Unified Generalizations of Hardy-Type Inequalities Through the Nabla Framework on Time Scales

Abstract

This research investigates innovative extensions of Hardy-type inequalities through the use of nabla Hölder’s and nabla Jensen’s inequalities, combined with the nabla chain rule and the characteristics of convex and submultiplicative functions. We extend these inequalities within a cohesive framework that integrates elements of both continuous and discrete calculus. Furthermore, our study revisits specific integral inequalities from the existing literature, showcasing the wide-ranging relevance of our results.

1. Introduction

Inequalities play an essential role in mathematical analysis and applications. Hardy’s inequality, introduced by G.H. Hardy in 1920, is a prominent example used extensively in functional analysis and differential equations. Hardy’s inequality in discrete form, as formulated in [1], is expressed as

$$\sum _{s=1}^{\infty }{\left({\displaystyle \frac{1}{s}}\sum _{k=1}^s\aleph \left(k\right)\right)}^q\le {\left({\displaystyle \frac{q}{q-1}}\right)}^q\sum _{s=1}^{\infty }{\aleph }^q\left(s\right),q>1,$$

(1)

where $\aleph \left(s\right)\ge 0$ for $s\ge 1$.

In 1925, Hardy extended this result to the integral form Theorem A [2], showing that if $\hslash \ge 0$ is integrable over $(0,\varsigma )$ for $\varsigma \in (0,\infty ),$ then

$${\int }_0^{\infty }{\left({\displaystyle \frac{1}{\varsigma }}{\int }_0^{\varsigma }\hslash \left(\vartheta \right)d\vartheta \right)}^{q}d\varsigma \le {\left({\displaystyle \frac{q}{q-1}}\right)}^q{\int }_0^{\infty }{\hslash }^q\left(\varsigma \right)d\varsigma ,q>1,$$

(2)

where ${\left(q/\left(q-1\right)\right)}^q$ is optimal in both discrete and integral cases. The earlier inequality continues to be applicable for variables a and b. That is, the inequality

$${\int }_a^b{\left({\displaystyle \frac{1}{\varsigma }}{\int }_0^{\varsigma }\hslash \left(\vartheta \right)d\vartheta \right)}^{q}d\varsigma \le {\left({\displaystyle \frac{q}{q-1}}\right)}^q{\int }_0^{\infty }{\hslash }^q\left(\varsigma \right)d\varsigma ,q>1.$$

(3)

is valid for $0<a<b<\infty$ (see [3]).

Since the introduction of these classical Hardy-type inequalities, many researchers have extended and generalized these results, as evidenced by works such as [4,5,6,7] and the books [8,9,10].

In [11], Hardy demonstrated that if $q>1$ and $\hslash \left(\vartheta \right)>0$ is a function that is integrable, then

$${\int }_0^{\infty }{\displaystyle \frac{1}{{\varsigma }^{\lambda }}}{\left({\int }_0^{\varsigma }\hslash \left(\vartheta \right)d\vartheta \right)}^{q}d\varsigma \le {\left({\displaystyle \frac{q}{\lambda -1}}\right)}^q{\int }_0^{\infty }{\displaystyle \frac{1}{{\varsigma }^{\lambda -q}}}{\hslash }^q\left(\varsigma \right)d\varsigma ,\mathrm{for}\lambda >1,$$

(4)

and

$${\int }_0^{\infty }{\displaystyle \frac{1}{{\varsigma }^{\lambda }}}{\left({\int }_{\varsigma }^{\infty }\hslash \left(\vartheta \right)d\vartheta \right)}^{q}d\varsigma \le {\left({\displaystyle \frac{q}{1-\lambda }}\right)}^q{\int }_0^{\infty }{\displaystyle \frac{1}{{\varsigma }^{\lambda -q}}}{\hslash }^q\left(\varsigma \right)d\varsigma ,\mathrm{for}\lambda <1.$$

(5)

Knopp’s contribution in 1929 established the following inequality involving an exponential function [12]:

$${\int }_0^{\infty }exp\left({\displaystyle \frac{1}{\varsigma }}{\int }_0^{\varsigma }ln\hslash \left(\eta \right)d\eta \right)d\varsigma \le e{\int }_0^{\infty }\hslash \left(\varsigma \right)d\varsigma ,$$

(6)

where $0<\hslash \in L^1\left({\mathbb{R}}_{+}\right)$ and e is optimal.

In [13], the authors extended (6) as follows:

$${\int }_0^{\infty }\Phi \left({\displaystyle \frac{1}{\varsigma }}{\int }_0^{\varsigma }\hslash \left(\eta \right)d\eta \right){\displaystyle \frac{d\varsigma }{\varsigma }}\le {\int }_0^{\infty }\Phi (\hslash \left(\varsigma \right)){\displaystyle \frac{d\varsigma }{\varsigma }}.$$

(7)

In [14], the authors extended (7) by incorporating two weight functions, showing the following:

$${\int }_0^{\ell }\Im \left(\varsigma \right)\Psi \left({\displaystyle \frac{1}{\varsigma }}{\int }_0^{\varsigma }\hslash \left(\eta \right)d\eta \right){\displaystyle \frac{d\varsigma }{\varsigma }}\le {\int }_0^{\ell }\vartheta \left(\varsigma \right)\Psi (\hslash \left(\varsigma \right)){\displaystyle \frac{d\varsigma }{\varsigma }},$$

(8)

where $0\le \Im :(0,\mathcal{l})\to \mathbb{R}$ s.t. $\varsigma \to \Im \left(\varsigma \right)/{\varsigma }^2$ is locally integrable on $(0,\mathcal{l})$, $\Psi$ is a convex function on $\left(\alpha ,\beta \right)$; $-\infty <\alpha \le \beta <\infty$, $\hslash :(0,\mathcal{l})\to \mathbb{R}$ s.t. $\hslash \left(\varsigma \right)\in \left(\alpha ,\beta \right)$ and

$$\vartheta \left(\varsigma \right)=\varsigma {\int }_{\varsigma }^{\mathcal{l}}{\displaystyle \frac{\Im \left(\varsigma \right)}{{\varsigma }^2}}d\varsigma ,\varsigma \in (0,\mathcal{l}).$$

In [15], Sulaiman established generalized inequalities of Hardy type and proved the following:

$${\int }_0^{\mathcal{l}}\Psi \left({\displaystyle \frac{{\int }_0^{\varsigma }\hslash \left(\eta \right)d\eta }{\varsigma }}\right)d\varsigma \le {\int }_0^{\mathcal{l}}\Psi \left(\hslash \left(\varsigma \right)\right)d\varsigma ,$$

(9)

for non-decreasing functions $\hslash ,\Psi \ge 0$ and $\mathcal{l}\in (0,\infty ].$ Additionally, for $q>1$ and $\hslash ,\Psi \ge 0$ such that $\Psi$ is convex, he demonstrated that

$${\int }_0^{\infty }{\Psi }^q\left({\displaystyle \frac{{\int }_0^{\varsigma }\hslash \left(\eta \right)d\eta }{\varsigma }}\right)d\varsigma \le {\left({\displaystyle \frac{q}{q-1}}\right)}^q{\int }_0^{\infty }{\Psi }^q\left(\hslash \left(\varsigma \right)\right)d\varsigma .$$

(10)

More recently, the theory of time scales has evolved into a thriving field of mathematics, introduced by Hilger in their doctoral thesis [16] with the goal of unifying discrete and continuous analyses [17]. This theory has been applied to various mathematical fields, including dynamic inequalities on time scales, which have gained significant interest. Among these are Hardy-type inequalities, which have seen substantial development. For instance, in [18], P. Řehák proposed a time scale formulation of Hardy’s inequality, stating that for a time scale $\mathbb{T},$ if $q>1,$ then

$${\int }_{\mathcal{l}}^{\infty }{\left({\displaystyle \frac{{\int }_{\mathcal{l}}^{\sigma \left(v\right)}\hslash \left(\eta \right)\Delta \eta }{\sigma \left(v\right)-\mathcal{l}}}\right)}^q\Delta v<{\left({\displaystyle \frac{q}{q-1}}\right)}^q{\int }_{\mathcal{l}}^{\infty }{\hslash }^q\left(v\right)\Delta v,\mathrm{for}v\in {[\mathcal{l},\infty )}_{\mathbb{T}},$$

(11)

unless $\hslash \equiv 0$. Additionally, if $\mu \left(v\right)/v\to 0$ as $v\to \infty ,$ then ${\left({\displaystyle \frac{q}{q-1}}\right)}^q$ is optimal. Notations used in the above inequality (11) and subsequent content are defined in Section 2.

In Theorem 7.1.3 [19], a Hardy-type inequality for a convex function on $\mathbb{T}$ was introduced:

$${\int }_b^{\mathcal{l}}\Im \left(v\right)\Psi \left({\displaystyle \frac{1}{\sigma \left(v\right)-b}}{\int }_b^{\sigma \left(v\right)}\hslash \left(\eta \right)\Delta \eta \right){\displaystyle \frac{\Delta v}{v-b}}\le {\int }_b^{\mathcal{l}}\vartheta \left(v\right)\Psi (\hslash \left(v\right)){\displaystyle \frac{\Delta v}{v-b}},$$

(12)

where $0\le \Im \in C_{rd}({[b,\mathcal{l})}_{\mathbb{T}},\mathbb{R})$, $\Psi :(ϵ,\epsilon )\to \mathbb{R}$ is continuous and convex, $\hslash \in C_{rd}({[b,\mathcal{l})}_{\mathbb{T}},\mathbb{R})$ is delta integrable with $\hslash \left(v\right)\in \left(ϵ,\epsilon \right)$ and

$$\vartheta \left(v\right)=(v-\beta ){\int }_v^{\mathcal{l}}{\displaystyle \frac{\Im \left(\eta \right)\Delta \eta }{(\eta -b)\left(\sigma \right(\eta )-b)}},\varsigma \in {[b,\mathcal{l})}_{\mathbb{T}}.$$

In [20], the authors introduced a time-scale form of (9) and (10) for delta calculus, respectively, as follows: for $a,\mathcal{l}\in \mathbb{T}$ and non-decreasing functions ℏ and $\Psi$ with $\hslash ,\Psi \ge 0$, then

$${\int }_a^{\mathcal{l}}\Psi \left({\displaystyle \frac{{\int }_a^{\sigma \left(\varsigma \right)}\hslash \left(\eta \right)\Delta \eta }{\sigma \left(\varsigma \right)-a}}\right)\Delta \varsigma \le {\int }_a^{\mathcal{l}}\Psi \left(\hslash \left(\varsigma \right)\right)\Delta \varsigma .$$

(13)

Additionally, if $\hslash ,\Psi \ge 0$ where $\Psi$ is convex, $q>1$ and assuming the existence of a positive constant $\mathcal{Q}$ satisfying $\sigma \left(\varsigma \right)-a\le \mathcal{Q}\left(\varsigma -a\right),$ then

$${\int }_a^{\infty }{\Psi }^p\left({\displaystyle \frac{{\int }_a^{\sigma \left(\varsigma \right)}\hslash \left(\eta \right)\Delta \eta }{\sigma \left(\varsigma \right)-a}}\right)\Delta \varsigma \le {\left({\displaystyle \frac{q}{q-1}}\right)}^q{\mathcal{Q}}^{{\displaystyle \frac{q-1}{q}}}{\int }_a^{\infty }{\Psi }^q\left(\hslash \left(\varsigma \right)\right)\Delta \varsigma .$$

(14)

Furthermore, within the same study [20], it was demonstrated that if $q>1$, $\hslash ,\Psi \ge 0$ with $\Psi$ being convex and submultiplicative and satisfying $\Psi \left(a\right)=0$, then

$${\int }_a^{\mathcal{l}}{\displaystyle \frac{{\left(\sigma \left(\varsigma \right)-a\right)}^{1-q}}{\Psi \left(\varsigma \right)\Psi \left(\sigma \left(\varsigma \right)-a\right)}}\Psi \left({\int }_a^{\sigma \left(\varsigma \right)}\hslash \left(\eta \right)\Delta \eta \right)\Delta \varsigma \le {\displaystyle \frac{1}{q-1}}{\int }_a^{\mathcal{l}}{\displaystyle \frac{\Psi \left(\hslash \left(\varsigma \right)\right)}{{\left(\varsigma -a\right)}^{q-1}\Psi \left(\varsigma \right)}}\Delta \varsigma .$$

(15)

In [21], the authors presented a nabla analogue of (13) and (14), respectively, as follows: for $a,\mathcal{l},\varsigma \in \mathbb{T}$ and non-decreasing functions ℏ and $\Psi$ with $\hslash ,\Psi \ge 0$, then

$${\int }_a^{\mathcal{l}}\Psi \left({\displaystyle \frac{{\int }_a^{\varsigma }\hslash \left(\eta \right)\nabla \eta }{\varsigma -a}}\right)\nabla \varsigma \le {\int }_a^{\mathcal{l}}\Psi \left(\hslash \left(\varsigma \right)\right)\nabla \varsigma .$$

(16)

Additionally, if $\hslash ,\Psi \ge 0$ where $\Psi$ is convex, $q>1$ and assuming the existence of a positive constant $\mathcal{Q}$ satisfying $\varsigma -a\le \mathcal{Q}\left(\rho \left(\varsigma \right)-a\right),$ then

$${\int }_a^{\infty }{\Psi }^p\left({\displaystyle \frac{{\int }_a^{\varsigma }\hslash \left(\eta \right)\Delta \eta }{\varsigma -a}}\right)\nabla \varsigma \le {\left({\displaystyle \frac{q}{q-1}}\right)}^q{\mathcal{Q}}^{{\displaystyle \frac{q-1}{q}}}{\int }_a^{\infty }{\Psi }^q\left(\hslash \left(\varsigma \right)\right)\nabla \varsigma .$$

(17)

Furthermore, within the same study [21], it was demonstrated that if $q>1$, $\hslash ,\Psi \ge 0$ with $\Psi$ being convex and submultiplicative and satisfying $\Psi \left(a\right)=0$, then

$${\int }_a^{\mathcal{l}}{\displaystyle \frac{{\left(\varsigma -a\right)}^{1-q}}{\Psi \left(\varsigma \right)\Psi \left(\varsigma -a\right)}}\Psi \left({\int }_a^{\varsigma }\hslash \left(\eta \right)\Delta \eta \right)\nabla \varsigma \le {\displaystyle \frac{1}{q-1}}{\int }_a^{\mathcal{l}}{\displaystyle \frac{\Psi \left(\hslash \left(\varsigma \right)\right)}{{\left(\rho \left(\varsigma \right)-a\right)}^{q-1}{\Psi }^{\rho }\left(\varsigma \right)}}\nabla \varsigma .$$

(18)

Recently, many authors have contributed to the development of Hardy-type inequalities on $\mathbb{T}$ (see, [22,23,24,25,26,27,28,29,30,31,32]). These developments highlight the significant growth in the research on dynamic inequalities within the time-scale framework, driven by their applications in various mathematical and applied contexts.

The objective of this document is to present novel extensions of the inequalities (16)–(18). These results will further extend existing inequalities and provide novel contributions within the framework of time-scale analysis, with a specific emphasis on convexity and its applications to Hardy-type inequalities.

The organization of this manuscript is outlined as follows: Section 1, presents a review of key ideas in time-scale calculus, introduces convex and submultiplicative functions, and highlights key properties relevant to our results. Section 2, presents our main findings, which, as special cases, reduce to the inequalities proved by Sulaiman [15] when $\mathbb{T}=\mathbb{R}$.

2. Preliminaries’ Overview

The content presented involves preliminaries on time scales and related calculus concepts, providing foundational definitions, theorems, and lemmas. Here is an overview:

A time scale $\mathbb{T}$, as described by Martin and Allan [33], is any non-empty closed subset of $\mathbb{R}$. The fundamental operators and functions include the following:

• Forward jump operator $\sigma$ is $\sigma \left(\varsigma \right):=inf\{\varrho \in \mathbb{T}:s>\varsigma \}.$
• Backward jump operator $\rho$ is $\rho \left(\varsigma \right):=sup\{\varrho \in \mathbb{T}:\varrho <\varsigma \}.$
• Backward graininess function v is $v\left(\varsigma \right):=\varsigma -\rho \left(\varsigma \right)\ge 0.$

For a function $\hslash :\mathbb{T}\to \mathbb{R}$, ${\hslash }^{\rho }\left(\varsigma \right)$ refers to $\hslash \left(\rho \right(\varsigma \left)\right).$ Points $\varsigma \in \mathbb{T}$ are classified based on their relationships with the following operators:

• Right-dense if $\sigma \left(\varsigma \right)=\varsigma ,$
• Left-dense if $\rho \left(\varsigma \right)=\varsigma ,$
• Right-scattered if $\sigma \left(\varsigma \right)>\varsigma ,$
• Left-scattered if $\rho \left(\varsigma \right)<\varsigma .$

The set ${\mathbb{T}}_k$ is given by

$${\mathbb{T}}_k=\left\{\begin{array}{c}\mathbb{T}\setminus [inf\mathbb{T},(\sigma (inf\mathbb{T})],\mathrm{if}inf\mathbb{T}<\infty ,\\ \hspace{1em} \mathbb{T},\mathrm{if}inf\mathbb{T}=-\infty .\end{array}\right.$$

In time-scale calculus, the interval ${\mathcal{L}}_{\mathbb{T}}$ is given by ${\mathcal{L}}_{\mathbb{T}}=\mathcal{L}\cap \mathbb{T}.$

2.1. ∇-Differentiability and Related Theorems

A function ℏ defined on $\mathbb{T}$ is considered ∇-differentiable at $\eta \in \mathbb{T}$ if, for every $\epsilon >0,$ there is a neighborhood W of $\eta$ such that for some $\beta ,$ the following condition is satisfied:

$$|\hslash (\rho \left(\eta \right))-\hslash (s)-\beta (\rho \left(\eta \right)-s\left)\right|\le \epsilon \left|\rho \right(\eta )-s|,s\in W.$$

In this case, we denote ${\hslash }^{\nabla }\left(\eta \right)=\beta$.

Theorems related to ∇-differentiation include the following:

• Sum Rule: If $\hslash ,\vartheta$ are ∇-differentiable at $s\in \mathbb{T}$, then

  $${(\hslash +\vartheta )}^{\nabla }\left(s\right)={\hslash }^{\nabla }\left(s\right)+{\vartheta }^{\nabla }\left(s\right).$$

  (19)
• Product Rule: The product $\hslash \vartheta$ is ∇-differentiable at s with

  $${(\hslash \vartheta )}^{\nabla }\left(s\right)={\hslash }^{\nabla }\left(s\right)\vartheta \left(s\right)+\hslash \left(\rho \left(s\right)\right){\vartheta }^{\nabla }\left(s\right)=\hslash \left(s\right){\vartheta }^{\nabla }\left(s\right)+{\hslash }^{\nabla }\left(s\right)\vartheta \left(\rho \left(s\right)\right).$$

  (20)
• Quotient Rule: If $\vartheta \left(s\right)\vartheta \left(\rho \right(s\left)\right)\ne 0$, then the quotient $\hslash /\vartheta$ is ∇-differentiable at s and

  $${\left({\displaystyle \frac{\hslash }{\vartheta }}\right)}^{\nabla }\left(s\right)={\displaystyle \frac{{\hslash }^{\nabla }\left(s\right)\vartheta \left(s\right)-\hslash \left(s\right){\vartheta }^{\nabla }\left(s\right)}{\vartheta \left(s\right)\vartheta \left(\rho \right(s\left)\right)}}.$$

  (21)
• Chain Rule: If $\vartheta :\mathbb{T}\to \mathbb{R}$ is continuous and ∇-differentiable, and $\hslash :\mathbb{R}\to \mathbb{R}$ is continuously differentiable, then

  $${\left(\hslash \circ \vartheta \right)}^{\nabla }\left(s\right)={\hslash }^{\prime }\left(\vartheta \left(c\right)\right){\vartheta }^{\nabla }\left(s\right);c\in [\rho \left(s\right),s].$$

  (22)

2.2. One-Dimensional Continuity and Integration on Time Scales

• $ld$–continuous: A function ℏ is $ld$–continuous if it is continuous at left-dense points and its right-sided limits exist at right-dense points. The set of all $ld$–continuous functions is denoted by ${\mathrm{C}}_{ld}(\mathbb{T},\mathbb{R})$, while the set of all continuous functions is denoted by $\mathrm{C}(\mathbb{T},\mathbb{R})$.
• ∇ integral: If $\mathcal{H}:\mathbb{T}\to \mathbb{R}$ is a ∇-antiderivative of $\hslash :{\mathbb{T}}_k\to \mathbb{R}$, then

  $${\mathcal{H}}^{\nabla }\left(\eta \right)=\hslash \left(\eta \right),\forall \eta \in {\mathbb{T}}_k.$$

Hence, the nabla integral of ℏ is defined by

$${\int }_{\mathcal{l}}^{\tau }\hslash \left(s\right)\nabla s=\mathcal{H}\left(\mathcal{l}\right)-\mathcal{H}\left(\tau \right),\forall \mathcal{l},\tau \in \mathbb{T}.$$

Lemma 1.

If $s_0,s\in \mathbb{T}$ and ${\hslash }^{\nabla }\left(s\right)$ is continuous, then

$${\left({\int }_{s_0}^s\hslash \left(\tau \right)\nabla \tau \right)}^{\nabla }=\hslash \left(s\right).$$

Theorems that establish the basic properties of integration on time scales include the following:

• If $b,\mathcal{l},\varsigma \in \mathbb{T}$, $\alpha ,\beta \in \mathbb{R}$ and $\hslash ,\vartheta \in {\mathrm{C}}_{ld}({[b,\mathcal{l}]}_{\mathbb{T}},\mathbb{R})$, then

  $${\int }_b^{\mathcal{l}}\left[\alpha \hslash \left(s\right)+\beta \vartheta \left(s\right)\right]\nabla s=\alpha {\int }_b^{\mathcal{l}}\hslash \left(s\right)\nabla s+\beta {\int }_b^{\mathcal{l}}\vartheta \left(s\right)\nabla s,$$

  (23)

  $${\int }_b^{\mathcal{l}}\hslash \left(s\right)\nabla s={\int }_b^{\varsigma }\hslash \left(s\right)\nabla s+{\int }_{\varsigma }^{\mathcal{l}}\hslash \left(s\right)\nabla s,$$

  (24)

  and if $\hslash \left(s\right)\ge 0$ $\forall s\in {[b,\mathcal{l}]}_{\mathbb{T}},$ then ${\int }_b^{\mathcal{l}}\hslash \left(s\right)\nabla s\ge 0.$
• Integration by Parts: If $b,\mathcal{l}\in \mathbb{T}$ and $u,w\in {\mathrm{C}}_{ld}{([b,\mathcal{l}]}_{\mathbb{T}}$, $\mathbb{R}),$ then

  $${\int }_b^{\mathcal{l}}{u}^{\nabla }\left(s\right)w\left(s\right)\nabla s={\left[u\left(s\right)w\left(s\right)\right]}_b^{\mathcal{l}}-{\int }_b^{\mathcal{l}}{u}^{\rho }\left(s\right)w^{\nabla }\left(s\right)\nabla s.$$

  (25)
• If $\hslash \in {\mathrm{C}}_{ld}(\mathbb{T},\mathbb{R})$ and $e\in \mathbb{T}$, then

  $${\int }_e^{\rho \left(e\right)}\hslash \left(s\right)\nabla s=-v\left(e\right)\hslash \left(e\right).$$

  (26)

2.3. Hölder’s Inequality, Convexity and Submultiplicative Functions, as well as Jensen’s Inequality

• Weighted Hölder’s inequality ([34]): If $b,\mathcal{l}\in \mathbb{T}$ and $h,\zeta ,\varpi \in C_{ld}{([b,\mathcal{l}]}_{\mathbb{T}},{\mathbb{R}}^{+}),$ then

  $${\int }_b^{\mathcal{l}}h\left(s\right)\zeta \left(s\right)\varpi \left(s\right)\nabla s\le {\left[{\int }_b^{\mathcal{l}}h\left(s\right){\zeta }^k\left(s\right)\nabla s\right]}^{{\displaystyle \frac{1}{k}}}{\left[{\int }_b^{\mathcal{l}}h\left(s\right){\varpi }^{\lambda }\left(s\right)\nabla s\right]}^{{\displaystyle \frac{1}{\lambda }}},$$

  (27)

  where $k>1,$ and $1/k+1/\lambda =1.$ The inequality (27) is reversed for $0<k<1$ or $k<0.$
• Jensen’s inequality ([34]): Let $b,\mathcal{l}\in \mathbb{T}$ and $\gamma ,\delta \in \mathbb{R}$. If $\hslash \in {\mathrm{C}}_{ld}({[b,\mathcal{l}]}_{\mathbb{T}},\mathbb{R})$, $\vartheta \in {\mathrm{C}}_{ld}({[b,\mathcal{l}]}_{\mathbb{T}},(\gamma ,\delta ))$ and $\Psi \in \mathrm{C}\left(\right(\gamma ,\delta ),\mathbb{R})$ is convex, then

  $$\Psi \left({\displaystyle \frac{1}{{\int }_b^{\mathcal{l}}\hslash \left(s\right)\nabla s}}{\int }_b^{\mathcal{l}}\hslash \left(s\right)\vartheta \left(s\right)\nabla s\right)\le {\displaystyle \frac{1}{{\int }_b^{\mathcal{l}}\hslash \left(s\right)\nabla s}}{\int }_b^{\mathcal{l}}\hslash \left(s\right)\Psi \left(\vartheta \left(s\right)\right)\nabla s.$$

  (28)
• Convex Function ([35]): A function $\Im :{\mathcal{L}}_{\mathbb{T}}\to \mathbb{R}$ is convex if

  $$\Im \left(\lambda \eta +\left(1-\lambda \right)\vartheta \right)\le \lambda \Im \left(\eta \right)+\left(1-\lambda \right)\Im \left(\vartheta \right),$$

  for $\eta ,\vartheta \in {\mathcal{L}}_{\mathbb{T}}$ and $\lambda \in \left[0,1\right]$ such that $\lambda \eta +\left(1-\lambda \right)\vartheta \in {\mathcal{L}}_{\mathbb{T}}.$
• Submultiplicative Function ([35]): A function $G:{\mathcal{L}}_{\mathbb{T}}\to {\mathbb{R}}^{+}$ is submultiplicative if

  $$G\left(\varsigma \tau \right)\le G\left(\varsigma \right)G\left(\tau \right),\forall \varsigma ,\tau \in {\mathcal{L}}_{\mathbb{T}}.$$
• Convexity Lemma ([35]): Let $\Im :{\mathcal{L}}_{\mathbb{T}}\to \mathbb{R}$ be a continuous function. If ${\Im }^{\nabla \nabla }$ exists on ${\mathcal{L}}_{\mathbb{T}}$ and ${\Im }^{\nabla \nabla }\ge 0,$ then ℑ is convex.

3. Main Results

To derive our primary conclusions, we first establish the following key lemma:

Lemma 2.

Let $b\in \mathbb{T}$ and $\Im \ge 0$ be a convex function such that $\Im \left(b\right)=0.$ Then, $\Im \left(\eta \right)/\left(\eta -b\right)$ is non-decreasing.

Proof. 

Applying the quotient rule for the ∇-derivative to $\Im \left(\eta \right)/\left(\eta -b\right),$ we have

$${\left({\displaystyle \frac{\Im \left(\eta \right)}{\eta -b}}\right)}^{\nabla }={\displaystyle \frac{\left(\eta -b\right){\Im }^{\nabla }\left(\eta \right)-\Im \left(\eta \right)}{\left(\eta -b\right)\left(\rho \left(\eta \right)-b\right)}}={\displaystyle \frac{B\left(\eta \right)}{\left(\eta -a\right)\left(\rho \left(\eta \right)-a\right)}},$$

(29)

where

$$B\left(\eta \right)=\left(\eta -b\right){\Im }^{\nabla }\left(\eta \right)-\Im \left(\eta \right).$$

(30)

Applying the product rule formula (20) to $\left(\eta -b\right){\Im }^{\nabla }\left(\eta \right)$ with $\hslash \left(\eta \right)=\eta -b$ and $\vartheta \left(\eta \right)={\Im }^{\nabla }\left(\eta \right),$ we have

$$\begin{array}{ccc}\hfill {\left[\left(\eta -b\right){\Im }^{\nabla }\left(\eta \right)\right]}^{\nabla }& =& {\left(\eta -b\right)}^{\nabla }{\Im }^{\nabla }\left(\eta \right)+\left(\rho \left(\eta \right)-b\right){\Im }^{\nabla \nabla }\left(\eta \right)\hfill \\ & =& {\Im }^{\nabla }\left(\eta \right)+\left(\rho \left(\eta \right)-b\right){\Im }^{\nabla \nabla }\left(\eta \right).\hfill \end{array}$$

(31)

From (30) and (31), we find

$$B^{\nabla }\left(\eta \right)={\left[\left(\eta -b\right){\Im }^{\nabla }\left(\eta \right)\right]}^{\nabla }-{\Im }^{\nabla }\left(\eta \right)=\left(\rho \left(\eta \right)-b\right){\Im }^{\nabla \nabla }\left(\eta \right).$$

(32)

Since ℑ is convex, it follows that ${\Im }^{\nabla \nabla }\left(\eta \right)\ge 0$. Thus, from (32), we conclude that $B^{\nabla }\left(\eta \right)\ge 0,$ meaning $B\left(\eta \right)$ is non-decreasing. Now, for $\eta \ge b$, we have $B\left(\eta \right)\ge B\left(b\right).$ Since $\Im \left(b\right)=0,$ it follows form (30) that $B\left(b\right)=0,$ implying $B\left(\eta \right)\ge 0.$ By substituting this into (29), we obtain

$${\left({\displaystyle \frac{\Im \left(\eta \right)}{\eta -b}}\right)}^{\nabla }\ge 0,$$

which shows that $\Im \left(\eta \right)/\left(\eta -b\right)$ is non-decreasing. □

With this lemma established, we can now state and prove the key results of this paper. Throughout this work, we will assume that all functions under consideration are ld-continuous and that the integrals involved exist. Below, we present a result that both broadens and enhances the result found in (16).

Theorem 1.

Let $a,\mathcal{l},\eta \in \mathbb{T},$ such that $a\in {[0,\infty )}_{\mathbb{T}}$, ϑ be a positive function and $\hslash ,\Im$ be non-negative functions such that $\hslash ,\Im$ are non-decreasing. Then, for $\mathcal{l}\in {(0,\infty ]}_{\mathbb{T}},$ we have

$${\int }_a^{\mathcal{l}}\Im \left({\displaystyle \frac{{\mathcal{H}}^{\rho }\left(\eta \right)}{{\Lambda }^{\rho }\left(\eta \right)}}\right)\nabla \eta \le {\int }_a^{\mathcal{l}}\Im \left(\hslash \left(\eta \right)\right)\nabla \eta ,$$

(33)

where

$$\mathcal{H}\left(\eta \right)={\int }_a^{\eta }\vartheta \left(s\right)\hslash \left(s\right)\nabla s\mathit{and}\Lambda \left(\eta \right)={\int }_a^{\eta }\vartheta \left(s\right)\nabla s.$$

(34)

Proof. 

Based on the assumptions of the theorem, we observe

$${\int }_a^{\mathcal{l}}\Im \left({\displaystyle \frac{{\mathcal{H}}^{\rho }\left(\eta \right)}{{\Lambda }^{\rho }\left(\eta \right)}}\right)\nabla \eta ={\int }_a^{\mathcal{l}}\Im \left({\displaystyle \frac{{\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\hslash \left(s\right)\nabla s}{{\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\nabla s}}\right)\nabla \eta .$$

(35)

Using Properties (23) and (26), we have

$$\begin{array}{ccc}\hfill {\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\hslash \left(s\right)\nabla s& =& {\int }_a^{\eta }\vartheta \left(s\right)\hslash \left(s\right)\nabla s+{\int }_{\eta }^{\rho \left(\eta \right)}\vartheta \left(s\right)\hslash \left(s\right)\nabla s\hfill \\ & =& {\int }_a^{\eta }\vartheta \left(s\right)\hslash \left(s\right)\nabla s-{\int }_{\rho \left(\eta \right)}^{\eta }\vartheta \left(s\right)\hslash \left(s\right)\nabla s\hfill \\ & =& {\int }_a^{\eta }\vartheta \left(s\right)\hslash \left(s\right)\nabla s-\nu \left(\eta \right)\vartheta \left(\eta \right)\hslash \left(\eta \right).\hfill \end{array}$$

(36)

Since $s\le \eta$ and with the use of the hypothesis of ℏ, we see $\hslash \left(s\right)\le \hslash \left(\eta \right),$ so (36) becomes

$$\begin{array}{ccc}\hfill {\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\hslash \left(s\right)\nabla s& \le & \hslash \left(\eta \right){\int }_a^{\eta }\vartheta \left(s\right)\nabla s-\hslash \left(\eta \right)\hfill \\ & =& \hslash \left(\eta \right)\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s-\nu \left(\eta \right)\vartheta \left(\eta \right)\right)\hfill \\ & =& \hslash \left(\eta \right)\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s+{\int }_{\eta }^{\rho \left(\eta \right)}\vartheta \left(s\right)\nabla s\right)\hfill \\ & =& \hslash \left(\eta \right){\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\nabla s.\hfill \end{array}$$

(37)

By dividing (37) by ${\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\nabla s$ and using the hypothesis of ℑ, we get

$$\Im \left({\displaystyle \frac{{\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\hslash \left(s\right)\nabla s}{{\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\nabla s}}\right)\le \Im \left(\hslash \left(\eta \right)\right).$$

(38)

By integrating (38) with respect to $\eta$ from a to ℓ, we observe

$${\int }_a^{\mathcal{l}}\Im \left({\displaystyle \frac{{\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\hslash \left(s\right)\nabla s}{{\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\nabla s}}\right)\nabla \eta \le {\int }_a^{\mathcal{l}}\Im \left(\hslash \left(\eta \right)\right)\nabla \eta .$$

(39)

By substituting (39) into (35), we get

$${\int }_a^{\mathcal{l}}\Im \left({\displaystyle \frac{{\mathcal{H}}^{\rho }\left(\eta \right)}{{\Lambda }^{\rho }\left(\eta \right)}}\right)\nabla \eta \le {\int }_a^{\mathcal{l}}\Im \left(\hslash \left(\eta \right)\right)\nabla \eta ,$$

which is (33). □

Corollary 1.

In Theorem 1, if $\mathbb{T}=\mathbb{R}$, then $\rho \left(\eta \right)=\eta$ and the inequality (33) becomes

$${\int }_a^{\mathcal{l}}\Im \left({\displaystyle \frac{{\int }_a^{\eta }\vartheta \left(s\right)\hslash \left(s\right)ds}{{\int }_a^{\eta }\vartheta \left(s\right)ds}}\right)d\eta \le {\int }_a^{\mathcal{l}}\Im \left(\hslash \left(\eta \right)\right)d\eta .$$

Particularly, for $\Im \left(s\right)=s^p$ with $p\ge 1,$ we get

$${\int }_a^{\mathcal{l}}{\left({\displaystyle \frac{{\int }_a^{\eta }\vartheta \left(s\right)\hslash \left(s\right)ds}{{\int }_a^{\eta }\vartheta \left(s\right)ds}}\right)}^{p}d\eta \le {\int }_a^{\mathcal{l}}{\hslash }^p\left(\eta \right)d\eta .$$

Remark 1.

If $\mathbb{T}=\mathbb{R}$, $\vartheta \left(t\right)=1$ and $a=0$ in Corollary 1, the result is consistent with Theorem 2.1 [15].

Corollary 2.

In Theorem 1, if $\mathbb{T}=\mathbb{N}$ and $a=1,$ then $\rho \left(\eta \right)=\eta +1$, and the inequality (33) becomes:

$$\sum _{\eta =1}^{\mathcal{l}-1}\Im \left({\displaystyle \frac{{\sum }_{s=1}^{\eta -1}\vartheta \left(s\right)\hslash \left(s\right)}{{\sum }_{s=1}^{\eta -1}\vartheta \left(s\right)}}\right)\le \sum _{\eta =1}^{\mathcal{l}-1}\Im \left(\hslash \left(\eta \right)\right).$$

In particular, by putting ℑ$\left(s\right)=s^p$ with $p\ge 1,$ we obtain

$$\sum _{\eta =1}^{\mathcal{l}-1}{\left({\displaystyle \frac{{\sum }_{s=1}^{\eta -1}\vartheta \left(s\right)\hslash \left(s\right)}{{\sum }_{s=1}^{\eta -1}\vartheta \left(s\right)}}\right)}^p\le \sum _{\eta =1}^{\mathcal{l}-1}{\hslash }^p\left(\eta \right).$$

The result below provides both a generalization and an enhancement of (17).

Theorem 2.

Assume $a,\eta \in \mathbb{T},$ s.t. $a\in {[0,\infty )}_{\mathbb{T}}$, $p>1$ and $\hslash ,\Im \ge 0$ with ℑ being convex. Let $\exists \mathcal{Q} > 0$ be a constant such that

$${\int }_a^{\eta }\vartheta \left(y\right)\nabla y\le \mathcal{Q}{\int }_a^{\rho \left(\eta \right)}\vartheta \left(y\right)\nabla y.$$

(40)

Thus,

$${\int }_a^{\infty }\vartheta \left(\eta \right){\Im }^p\left({\displaystyle \frac{\mathcal{H}\left(\eta \right)}{\Lambda \left(\eta \right)}}\right)\nabla \eta \le {\left({\displaystyle \frac{p}{p-1}}\right)}^p{\mathcal{Q}}^{{\displaystyle \frac{p-1}{p}}}{\int }_a^{\infty }\vartheta \left(\eta \right){\Im }^p\left(\hslash \left(\eta \right)\right)\nabla \eta ,$$

(41)

where $\mathcal{H}\left(\eta \right)$ and $\Lambda \left(\eta \right)$ are defined as in (34).

Proof. 

From the assumptions of the theorem, we start with

$${\int }_a^{\infty }\vartheta \left(\eta \right){\Im }^p\left({\displaystyle \frac{\mathcal{H}\left(\eta \right)}{\Lambda \left(\eta \right)}}\right)\nabla \eta ={\int }_a^{\infty }\vartheta \left(\eta \right){\left[\Im \left({\displaystyle \frac{{\int }_a^{\eta }\vartheta \left(s\right)\hslash \left(s\right)\nabla s}{{\int }_a^{\eta }\vartheta \left(s\right)\nabla s}}\right)\right]}^p\nabla \eta .$$

(42)

Using (28), we have

$$\Im \left({\displaystyle \frac{{\int }_a^{\eta }\vartheta \left(s\right)\hslash \left(s\right)\nabla s}{{\int }_a^{\eta }\vartheta \left(s\right)\nabla s}}\right)\le {\displaystyle \frac{{\int }_a^{\eta }\vartheta \left(s\right)\Im \left(\hslash \left(s\right)\right)\nabla s}{{\int }_a^{\eta }\vartheta \left(s\right)\nabla s}}.$$

(43)

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

$$\begin{array}{cc}\hfill & {\int }_a^{\infty }\vartheta \left(\eta \right){\Im }^p\left({\displaystyle \frac{\mathcal{H}\left(\eta \right)}{\Lambda \left(\eta \right)}}\right)\nabla \eta \hfill \\ \hfill & \le {\int }_a^{\infty }\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{-p}{\left({\int }_a^{\eta }\vartheta \left(s\right)\Im \left(\hslash \left(s\right)\right)\nabla s\right)}^p\nabla \eta \hfill \\ \hfill & ={\int }_a^{\infty }\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{-p}\hfill \\ \hfill & \times {\left({\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}}{\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{1}{p}}}\Im \left(\hslash \left(s\right)\right)\nabla s\right)}^p\nabla \eta .\hfill \end{array}$$

(44)

Applying weighted Hölder’s inequality to ${\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}}{\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{1}{p}}}$$\Im \left(\hslash \left(s\right)\right)\nabla s$ with indices $p>1,$$p/\left(p-1\right),$ we get

$$\begin{array}{ccc}& & {\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}}{\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{1}{p}}}\Im \left(\hslash \left(s\right)\right)\nabla s\hfill \\ & \le & {\left({\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}}\nabla s\right)}^{{\displaystyle \frac{p-1}{p}}}\hfill \\ & & \times {\left({\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}}\left({\int }_a^s\vartheta \left(y\right)\nabla y\right){\Im }^p\left(\hslash \left(s\right)\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =& {\left({\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}}\nabla s\right)}^{{\displaystyle \frac{p-1}{p}}}\hfill \\ & & \times {\left({\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{p-1}{p}}}{\Im }^p\left(\hslash \left(s\right)\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

(45)

By substituting (45) into (44), we get

$$\begin{array}{cc}\hfill & {\int }_a^{\infty }\vartheta \left(\eta \right){\Im }^p\left({\displaystyle \frac{\mathcal{H}\left(\eta \right)}{\Lambda \left(\eta \right)}}\right)\nabla \eta \hfill \\ \hfill & \le {\int }_a^{\infty }\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{-p}{\left({\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}}\nabla s\right)}^{p-1}\hfill \\ \hfill & \times \left({\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{p-1}{p}}}{\Im }^p\left(\hslash \left(s\right)\right)\nabla s\right)\nabla \eta .\hfill \end{array}$$

(46)

By applying (22) on ${\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}+1},$ we find

$${\left[{\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}+1}\right]}^{\nabla }={\displaystyle \frac{p-1}{p}}{\left({\int }_a^{\xi }\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}}\vartheta \left(s\right),\xi \in \left[\rho \left(s\right),s\right].$$

Since $\rho \left(s\right)\le \xi \le s$, $p>1,$ we observe ${\left({\int }_a^{\xi }\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}}\ge {\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}}$, and then we have

$$\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}}\le {\displaystyle \frac{p}{p-1}}{\left[{\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}+1}\right]}^{\nabla }.$$

(47)

By integrating (47) over s from a to $\eta$, we have

$$\begin{array}{ccc}& & {\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}}\nabla t\hfill \\ & \le & {\displaystyle \frac{p}{p-1}}{\int }_a^{\eta }{\left[{\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{-1}{p}}+1}\right]}^{\nabla }\nabla s\hfill \\ & =& {\displaystyle \frac{p}{p-1}}{\left({\int }_a^{\eta }\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{p-1}{p}}}.\hfill \end{array}$$

(48)

By substituting (48) into (46), we observe

$$\begin{array}{cc}\hfill & {\int }_a^{\infty }\vartheta \left(\eta \right){\Im }^p\left({\displaystyle \frac{\mathcal{H}\left(\eta \right)}{\Lambda \left(\eta \right)}}\right)\nabla \eta \hfill \\ \hfill & \le {\left({\displaystyle \frac{p}{p-1}}\right)}^{p-1}{\int }_a^{\infty }\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-2}\hfill \\ \hfill & \times \left({\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{p-1}{p}}}{\Im }^p\left(\hslash \left(s\right)\right)\nabla s\right)\nabla \eta .\hfill \end{array}$$

(49)

By applying (25) to

$${\int }_a^{\infty }\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-2}\left({\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{p-1}{p}}}{\Im }^p\left(\hslash \left(s\right)\right)\nabla s\right)\nabla \eta ,$$

with $u^{\nabla }\left(\eta \right)=\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-2}$ and $w\left(\eta \right)={\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{p-1}{p}}}{\Im }^p\left(\hslash \left(s\right)\right)\nabla s,$ we obtain

$$\begin{array}{ccc}& & {\int }_a^{\infty }\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-2}\left({\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{p-1}{p}}}{\Im }^p\left(\hslash \left(s\right)\right)\nabla s\right)\nabla \eta \hfill \\ & =& {\left.u\left(\eta \right)\left({\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{p-1}{p}}}{\Im }^p\left(\hslash \left(s\right)\right)\nabla s\right)\right|}_a^{\infty }\hfill \\ & & -{\int }_a^{\infty }{u}^{\rho }\left(\eta \right)\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{p-1}{p}}}{\Im }^p\left(\hslash \left(\eta \right)\right)\nabla \eta ,\hfill \end{array}$$

(50)

where $u\left(\eta \right)=-{\int }_{\eta }^{\infty }\vartheta \left(y\right){\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-2}\nabla y.$ Since ${lim}_{\eta \to \infty }u\left(\eta \right)=0,$ from (50), we get

$$\begin{array}{ccc}& & {\int }_a^{\infty }\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-2}\left({\int }_a^{\eta }\vartheta \left(s\right){\left({\int }_a^s\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{p-1}{p}}}{\Im }^p\left(\hslash \left(s\right)\right)\nabla s\right)\nabla \eta \hfill \\ & =& {\int }_a^{\infty }\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{p-1}{p}}}{\Im }^p\left(\hslash \left(\eta \right)\right)\left[{\int }_{\rho \left(\eta \right)}^{\infty }\vartheta \left(y\right){\left({\int }_a^y\vartheta \left(s\right)\nabla t\right)}^{{\displaystyle \frac{1}{p}}-2}\nabla y\right]\nabla \eta .\hfill \end{array}$$

(51)

By substituting (51) into (49), we obtain

$$\begin{array}{cc}\hfill & {\int }_a^{\infty }\vartheta \left(\eta \right){\Im }^p\left({\displaystyle \frac{\mathcal{H}\left(\eta \right)}{\Lambda \left(\eta \right)}}\right)\nabla \eta \hfill \\ \hfill & \le {\left({\displaystyle \frac{p}{p-1}}\right)}^{p-1}{\int }_a^{\infty }\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{p-1}{p}}}{\Im }^p\left(\hslash \left(\eta \right)\right)\hfill \\ \hfill & \times \left[{\int }_{\rho \left(\eta \right)}^{\infty }\vartheta \left(y\right){\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-2}\nabla y\right]\nabla \eta .\hfill \end{array}$$

(52)

By applying the property (22) to ${\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-1},$ we observe

$${\left[{\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-1}\right]}^{\nabla }={\displaystyle \frac{1-p}{p}}{\left({\int }_a^{\xi }\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-2}\vartheta \left(y\right),\xi \in \left[\rho \left(y\right),y\right].$$

(53)

Since $\rho \left(y\right)\le \xi \le y$ and $(1/p)-2<0,$ we have

$${\left({\int }_a^{\xi }\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-2}\ge {\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-2},$$

and then we have from (53) that

$$\vartheta \left(y\right){\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-2}\le {\displaystyle \frac{p}{1-p}}{\left[{\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-1}\right]}^{\nabla }.$$

By integrating this over y from $\rho \left(\eta \right)$ to $\infty ,$ we get

$$\begin{array}{ccc}& & {\int }_{\rho \left(\eta \right)}^{\infty }\vartheta \left(y\right){\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-2}\nabla y\hfill \\ & \le & {\displaystyle \frac{p}{1-p}}{\int }_{\rho \left(\eta \right)}^{\infty }{\left[{\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-1}\right]}^{\nabla }\nabla y\hfill \\ & =& {\displaystyle \frac{p}{p-1}}\left[{\left({\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-1}-{\left({\int }_a^{\infty }\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-1}\right]\hfill \\ & \le & {\displaystyle \frac{p}{p-1}}{\left({\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-1}.\hfill \end{array}$$

(54)

By substituting (54) into (52), we find

$$\begin{array}{cc}\hfill & {\int }_a^{\infty }\vartheta \left(\eta \right){\Im }^p\left({\displaystyle \frac{\mathcal{H}\left(\eta \right)}{\Lambda \left(\eta \right)}}\right)\nabla \eta \hfill \\ \hfill & \le {\left({\displaystyle \frac{p}{p-1}}\right)}^p{\int }_a^{\infty }\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(y\right)\nabla y\right)}^{{\displaystyle \frac{p-1}{p}}}{\Im }^p\left(\hslash \left(\eta \right)\right){\left({\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\nabla s\right)}^{{\displaystyle \frac{1}{p}}-1}\nabla \eta .\hfill \end{array}$$

Using (40), we obtain

$${\int }_a^{\infty }\vartheta \left(\eta \right){\Im }^p\left({\displaystyle \frac{\mathcal{H}\left(\eta \right)}{\Lambda \left(\eta \right)}}\right)\nabla \eta \le {\left({\displaystyle \frac{p}{p-1}}\right)}^p{\mathcal{Q}}^{{\displaystyle \frac{p-1}{p}}}{\int }_a^{\infty }\vartheta \left(\eta \right){\Im }^p\left(\hslash \left(\eta \right)\right)\nabla \eta ,$$

which is (41). □

Corollary 3.

In Theorem 2, if $\mathbb{T}=\mathbb{R}$, then $\rho \left(\eta \right)=\eta .$ In this case, the condition (40) holds with equality for $\mathcal{Q}=1.$ Consequently, the inequality (41) becomes

$${\int }_a^{\infty }\vartheta \left(\eta \right){\Im }^p\left({\displaystyle \frac{{\int }_a^{\eta }\vartheta \left(s\right)\hslash \left(s\right)ds}{{\int }_a^{\eta }\vartheta \left(s\right)ds}}\right)d\eta \le {\left({\displaystyle \frac{p}{p-1}}\right)}^p{\int }_a^{\infty }\vartheta \left(\eta \right){\Im }^p\left(\hslash \left(\eta \right)\right)d\eta .$$

Remark 2.

If $a=0$ and $\vartheta \left(t\right)=1$ in Corollary 3, then we obtain (10), which is the result established in [15] by Sulaiman.

Corollary 4.

In Theorem 2, if $\mathbb{T}=\mathbb{N}$, $a=1,$ then $\rho \left(\eta \right)=\eta +1$, and the condition (40) becomes

$$\sum _{y=1}^{\eta }\vartheta \left(y\right)\le \mathcal{Q}\sum _{y=1}^{\eta -1}\vartheta \left(y\right).$$

Consequently, the inequality (41) can be expressed as

$$\sum _{\eta =1}^{\infty }\vartheta \left(\eta \right){\Im }^p\left({\displaystyle \frac{{\sum }_{s=1}^{\eta -1}\vartheta \left(s\right)\hslash \left(s\right)}{{\sum }_{s=1}^{\eta -1}\vartheta \left(s\right)}}\right)\le {\left({\displaystyle \frac{p}{p-1}}\right)}^p{\mathcal{Q}}^{{\displaystyle \frac{p-1}{p}}}\sum _{\eta =1}^{\infty }\vartheta \left(\eta \right){\Im }^p\left(\hslash \left(\eta \right)\right).$$

The subsequent result presents both an extension and a refinement of (18).

Theorem 3.

Assume $a,\mathcal{l},\eta \in \mathbb{T}$ such that $a\in {[0,\infty )}_{\mathbb{T}}$ and $p>1.$ If $\hslash ,\Im \ge 0$ such that ℑ is a convex and submultiplicative function with $\Im \left(a\right)=0$, then for $\mathcal{l}\in {(0,\infty ]}_{\mathbb{T}},$ we have

$${\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left(\Lambda \left(\eta \right)\right)}^{1-p}}{\Im \left(\eta \right)\Im \left(\Lambda \left(\eta \right)\right)}}\Im \left(\mathcal{H}\left(\eta \right)\right)\nabla \eta \le {\displaystyle \frac{1}{p-1}}{\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right)\Im \left(\hslash \left(\eta \right)\right)}{{\Im }^{\rho }\left(\eta \right)}}{\left({\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\nabla s\right)}^{1-p}\nabla \eta ,$$

(55)

where $\mathcal{H}\left(\eta \right)$ and $\Lambda \left(\eta \right)$ are defined as in (34).

Proof. 

First, note that

$$\begin{array}{ccc}& & {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left(\Lambda \left(\eta \right)\right)}^{1-p}}{\Im \left(\eta \right)\Im \left(\Lambda \left(\eta \right)\right)}}\Im \left(\mathcal{H}\left(\eta \right)\right)\nabla \eta \hfill \\ & =& {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{1-p}}{\Im \left(\eta \right)\Im \left(\Lambda \left(\eta \right)\right)}}\Im \left({\int }_a^{\eta }\vartheta \left(s\right)\hslash \left(s\right)\nabla s\right)\nabla \eta \hfill \\ & =& {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{1-p}}{\Im \left(\eta \right)\Im \left(\Lambda \left(\eta \right)\right)}}\Im \left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s.{\displaystyle \frac{{\int }_a^{\eta }\vartheta \left(s\right)\hslash \left(s\right)\nabla s}{{\int }_a^{\eta }\vartheta \left(s\right)\nabla s}}\right)\nabla \eta .\hfill \end{array}$$

(56)

Using the hypothesis of ℑ, we have from the above equation that

$$\begin{array}{ccc}& & {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left(\Lambda \left(\eta \right)\right)}^{1-p}}{\Im \left(\eta \right)\Im \left(\Lambda \left(\eta \right)\right)}}\Im \left(\mathcal{H}\left(\eta \right)\right)\nabla \eta \hfill \\ & \le & {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{1-p}}{\Im \left(\eta \right)\Im \left(\Lambda \left(\eta \right)\right)}}\Im \left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)\Im \left({\displaystyle \frac{{\int }_a^{\eta }\vartheta \left(s\right)\hslash \left(s\right)\nabla s}{{\int }_a^{\eta }\vartheta \left(s\right)\nabla s}}\right)\nabla \eta \hfill \\ & =& {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{1-p}}{\Im \left(\eta \right)}}\Im \left({\displaystyle \frac{{\int }_a^{\eta }\vartheta \left(s\right)\hslash \left(s\right)\nabla s}{{\int }_a^{\eta }\vartheta \left(s\right)\nabla s}}\right)\nabla \eta .\hfill \end{array}$$

(57)

Using (28), we find

$$\Im \left({\displaystyle \frac{{\int }_a^{\eta }\vartheta \left(s\right)\hslash \left(s\right)\nabla s}{{\int }_a^{\eta }\vartheta \left(s\right)\nabla s}}\right)\le {\displaystyle \frac{1}{{\int }_a^{\eta }\vartheta \left(s\right)\nabla s}}{\int }_a^{\eta }\vartheta \left(s\right)\Im \left(\hslash \left(s\right)\right)\nabla s.$$

(58)

By substituting (58) into (57), we obtain

$$\begin{array}{ccc}& & {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left(\Lambda \left(\eta \right)\right)}^{1-p}}{\Im \left(\eta \right)\Im \left(\Lambda \left(\eta \right)\right)}}\Im \left(\mathcal{H}\left(\eta \right)\right)\nabla \eta \hfill \\ & \le & {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{-p}}{\Im \left(\eta \right)}}\left({\int }_a^{\eta }\vartheta \left(s\right)\Im \left(\hslash \left(s\right)\right)\nabla s\right)\nabla \eta .\hfill \end{array}$$

(59)

Applying formula (25) to ${\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{-p}}{\Im \left(\eta \right)}}\left({\int }_a^{\eta }\vartheta \left(s\right)\Im \left(\hslash \left(s\right)\right)\nabla s\right)\nabla \eta ,$ with

$$u^{\nabla }\left(\eta \right)={\displaystyle \frac{\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{-p}}{\Im \left(\eta \right)}}\mathrm{and}w\left(\eta \right)={\int }_a^{\eta }\vartheta \left(s\right)\Im \left(\hslash \left(s\right)\right)\nabla s,$$

we get

$$\begin{array}{ccc}& & {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{-p}}{\Im \left(\eta \right)}}\left({\int }_a^{\eta }\vartheta \left(s\right)\Im \left(\hslash \left(s\right)\right)\nabla s\right)\nabla \eta \hfill \\ & =& {\left.u\left(\eta \right)\left({\int }_a^{\eta }\vartheta \left(s\right)\Im \left(\hslash \left(s\right)\right)\nabla s\right)\right|}_a^{\mathcal{l}}-{\int }_a^{\mathcal{l}}{u}^{\rho }\left(\eta \right)\vartheta \left(\eta \right)\Im \left(\hslash \left(\eta \right)\right)\nabla \eta ,\hfill \end{array}$$

(60)

where

$$u\left(\eta \right)=-{\int }_{\eta }^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(y\right){\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{-p}}{\Im \left(y\right)}}\nabla y.$$

Since $u\left(\mathcal{l}\right)=0,$ we understand from (60) that

$$\begin{array}{ccc}& & {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{-p}}{\Im \left(\eta \right)}}\left({\int }_a^{\eta }\vartheta \left(s\right)\Im \left(\hslash \left(s\right)\right)\nabla s\right)\nabla \eta \hfill \\ & =& {\int }_a^{\mathcal{l}}\vartheta \left(\eta \right)\Im \left(\hslash \left(\eta \right)\right)\left({\int }_{\rho \left(\eta \right)}^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(y\right){\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{-p}}{\Im \left(y\right)}}\nabla y\right)\nabla \eta \hfill \\ & =& {\int }_a^{\mathcal{l}}\vartheta \left(\eta \right)\Im \left(\hslash \left(\eta \right)\right)\left({\int }_{\rho \left(\eta \right)}^{\mathcal{l}}{\displaystyle \frac{1}{y-a}}.{\displaystyle \frac{\vartheta \left(y\right){\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{-p}}{\left(\frac{\Im \left(y\right)}{y-a}\right)}}\nabla y\right)\nabla \eta .\hfill \end{array}$$

(61)

Since $y\ge \rho \left(\eta \right),$ we observe that $1/(y-a)\le 1/\left(\rho \right(\eta )-a)$; thus,

$$\begin{array}{ccc}& & {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{-p}}{\Im \left(\eta \right)}}\left({\int }_a^{\eta }\vartheta \left(s\right)\Im \left(\hslash \left(s\right)\right)\nabla s\right)\nabla \eta \hfill \\ & \le & {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right)\Im \left(\hslash \left(\eta \right)\right)}{\rho \left(\eta \right)-a}}\left({\int }_{\rho \left(\eta \right)}^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(y\right){\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{-p}}{\frac{\Im \left(y\right)}{y-a}}}\nabla y\right)\nabla \eta .\hfill \end{array}$$

(62)

By applying Lemma 2 where $\Im \left(y\right)/\left(y-a\right)$ is non-decreasing, considering $y\ge \rho \left(\eta \right)$, we obtain

$${\displaystyle \frac{\Im \left(y\right)}{y-a}}\ge {\displaystyle \frac{{\Im }^{\rho }\left(\eta \right)}{\rho \left(\eta \right)-a}}.$$

(63)

By substituting (63) into (62), we observe

$$\begin{array}{ccc}& & {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{-p}}{\Im \left(\eta \right)}}\left({\int }_a^{\eta }\vartheta \left(s\right)\Im \left(\hslash \left(s\right)\right)\nabla s\right)\nabla \eta \hfill \\ & =& {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right)\Im \left(\hslash \left(\eta \right)\right)}{{\Im }^{\rho }\left(\eta \right)}}\left({\int }_{\rho \left(\eta \right)}^{\mathcal{l}}\vartheta \left(y\right){\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{-p}\nabla y\right)\nabla \eta .\hfill \end{array}$$

(64)

By applying (22) on ${\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{1-p},$ we have

$${\displaystyle \frac{1}{1-p}}{\left({\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{-p+1}\right)}^{\nabla }={\left({\int }_a^{\xi }\vartheta \left(s\right)\nabla s\right)}^{-p}\vartheta \left(y\right),\xi \in \left[\rho \left(y\right),y\right].$$

Since $\rho \left(y\right)\le \xi \le y$ and $p>1,$ we get

$$\vartheta \left(y\right){\left({\int }_a^{\xi }\vartheta \left(s\right)\nabla s\right)}^{-p}\ge \vartheta \left(y\right){\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{-p},$$

and then

$$\vartheta \left(y\right){\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{-p}\le {\displaystyle \frac{1}{1-p}}{\left({\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{-p+1}\right)}^{\nabla }.$$

Integrating the preceding inequality w.r.t y from $\rho \left(\eta \right)$ to ℓ yields

$$\begin{array}{ccc}& & {\int }_{\rho \left(\eta \right)}^{\mathcal{l}}\vartheta \left(y\right){\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{-p}\nabla y\hfill \\ & \le & {\displaystyle \frac{1}{1-p}}{\int }_{\rho \left(\eta \right)}^{\mathcal{l}}{\left({\left({\int }_a^y\vartheta \left(s\right)\nabla s\right)}^{-p+1}\right)}^{\nabla }\nabla y\hfill \\ & =& {\displaystyle \frac{1}{p-1}}\left[{\left({\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\nabla s\right)}^{-p+1}-{\left({\int }_a^{\mathcal{l}}\vartheta \left(s\right)\nabla s\right)}^{-p+1}\right]\hfill \\ & \le & {\displaystyle \frac{1}{p-1}}{\left({\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\nabla s\right)}^{-p+1}.\hfill \end{array}$$

(65)

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

$$\begin{array}{ccc}& & {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)\nabla s\right)}^{-p}}{\Im \left(\eta \right)}}\left({\int }_a^{\eta }\vartheta \left(s\right)\Im \left(\hslash \left(s\right)\right)\nabla t\right)\nabla \eta \hfill \\ & =& {\displaystyle \frac{1}{p-1}}{\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right)\Im \left(\hslash \left(\eta \right)\right)}{{\Im }^{\rho }\left(\eta \right)}}{\left({\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\nabla s\right)}^{1-p}\nabla \eta .\hfill \end{array}$$

(66)

By inserting (66) into (59), we derive

$$\begin{array}{ccc}& & {\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left(\Lambda \left(\eta \right)\right)}^{1-p}}{\Im \left(\eta \right)\Im \left(\Lambda \left(\eta \right)\right)}}\Im \left(\mathcal{H}\left(\eta \right)\right)\nabla \eta \hfill \\ & \le & {\displaystyle \frac{1}{p-1}}{\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right)\Im \left(\hslash \left(\eta \right)\right)}{{\Im }^{\rho }\left(\eta \right)}}{\left({\int }_a^{\rho \left(\eta \right)}\vartheta \left(s\right)\nabla s\right)}^{1-p}\nabla \eta ,\hfill \end{array}$$

which is (55). □

Corollary 5.

In Theorem 3, setting $\mathbb{T}=\mathbb{R}$, then $\rho \left(\eta \right)=\eta$ and (55), which simplifies to

$${\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right){\left({\int }_a^{\eta }\vartheta \left(s\right)ds\right)}^{1-p}}{\Im \left(\eta \right)\Im \left({\int }_a^{\eta }\vartheta \left(s\right)ds\right)}}\Im \left({\int }_a^{\eta }\vartheta \left(s\right)\hslash \left(s\right)ds\right)d\eta \le {\displaystyle \frac{1}{p-1}}{\int }_a^{\mathcal{l}}{\displaystyle \frac{\vartheta \left(\eta \right)\Im \left(\hslash \left(\eta \right)\right)}{{\Im }^{\rho }\left(\eta \right)}}{\left({\int }_a^{\eta }\vartheta \left(s\right)ds\right)}^{1-p}d\eta .$$

Corollary 6.

In Theorem 3, by setting $\mathbb{T}=\mathbb{N}$ and $a=1,$ then $\rho \left(\eta \right)=\eta +1$ and (55), which becomes

$$\sum _{\eta =1}^{\mathcal{l}-1}{\displaystyle \frac{\vartheta \left(\eta \right){\left({\sum }_{s=1}^{\eta -1}\vartheta \left(s\right)\right)}^{1-p}}{\Im \left(\eta \right)\Im \left({\sum }_{s=1}^{\eta -1}\vartheta \left(s\right)\right)}}\Im \left(\sum _{s=1}^{\eta -1}\vartheta \left(s\right)\hslash \left(s\right)\right)\le {\displaystyle \frac{1}{p-1}}\sum _{\eta =1}^{\mathcal{l}-1}{\displaystyle \frac{\vartheta \left(\eta \right)\Im \left(\hslash \left(\eta \right)\right)}{{\Im }^{\rho }\left(\eta \right)}}{\left(\sum _{s=1}^{\eta -1}\vartheta \left(\eta \right)\right)}^{1-p}.$$

4. Conclusions

In this study, we introduced novel generalizations of Hardy-type dynamic inequalities using the framework of nabla calculus. Notably, we derived novel results for specific cases where $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{N},$ offering new insights into both continuous and discrete settings. In our next paper, we will extend these inequalities within the frameworks of conformable fractional calculus and diamond-alpha calculus on $\mathbb{T}$. In addition, we plan to extend the inequalities explored in this research to multiple dimensions. Furthermore, we aim to connect these generalizations to applications of fractional differential equations, focusing particularly on the qualitative behaviors of solutions, such as stability and convergence, while considering practical applications in mathematical and physical models. We anticipate that these findings will open new avenues for a more comprehensive understanding of fractional dynamics across different time scales.