}}{▽}_{\alpha }\theta \ge \left(\frac{M^{r+1-\alpha }}{r-\alpha }\right){\displaystyle {\int }_a^s}{\displaystyle D_{\alpha }^{▽}}\left({\Phi }^{\alpha -r}(s)\right){▽}_{\alpha }\theta \hfill \\ & =& \left(\frac{M^{r+1-\alpha }}{r-\alpha }\right){\Phi }^{\alpha -r}(s)\ge \left(\frac{M^{r+1-\alpha }}{r-\alpha }\right){\left[{\Phi }^{\rho }(s)\right]}^{\alpha -r}.\hfill \end{array}$$

(43)

By substituting (43) into (42), we obtain

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){({\overline{\Omega }}^{\rho }(s))}^p}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{▽}_{\alpha }s\ge \left(\frac{p{M}^{r+1-\alpha }}{r-\alpha }\right){\displaystyle {\int }_a^{\infty }}\frac{\chi (s)\eta (s){\left[{\Phi }^{\rho }(s)\right]}^{\alpha -r}}{{({\overline{\Omega }}^{\rho }(s))}^{1-p}}{▽}_{\alpha }s.$$

(44)

Raising (44) to the factor $p,$ we obtain

$${\left({\displaystyle {\int }_a^{\infty }}\frac{\chi (s){({\overline{\Omega }}^{\rho }(s))}^p}{{\left[{\Phi }^{\rho }(s)\right]}^{r-\alpha +1}}{▽}_{\alpha }s\right)}^p\ge {\left(\frac{p{M}^{r+1-\alpha }}{r-\alpha }\right)}^p{\left({\displaystyle {\int }_a^{\infty }}{\left(\frac{{\chi }^p(s){\eta }^p(s){\left[{\Phi }^{\rho }(s)\right]}^{p(\alpha -r)}}{{({\overline{\Omega }}^{\rho }(s))}^{p(1-p)}}\right)}^{\frac{1}{p}}{▽}_{\alpha }s\right)}^p.$$

(45)

The remainder of the proof is analogous to the proof of Theorem 3 and is thus omitted. □

Remark 4.

If we take $\alpha =1$ in Theorem 4, then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{\left[{\Phi }^{\rho }(s)\right]}^r}{({\overline{\Omega }}^{\rho }(s))}^p▽s\ge {\left(\frac{p{M}^r}{r-1}\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\left[{\Phi }^{\rho }(s)\right]}^{p-r}{\eta }^p(s)▽s,$$

(46)

where $a\in {[0,\infty )}_{\mathbb{T}},$ $0<p<1,$ $r\ge 1,$ $\overline{\Omega }(s)={\displaystyle {\int }_s^{\infty }}\chi (t)\eta (t)▽t$ and $\Phi (s)={\displaystyle {\int }_s^{\infty }}\varkappa (t)▽t$ such that

$$M=\underset{s\in \mathbb{T}}{inf}\frac{\Phi (s)}{{\Phi }^{\rho }(s)}>0,$$

which is (18).

Corollary 3.

If $\mathbb{T}=\mathbb{R}$ $(i.e.$ $\rho (u)=u)$ in Theorem 4, then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (u){(\overline{\Omega }(u))}^p}{{\Phi }^{r-\alpha +1}(u)}{u}^{\alpha -1}{d}_{\alpha }u\ge {\left(\frac{p}{r-\alpha }\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (u){\Phi }^{{}^{p-r+\alpha -1}}(u){\eta }^p(u)u^{\alpha -1}{d}_{\alpha }u,$$

(47)

where $0<p<\alpha ,$ $r\ge \alpha ,$ $\alpha \in \left(0,1\right],$ $\Phi (u)={\displaystyle {\int }_u^{\infty }}\varkappa (\theta ){\theta }^{\alpha -1}{d}_{\alpha }\theta ,$ $\overline{\Omega }(u)={\displaystyle {\int }_u^{\infty }}\varkappa (\theta )\eta (\theta ){\theta }^{\alpha -1}{d}_{\alpha }\theta$ and

$$M=\underset{u\in \mathbb{R}}{inf}\frac{\Phi (u)}{\Phi (u)}=1.$$

This is a new result for a conformable continuous calculus.

Remark 5.

If $\alpha =1$ and $a=1,$ then (47) becomes

$${\displaystyle {\int }_1^{\infty }}\frac{\chi (u)}{{\Phi }^r(u)}{(\overline{\Omega }(u))}^{p}du\ge {\left(\frac{p}{r-1}\right)}^p{\displaystyle {\int }_1^{\infty }}\chi (u){\Phi }^{{}^{p-r}}(u){\eta }^p(u)du,$$

which is Remark 3.5 in [12].

Corollary 4.

In (39), if $\mathbb{T}=\mathbb{Z}$ $(i.e.$ $\rho (s)=s-1)$ and $a=1,$ then

$${\displaystyle \sum _{i=2}^{\infty }}\frac{\chi (i){(\overline{\Omega }(i-1))}^p}{{\Phi }^{{}^{r-\alpha +1}}(i-1)}{i}^{\alpha -1}\ge {\left(\frac{p{M}^{r-\alpha +1}}{r-\alpha }\right)}^p{\displaystyle \sum _{i=2}^{\infty }}\chi (i){\Phi }^{{}^{p-r+\alpha -1}}(i-1){\eta }^p(i)i^{\alpha -1},$$

(48)

where $0<p<\alpha ,$ $r\ge \alpha ,$ $\alpha \in \left(0,1\right],$ $\overline{\Omega }(i)={\displaystyle {\displaystyle \sum _{j=2}^{\infty }}}\eta (j)\chi (j)j^{\alpha -1}$ and

$$M=\underset{i\in \mathbb{Z}}{inf}\frac{\Phi (i)}{\Phi (i-1)}.$$

This is a new result for a discrete calculus.

Remark 6.

If $\alpha =1$ in Corollary 4, then we obtain Remark 3.6 in [12].

Theorem 5.

Suppose that $\mathbb{T}$ is a time scale with $a\in {[0,\infty )}_{\mathbb{T}}$ and $\alpha \in \left(0,1\right].$In addition, let χ and η be nonnegative ld-continuous and α-nabla fractional differentiable functions on $a\in {[0,\infty )}_{\mathbb{T}}.$ Assume that $\overline{\Phi }(s)={\displaystyle {\int }_a^s}\chi (t){▽}_{\alpha }t$ and $\overline{\Omega }(s)={\displaystyle {\int }_s^{\infty }}\chi (t)\eta (t){▽}_{\alpha }t$. If $0<p<\alpha$ and $r\le \alpha -1,$ then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{\left({\overline{\Omega }}^{\rho }(s)\right)}^p{▽}_{\alpha }s\ge {\left(\frac{p}{\alpha -r}\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\eta }^p(s){(\overline{\Phi }(s))}^{p-r+\alpha -1}{▽}_{\alpha }s.$$

(49)

Proof. 

Using (24) on

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{\left({\overline{\Omega }}^{\rho }(s)\right)}^p{▽}_{\alpha }s,$$

with ${\xi }^{\rho }(s)={\left({\overline{\Omega }}^{\rho }(s)\right)}^p$ and ${\displaystyle D_{\alpha }^{▽}}v(s)=\chi (s)/{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}},$ we obtain

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{\left({\overline{\Omega }}^{\rho }(s)\right)}^p{▽}_{\alpha }s={\displaystyle {\left.v(s){\overline{\Omega }}^p(s)\right|}_a^{\infty }}+{\displaystyle {\int }_a^{\infty }}(-v(s))({\displaystyle D_{\alpha }^{▽}}{\overline{\Omega }}^p(s)){▽}_{\alpha }s,$$

where $v(s)={\displaystyle {\int }_a^s}\chi (t)/{(\overline{\Phi }(t))}^{{}^{r-\alpha +1}}{▽}_{\alpha }t$. This with $\overline{\Omega }(\infty )=0$ and $v(a)=0$ imply that

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{\left({\overline{\Omega }}^{\rho }(s)\right)}^p{▽}_{\alpha }s={\displaystyle {\int }_a^{\infty }}(-v(s))({\displaystyle D_{\alpha }^{▽}}{\overline{\Omega }}^p(s)){▽}_{\alpha }s.$$

(50)

By utilizing (22) and ${\displaystyle D_{\alpha }^{▽}}\overline{\Omega }(s)=-\chi (s)\eta (s)\le 0$, we obtain

$$\begin{array}{ccc}\hfill -{\displaystyle D_{\alpha }^{▽}}\left({\overline{\Omega }}^p(s)\right)& =& -p{\overline{\Omega }}^{p-1}(d){\displaystyle D_{\alpha }^{▽}}\overline{\Omega }(s),\mathrm{where}d\in [\rho (s),s]\hfill \\ & =& \frac{p\chi (s)\eta (s)}{{\left({\overline{\Omega }}^{\rho }(d)\right)}^{1-p}}\ge \frac{p\chi (s)\eta (s)}{{\left({\overline{\Omega }}^{\rho }(s)\right)}^{1-p}}.\hfill \end{array}$$

(51)

By substituting (51) into (50) and using ${\displaystyle D_{\alpha }^{▽}}\xi (s)\ge 0$, we have

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{\left({\overline{\Omega }}^{\rho }(s)\right)}^p{▽}_{\alpha }s\ge p{\displaystyle {\int }_a^{\infty }}v(s)\frac{\chi (s)\eta (s)}{{\left({\overline{\Omega }}^{\rho }(s)\right)}^{1-p}}{▽}_{\alpha }s.$$

(52)

Additionally, according to (22) and ${\displaystyle D_{\alpha }^{▽}}\overline{\Phi }(s)=\chi (s)\ge 0,$ we have

$$\begin{array}{ccc}\hfill {\displaystyle D_{\alpha }^{▽}}\left({(\overline{\Phi }(s))}^{\alpha -r}\right)& =& (\alpha -r){\overline{\Phi }}^{\alpha -r-1}(d){\displaystyle D_{\alpha }^{▽}}\overline{\Phi }(s)=\frac{\alpha -r}{{(\overline{\Phi }(d))}^{{}^{r-\alpha +1}}}{\displaystyle D_{\alpha }^{▽}}\overline{\Phi }(s)\hfill \\ & \le & \frac{\alpha -r}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{\displaystyle D_{\alpha }^{▽}}\overline{\Phi }(s)=\frac{\alpha -r}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}\chi (s).\hfill \end{array}$$

This implies that

$$v(s)={\displaystyle {\int }_a^s}\frac{\chi (t)}{{(\overline{\Phi }(t))}^{{}^{r-\alpha +1}}}{▽}_{\alpha }t\ge \frac{1}{\alpha -r}{\displaystyle {\int }_a^s}{\displaystyle D_{\alpha }^{▽}}{\left(\overline{\Phi }(t)\right)}^{{}^{\alpha -r}}{▽}_{\alpha }t=\frac{1}{\alpha -r}{(\overline{\Phi }(s))}^{\alpha -r}.$$

(53)

By substituting (53) into (52), we obtain

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\left({\overline{\Omega }}^{\rho }(s)\right)}^p}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{▽}_{\alpha }s\ge \frac{p}{\alpha -r}{\displaystyle {\int }_a^{\infty }}{(\overline{\Phi }(s))}^{\alpha -r}\frac{\chi (s)\eta (s)}{{\left({\overline{\Omega }}^{\rho }(s)\right)}^{1-p}}{▽}_{\alpha }s.$$

(54)

Raising (54) to the factor $p,$ we obtain

$${\left({\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\left({\overline{\Omega }}^{\rho }(s)\right)}^p}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{▽}_{\alpha }s\right)}^p\ge {\left(\frac{p}{\alpha -r}\right)}^p{\left({\displaystyle {\int }_a^{\infty }}{\left(\frac{{\chi }^p(s){\eta }^p(s){(\overline{\Phi }(s))}^{p(\alpha -r)}}{{\left({\overline{\Omega }}^{\rho }(s)\right)}^{p(1-p)}}\right)}^{\frac{1}{p}}{▽}_{\alpha }s\right)}^p.$$

(55)

The remainder of the proof is analogous to the proof of Theorem 3 and is thus omitted. □

Remark 7.

If $\alpha =1$ in Theorem 5, then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{(\overline{\Phi }(s))}^{{}^r}}{\left({\overline{\Omega }}^{\rho }(s)\right)}^p▽s\ge {\left(\frac{p}{1-r}\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\eta }^p(s){(\overline{\Phi }(s))}^{p-r}▽s,$$

(56)

where $a\in {[0,\infty )}_{\mathbb{T}},$ $r\le 0<p<1,$

$$\overline{\Phi }(s)={\displaystyle {\int }_a^s}\chi (t)▽t\mathit{and}\overline{\Omega }(s)={\displaystyle {\int }_s^{\infty }}\chi (t)\eta (t)▽t,$$

which is (19).

Corollary 5.

In (49), if $\mathbb{T}=\mathbb{R}$ $(i.e.$ $\rho (s)=s),$ then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\left(\overline{\Omega }(s)\right)}^p}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{s}^{\alpha -1}{d}_{\alpha }s\ge {\left(\frac{p}{\alpha -r}\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\eta }^p(s){(\overline{\Phi }(s))}^{p-r+\alpha -1}{s}^{\alpha -1}{d}_{\alpha }s,$$

(57)

where $z\in [0,\infty ),$ $0<p<\alpha ,$ $r\le \alpha -1,$ $\alpha \in \left(0,1\right],$

$$\overline{\Phi }(s)={\displaystyle {\int }_a^s}\chi (\theta ){\theta }^{\alpha -1}{d}_{\alpha }\theta \mathit{and}\overline{\Omega }(s)={\displaystyle {\int }_s^{\infty }}\chi (\theta )\eta (\theta ){\theta }^{\alpha -1}{d}_{\alpha }\theta .$$

This is a new result for a conformable continuous calculus.

Remark 8.

If $\alpha =1$ and $a=1,$ then (57) becomes

$${\displaystyle {\int }_1^{\infty }}\frac{\chi (s)}{{(\overline{\Phi }(s))}^{{}^r}}{\left(\overline{\Omega }(s)\right)}^{p}ds\ge {\left(\frac{p}{1-r}\right)}^p{\displaystyle {\int }_1^{\infty }}\chi (s){\eta }^p(s){(\overline{\Phi }(s))}^{p-r}ds,$$

(58)

where

$$\overline{\Phi }(s)={\displaystyle {\int }_1^s}\chi (\theta )d\theta \mathit{and}\overline{\Omega }(s)={\displaystyle {\int }_s^{\infty }}\chi (\theta )\eta (\theta )d\theta ,$$

which is Remark 3.8 in [12].

Corollary 6.

In (49), if $\mathbb{T}=\mathbb{Z}$ $(i.e.$ $\rho (s)=s-1)$ and $a=1,$ then

$${\displaystyle \sum _{i=2}^{\infty }}\frac{\chi (i){\left(\overline{\Omega }(i-1)\right)}^p}{{(\overline{\Phi }(i))}^{{}^{r-\alpha +1}}}{i}^{\alpha -1}\ge {\left(\frac{p}{\alpha -r}\right)}^p{\displaystyle \sum _{i=2}^{\infty }}\chi (i){\eta }^p(i){(\overline{\Phi }(i))}^{p-r+\alpha -1}{i}^{\alpha -1},$$

(59)

where $0<p<\alpha ,$ $r\le \alpha -1,$ $\alpha \in \left(0,1\right]$ and

$$\overline{\Phi }(i)={\displaystyle \sum _{r=2}^i}\chi (r)r^{\alpha -1}.$$

This is a new result for a discrete calculus.

Remark 9.

For $\alpha =1$ in (59), then we have ([12] Remark 3.9).

Theorem 6.

Suppose that $\mathbb{T}$ is a time scale with $a\in {[0,\infty )}_{\mathbb{T}}$ and $\alpha \in \left(0,1\right].$In addition, let χ and η be nonnegative ld-continuous and α-nabla fractional differentiable functions on $a\in {[0,\infty )}_{\mathbb{T}}.$ Define $\Omega (s)={\displaystyle {\int }_a^s}\chi (t)\eta (t){▽}_{\alpha }t$ and $\overline{\Phi }(s)={\displaystyle {\int }_a^s}\chi (t){▽}_{\alpha }t$ such that

$$L=\underset{s\in \mathbb{T}}{inf}\frac{{\overline{\Phi }}^{\rho }(s)}{\overline{\Phi }(s)}>0.$$

(60)

If $0<p<\alpha$ and $r\ge \alpha ,$ then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^{{}^p}(s)}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{▽}_{\alpha }s\ge {\left(\frac{p{L}^{\alpha -r}}{r-\alpha }\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\eta }^p(s){(\overline{\Phi }(s))}^{p-r+\alpha -1}{▽}_{\alpha }s.$$

(61)

Proof. 

Using (23) on

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{\Omega }^{{}^p}(s){▽}_{\alpha }s,$$

with $\xi (s)={\Omega }^{{}^p}(s)$ and ${\displaystyle D_{\alpha }^{▽}}v(s)=\chi (s)/{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}},$ we have

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^{{}^p}(s)}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{▽}_{\alpha }s={\displaystyle {\left.v(s){\Omega }^{{}^p}(s)\right|}_a^{\infty }}+{\displaystyle {\int }_a^{\infty }}(-v^{\rho }(s))({\displaystyle D_{\alpha }^{▽}}\left({\Omega }^{{}^p}(s)\right){▽}_{\alpha }s,$$

where $v(s)=-{\displaystyle {\int }_s^{\infty }}\chi (t)/{(\overline{\Phi }(t))}^{{}^{r-\alpha +1}}{▽}_{\alpha }t.$ This with $v(\infty )=0$ and $\Omega (a)=0$ imply that

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^{{}^p}(s)}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{▽}_{\alpha }s={\displaystyle {\int }_a^{\infty }}(-v^{\rho }(s))({\displaystyle D_{\alpha }^{▽}}\left({\Omega }^{{}^p}(s)\right){▽}_{\alpha }s.$$

(62)

By utilizing (22) and ${\displaystyle D_{\alpha }^{▽}}\Omega (s)=\chi (s)\eta (s)\ge 0$, we obtain

$$\begin{array}{ccc}\hfill {\displaystyle D_{\alpha }^{▽}}\left({\Omega }^{{}^p}(s)\right)& =& p{\Omega }^{p-1}(d){\displaystyle D_{\alpha }^{▽}}\Omega (s),\mathrm{where}d\in [\rho (s),s]\hfill \\ & =& \frac{p\chi (s)\eta (s)}{{\Omega }^{1-p}(d)}\ge \frac{p\chi (s)\eta (s)}{{\Omega }^{1-p}(s)}.\hfill \end{array}$$

(63)

By substituting (63) into (62) and using ${\displaystyle D_{\alpha }^{\Delta }}\overline{\Phi }(s)\ge 0$, we have

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s)}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{\Omega }^{{}^p}(s){▽}_{\alpha }s\ge p{\displaystyle {\int }_a^{\infty }}(-v^{\rho }(s))\frac{\chi (s)\eta (s)}{{\Omega }^{1-p}(s)}{▽}_{\alpha }s.$$

(64)

Again by (22) and ${\displaystyle D_{\alpha }^{▽}}\overline{\Phi }(s)=\chi (s)\ge 0$, we have

$$\begin{array}{ccc}\hfill {\displaystyle D_{\alpha }^{▽}}\left({\overline{\Phi }}^{\alpha -r}(s)\right)& =& (\alpha -r){\overline{\Phi }}^{\alpha -r-1}(d){\displaystyle D_{\alpha }^{▽}}\overline{\Phi }(s)=\frac{\alpha -r}{{(\overline{\Phi }(d))}^{r+1-\alpha }}{\displaystyle D_{\alpha }^{▽}}\overline{\Phi }(s)\hfill \\ & \le & \frac{\alpha -r}{{(\overline{\Phi }(s))}^{r+1-\alpha }}{\displaystyle D_{\alpha }^{▽}}\overline{\Phi }(s)\le \frac{\alpha -r}{{(\overline{\Phi }(s))}^{r+1-\alpha }}\chi (s),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill -v^{\rho }(s)& =& {\displaystyle {\int }_{\rho (s)}^{\infty }}\frac{\chi (t)}{{(\overline{\Phi }(t))}^{r+1-\alpha }}{▽}_{\alpha }t\hfill \\ & \ge & \frac{1}{\alpha -r}{\displaystyle {\int }_{\rho (s)}^{\infty }}{\displaystyle D_{\alpha }^{▽}}{\left(\overline{\Phi }(t)\right)}^{\alpha -r}{▽}_{\alpha }t\hfill \end{array}$$

(65)

$$\begin{array}{ccc}& =& \frac{{({\overline{\Phi }}^{\rho }(s))}^{\alpha -r}}{r-\alpha }=\left(\frac{1}{r-\alpha }\right)\frac{{({\overline{\Phi }}^{\rho }(s))}^{\alpha -r}{(\overline{\Phi }(s))}^{\alpha -r}}{{(\overline{\Phi }(s))}^{\alpha -r}}\hfill \\ & \ge & \left(\frac{L^{\alpha -r}}{r-\alpha }\right){(\overline{\Phi }(s))}^{\alpha -r}.\hfill \end{array}$$

By substituting (65) into (64), we obtain

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{▽}_{\alpha }s\ge \left(\frac{p{L}^{\alpha -r}}{r-\alpha }\right){\displaystyle {\int }_a^{\infty }}\frac{\chi (s)\eta (s)}{{(\overline{\Phi }(s))}^{r-\alpha }{\Omega }^{1-p}(s)}{▽}_{\alpha }s.$$

(66)

Raising (66) to the factor $p,$ we obtain

$${\left({\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{▽}_{\alpha }s\right)}^p\ge {\left(\frac{p{L}^{\alpha -r}}{r-\alpha }\right)}^p{\left({\displaystyle {\int }_a^{\infty }}{\left(\frac{{\chi }^p(s){\eta }^p(s)}{{(\overline{\Phi }(s))}^{p(r-\alpha )}{\Omega }^{p(1-p)}(s)}\right)}^{\frac{1}{p}}{▽}_{\alpha }s\right)}^p.$$

(67)

The remainder of the proof is analogous to the proof of Theorem 3 and is thus omitted. □

Remark 10.

If $\alpha =1$ in Theorem 6, then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{(\overline{\Phi }(s))}^r}▽s\ge {\left(\frac{p{L}^{1-r}}{r-1}\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\eta }^p(s){(\overline{\Phi }(s))}^{p-r}▽s,$$

(68)

where $z\in {[0,\infty )}_{\mathbb{T}},$ $0<p<1,$ $r\ge 1,$ $\overline{\Phi }(s)={\displaystyle {\int }_a^s}\chi (\theta )▽\theta$ and $\Omega (s)={\displaystyle {\int }_a^s}\chi (\theta )\eta (\theta )▽\theta$ such that

$$L:=\underset{s\in \mathbb{T}}{inf}\frac{{\overline{\Phi }}^{\rho }(s)}{\overline{\Phi }(s)}>0,$$

which is (20).

Corollary 7.

If $\mathbb{T}=\mathbb{R}$ $(i.e.$ $\rho (s)=s)$ in Theorem 6, then

$${\displaystyle {\int }_a^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{(\overline{\Phi }(s))}^{{}^{r-\alpha +1}}}{s}^{\alpha -1}{d}_{\alpha }s\ge {\left(\frac{p}{r-\alpha }\right)}^p{\displaystyle {\int }_a^{\infty }}\chi (s){\eta }^p(s){(\overline{\Phi }(s))}^{p-r+\alpha -1}{s}^{\alpha -1}{d}_{\alpha }s,$$

(69)

where $0<p<\alpha ,$ $r\ge \alpha ,$ $\alpha \in \left(0,1\right],$ $a\in [0,\infty ),\Omega (s)={\displaystyle {\int }_a^s}\chi (y)\eta (y)y^{\alpha -1}{d}_{\alpha }y$ and $\overline{\Phi }(s)={\displaystyle {\int }_a^s}\chi (y)y^{\alpha -1}{d}_{\alpha }y$ such that

$$L=\underset{s\in \mathbb{R}}{inf}\frac{\overline{\Phi }(s)}{\overline{\Phi }(s)}=1.$$

This is a new result for a conformable continuous calculus.

Remark 11.

If $\alpha =1$ and $a=1,$ then (69) becomes

$${\displaystyle {\int }_1^{\infty }}\frac{\chi (s){\Omega }^p(s)}{{(\overline{\Phi }(s))}^{{}^r}}ds\ge {\left(\frac{p}{r-1}\right)}^p{\displaystyle {\int }_1^{\infty }}\chi (s){\eta }^p(s){(\overline{\Phi }(s))}^{p-r}ds,$$

where

$$\overline{\Phi }(s)={\displaystyle {\int }_1^s}\chi (y)dy\mathit{and}\Omega (s)={\displaystyle {\int }_1^s}\chi (y)\eta (y)dy,$$

which is Remark 3.11 in [12].

Corollary 8.

In (60), if $\mathbb{T}=\mathbb{Z}$ $(i.e.$ $\rho (s)=s-1)$ and $a=1,$ then

$${\displaystyle \sum _{i=2}^{\infty }}\frac{\chi (i)i^{\alpha -1}}{{(\overline{\Phi }(i))}^{{}^{r-\alpha +1}}}{\left({\displaystyle \sum _{j=2}^i}\chi (j)\eta (j)j^{\alpha -1}\right)}^p\ge {\left(\frac{p{L}^{\alpha -r}}{r-\alpha }\right)}^p{\displaystyle \sum _{i=2}^{\infty }}\chi (i){\eta }^p(i){(\overline{\Phi }(i))}^{p-r+\alpha -1}{i}^{\alpha -1},$$

(70)

where $0<p<\alpha ,$ $r\ge \alpha ,$ $\alpha \in \left(0,1\right],$ $\overline{\Phi }(i)={\displaystyle \sum _{j=2}^i}\chi (j)j^{\alpha -1}$ and

$$L=\underset{s\in \mathbb{Z}}{inf}\frac{\overline{\Phi }(i-1)}{\overline{\Phi }(i)},$$

This is a new result for a discrete calculus.

Remark 12.

For $\alpha =1$ in (70), then we have [12] (Remark 3.12).

4. Conclusions

In this work, with the help of a simple consequence of Keller’s chain rule and Hölder inequality for the $\alpha$-conformable nabla calculus on time scales, we generalized several of Bennett and Leindler-type inequalities. We also expand our inequalities to a discrete and continuous calculus in order to derive some more inequalities as specific cases. By utilising Specht’s ratio and Kantorovich’s ratio in subsequent work, we will proceed to generalise more fractional dynamic inequalities. Additionally, utilising the findings from this study, researchers may be able to derive other generalisations for the dynamic Hardy inequality and its related inequalities.