Dynamic Hardy–Copson-Type Inequalities via (γ,a)-Nabla-Conformable Derivatives on Time Scales

Abstract

We prove new Hardy–Copson-type $(\gamma ,a)$-nabla fractional dynamic inequalities on time scales. Our results are proven by using Keller’s chain rule, the integration by parts formula, and the dynamic Hölder inequality on time scales. When $\gamma =1$, then we obtain some well-known time-scale inequalities due to Hardy. As special cases, we obtain new continuous and discrete inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

1. Introduction

A great number of dynamic Hardy type inequalities on time scales have been established by many researchers who were motivated by some applications [1,2,3]. Additionally, over several decades, many generalizations, extensions, and refinements of other types of dynamic inequalities have been studied; we refer the reader to the papers [4,5,6,7,8].

In 1920, Hardy [9] established the following renowned discrete inequality.

Theorem 1.

If ${\left\{a\left(n\right)\right\}}_{n=0}^{\infty }$ is a non-negative real sequence and $p>1$, then

$$\sum _{n=1}^{\infty }{\left(\frac{1}{n}\sum _{\ell =1}^{n}a(\ell )\right)}^p\le {\left(\frac{p}{p-1}\right)}^p\sum _{n=1}^{\infty }{a}^p\left(n\right).$$

(1)

In 1925, using the calculus of variations, Hardy himself in [10] gave the integral analogous of inequality (1) as follows:

Theorem 2.

If f is a non-negative continuous function on $[0,\infty )$ and $p>1$, then

$${\int }_0^{\infty }{\left(\frac{1}{\vartheta }{\int }_0^{\vartheta }f\left(\varrho \right)d\varrho \right)}^{p}d\vartheta \le {\left(\frac{p}{p-1}\right)}^p{\int }_0^{\infty }{f}^p\left(\vartheta \right)d\vartheta .$$

(2)

Copson [11] improved inequality (1) by replacing the arithmetic mean of a sequence by a weighted arithmetic mean in the following manner: Let $b(\ell )⩾0$ and $\omega (\ell )⩾0$ for all ℓ. If $p>1,$ $c>1,$ then

$$\sum _{\ell =1}^{\infty }\frac{\omega (\ell )}{{\left[\overline{A}(\ell )\right]}^c}{\left(\sum _{j=1}^{\ell }b\left(j\right)\omega \left(j\right)\right)}^p⩽{\left(\frac{p}{c-1}\right)}^p\sum _{\ell =1}^{\infty }\omega (\ell ){\left[\overline{A}(\ell )\right]}^{p-c}{b}^p(\ell ),$$

(3)

where $\overline{A}(\ell )={\sum }_{j=1}^{\ell }\omega \left(j\right),$ and if $0⩽c<1<p,$ then

$$\sum _{\ell =1}^{\infty }\frac{\omega (\ell )}{{\left[\overline{A}(\ell )\right]}^c}{\left(\sum _{j=1}^{\infty }b\left(j\right)\omega \left(j\right)\right)}^p⩽{\left(\frac{p}{1-c}\right)}^p\sum _{\ell =1}^{\infty }\omega (\ell ){\left[\overline{A}(\ell )\right]}^{p-c}{b}^p(\ell ).$$

(4)

Hardy–Copson inequalities have been developed by Bennett as follows: Let $b(\ell )⩾0$ and $\omega (\ell )⩾0$ for all ℓ and ${\sum }_{j=1}^{\infty }\omega \left(j\right)<\infty$. If $0⩽c<1<p,$ then

$$\sum _{\ell =1}^{\infty }\frac{\omega (\ell )}{{\left[\overline{A}(\ell )\right]}^c}{\left(\sum _{j=1}^{\ell }b\left(j\right)\omega \left(j\right)\right)}^p⩽{\left(\frac{p}{1-c}\right)}^p\sum _{\ell =1}^{\infty }\omega (\ell ){\left[\overline{A}(\ell )\right]}^{p-c}{b}^p(\ell ),$$

(5)

and

$$\sum _{\ell =1}^{\infty }\frac{\omega (\ell )}{{\left[A(\ell )\right]}^c}{\left(\sum _{j=1}^{\infty }b\left(j\right)\omega \left(j\right)\right)}^p⩽{\left(\frac{p}{1-c}\right)}^p\sum _{\ell =1}^{\infty }\omega (\ell ){\left[A(\ell )\right]}^{p-c}{b}^p(\ell ),$$

(6)

where $\overline{A}(\ell )={\sum }_{j=1}^{\ell }\omega \left(j\right),$ and $A(\ell )={\sum }_{j=\ell }^{\infty }\omega \left(j\right)$.

If $1<c⩽p,$ then

$$\sum _{\ell =1}^{\infty }\frac{\omega (\ell )}{{\left[\overline{A}(\ell )\right]}^c}{\left(\sum _{j=1}^{\ell }b\left(j\right)\omega \left(j\right)\right)}^p⩽{\left(\frac{p}{1-c}\right)}^p\sum _{\ell =1}^{\infty }\omega (\ell ){\left[\overline{A}(\ell )\right]}^{p-c}{b}^p(\ell ),$$

(7)

and

$$\sum _{\ell =1}^{\infty }\frac{\omega (\ell )}{{\left[A(\ell )\right]}^c}{\left(\sum _{j=1}^{\infty }b\left(j\right)\omega \left(j\right)\right)}^p⩽{\left(\frac{p}{1-c}\right)}^p\sum _{\ell =1}^{\infty }\omega (\ell ){\left[A(\ell )\right]}^{p-c}{b}^p(\ell ).$$

(8)

Then, the continuous version of the inequalities (3) (or (7)) and (6) (or (8)) established by Copson [12], respectively, are as follows: Let $\omega \left(\varsigma \right)$ and $f\left(\varsigma \right)$ be non-negative functions and $\overline{A}\left(\varsigma \right)={\int }_0^{\varsigma }\omega \left(\varrho \right)d\varrho ,$$B\left(\varsigma \right)={\int }_0^{\varsigma }\omega \left(\varrho \right)f\left(\varrho \right)d\varrho$, $\overline{B}\left(\varsigma \right)={\int }_{\varsigma }^{\infty }\omega \left(\varrho \right)f\left(\varrho \right)d\varrho$. If $1<c,1⩽p,1<b⩽\infty$, then

$${\int }_0^b\frac{\omega \left(\varsigma \right)}{{\left[\overline{A}\left(\varsigma \right)\right]}^c}{\left[B\left(\varsigma \right)\right]}^{p}dt⩽{\left(\frac{p}{c-1}\right)}^p{\int }_0^b\omega \left(\varsigma \right){\left[\overline{A}\left(\varsigma \right)\right]}^{p-c}{f}^p\left(\varsigma \right)dt,$$

(9)

and if $c<1⩽p,$ $a>0$, then

$${\int }_a^{\infty }\frac{\omega \left(\varsigma \right)}{{\left[\overline{A}\left(\varsigma \right)\right]}^c}{\left[\overline{B}\left(\varsigma \right)\right]}^{p}dt⩽{\left(\frac{p}{1-c}\right)}^p{\int }_a^{\infty }\omega \left(\varsigma \right){\left[\overline{A}\left(\varsigma \right)\right]}^{p-c}{f}^p\left(\varsigma \right)dt.$$

(10)

Other refinements of continuous Hardy–Copson inequalities, which are generalizations of (9) and (10), respectively, have been introduced by Pacarić and Hanjs as follows: Let $\omega \left(\varsigma \right)$ and $f\left(\varsigma \right)$ be non-negative functions and $\overline{A}\left(\varsigma \right)={\int }_0^{\varsigma }\omega \left(\varrho \right)d\varrho ,$$B\left(\varsigma \right)={\int }_0^{\varsigma }\omega \left(\varrho \right)f\left(\varrho \right)d\varrho$, $\overline{B}\left(\varsigma \right)={\int }_{\varsigma }^{\infty }\omega \left(\varrho \right)f\left(\varrho \right)d\varrho$.

If $\xi ⩾0,$ $p>1,$ $\xi +c>1$ then

$${\int }_0^{\infty }\frac{\omega \left(\varsigma \right)}{{\left[\overline{A}\left(\varsigma \right)\right]}^c}{\left[B\left(\varsigma \right)\right]}^{p+\xi }dt⩽{\left(\frac{p+\xi }{c+\xi -1}\right)}^p{\int }_0^{\infty }\frac{\omega \left(\varsigma \right)f^p\left(\varsigma \right){\left[B\left(\varsigma \right)\right]}^{\xi }}{{\left[\overline{A}\left(\varsigma \right)\right]}^{\xi +c-p}}dt,$$

(11)

and if $\xi ⩾0,$ $p>1,$ $\xi +c<1$ then

$${\int }_0^{\infty }\frac{\omega \left(\varsigma \right)}{{\left[\overline{A}\left(\varsigma \right)\right]}^c}{\left[\overline{B}\left(\varsigma \right)\right]}^{p}dt⩽{\left(\frac{p+\xi }{1-c-\xi }\right)}^p{\int }_0^{\infty }\frac{\omega \left(\varsigma \right)f^p\left(\varsigma \right){\left[\overline{B}\left(\varsigma \right)\right]}^{\xi }}{{\left[\overline{A}\left(\varsigma \right)\right]}^{c+\xi -p}}dt.$$

(12)

A time scale $\mathbb{T}$ is an arbitrary non-empty closed subset of the real numbers $\mathbb{R}$. We assume throughout that $\mathbb{T}$ has the topology that it inherits from the standard topology on the real number $\mathbb{R}$. For more details on time-scales calculus, see [13,14]. We define the forward jump operator $\sigma :\mathbb{T}\to \mathbb{T}$ by

$$\sigma \left(\varsigma \right):=inf\{\varrho \in \mathbb{T}:\varrho >\varsigma \},\varsigma \in \mathbb{T},$$

(13)

and the backward jump operator $\rho :\mathbb{T}:\to \mathbb{T}$ is defined by

$$\rho \left(\varsigma \right):=sup\{\varrho \in \mathbb{T}:\varrho <\varsigma \},\varsigma \in \mathbb{T}.$$

(14)

In the previous two definitions, we set $inf\varnothing =sup\mathbb{T}$ (i.e., if $\varsigma$ is the minimum of $\mathbb{T}$, then $\sigma \left(\varsigma \right)=\varsigma$) and $sup\varnothing =inf\mathbb{T}$ (i.e., if $\varsigma$ is the maximum of $\varsigma$, then $\rho \left(\varsigma \right)=\varsigma$), where ∅ is the empty set.

We introduce the nabla derivative of a function $f:\mathbb{T}\to \mathbb{R}$ at a point $t\in {\mathbb{T}}_{\kappa }$, as follows:

Definition 1.

Let $f:\mathbb{T}\to \mathbb{R}$ be a function, and let $t\in {\mathbb{T}}_{\kappa }$. We define $f^{\nabla }\left(t\right)$ as the real number (provided it exists) with the property that for any $ϵ>0$, there exists a neighborhood N of t (i.e., $N={(t-\delta ,t+\delta )}_{\mathbb{T}}$ for some $\delta >0$), such that

$$|[f^{\rho }\left(t\right)-f\left(s\right)]-f^{\nabla }\left(t\right)[\rho \left(t\right)-s]|\le ϵ|\rho \left(t\right)-s|foreverys\in N.$$

We say that $f^{\nabla }\left(t\right)$ is the nabla derivative of f at t.

Definition 2.

We say that a function $F:\mathbb{T}\to \mathbb{R}$ is a nabla antiderivative of $f:\mathbb{T}\to \mathbb{R}$ if $F^{\nabla }\left(t\right)=f\left(t\right)$ for all $t\in {\mathbb{T}}_{\kappa }$. In this case, the nabla integral of f is defined by

$${\int }_a^{t}f\left(t\right)\nabla \tau =F\left(t\right)-F\left(a\right)forallt\in {\mathbb{T}}_{\kappa }.$$

Now, we introduce the set of all ld-continuous functions in order to find a class of functions that have nabla antiderivatives.

Definition 3

(Ld-Continuous Function). We say that the function $f:\mathbb{T}\to \mathbb{R}$ is ld-continuous if it is continuous at all left-dense points of $\mathbb{T}$ and its right-sided limits exist (finite) at all right-dense points of $\mathbb{T}$.

Theorem 3

(Existence of Nabla Antiderivatives). Every ld-continuous function possess a nabla antiderivative.

In 2021, Kayar et al. [15], established the time-scale version unification of discrete and continuous Bennett–Leindler inequalities (10) and (11), as in the following theorem.

Theorem 4.

Assume that $\mathbb{T}$ is a time scale with $a\in [0,\infty )$. Moreover, suppose that ϕ and g are non-negative, ld-continuous, ∇-differentiable, and locally nabla integrable functions. Set

$$\varpi \left(\varsigma \right)={\int }_r^{\varsigma }\varphi \left(\varrho \right)\nabla \varrho \Psi \left(\varsigma \right)={\int }_r^{\varsigma }\varphi \left(\varrho \right)g\left(\varrho \right)\nabla \varrho .$$

Assume that $\Psi (\infty )<\infty$ and ${\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta }}}<\infty$. Suppose that there exists $M>0$, such that ${\displaystyle \frac{\overline{\varpi }\left(\varsigma \right)}{{\overline{\varpi }}^{\rho }\left(\varsigma \right)}}⩽M$ for $\varsigma \in {(a,\infty )}_{\mathbb{T}}$. If $c⩾0,$ $p>1$, and $c+\theta >1$ are real constants, then

(i)

${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p}}{{\varpi }^{c+\theta }\left(\varsigma \right)}\nabla \varsigma ⩽{\displaystyle \frac{c+p}{c+\theta -1}}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-1}\left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -1}}}\nabla \varsigma .$

(ii)

${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{c+p}\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta }}\nabla \varsigma ⩽{\left(\frac{c+p}{c+\theta -1}\right)}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\mathsf{\Psi }}^c\left(\varsigma \right){\left[\varpi \left(\varsigma \right)\right]}^{(c+\theta )(p-1)}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{(c+\theta -1)p}}\nabla \varsigma .$

(iii)

${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{c+p}\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta }}\nabla \varsigma ⩽{\left(\frac{c+p}{c+\theta -1}\right)}^p{M}^{(c+\theta )(p-1)}{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right)\left[{\mathsf{\Psi }}^c\left(\varsigma \right)\right]}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{(c+\theta -p)}}\nabla \varsigma .$

(iv)

${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p}}{{\varpi }^{c+\theta }\left(\varsigma \right)}\nabla \varsigma ⩽{\left[{\displaystyle \frac{c+p}{c+\theta -1}}{M}^{c+\theta -1}\right]}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right)\left[{\mathsf{\Psi }}^c\left(\varsigma \right)\right]}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p}}\nabla \varsigma .$

(v)

${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta }}\nabla \varsigma ⩽{\displaystyle \frac{c+p}{c+\theta -1}}{M}^{c+\theta }{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-1}\left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -1}}}\nabla \varsigma .$

(vi)

${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{c+p}\left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta }}\nabla \varsigma ⩽{\left(M^{c+\theta }\frac{c+p}{c+\theta -1}\right)}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right)\left[{\mathsf{\Psi }}^c\left(\varsigma \right)\right]}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -p}}\nabla \varsigma .$

Rahmat introduced a new nabla conformable calculus in his recent paper [16]. He presents the following basic definitions and concepts.

Definition 4.

Let $[\varrho ,\varsigma ]\subset \mathbb{T}$ and $\varrho <\varsigma$. The generalized time-scale power function ${\widehat{G}}_n:\mathbb{T}\times \mathbb{T}⟶{\mathbb{R}}^{+}$ for $n\in {\mathbb{N}}_0$ is defined by

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

(15)

and its inverse function ${\widehat{G}}_{-n}:\mathbb{T}\times \mathbb{T}⟶{\mathbb{R}}^{+}$ is then given by

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

(16)

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

Definition 5.

Given a function $f:\mathbb{T}⟶\mathbb{R}$ and $a\in \mathbb{T},$ f is $(\gamma ,a)$-nabla differentiable at $\varsigma >a,$ if it is nabla differentiable at ς, and its $(\gamma ,a)$-nabla derivative is defined by

$${\nabla }_a^{\gamma }f\left(\varsigma \right)={\widehat{G}}_{1-\gamma }(\varsigma ,a)f^{\nabla }\left(\varsigma \right)\varsigma >a,$$

where the function ${\widehat{G}}_{1-\gamma }(\varsigma ,a)$, as defined in (15). If ${\nabla }_a^{\gamma }\left[f\left(\varsigma \right)\right]$ exists in some interval ${(a,a+ϵ)}_{\mathbb{T}},ϵ>0,$ then we define

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

if the ${lim}_{\varsigma ⟶a^{+}}{\nabla }_a^{\gamma }\left[f\left(\varsigma \right)\right]$ exists. Moreover, we call f as $(\gamma ,a)$-nabla differentiable on ${\mathbb{T}}_k$ $(a\in {\mathbb{T}}_k)$, provided ${\nabla }_a^{\gamma }\left[f\left(\varsigma \right)\right]$ exists for all $\varsigma \in {\mathbb{T}}_k$. The function ${\nabla }_a^{\gamma }:{\mathbb{T}}_k⟶\mathbb{R}$ is then called the $(\gamma ,a)$-nabla derivative of f on ${\mathbb{T}}_k$.

Next, we provide the $(\gamma ,a)$-nabla derivatives of sums, products, and quotients of $(\gamma ,a)$-nabla differentiable functions.

Theorem 5.

Assume $f,g:\mathbb{T}⟶\mathbb{R}$ are $(\gamma ,a)$-nabla differentiable at $t\in {\mathbb{T}}_k,$ $t>a$. Then:

(i)

The sum $f+g:\mathbb{T}⟶\mathbb{R}$ is $(\gamma ,a)$-nabla differentiable at t, with

$${\nabla }_a^{\gamma }(f+g)\left(t\right)={\nabla }_a^{\gamma }f\left(t\right)+{\nabla }_a^{\gamma }g\left(t\right).$$

(ii)

For all $k\in \mathbb{R},$ then $kf:\mathbb{T}⟶\mathbb{R}$ is $(\gamma ,a)$-nabla differentiable at t with

$${\nabla }_a^{\gamma }\left(kf\right)\left(t\right)=k{\nabla }_a^{\gamma }f\left(t\right).$$

(iii)

The product $fg:\mathbb{T}⟶\mathbb{R}$ is $(\gamma ,a)$-nabla differentiable at t, with

$${\nabla }_a^{\gamma }\left(fg\right)=\left[{\nabla }_a^{\gamma }f\left(t\right)\right]g\left(t\right)+f^{\rho }\left(t\right)\left[{\nabla }_a^{\gamma }g\left(t\right)\right].$$

(17)

(iv)

If $g\left(t\right)g^{\rho }\left(t\right)\ne 0,$ then $f/g$ is $(\gamma ,a)$-nabla differentiable at t, with

$${\nabla }_a^{\gamma }\left(\frac{f}{g}\right)\left(t\right)=\frac{\left[{\nabla }_a^{\gamma }f\left(t\right)\right]g\left(t\right)-f\left(t\right)\left[{\nabla }_a^{\gamma }g\left(t\right)\right]}{g\left(t\right)g^{\rho }\left(t\right)}.$$

Lemma 1.

Let $g\in C_{ld}^{\nabla }\left(\mathbb{T}\right)$ and assume that $f:\mathbb{R}\to \mathbb{R}$ is a continuously differentiable function. Then, $(f\circ g):\mathbb{T}\to \mathbb{R}$ is $(\gamma ,a)$-nabla differentiable and satisfies

$${\nabla }_a^{\gamma }(f\circ g)\left(\varsigma \right)=\left\{{\int }_0^1{f}^{\prime }\left(g\left(\varsigma \right)-h\nu \left(\varsigma \right)g^{\nabla }\left(\varsigma \right)\right)dh\right\}{\nabla }_a^{\gamma }g\left(\varsigma \right).$$

(18)

Lemma 2.

Let $\gamma \in (0,1]$. Assume $\xi :\mathbb{T}\to \mathbb{R}$ is continuous and $(\gamma ,a)$-nabla differentiable of order γ at $\varsigma \in {\mathbb{T}}_k,$ where $\varsigma >a$ and $\varpi :\mathbb{R}\to \mathbb{R}$ is continuously differentiable. Then, there is c in the real interval $\left[\rho \right(\varsigma ),\varsigma ]$, such that

$${\nabla }_a^{\gamma }(\varpi \circ \xi )\left(\varsigma \right)={\varpi }^{\prime }\left(\xi \left(c\right)\right){\nabla }_a^{\gamma }\left(\xi \left(\varsigma \right)\right).$$

(19)

Lemma 3.

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

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

(20)

Definition 6.

Assume that $0<\gamma ⩽1,$ $a,$ ${\varsigma }_1,$ ${\varsigma }_2\in \mathbb{T},$ $a⩽{\varsigma }_1⩽{\varsigma }_2$, and $f\in C_{ld}\left(\mathbb{T}\right),$ then we say that f is $(\gamma ,a)$-nabla integrable on interval $[{\varsigma }_1,{\varsigma }_2]$ if the following integral:

$${\nabla }_a^{-\gamma }f\left(\varsigma \right)={\int }_{{\varsigma }_1}^{{\varsigma }_2}f\left(\tau \right){\nabla }_a^{\gamma }\tau ={\int }_{{\varsigma }_1}^{{\varsigma }_2}f\left(\tau \right){\widehat{G}}_{\gamma -1}({\sigma }^{\gamma -1}\left(\tau \right),a)\nabla \tau .$$

exists and is finite.

For the case $\mathbb{T}=\mathbb{R},$ we have the classical conformable integral as defined in [17], namely

$${\int }_a^{\varsigma }f\left(\tau \right){\nabla }_a^{\gamma }\tau ={\int }_a^{\varsigma }f\left(\tau \right){(\tau -a)}^{\gamma -1}d\tau .$$

(21)

For $\mathbb{T}=h\mathbb{Z},$ $h>0,$ we have a new conformable fractional h-sum given by

$${\int }_a^{\varsigma }f\left(\tau \right){\nabla }_a^{\gamma }\tau =\sum _{\tau \in (a,\varsigma ]}hf\left(\tau \right){\left({\rho }^{\gamma -1}\left(\tau \right)-a\right)}_h^{(\gamma -1)}.$$

(22)

For $\mathbb{T}=q^{{\mathbb{N}}_0},$ we have a new conformable fractional q-sum given by

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

(23)

Theorem 6.

Let $\gamma \in (0,1]$ and $a\in \mathbb{T}$. Then, for any ld-continuous function $f:\mathbb{T}⟶\mathbb{R},$ there exists a function $F:\mathbb{T}⟶\mathbb{R}$, such that

$${\nabla }_a^{\gamma }F\left(\varsigma \right)=f\left(\varsigma \right)forall\varsigma \in {\mathbb{T}}_k.$$

The function F is called an $(\gamma ,a)$-nabla antiderivative of f.

Lemma 4

([18]) (Hölder inequality for the $\gamma$-nabla derivative). Let $d,b\in \mathbb{T}$, where $b>d$. If $\gamma \in (0,1]$ and $\varpi ,\xi :\mathbb{T}⟶\mathbb{R},$ then

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

(24)

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

Here, we prove the new Hardy–Copson-type dynamic via the $(\gamma ,a)$-nabla-conformable calculus on time scales. Our inequalities have a completely new form and may be considered as extensions of [15]. As special cases, we obtain some new continuous and discrete inequalities of the Hardy-type, generalizing those obtained in the literature.

The paper is organized as follows. The original results are given and proven in Section 2. In Section 3, we stated the discussion of the results. We end with Section 4, the conclusion.

Now, we are ready to state and to prove our main results. Throughout the following, we assumed that the time-scale $\mathbb{T}$ is unbounded above.

2. Main Results

Theorem 7.

Assume that $\mathbb{T}$ is a time scale with $0\le r\in \mathbb{T}$ and $\varsigma ⩾a$. Moreover, suppose that g and ϕ are non-negative $(\gamma ,a)$-nabla fractional differentiable and locally integrable functions on ${[r,\infty )}_{\mathbb{T}}$. Set $\varpi \left(\varsigma \right)={\int }_r^{\varsigma }\varphi \left(\varrho \right){\nabla }_a^{\gamma }\varrho$ and $\mathsf{\Psi }\left(\varsigma \right)={\int }_r^{\varsigma }\varphi \left(\varrho \right)g\left(\varrho \right){\nabla }_a^{\gamma }\varrho ,$ and assume that $\mathsf{\Psi }(\infty )<\infty$ and ${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)}{\varpi \left(\varsigma \right)}{\nabla }_a^{\gamma }\varsigma <\infty$. Suppose that there exists $\Lambda >0$, such that $\frac{\varpi \left(\varsigma \right)}{{\varpi }^{\rho }\left(\varsigma \right)}⩽\mathrm{\Lambda }$ for $\varsigma \in {[r,\infty )}_{\mathbb{T}}$. If $c⩾0,$ $p>1,$ $c+\theta >1$, and $0<\gamma ⩽1$ are real constants, then

(i)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{\nabla }_a^{\gamma }\varsigma ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

(25)

(ii)

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\left(\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}{\left[\varpi \left(\varsigma \right)\right]}^{(c+\theta -\gamma +1)(p-1)}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{(c+\theta -\gamma )p}}{\nabla }_a^{\gamma }\varsigma .$$

(26)

(iii)

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\left(\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p{\mathrm{\Lambda }}^{(c+\theta -\gamma +1)(p-1)}{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{(c+\theta -p-\gamma +1)}}{\nabla }_a^{\gamma }\varsigma .$$

(27)

(iv)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{\nabla }_a^{\gamma }\varsigma ⩽{\left[{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma }\right]}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{1+c-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

(28)

(v)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma +1}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

(29)

(vi)

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\left({\mathrm{\Lambda }}^{c+\theta -\gamma +1}\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}{\nabla }_a^{\gamma }\varsigma .$$

(30)

Proof. 

(i)

By using the integration by parts to the left-hand side of inequality (25), we have ${\nabla }_a^{\gamma }u\left(\varsigma \right)=\frac{\varphi \left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}$ and $v\left(\varsigma \right)={\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1};$ it becomes

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{\nabla }_a^{\gamma }\varsigma =u\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right){|}_r^{\infty }+{\int }_r^{\infty }-u^{\rho }\left(\varsigma \right)\left[{\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)\right]{\nabla }_a^{\gamma }\varsigma \end{array}$$

(31)

where $u\left(\varsigma \right)=-{\int }_{\varsigma }^{\infty }\frac{\varphi \left(\varrho \right)}{{\left[\varpi \left(\varrho \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varrho$. By using $\mathsf{\Psi }\left(r\right)=0$ and $u(\infty )=0,$ one can have

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{\nabla }_a^{\gamma }\varsigma ={\int }_r^{\infty }-u^{\rho }\left(\varsigma \right)\left[{\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)\right]{\nabla }_a^{\gamma }\varsigma .$$

(32)

By chain rule, and using ${\nabla }_a^{\gamma }\mathsf{\Psi }\left(\varsigma \right)=\varphi \left(\varsigma \right)g\left(\varsigma \right)⩾0,$ we obtain

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)& =& (c+p-\gamma +1){\mathsf{\Psi }}^{c+p-\gamma }\left(d\right){\nabla }_a^{\gamma }\mathsf{\Psi }\left(\varsigma \right)=(c+p-\gamma +1)\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(d\right)\hfill \\ & ⩽& (c+p-\gamma +1)\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(\varsigma \right).\hfill \end{array}$$

(33)

Since ${\nabla }_a^{\gamma }\varpi \left(\varsigma \right)=\varphi \left(\varsigma \right)⩾0$ and $c+\theta >1,$ by chain rule, we get

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varsigma \right)\right]& =& {\int }_0^1{\displaystyle \frac{(\gamma -c-\theta ){\nabla }_a^{\gamma }\varpi \left(\varsigma \right)}{{[(1-h)\varpi \left(\varsigma \right)+h{\varpi }^{\rho }\left(\varsigma \right)]}^{c+\theta -\gamma +1}}}dh={\int }_0^1{\displaystyle \frac{-(c+\theta -\gamma )\varphi \left(\varsigma \right)}{{[(1-h)\varpi \left(\varsigma \right)+h{\varpi }^{\rho }\left(\varsigma \right)]}^{c+\theta -\gamma +1}}}dh\hfill \\ \hfill & ⩽& -(c+\theta -\gamma ){\int }_0^1{\displaystyle \frac{\varphi \left(\varsigma \right)}{{[(1-h)\varpi \left(\varsigma \right)+h\varpi \left(\varsigma \right)]}^{c+\theta -\gamma +1}}}dh=-(c+\theta -\gamma ){\displaystyle \frac{\varphi \left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}}\hfill \\ \hfill {\displaystyle \frac{\varphi \left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}}& ⩽& -{\displaystyle \frac{{\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varsigma \right)\right]}{(c+\theta -\gamma )}}.\hfill \end{array}$$

(34)

Therefore,

$$\begin{array}{ccc}\hfill -u^{\rho }\left(\varsigma \right)& =& {\int }_{\rho \left(\varsigma \right)}^{\infty }{\displaystyle \frac{\varphi \left(\varrho \right)}{{\left[\varpi \left(\varrho \right)\right]}^{c+\theta -\gamma +1}}}{\nabla }_a^{\gamma }\varrho ⩽{\int }_{\rho \left(\varsigma \right)}^{\infty }-{\displaystyle \frac{{\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varrho \right)\right]}{(c+\theta -\gamma )}}{\nabla }_a^{\gamma }\varrho ={\displaystyle \frac{-{\varpi }^{\gamma -c-\theta }(\infty )+{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{\gamma -c-\theta }}{c+\theta -\gamma }}\hfill \\ & ⩽& {\displaystyle \frac{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{\gamma -c-\theta }}{c+\theta -\gamma }}.\hfill \end{array}$$

(35)

Substituting (33) and (35) into (32) leads to

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{\nabla }_a^{\gamma }\varsigma ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma ,\end{array}$$

which is the desired result (25).

(ii)

Since

$$\begin{array}{ccc}& & {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma }}{\nabla }_a^{\gamma }\varsigma \hfill \\ & & ={\int }_r^{\infty }\frac{{\varphi }^{\frac{1}{p}}\left(\varsigma \right)g\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{\frac{c-\gamma +1}{p}}{\left[\varpi \left(\varsigma \right)\right]}^{\frac{(c+\theta -\gamma +1)(p-1)}{p}}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma }}\hfill \\ & & \times {\left[\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}\right]}^{\frac{p-1}{p}}{\nabla }_a^{\gamma }\varsigma .\hfill \end{array}$$

(36)

Applying the Hölder inequality with the constants p and $p/p-1$ to the right-hand side of above equation leads to the inequality

$$\begin{array}{ccc}& & {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma \hfill \\ & & ⩽\frac{(1+c+p-\gamma )}{(c+\theta -\gamma )}{\left({\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}{\left[\varpi \left(\varsigma \right)\right]}^{(c+\theta -\gamma +1)(p-1)}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{(c+\theta -\gamma )p}}{\nabla }_a^{\gamma }\varsigma \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma \right)}^{\frac{p-1}{p}}\hfill \end{array}$$

(37)

therefore,

$$\begin{array}{ccc}& & {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\left(\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p\hfill \\ & & \times {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}{\left[\varpi \left(\varsigma \right)\right]}^{(c+\theta -\gamma +1)(p-1)}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{(c+\theta -\gamma )p}}{\nabla }_a^{\gamma }\varsigma ,\hfill \end{array}$$

which is the desired inequality (26).

(iii)

To obtain inequality (27), we use

$$\frac{{\left[\varpi \left(\varsigma \right)\right]}^{(c+\theta -\gamma +1)(p-1)}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{(c+\theta -\gamma )p}}={\left(\frac{\varpi \left(\varsigma \right)}{{\varpi }^{\rho }\left(\varsigma \right)}\right)}^{(c+\theta -\gamma +1)(p-1)}\frac{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{-c-\theta }}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{-p-\gamma +1}}=\frac{{\mathrm{\Lambda }}^{(c+\theta -\gamma +1)(p-1)}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{(c+\theta -p-\gamma +1)}}$$

in inequality (26), and then inequality (26) becomes

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\left(\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p{\mathrm{\Lambda }}^{(c+\theta -\gamma +1)(p-1)}{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{(c+\theta -p-\gamma +1)}}{\nabla }_a^{\gamma }\varsigma .$$

(iv)

In order to obtain inequality (28), we use inequality (25) and the constant $\mathrm{\Lambda }$ as follows.

$$\begin{array}{ccc}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{\nabla }_a^{\gamma }\varsigma & ⩽& {\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma \hfill \\ \hfill & ⩽& {\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma }{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma .\hfill \end{array}$$

Now, by using the Hölder inequality with the constants $p,$ $p/p-1,$ we get that

$$\begin{array}{ccc}\hfill & & {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{\nabla }_a^{\gamma }\varsigma \hfill \\ & & ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma }{\int }_r^{\infty }\left[\frac{{\varphi }^{\frac{1}{p}}g\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{\frac{1+c-\gamma }{p}}}{{\varpi }^{\frac{c+\theta -p-\gamma +1}{p}}\left(\varsigma \right)}\right]{\left[\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{1+c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}\right]}^{\frac{p-1}{p}}{\nabla }_a^{\gamma }\varsigma \hfill \end{array}$$

thus,

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{\nabla }_a^{\gamma }\varsigma ⩽{\left[{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma }\right]}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{1+c-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

(v)

By using the integration by parts to the left-hand side of inequality (29), we have ${\nabla }_a^{\gamma }u\left(\varsigma \right)=\frac{\varphi \left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}$ and $v\left(\varsigma \right)={\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1},$ which becomes

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma =u\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right){|}_r^{\infty }+{\int }_r^{\infty }-u^{\rho }\left(\varsigma \right)\left[{\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)\right]{\nabla }_a^{\gamma }\varsigma \end{array}$$

(38)

where $u\left(\varsigma \right)=-{\int }_{\varsigma }^{\infty }\frac{\varphi \left(\varrho \right)}{{\left[{\varpi }^{\rho }\left(\varrho \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varrho$. By using $\mathsf{\Psi }\left(r\right)=0$ and $u(\infty )=0,$ one can have

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ={\int }_r^{\infty }-u^{\rho }\left(\varsigma \right)\left[{\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)\right]{\nabla }_a^{\gamma }\varsigma .$$

(39)

By following the procedure in the proof of $\left(i\right)$, we arrive at inequality (34) as

$$\begin{array}{c}\hfill {\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varsigma \right)\right]⩽-(c+\theta -\gamma ){\displaystyle \frac{\varphi \left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}}⩽-(c+\theta -\gamma ){\displaystyle \frac{\varphi \left(\varsigma \right)}{{\mathrm{\Lambda }}^{c+\theta -\gamma +1}{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}}\end{array}$$

which implies ${\displaystyle \frac{\varphi \left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}}⩽-{\mathrm{\Lambda }}^{c+\theta -\gamma +1}{\displaystyle \frac{{\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varsigma \right)\right]}{c+\theta -\gamma }}$. Then, we obtain that

$$\begin{array}{ccc}\hfill -u^{\rho }\left(\varsigma \right)& =& {\int }_{\rho \left(\varsigma \right)}^{\infty }{\displaystyle \frac{\varphi \left(\varrho \right)}{{\left[{\varpi }^{\rho }\left(\varrho \right)\right]}^{c+\theta -\gamma +1}}}{\nabla }_a^{\gamma }\varrho ⩽{\int }_{\rho \left(\varsigma \right)}^{\infty }-{\mathrm{\Lambda }}^{c+\theta -\gamma +1}{\displaystyle \frac{{\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varrho \right)\right]}{(c+\theta -\gamma )}}{\nabla }_a^{\gamma }\varrho \hfill \\ \hfill & =& {\mathrm{\Lambda }}^{c+\theta -\gamma +1}{\displaystyle \frac{-{\varpi }^{\gamma -c-\theta }(\infty )+{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{\gamma -c-\theta }}{c+\theta -\gamma }}\hfill \\ & ⩽& {\mathrm{\Lambda }}^{c+\theta -\gamma +1}{\displaystyle \frac{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{\gamma -c-\theta }}{c+\theta -\gamma }}.\hfill \end{array}$$

(40)

Substituting (33) and (40) into (39) leads to

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma +1}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

which is the desired result (29).

(vi)

Since

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma }}{\nabla }_a^{\gamma }\varsigma ={\int }_r^{\infty }\frac{{\varphi }^{\frac{1}{p}}\left(\varsigma \right)g\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{\frac{c-\gamma +1}{p}}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{\frac{c+\theta -p-\gamma +1}{p}}}{\left[\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}\right]}^{\frac{p-1}{p}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

(41)

Applying the Hölder inequality with the constants p and $p/p-1$ to the right-hand side of the above equation leads to the inequality

$$\begin{array}{ccc}& & {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma \hfill \\ \hfill & & ⩽{\mathrm{\Lambda }}^{c+\theta -\gamma +1}\frac{(1+c+p-\gamma )}{(c+\theta -\gamma )}{\left({\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}{\nabla }_a^{\gamma }\varsigma \right)}^{\frac{1}{p}}\hfill \end{array}$$

(42)

$$\begin{array}{ccc}& & \times {\left({\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma \right)}^{\frac{p-1}{p}}\hfill \\ \hfill \end{array}$$

(43)

therefore,

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\left({\mathrm{\Lambda }}^{c+\theta -\gamma +1}\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}{\nabla }_a^{\gamma }\varsigma ,$$

which is the desired inequality (30).

□

Remark 1.

In Theorem 7, if we take $\gamma =1$, then we get Theorem 4.

Corollary 1.

In Theorem 7, if we take $\mathbb{T}=\mathbb{R}$ and $\mathrm{\Lambda }=1$, then we get

(i)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{(\varsigma -a)}^{\gamma -1}dt⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{(\varsigma -a)}^{\gamma -1}dt.\end{array}$$

(44)

(ii)

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{(\varsigma -a)}^{\gamma -1}dt⩽{\left(\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}{(\varsigma -a)}^{\gamma -1}dt.$$

(45)

where

$$\varpi \left(\varsigma \right)={\int }_r^{\varsigma }\varphi \left(\varrho \right){(\varrho -a)}^{1-\gamma }d\varrho ,\mathit{and}\mathsf{\Psi }\left(\varsigma \right)={\int }_r^{\varsigma }\varphi \left(\varrho \right)g\left(\varrho \right){(\varrho -a)}^{1-\gamma }d\varrho .$$

Remark 2.

In inequality (45), if we take $r=0,$ $\gamma =1,$ and $c=0$, we get inequality (9).

Remark 3.

In inequality (45), if we take $r=0,$ and $\gamma =1,$ we get inequality (11).

Corollary 2.

In Theorem 7, if we take $\mathbb{T}=h\mathbb{Z}$, then we get

(i)

$$\begin{array}{c}\hfill \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}\sum _{\varsigma =\frac{r}{h}}^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi (ht-h)\right]}^{c+\theta -\gamma }}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\end{array}$$

(ii)

$$\begin{array}{ccc}\hfill & & \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\hfill \\ & & ⩽{\left(\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p\sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}{\left[\varpi \left(\varsigma \right)\right]}^{(c+\theta -\gamma +1)(p-1)}}{{\left[\varpi (ht-h)\right]}^{c+\theta -\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\hfill \end{array}$$

(iii)

$$\begin{array}{ccc}& & \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p}\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta }}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\hfill \\ & & ⩽{\left(\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p{\mathrm{\Lambda }}^{(c+\theta -\gamma +1)(p-1)}\sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[\varpi (ht-h)\right]}^{(c+\theta -p-\gamma +1)}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\hfill \end{array}$$

(iv)

$$\begin{array}{ccc}\hfill & & \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\hfill \\ & & ⩽{\left[{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma }\right]}^p\sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{1+c-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\hfill \end{array}$$

(v)

$$\begin{array}{ccc}\hfill & & \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[\varpi (ht-h)\right]}^{c+\theta -\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\hfill \\ & & ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma +1}\sum _{\varsigma =\frac{r}{h}}^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi (ht-h)\right]}^{c+\theta -\gamma }}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\hfill \end{array}$$

(vi)

$$\begin{array}{ccc}& & \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi (ht-h)\right]}^{c+\theta -\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\hfill \\ & & ⩽{\left({\mathrm{\Lambda }}^{c+\theta -\gamma +1}\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p\sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[\varpi (ht-h)\right]}^{c+\theta -p-\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\hfill \end{array}$$

where

$$\varpi \left(\varsigma \right)=h\sum _{\varsigma =\frac{r}{h}}^{\frac{\varsigma }{h}-1}\varphi \left(\varrho \right){({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)},\mathit{and}\mathsf{\Psi }\left(\varsigma \right)=h\sum _{\varsigma =\frac{r}{h}}^{\frac{\varsigma }{h}-1}\varphi \left(\varrho \right)g\left(\varrho \right){({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.$$

Theorem 8.

Assume that $\mathbb{T}$ is a time scale with $0\le r\in \mathbb{T}$ and $\varsigma ⩾a$. Moreover, suppose that g and ϕ are non-negative $(\gamma ,a)$-nabla fractional differentiable and locally integrable functions on ${[r,\infty )}_{\mathbb{T}}$. Set $\varpi \left(\varsigma \right)={\int }_{\varsigma }^{\infty }\varphi \left(\varrho \right){\nabla }_a^{\gamma }\varrho$ and $\mathsf{\Psi }\left(\varsigma \right)={\int }_{\varsigma }^{\infty }\varphi \left(\varrho \right)g\left(\varrho \right){\nabla }_a^{\gamma }\varrho ,$ and assume that $\mathsf{\Psi }\left(r\right)<\infty$ and ${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)}{{\varpi }^{\rho }\left(\varsigma \right)}{\nabla }_a^{\gamma }\varsigma <\infty$. Suppose that there exists $\mathrm{\Lambda }>0$, such that $\frac{{\varpi }^{\rho }\left(\varsigma \right)}{\varpi \left(\varsigma \right)}⩽\mathrm{\Lambda }$ for $\varsigma \in {[r,\infty )}_{\mathbb{T}}$. If $c⩾0,$ $p>1,$ $c+\theta >1$, and $0<\gamma ⩽1$ are real constants, then

(i)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

(46)

(ii)

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{1+c+p-\gamma }}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\left(\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c-\gamma +1}{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{(c+\theta -\gamma +1)(p-1)}}{{\left[\varpi \left(\varsigma \right)\right]}^{(c+\theta -\gamma )p}}{\nabla }_a^{\gamma }\varsigma .$$

(47)

(iii)

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{1+c+p-\gamma }}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\left(\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p{\mathrm{\Lambda }}^{(c+\theta -\gamma +1)(p-1)}{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}{\nabla }_a^{\gamma }\varsigma .$$

(48)

(iv)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\left[{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma }\right]}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{1+c-\gamma }}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

(49)

(v)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma +1}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

(50)

(vi)

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{1+c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\left({\mathrm{\Lambda }}^{c+\theta -\gamma +1}\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}{\nabla }_a^{\gamma }\varsigma .$$

(51)

Proof. 

(i)

By using the integration by parts to the left-hand side of inequality (46), we have ${\nabla }_a^{\gamma }u\left(\varsigma \right)=\frac{\varphi \left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}$ and $v\left(\varsigma \right)={\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1};$ it becomes

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma =u\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right){|}_r^{\infty }+{\int }_r^{\infty }-u\left(\varsigma \right)\left[{\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)\right]{\nabla }_a^{\gamma }\varsigma \end{array}$$

(52)

where $u\left(\varsigma \right)={\int }_r^{\varsigma }\frac{\varphi \left(\varrho \right)}{{\left[{\varpi }^{\rho }\left(\varrho \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varrho$. By using $\mathsf{\Psi }(\infty )=0$ and $u\left(r\right)=0,$ one can have

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ={\int }_r^{\infty }-u\left(\varsigma \right)\left[{\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)\right]{\nabla }_a^{\gamma }\varsigma .$$

(53)

By chain rule, and by using ${\nabla }_a^{\gamma }\mathsf{\Psi }\left(\varsigma \right)=-\varphi \left(\varsigma \right)g\left(\varsigma \right)⩽0,$ we obtain

$$\begin{array}{ccc}\hfill -{\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)& =& -(c+p-\gamma +1){\mathsf{\Psi }}^{c+p-\gamma }\left(d\right){\nabla }_a^{\gamma }\mathsf{\Psi }\left(\varsigma \right)=(c+p-\gamma +1)\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(d\right)\hfill \\ & ⩽& (c+p-\gamma +1)\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma }.\hfill \end{array}$$

(54)

Since ${\nabla }_a^{\gamma }\varpi \left(\varsigma \right)=-\varphi \left(\varsigma \right)⩽0$ and $c+\theta >1,$ by chain rule, we get

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varsigma \right)\right]& =& {\int }_0^1{\displaystyle \frac{(\gamma -c-\theta ){\nabla }_a^{\gamma }\varpi \left(\varsigma \right)}{{[(1-h)\varpi \left(\varsigma \right)+h{\varpi }^{\rho }\left(\varsigma \right)]}^{c+\theta -\gamma +1}}}dh={\int }_0^1{\displaystyle \frac{(c+\theta -\gamma )\varphi \left(\varsigma \right)}{{[(1-h)\varpi \left(\varsigma \right)+h{\varpi }^{\rho }\left(\varsigma \right)]}^{c+\theta -\gamma +1}}}dh\hfill \\ \hfill & ⩾& (c+\theta -\gamma ){\int }_0^1{\displaystyle \frac{\varphi \left(\varsigma \right)}{{[(1-h){\varpi }^{\rho }\left(\varsigma \right)+h{\varpi }^{\rho }\left(\varsigma \right)]}^{c+\theta -\gamma +1}}}dh=(c+\theta -\gamma ){\displaystyle \frac{\varphi \left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}}\hfill \\ \hfill {\displaystyle \frac{\varphi \left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}}& ⩽& {\displaystyle \frac{{\nabla }_a^{\gamma }{\left[\varpi \left(\varsigma \right)\right]}^{\gamma -c-\theta }}{c+\theta -\gamma }}.\hfill \end{array}$$

(55)

Therefore,

$$\begin{array}{ccc}\hfill u\left(\varsigma \right)& =& {\int }_r^{\varsigma }{\displaystyle \frac{\varphi \left(\varrho \right)}{{\left[{\varpi }^{\rho }\left(\varrho \right)\right]}^{c+\theta -\gamma +1}}}{\nabla }_a^{\gamma }\varrho ⩽{\int }_r^{\varsigma }{\displaystyle \frac{{\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varrho \right)\right]}{(c+\theta -\gamma )}}{\nabla }_a^{\gamma }\varrho ={\displaystyle \frac{-{\varpi }^{\gamma -c-\theta }\left(r\right)+{\left[\varpi \left(\varsigma \right)\right]}^{\gamma -c-\theta }}{c+\theta -\gamma }}\hfill \\ & ⩽& {\displaystyle \frac{{\left[\varpi \left(\varsigma \right)\right]}^{\gamma -c-\theta }}{c+\theta -\gamma }}.\hfill \end{array}$$

(56)

Substituting (54) and (56) into (53) leads to

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma ,\end{array}$$

which is the desired result (46).

(ii)

Since

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}{\nabla }_a^{\gamma }\varsigma ={\int }_r^{\infty }\frac{{\varphi }^{\frac{1}{p}}\left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{\frac{c-\gamma +1}{p}}{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{\frac{(c+\theta -\gamma +1)(p-1)}{p}}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}{\left[\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{1+c+p-\gamma }}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}\right]}^{\frac{p-1}{p}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

(57)

Applying the Hölder inequality with the constants p and $p/p-1$ to the right-hand side of the above equation leads to inequality

$$\begin{array}{ccc}& & {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{1+c+p-\gamma }}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma \hfill \\ \hfill & & ⩽\frac{(1+c+p-\gamma )}{(c+\theta -\gamma )}{\left({\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{(c+\theta -\gamma +1)(p-1)}}{{\left[\varpi \left(\varsigma \right)\right]}^{(c+\theta -\gamma )p}}{\nabla }_a^{\gamma }\varsigma \right)}^{\frac{1}{p}}\hfill \end{array}$$

(58)

$$\begin{array}{ccc}& & \times {\left({\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{1+c+p-\gamma }}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma \right)}^{\frac{p-1}{p}}\hfill \\ \hfill \end{array}$$

(59)

therefore,

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{1+c+p-\gamma }}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\left(\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c-\gamma +1}{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{(c+\theta -\gamma +1)(p-1)}}{{\left[\varpi \left(\varsigma \right)\right]}^{(c+\theta -\gamma )p}}{\nabla }_a^{\gamma }\varsigma ,$$

which is the desired inequality (47).

(iii)

To obtain inequality (48), we use

$$\frac{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{(c+\theta -\gamma +1)(p-1)}}{{\left[\varpi \left(\varsigma \right)\right]}^{(c+\theta -\gamma )p}}={\left(\frac{{\varpi }^{\rho }\left(\varsigma \right)}{\varpi \left(\varsigma \right)}\right)}^{(c+\theta -\gamma +1)(p-1)}\frac{{\left[\varpi \left(\varsigma \right)\right]}^{-c-\theta }}{{\left[\varpi \left(\varsigma \right)\right]}^{-p-\gamma +1}}=\frac{{\mathrm{\Lambda }}^{(c+\theta -\gamma +1)(p-1)}}{{\left[\varpi \left(\varsigma \right)\right]}^{(c+\theta -p-\gamma +1)}}$$

in inequality (47). Then, the desired inequality (48) can be proven directly.

(iv)

In order to obtain inequality (49), we use inequality (46) and the constant $\mathrm{\Lambda }$ as follows.

$$\begin{array}{ccc}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma & ⩽& {\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma \hfill \\ \hfill & ⩽& {\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma }{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma }}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma .\hfill \end{array}$$

Now, by using the Hölder inequality with the constants $p,$ $p/p-1,$ we get that

$$\begin{array}{ccc}\hfill & & {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma \hfill \\ & & ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma }{\int }_r^{\infty }\left[\frac{{\varphi }^{\frac{1}{p}}g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{\frac{1+c-\gamma }{p}}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{\frac{c+\theta -p-\gamma +1}{p}}}\right]{\left[\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{1+c+p-\gamma }}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}\right]}^{\frac{p-1}{p}}{\nabla }_a^{\gamma }\varsigma \hfill \end{array}$$

thus,

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\left[{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma }\right]}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{1+c-\gamma }}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

(v)

By using the integration by parts to the left-hand side of inequality (50), we have ${\nabla }_a^{\gamma }u\left(\varsigma \right)=\frac{\varphi \left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}$ and $v\left(\varsigma \right)={\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1};$ it becomes

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma =u\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right){|}_r^{\infty }+{\int }_r^{\infty }-u\left(\varsigma \right)\left[{\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)\right]{\nabla }_a^{\gamma }\varsigma \end{array}$$

(60)

where $u\left(\varsigma \right)={\int }_r^{\varsigma }\frac{\varphi \left(\varrho \right)}{{\left[\varpi \left(\varrho \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varrho$. By using $\mathsf{\Psi }(\infty )=0$ and $u\left(r\right)=0,$ one can have

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ={\int }_r^{\infty }-u\left(\varsigma \right)\left[{\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)\right]{\nabla }_a^{\gamma }\varsigma .$$

(61)

By following the procedure in the proof of $\left(i\right)$, we arrive at inequality (55) as

$$\begin{array}{c}\hfill {\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varsigma \right)\right]⩾(c+\theta -\gamma ){\displaystyle \frac{\varphi \left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}}⩾(c+\theta -\gamma ){\displaystyle \frac{\varphi \left(\varsigma \right)}{{\mathrm{\Lambda }}^{c+\theta -\gamma +1}{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}}\end{array}$$

which implies ${\displaystyle \frac{\varphi \left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}}⩽{\mathrm{\Lambda }}^{c+\theta -\gamma +1}{\displaystyle \frac{{\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varsigma \right)\right]}{c+\theta -\gamma }}$. Then, we obtain that

$$\begin{array}{ccc}\hfill u\left(\varsigma \right)& =& {\int }_r^{\varsigma }{\displaystyle \frac{\varphi \left(\varrho \right)}{{\left[\varpi \left(\varrho \right)\right]}^{c+\theta -\gamma +1}}}{\nabla }_a^{\gamma }\varrho ⩽{\int }_r^{\varsigma }{\mathrm{\Lambda }}^{c+\theta -\gamma +1}{\displaystyle \frac{{\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varrho \right)\right]}{(c+\theta -\gamma )}}{\nabla }_a^{\gamma }\varrho \hfill \\ \hfill & =& {\mathrm{\Lambda }}^{c+\theta -\gamma +1}{\displaystyle \frac{{\varpi }^{\gamma -c-\theta }\left(r\right)+{\left[\varpi \left(\varsigma \right)\right]}^{\gamma -c-\theta }}{c+\theta -\gamma }}\hfill \\ & ⩽& {\mathrm{\Lambda }}^{c+\theta -\gamma +1}{\displaystyle \frac{{\left[\varpi \left(\varsigma \right)\right]}^{\gamma -c-\theta }}{c+\theta -\gamma }}.\hfill \end{array}$$

(62)

Substituting (54) and (62) into (61) leads to

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma +1}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

which is the desired result (50).

(vi)

Since,

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}{\nabla }_a^{\gamma }\varsigma ={\int }_r^{\infty }\frac{{\varphi }^{\frac{1}{p}}\left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{\frac{c-\gamma +1}{p}}}{{\left[\varpi \left(\varsigma \right)\right]}^{\frac{c+\theta -p-\gamma +1}{p}}}{\left[\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{1+c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}\right]}^{\frac{p-1}{p}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

(63)

Applying the Hölder inequality with the constants p and $p/p-1$ to the right-hand side of the above equation leads to the inequality

$$\begin{array}{ccc}& & {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{1+c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma \hfill \\ \hfill & & ⩽{\mathrm{\Lambda }}^{c+\theta -\gamma +1}\frac{(1+c+p-\gamma )}{(c+\theta -\gamma )}{\left({\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}{\nabla }_a^{\gamma }\varsigma \right)}^{\frac{1}{p}}\hfill \end{array}$$

(64)

$$\begin{array}{ccc}& & \times {\left({\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{1+c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma \right)}^{\frac{p-1}{p}}\hfill \end{array}$$

(65)

therefore,

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{1+c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\left({\mathrm{\Lambda }}^{c+\theta -\gamma +1}\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}{\nabla }_a^{\gamma }\varsigma ,$$

which is the desired inequality (51).

□

Corollary 3.

In Theorem 8, if we take $\gamma =1$, then we get

(i)

${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta }}\nabla \varsigma ⩽{\displaystyle \frac{c+p}{c+\theta -1}}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -1}}}\nabla \varsigma .$

(ii)

${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta }}\nabla \varsigma ⩽{\left(\frac{c+p}{c+\theta -1}\right)}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^c{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{(c+\theta )(p-1)}}{{\left[\varpi \left(\varsigma \right)\right]}^{(c+\theta -1)p}}\nabla \varsigma .$

(iii)

${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta }}\nabla \varsigma ⩽{\left(\frac{c+p}{c+\theta -1}\right)}^p{\mathrm{\Lambda }}^{(c+\theta )(p-1)}{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^c}{{\left[\varpi \left(\varsigma \right)\right]}^{(c+\theta -p)}}\nabla \varsigma .$

(iv)

${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta }}\nabla \varsigma ⩽{\left[{\displaystyle \frac{c+p}{c+\theta -1}}{\mathrm{\Lambda }}^{c+\theta }\right]}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right)\left[{\mathsf{\Psi }}^c\left(\varsigma \right)\right]}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -p}}\nabla \varsigma .$

(v)

${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta }}\nabla \varsigma ⩽{\displaystyle \frac{c+p}{c+\theta -1}}{\mathrm{\Lambda }}^{c+\theta }{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -1}}}\nabla \varsigma .$

(vi)

${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{c+p}\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta }}\nabla \varsigma ⩽{\left({\mathrm{\Lambda }}^{c+\theta }\frac{c+p}{c+\theta -1}\right)}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^c}{\left[{\varpi }^{(}\varsigma \right){]}^{c+\theta -p}}\nabla \varsigma .$

where

$$\varpi \left(\varsigma \right)={\int }_{\varsigma }^{\infty }\varphi \left(\varrho \right)\nabla \varrho ,\mathit{and}\mathsf{\Psi }\left(\varsigma \right)={\int }_{\varsigma }^{\infty }\varphi \left(\varrho \right)g\left(\varrho \right)\nabla \varrho .$$

which is Theorem 3.8 in [15].

Corollary 4.

In Theorem 8, if we take $\mathbb{T}=\mathbb{R}$ and $\mathrm{\Lambda }=1$, then we get

(i)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{(\varsigma -a)}^{\gamma -1}dt⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{(\varsigma -a)}^{\gamma -1}dt.\end{array}$$

(ii)

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\mathsf{\Psi }}^{1+c+p-\gamma }\left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{(\varsigma -a)}^{\gamma -1}dt⩽{\left(\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p{\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}{(\varsigma -a)}^{\gamma -1}dt.$$

(66)

where

$$\varpi \left(\varsigma \right)={\int }_r^{\varsigma }\varphi \left(\varrho \right){(\varrho -a)}^{1-\gamma }d\varrho ,\mathit{and}\mathsf{\Psi }\left(\varsigma \right)={\int }_r^{\varsigma }\varphi \left(\varrho \right)g\left(\varrho \right){(\varrho -a)}^{1-\gamma }d\varrho .$$

Corollary 5.

In Theorem 8, if we take $\mathbb{T}=h\mathbb{Z}$, then we get

(i)

$$\begin{array}{ccc}\hfill & & \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }(ht-h)\right]}^{c+p-\gamma +1}}{{\left[\varpi (ht-h)\right]}^{c+\theta -\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\hfill \\ & & ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}\sum _{\varsigma =\frac{r}{h}}^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[\mathsf{\Psi }(ht-h)\right]}^{c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\hfill \end{array}$$

(ii)

$$\begin{array}{ccc}& & \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }(ht-h)\right]}^{1+c+p-\gamma }}{{\left[\varpi (ht-h)\right]}^{c+\theta -\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\hfill \\ & & ⩽{\left(\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p\sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }(ht-h)\right]}^{c-\gamma +1}{\left[\varpi (ht-h)\right]}^{(c+\theta -\gamma +1)(p-1)}}{{\left[\varpi \left(\varsigma \right)\right]}^{(c+\theta -\gamma )p}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\hfill \end{array}$$

(iii)

$$\begin{array}{ccc}& & \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }(ht-h)\right]}^{1+c+p-\gamma }}{{\left[\varpi (ht-h)\right]}^{c+\theta -\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\hfill \\ & & ⩽{\left(\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p{\mathrm{\Lambda }}^{(c+\theta -\gamma +1)(p-1)}\sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }(ht-h)\right]}^{c-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{(c+\theta -p-\gamma +1)}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\hfill \end{array}$$

(iv)

$$\begin{array}{ccc}\hfill & & \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }(ht-h)\right]}^{c+p-\gamma +1}}{{\left[\varpi (ht-h)\right]}^{c+\theta -\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\hfill \\ & & ⩽{\left[{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma }\right]}^p\sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{1+c-\gamma }}{{\left[\varpi (ht-h)\right]}^{c+\theta -p-\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\hfill \end{array}$$

(v)

$$\begin{array}{ccc}\hfill & & \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }(ht-h)\right]}^{c+p-\gamma +1}}{{\left[\varpi (ht-h)\right]}^{c+\theta -\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\hfill \\ & & ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\mathrm{\Lambda }}^{c+\theta -\gamma +1}\sum _{\varsigma =\frac{r}{h}}^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[\mathsf{\Psi }(ht-h)\right]}^{c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\hfill \end{array}$$

(vi)

$$\begin{array}{ccc}& & \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }(ht-h)\right]}^{1+c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\hfill \\ & & ⩽{\left({\mathrm{\Lambda }}^{c+\theta -\gamma +1}\frac{1+c+p-\gamma }{c+\theta -\gamma }\right)}^p\sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }(ht-h)\right]}^{c-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\hfill \end{array}$$

where

$$\varpi \left(\varsigma \right)=h\sum _{\varrho =\frac{\varsigma }{h}}^{\infty }\varphi \left(\varrho \right){({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)},\mathit{and}\mathsf{\Psi }\left(\varsigma \right)=h\sum _{\varrho =\frac{\varsigma }{h}}^{\infty }\varphi \left(\varrho \right)g\left(\varrho \right){({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.$$

Theorem 9.

Assume that $\mathbb{T}$ is a time scale with $0\le r\in \mathbb{T}$ and $\varsigma ⩾a$. Moreover, suppose that g and ϕ are non-negative $(\gamma ,a)$-nabla fractional differentiable and locally integrable functions on ${[r,\infty )}_{\mathbb{T}}$. Set $\varpi \left(\varsigma \right)={\int }_r^{\varsigma }\varphi \left(\varrho \right){\nabla }_a^{\gamma }\varrho$ and $\mathsf{\Psi }\left(\varsigma \right)={\int }_{\varsigma }^{\infty }\varphi \left(\varrho \right)g\left(\varrho \right){\nabla }_a^{\gamma }\varrho ,$ and assume that $\mathsf{\Psi }\left(r\right)<\infty$ and ${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma <\infty$. If $c⩾0,$ $p>1,$ $0<c+\theta <\gamma$ and $0<\gamma ⩽1$ are real constants, then

(i)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{\nabla }_a^{\gamma }\varsigma ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

(67)

(ii)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{\nabla }_a^{\gamma }\varsigma ⩽{\left[{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}\right]}^p{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

(68)

Proof. 

(i)

By using the integration by parts to the left-hand side of inequality (67), we have ${\nabla }_a^{\gamma }u\left(\varsigma \right)=\frac{\varphi \left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}$ and $v\left(\varsigma \right)={\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1},$ and it becomes

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{\nabla }_a^{\gamma }\varsigma =u\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right){|}_r^{\infty }+{\int }_r^{\infty }-u\left(\varsigma \right)\left[{\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)\right]{\nabla }_a^{\gamma }\varsigma \end{array}$$

(69)

where $u\left(\varsigma \right)={\int }_r^{\varsigma }\frac{\varphi \left(\varrho \right)}{{\left[\varpi \left(\varrho \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varrho$. By using $\mathsf{\Psi }(\infty )=0$ and $u\left(r\right)=0,$ one can have

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{\nabla }_a^{\gamma }\varsigma ={\int }_r^{\infty }-u\left(\varsigma \right)\left[{\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)\right]{\nabla }_a^{\gamma }\varsigma .$$

(70)

By chain rule, and by using ${\nabla }_a^{\gamma }\mathsf{\Psi }\left(\varsigma \right)=-\varphi \left(\varsigma \right)g\left(\varsigma \right)⩽0,$ we obtain

$$\begin{array}{ccc}\hfill -{\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)& =& (c+p-\gamma +1){\mathsf{\Psi }}^{c+p-\gamma }\left(d\right){\nabla }_a^{\gamma }\mathsf{\Psi }\left(\varsigma \right)=(c+p-\gamma +1)\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(d\right)\hfill \\ & ⩽& (c+p-\gamma +1)\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma }.\hfill \end{array}$$

(71)

Since ${\nabla }_a^{\gamma }\varpi \left(\varsigma \right)=\varphi \left(\varsigma \right)⩾0$ and $0<c+\theta <\gamma ,$ by chain rule, we get

$$\begin{array}{ccc}\hfill & & {\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varsigma \right)\right]\hfill \\ & & ={\int }_0^1{\displaystyle \frac{(\gamma -c-\theta ){\nabla }_a^{\gamma }\varpi \left(\varsigma \right)}{{[(1-h)\varpi \left(\varsigma \right)+h{\varpi }^{\rho }\left(\varsigma \right)]}^{c+\theta -\gamma +1}}}dh={\int }_0^1{\displaystyle \frac{(\gamma -c-\theta )\varphi \left(\varsigma \right)}{{[(1-h)\varpi \left(\varsigma \right)+h{\varpi }^{\rho }\left(\varsigma \right)]}^{c+\theta -\gamma +1}}}dh\hfill \end{array}$$

(72)

$$\begin{array}{ccc}& & ⩾(\gamma -c-\theta ){\int }_0^1{\displaystyle \frac{\varphi \left(\varsigma \right)}{{[(1-h)\varpi \left(\varsigma \right)+h\varpi \left(\varsigma \right)]}^{c+\theta -\gamma +1}}}dh=(\gamma -c-\theta ){\displaystyle \frac{\varphi \left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\varphi \left(\varsigma \right)}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}}& ⩽& {\displaystyle \frac{{\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varsigma \right)\right]}{(\gamma -c-\theta )}}.\hfill \end{array}$$

(73)

Therefore,

$$\begin{array}{ccc}\hfill u\left(\varsigma \right)& =& {\int }_r^{\varsigma }{\displaystyle \frac{\varphi \left(\varrho \right)}{{\left[\varpi \left(\varrho \right)\right]}^{c+\theta -\gamma +1}}}{\nabla }_a^{\gamma }\varrho ⩽{\int }_r^{\varsigma }{\displaystyle \frac{{\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varrho \right)\right]}{\gamma -c-\theta }}{\nabla }_a^{\gamma }\varrho ={\displaystyle \frac{-{\varpi }^{\gamma -c-\theta }\left(r\right)+{\left[\varpi \left(\varsigma \right)\right]}^{\gamma -c-\theta }}{\gamma -c-\theta }}\hfill \\ & ⩽& {\displaystyle \frac{{\left[\varpi \left(\varsigma \right)\right]}^{\gamma -c-\theta }}{\gamma -c-\theta }}.\hfill \end{array}$$

(74)

Substituting (71) and (74) into (70) leads to

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{\nabla }_a^{\gamma }\varsigma ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma ,\end{array}$$

which is the desired result (67).

(ii)

Employing the same procedure of the proof of $\left(ii\right)$ of Theorem 7, we obtain the desired result (68).

□

Corollary 6.

In Theorem 9, if we take $\gamma =1$, we get that

(i)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p}}{{\varpi }^{c+\theta }\left(\varsigma \right)}\nabla \varsigma ⩽{\displaystyle \frac{c+p}{c+\theta -1}}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p-1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -1}}}\nabla \varsigma .\end{array}$$

(ii)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^{c+p}}{{\varpi }^{c+\theta }\left(\varsigma \right)}\nabla \varsigma ⩽{\left[{\displaystyle \frac{c+p}{c+\theta -1}}\right]}^p{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[{\mathsf{\Psi }}^{\rho }\left(\varsigma \right)\right]}^c}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p}}}\nabla \varsigma .\end{array}$$

where

$$\varpi \left(\varsigma \right)={\int }_r^{\varsigma }\varphi \left(\varrho \right)\nabla \varrho \mathit{and}\mathsf{\Psi }\left(\varsigma \right)={\int }_{\varsigma }^{\infty }\varphi \left(\varrho \right)g\left(\varrho \right)\nabla \varrho$$

which is Theorem 3.13 in [15].

Corollary 7.

In Theorem 9, if we take $\mathbb{T}=\mathbb{R}$, we get that

(i)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{(\varsigma -a)}^{\gamma -1}dt⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{(\varsigma -a)}^{\gamma -1}dt.\end{array}$$

(ii)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{(\varsigma -a)}^{\gamma -1}dt⩽{\left[{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}\right]}^p{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}}{(\varsigma -a)}^{\gamma -1}dt.\end{array}$$

where

$$\varpi \left(\varsigma \right)={\int }_r^{\varsigma }\varphi \left(\varrho \right){(\varrho -a)}^{\gamma -1}dt\mathit{and}\mathsf{\Psi }\left(\varsigma \right)={\int }_{\varsigma }^{\infty }\varphi \left(\varrho \right)g\left(\varrho \right){(\varrho -a)}^{\gamma -1}d\varrho$$

Corollary 8.

In Theorem 9, if we take $\mathbb{T}=h\mathbb{Z}$, we get that

(i)

$$\begin{array}{ccc}\hfill & & \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }(ht-h)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\hfill \\ & & ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}\sum _{\varsigma =\frac{r}{h}}^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[\mathsf{\Psi }(ht-h)\right]}^{c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\hfill \end{array}$$

(ii)

$$\begin{array}{ccc}\hfill & & \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }(ht-h)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\hfill \\ & & ⩽{\left[{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}\right]}^p\sum _{\varsigma =\frac{r}{h}}^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }(ht-h)\right]}^{c-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\hfill \end{array}$$

where

$$\varpi \left(\varsigma \right)=h\sum _{\varrho =\frac{r}{h}}^{\frac{\varsigma }{h}-1}\varphi \left(\varrho \right){({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\mathit{and}\mathsf{\Psi }\left(\varsigma \right)=h\sum _{\varrho =\frac{r}{h}}^{\infty }\varphi \left(\varrho \right)g\left(\varrho \right){({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.$$

Theorem 10.

Assume that $\mathbb{T}$ is a time scale with $0\le r\in \mathbb{T}$ and $\varsigma ⩾a$. Moreover, suppose that g and ϕ are non-negative $(\gamma ,a)$-nabla fractional differentiable and locally integrable functions on ${[r,\infty )}_{\mathbb{T}}$. Set $\varpi \left(\varsigma \right)={\int }_{\varsigma }^{\infty }\varphi \left(\varrho \right){\nabla }_a^{\gamma }\varrho$ and $\mathsf{\Psi }\left(\varsigma \right)={\int }_r^{\varsigma }\varphi \left(\varrho \right)g\left(\varrho \right){\nabla }_a^{\gamma }\varrho ,$ and assume that $\mathsf{\Psi }(\infty )<\infty$ and ${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma <\infty$. If $c⩾0,$ $p>1,$ $0<c+\theta <\gamma$ and $0<\gamma ⩽1$ are real constants, then

(i)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma }}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

(75)

(ii)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\left({\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}\right)}^p{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

(76)

Proof. 

(i)

By using the integration by parts to the left-hand side of inequality (75), we have ${\nabla }_a^{\gamma }u\left(\varsigma \right)=\frac{\varphi \left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}$ and $v\left(\varsigma \right)={\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1};$ it becomes

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma =u\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right){|}_r^{\infty }+{\int }_r^{\infty }-u^{\rho }\left(\varsigma \right)\left[{\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)\right]{\nabla }_a^{\gamma }\varsigma \end{array}$$

(77)

where $u\left(\varsigma \right)=-{\int }_{\varsigma }^{\infty }\frac{\varphi \left(\varrho \right)}{{\left[{\varpi }^{\rho }\left(\varrho \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varrho$. By using $\mathsf{\Psi }\left(r\right)=0$ and $u(\infty )=0,$ one can have

$${\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ={\int }_r^{\infty }-u^{\rho }\left(\varsigma \right)\left[{\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)\right]{\nabla }_a^{\gamma }\varsigma .$$

(78)

By chain rule, and using ${\nabla }_a^{\gamma }\mathsf{\Psi }\left(\varsigma \right)=\varphi \left(\varsigma \right)g\left(\varsigma \right)⩾0,$ we obtain

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }{\mathsf{\Psi }}^{c+p-\gamma +1}\left(\varsigma \right)& =& (c+p-\gamma +1){\mathsf{\Psi }}^{c+p-\gamma }\left(d\right){\nabla }_a^{\gamma }\mathsf{\Psi }\left(\varsigma \right)=(c+p-\gamma +1)\varphi \left(\varsigma \right)g\left(\varsigma \right){\mathsf{\Psi }}^{c+p-\gamma }\left(d\right)\hfill \\ & ⩽& (c+p-\gamma +1)\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma }.\hfill \end{array}$$

(79)

Since ${\nabla }_a^{\gamma }\varpi \left(\varsigma \right)=-\varphi \left(\varsigma \right)⩽0$ and $0<c+\theta <\gamma ,$ by chain rule, we get

$$\begin{array}{ccc}\hfill {\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varsigma \right)\right]& =& {\int }_0^1{\displaystyle \frac{(\gamma -c-\theta ){\nabla }_a^{\gamma }\varpi \left(\varsigma \right)}{{[(1-h)\varpi \left(\varsigma \right)+h{\varpi }^{\rho }\left(\varsigma \right)]}^{c+\theta -\gamma +1}}}dh={\int }_0^1{\displaystyle \frac{(\gamma -c-\theta )\varphi \left(\varsigma \right)}{{[(1-h)\varpi \left(\varsigma \right)+h{\varpi }^{\rho }\left(\varsigma \right)]}^{c+\theta -\gamma +1}}}dh\hfill \\ \hfill & ⩽& -(\gamma -c-\theta ){\int }_0^1{\displaystyle \frac{\varphi \left(\varsigma \right)}{{[(1-h){\varpi }^{\rho }\left(\varsigma \right)+h{\varpi }^{\rho }\left(\varsigma \right)]}^{c+\theta -\gamma +1}}}dh=-(\gamma -c-\theta ){\displaystyle \frac{\varphi \left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}}\hfill \\ \hfill {\displaystyle \frac{\varphi \left(\varsigma \right)}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}}& ⩽& -{\displaystyle \frac{{\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varsigma \right)\right]}{(\gamma -c-\theta )}}.\hfill \end{array}$$

(80)

Therefore,

$$\begin{array}{ccc}\hfill -u^{\rho }\left(\varsigma \right)& =& {\int }_{\rho \left(\varsigma \right)}^{\infty }{\displaystyle \frac{\varphi \left(\varrho \right)}{{\left[{\varpi }^{\rho }\left(\varrho \right)\right]}^{c+\theta -\gamma +1}}}{\nabla }_a^{\gamma }\varrho ⩽{\int }_{\rho \left(\varsigma \right)}^{\infty }-{\displaystyle \frac{{\nabla }_a^{\gamma }\left[{\varpi }^{\gamma -c-\theta }\left(\varrho \right)\right]}{(\gamma -c-\theta )}}{\nabla }_a^{\gamma }\varrho ={\displaystyle \frac{-{\varpi }^{\gamma -c-\theta }(\infty )+{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{\gamma -c-\theta }}{\gamma -c-\theta }}\hfill \\ & ⩽& {\displaystyle \frac{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{\gamma -c-\theta }}{\gamma -c-\theta }}.\hfill \end{array}$$

(81)

Substituting (79) and (81) into (78) leads to

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{\nabla }_a^{\gamma }\varsigma ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma }}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{\nabla }_a^{\gamma }\varsigma .\end{array}$$

which is the desired result (75).

(ii)

Employing the same procedure of the proof of $\left(ii\right)$ of Theorem 7, we obtain the desired result (76).

□

Corollary 9.

In Theorem 10, if we take $\gamma =1$, we get that

(i)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta }}\nabla \varsigma ⩽{\displaystyle \frac{c+p}{c+\theta -1}}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-1}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -1}}}\nabla \varsigma .\end{array}$$

(ii)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p}}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta }}\nabla \varsigma ⩽{\left[{\displaystyle \frac{c+p}{c+\theta -1}}\right]}^p{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^c}{{\left[{\varpi }^{\rho }\left(\varsigma \right)\right]}^{c+\theta -p}}}\nabla \varsigma .\end{array}$$

where

$$\varpi \left(\varsigma \right)={\int }_r^{\varsigma }\varphi \left(\varrho \right)\nabla \varrho \mathit{and}\mathsf{\Psi }\left(\varsigma \right)={\int }_{\varsigma }^{\infty }\varphi \left(\varrho \right)g\left(\varrho \right)\nabla \varrho$$

which is Theorem 3.19 in [15].

Corollary 10.

In Theorem 10, if we take $\mathbb{T}=\mathbb{R}$, we get that

(i)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{(\varsigma -a)}^{\gamma -1}dt⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma }}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma }}}{(\varsigma -a)}^{\gamma -1}dt.\end{array}$$

(ii)

$$\begin{array}{c}\hfill {\int }_r^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\varpi }^{c+\theta -\gamma +1}\left(\varsigma \right)}{(\varsigma -a)}^{\gamma -1}dt⩽{\left[{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}\right]}^p{\int }_r^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -p-\gamma +1}}}{(\varsigma -a)}^{\gamma -1}dt.\end{array}$$

where

$$\varpi \left(\varsigma \right)={\int }_{\varsigma }^{\infty }\varphi \left(\varrho \right){(\varrho -a)}^{\gamma -1}dt\mathit{and}\mathsf{\Psi }\left(\varsigma \right)={\int }_r^{\varsigma }\varphi \left(\varrho \right)g\left(\varrho \right){(\varrho -a)}^{\gamma -1}d\varrho$$

Corollary 11.

In Theorem 10, if we take $\mathbb{T}=h\mathbb{Z}$, we get that

(i)

$$\begin{array}{ccc}\hfill & & \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[\varpi \left(\varsigma \right)\right]}^{c+\theta -\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\hfill \\ & & ⩽{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}\sum _{\varsigma =\frac{r}{h}}^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma }}{{\left[\varpi (ht-h)\right]}^{c+\theta -\gamma }}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\hfill \end{array}$$

(ii)

$$\begin{array}{ccc}\hfill & & \sum _{\varsigma =\frac{r}{h}}^{\infty }\frac{\varphi \left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c+p-\gamma +1}}{{\left[\varpi (ht-h)\right]}^{c+\theta -\gamma +1}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\hfill \\ & & ⩽{\left[{\displaystyle \frac{c+p-\gamma +1}{c+\theta -\gamma }}\right]}^p\sum _{\varsigma =\frac{r}{h}}^{\infty }{\displaystyle \frac{\varphi \left(\varsigma \right)g^p\left(\varsigma \right){\left[\mathsf{\Psi }\left(\varsigma \right)\right]}^{c-\gamma +1}}{{\left[\varpi (ht-h)\right]}^{c+\theta -p-\gamma +1}}}{({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.\hfill \end{array}$$

where

$$\varpi \left(\varsigma \right)=h\sum _{\varrho =\frac{\varsigma }{h}}^{\infty }\varphi \left(\varrho \right){({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}\mathit{and}\mathsf{\Psi }\left(\varsigma \right)=h\sum _{\varrho =\frac{\varsigma }{h}}^{\frac{\varsigma }{h}-1}\varphi \left(\varrho \right)g\left(\varrho \right){({\rho }^{\gamma -1}\left(\varsigma \right)-a)}_h^{(\gamma -1)}.$$

3. Discussion

In this work, firstly, we began with the Introduction section, which contains a brief recall of the necessary results of the Hardy inequalities and the notions of time-scale calculus. Then, the original results are then given and proven in Section 2, since in the first theorem we established seven new inequalities that may be considered as a generalization of Theorem 4. Additionally, we obtain Inequalities (9) and (11) as a special case of our main results. After that, in the second theorem, we proved six new inequalities that may be considered as a generalization of Theorem 3.8 in [15]. In the last theorem, we investigated two main results, which generalize Theorem 3.19 in [15] After each theorem, we discussed the cases of time scales $\mathbb{T}=\mathbb{R}$, $\mathbb{T}=\mathbb{Z}$, and $\mathbb{T}=h\mathbb{Z}$.

4. Conclusions

In this manuscript, by employing the $(\gamma ,a)$-nabla-conformable fractional calculus on the time scales of Rahmat et al. [16], several new Hardy-type inequalities were proven. The results extend several dynamic inequalities known in the literature, being new, even in the discrete and continuous settings.