Advanced Generalizations of Weighted Opial-Type Inequalities in the Framework of Time Scale Calculus

Abstract

This work presents refined and generalized forms of weighted Opial-type inequalities within the framework of time scale calculus. The proofs rely on several algebraic techniques, together with Hölder’s inequality and Keller’s chain rule. These results extend the classical Opial-type inequalities by embedding them into the time scale setting, which unifies both continuous and discrete analyses. Consequently, various integral and discrete inequalities emerge as particular cases of our main results, thereby broadening the applicability of Opial-type inequalities to dynamic systems and discrete models.

1. Introduction

The study of inequalities has long played a central role in mathematical analysis, particularly in the fields of differential and integral inequalities. This study is motivated by the need to extend known inequalities and their applications to dynamic equations on time scales, bridging the gap between continuous and discrete analysis. The results presented herein improve upon existing literature by providing more general conditions under which these inequalities hold, including the incorporation of weight functions and fractional-order terms. These findings offer an advantage in modeling dynamic systems that operate across mixed continuous–discrete time domains, enabling more accurate estimates and broader applicability in mathematical and engineering contexts. Classical analytic inequalities and their sharp constants have been extensively studied in the literature; a comprehensive treatment is provided in the monograph by Mitrinović and Vasić [1], while sharp constants and refinements for several integral inequalities of Opial type were investigated by Milovanović and Milovanović [2]. A seminal result in this area was established by Opial in [3], who proved the classical inequality:

$${\int }_0^{\mathit{𝚥}}\left|\chi \left(\vartheta \right)\right||{\chi }^{\prime }\left(\vartheta \right)|d\vartheta \le {\displaystyle \frac{𝚥}{4}}{\int }_0^{𝚥}{\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{2}d\vartheta ,$$

(1)

where $\chi \in C^1[0,𝚥],$ $𝚥>0,$ with $\chi \left(0\right)=\chi \left(𝚥\right)=0$ and $\chi \left(\vartheta \right)>0$ for $0<\vartheta <𝚥.$ The constant ${\displaystyle \frac{𝚥}{4}}$ is known to be sharp.

Subsequent work has focused on generalizing this result. For instance, Hua [4] extended inequality (1) to the power-weighted form:

$${\int }_0^{𝚥}{\left|\chi \left(\vartheta \right)\right|}^p|{\chi }^{\prime }\left(\vartheta \right)|d\vartheta \le {\displaystyle \frac{{𝚥}^p}{p+1}}{\int }_0^{𝚥}{\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{p+1}d\vartheta ,$$

(2)

for absolutely continuous functions $\chi$ with $\chi \left(0\right)=0$, and for any positive integer p.

Yang [5] introduced further refinements, proving that for $p\ge 0$ and $𝓁\ge 1,$

$${\int }_0^{𝚥}{\left|\chi \left(\vartheta \right)\right|}^p|{\chi }^{\prime }{\left(\vartheta \right)|}^{𝓁}d\vartheta \le \left({\displaystyle \frac{𝓁}{p+𝓁}}\right){𝚥}^p{\int }_0^{𝚥}{\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{p+𝓁}d\vartheta .$$

(3)

Additional variants were presented in the same work [5] subject to different boundary conditions:

$${\int }_0^{𝚥}{\left|\chi \left(\vartheta \right)\right|}^p|{\chi }^{\prime }{\left(\vartheta \right)|}^{𝓁}d\vartheta \le \left({\displaystyle \frac{𝓁}{p+𝓁}}\right){𝚥}^p{\int }_0^{𝚥}{\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{p+𝓁}d\vartheta ,\chi \left(𝚥\right)=0.$$

(4)

and

$${\int }_0^{𝚥}{\left|\chi \left(\vartheta \right)\right|}^p|{\chi }^{\prime }{\left(\vartheta \right)|}^{𝓁}d\vartheta \le \left({\displaystyle \frac{𝓁}{p+𝓁}}\right){\left({\displaystyle \frac{𝚥}{2}}\right)}^p{\int }_0^{𝚥}{\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{p+𝓁}d\vartheta ,\chi \left(0\right)=\chi \left(𝚥\right)=0.$$

(5)

In a broader context, Yang [6] established weighted versions of these inequalities involving positive, bounded monotonic weight functions $\eta .$ For instance, if $\eta \left(\vartheta \right)>0$ is non-increasing on $[𝚤,m],$ and $\chi$ is absolutely continuous with $\chi \left(𝚤\right)=0,$ then

$${\int }_{𝚤}^m{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p|{\chi }^{\prime }{\left(\vartheta \right)|}^{𝓁}d\vartheta \le {\displaystyle \frac{𝓁}{p+𝓁}}{(m-𝚤)}^p{\int }_{𝚤}^m\eta \left(\vartheta \right){\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{p+𝓁}d\vartheta ,(p\ge 0;𝓁\ge 1).$$

(6)

Analogous results were derived for non-decreasing weights over $[m,𝚥]$ and for the full interval $[𝚤,𝚥]$ under suitable boundary conditions:

$${\int }_m^{𝚥}{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p|{\chi }^{\prime }{\left(\vartheta \right)|}^{𝓁}d\vartheta \le {\displaystyle \frac{𝓁}{p+𝓁}}{(𝚥-m)}^p{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{p+𝓁}d\vartheta ,\chi \left(𝚥\right)=0.$$

(7)

$${\int }_{𝚤}^{𝚥}{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p|{\chi }^{\prime }{\left(\vartheta \right)|}^{𝓁}d\vartheta \le {\displaystyle \frac{𝓁}{p+𝓁}}{(𝚥-𝚤)}^p{\int }_{𝚤}^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{p+𝓁}d\vartheta ,\chi \left(𝚤\right)=\chi \left(𝚥\right)=0.$$

(8)

More recently, motivated by the unified framework offered by time scales theory (details regarding the definitions and notation used here are provided in Section 2), the authors in [7] extended Opial’s inequality (1) to the delta-calculus setting as follows: for a $\Delta$-differentiable function $\chi :[0,\hslash ]\cap \mathbb{T}⟶\mathbb{R},$ $\hslash >0$ with $\chi \left(0\right)=0$, they proved

$${\int }_0^{\hslash }\left|\chi \left(\epsilon \right)+{\chi }^{\sigma }\left(\epsilon \right)\right|\left|{\chi }^{\Delta }\left(\epsilon \right)\right|\Delta \epsilon \le \hslash {\int }_0^{\hslash }{\left|{\chi }^{\Delta }\left(\epsilon \right)\right|}^2\Delta \epsilon .$$

(9)

In a subsequent work, the authors in [8] provided generalizations of inequality (9): Let $\chi :{[𝚤,𝚥]}_{\mathbb{T}}⟶\mathbb{R},$$(𝚥>𝚤\ge 0)$ be $\Delta$-differentiable with $\chi \left(𝚤\right)=0,$ and suppose that $p\ge 0,$ $𝓁\ge 1$, and $\eta \in {\mathrm{C}}_{rd}{([𝚤,m]}_{\mathbb{T}},{\mathbb{R}}^{+})$ is non-increasing. Then,

$${\int }_{𝚤}^m{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p|{\chi }^{\Delta }{\left(\vartheta \right)|}^{𝓁}\Delta \vartheta \le {\displaystyle \frac{𝓁}{p+𝓁}}{(m-𝚤)}^p{\int }_{𝚤}^m\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{p+𝓁}\Delta \vartheta .$$

(10)

Assuming $\chi \left(𝚥\right)=0$ and $\eta \in {\mathrm{C}}_{rd}{([m,𝚥]}_{\mathbb{T}},{\mathbb{R}}^{+})$ is non-increasing, they also showed:

$${\int }_m^{𝚥}{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p|{\chi }^{\Delta }{\left(\vartheta \right)|}^{𝓁}\Delta \vartheta \le {\displaystyle \frac{𝓁}{p+𝓁}}{(𝚥-m)}^p{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{p+𝓁}\Delta \vartheta .$$

(11)

Under the conditions $\chi \left(𝚤\right)=\chi \left(𝚥\right)=0,$ and assuming $\eta$ is non-increasing on both subintervals ${[𝚤,m]}_{\mathbb{T}}$ and ${[m,𝚥]}_{\mathbb{T}}$ for some $m\in {[𝚤,𝚥]}_{\mathbb{T}}$, they derived the combined estimate

$$\begin{array}{ccc}& & {\int }_{𝚤}^{𝚥}{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p{\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & \le & {\displaystyle \frac{𝓁}{p+𝓁}}{(m-𝚤)}^p{\int }_{𝚤}^m\eta \left(\vartheta \right)|{\chi }^{\Delta }{\left(\vartheta \right)|}^{p+𝓁}\Delta \vartheta +{\displaystyle \frac{𝓁}{p+𝓁}}{(𝚥-m)}^p{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{p+𝓁}\Delta \vartheta .\hfill \end{array}$$

(12)

For a broader overview of further developments and significant refinements of related inequalities and other dynamic inequalities, see [9,10,11,12,13]. Under the framework of time scales, this paper systematically generalizes weighted Opial-type inequalities and proposes a new form that allows for non-zero boundary conditions and non-increasing weight functions. The primary aim of this article is to develop new dynamic Opial-type inequalities on time scales, enriching the existing literature by offering novel generalizations that encompass both the continuous case ($\mathbb{T}=\mathbb{R}$) and the discrete case ($\mathbb{T}=\mathbb{N}$) as special instances. This unified approach not only extends classical results, which are usually treated separately, but also provides more flexible formulations that enhance the applicability of Opial-type inequalities to dynamic equations and hybrid models. The present article is arranged as follows: Section 2 outlines the essential preliminaries and fundamental tools on time scales. Section 3 is devoted to the main theorems, their proofs, as well as some particular cases and remarks that illustrate the significance and applicability of the obtained results.

2. Preliminaries

We begin by outlining essential concepts and tools from time scale theory, which serve as the foundation for the subsequent results.

A time scale $\mathbb{T}$ is any nonempty closed subset of $\mathbb{R}$. For a point $\varrho \in \mathbb{T}$, we define $\sigma :\mathbb{T}\to \mathbb{T},$ $\rho :\mathbb{T}\to \mathbb{T}$ and $\mu :\mathbb{T}\to \mathbb{R}$ as $\sigma \left(\varrho \right):=inf\{j\in \mathbb{T}:j>\varrho \},$ $\rho \left(\varrho \right):=sup\{j\in \mathbb{T}:j<\varrho \}$ and $\mu \left(\varrho \right):=\sigma \left(\varrho \right)-\varrho \ge 0.$ These components are referred to as the forward jump operator, the backward jump operator and the forward graininess function, respectively. According to these definitions, a point $\varrho \in \mathbb{T}$ where $inf\mathbb{T}<\varrho <sup\mathbb{T},$ is

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

Next, we define the set ${\mathbb{T}}^k$ as follows: If $\mathbb{T}\mathrm{has}\mathrm{a}\mathrm{left}\text{-}\mathrm{scattered}\mathrm{maximum}y,$ then ${\mathbb{T}}^k=\mathbb{T}/\left\{y\right\}.$ $\mathrm{Otherwise},{\mathbb{T}}^k=\mathbb{T}.$

Let $\vartheta :\mathbb{T}\to \mathbb{R}$ be a function. The composition ${\vartheta }^{\sigma }\left(\varrho \right)$ is understood as $\vartheta \left(\sigma \right(\varrho \left)\right),$$\forall \varrho \in \mathbb{T}.$ A function $\vartheta$ is right-dense continuous ($rd$-continuous) if it is continuous at all right-dense points and possesses finite left-sided limits at left-dense points. The space of such functions is denoted ${\mathrm{C}}_{rd}(\mathbb{T},\mathbb{R})$.

Definition 1

([14,15]). The delta derivative ${\vartheta }^{\Delta }\left(\varrho \right)$ of a function $\vartheta :\mathbb{T}\to \mathbb{R}$ at $\varrho \in {\mathbb{T}}^k$ exists if, for any $\epsilon >0,$ there is a neighborhood $\mathcal{U}$ of ϱ such that

$$\left|\vartheta \left(\sigma \left(\varrho \right)\right)-\vartheta \left(z\right)-{\vartheta }^{\Delta }\left(\varrho \right)[\sigma \left(\varrho \right)-z]\right|\le \epsilon \left|\sigma \left(\varrho \right)-z\right|,\forall z\in \mathcal{U}.$$

If this condition holds at each point in ${\mathbb{T}}^k,$ we say ϑ is delta-differentiable on ${\mathbb{T}}^k.$

Theorem 1

(Chain Rule [14,15]). Let $\vartheta :\mathbb{R}\to \mathbb{R}$ be continuous, $\vartheta :\mathbb{T}\to \mathbb{R}$ be Δ-differentiable on ${\mathbb{T}}^k,$ and let $\Im :\mathbb{R}\to \mathbb{R}$ be continuously differentiable. Then, for all $\varrho \in {\mathbb{T}}^k,$ we have

$${\left(\Im \circ \vartheta \right)}^{\Delta }\left(\varrho \right)={\Im }^{\prime }\left(\vartheta \left(c\right)\right){\vartheta }^{\Delta }\left(\varrho \right)\mathrm{for}c\in [\varrho ,\sigma \left(\varrho \right)].$$

(13)

We next introduce the concept of delta integral and summarize its key properties.

Definition 2

([14,15]). A function G is called a Δ-antiderivative of $\vartheta :\mathbb{T}\to \mathbb{R}$ if

$$G^{\Delta }\left(v\right)=\vartheta \left(v\right)\forall v\in {\mathbb{T}}^k.$$

The delta integral of ϑ from u to w is then defined by

$${\int }_u^w\vartheta \left(v\right)\Delta v:=G\left(w\right)-G\left(u\right).$$

Theorem 2

([14,15]). Let $w_0,w\in \mathbb{T}$. Then

$${\left({\int }_{w_0}^w\vartheta \left(v\right)\Delta v\right)}^{\Delta }=\vartheta \left(w\right).$$

(14)

Theorem 3

([14,15]). Let $a,w_0,w\in \mathbb{T},$ and let $c,l\in \mathbb{R}.$ Suppose $\Im ,\vartheta \in {\mathrm{C}}_{rd}{([a,c]}_{\mathbb{T}},$$\mathbb{R})$. Then

1. 

${\int }_a^{w_0}\left[c\Im \left(v\right)+l\vartheta \left(v\right)\right]\Delta v=c{\int }_a^{w_0}\Im \left(v\right)\Delta v+l{\int }_a^{w_0}\vartheta \left(v\right)\Delta v$;

2. 

${\int }_a^{w_0}\Im \left(v\right)\Delta v={\int }_a^w\Im \left(v\right)\Delta v+{\int }_w^{w_0}\Im \left(v\right)\Delta v$;

3. 

If $\Im \left(v\right)\ge 0;$ $\forall v\in {[a,w_0]}_{\mathbb{T}},$ then ${\int }_a^{w_0}\Im \left(v\right)\Delta v\ge 0.$

Moreover, we highlight two key special cases:

• If $\mathbb{T}=\mathbb{R}$, then

  $$\sigma \left(v\right)=v,\mu \left(v\right)=0,{\Im }^{\Delta }\left(v\right)={\Im }^{\prime }\left(v\right)\mathrm{and}{\int }_a^b\Im \left(v\right)\Delta v={\int }_a^b\Im \left(v\right)dv.$$

  (15)

• If $\mathbb{T}=\mathbb{Z}$, then

  $$\sigma \left(v\right)=v+1,\mu \left(v\right)=1,{\Im }^{\Delta }\left(v\right)=\Delta \Im \left(v\right)\mathrm{and}{\int }_a^b\Im \left(v\right)\Delta v=\sum _{v=a}^{b-1}\Im \left(v\right).$$

  (16)

Lemma 1

(Hölder’s Inequality [15,16]). If $a,c\in \mathbb{T}$ and $\Im ,\mathrm{{\rm Y}}\in {\mathrm{C}}_{rd}{([a,c]}_{\mathbb{T}},$${\mathbb{R}}^{+}),$ then

$${\int }_a^c\Im \left(\delta \right)\mathrm{{\rm Y}}\left(\delta \right)\Delta \delta \le {\left[{\int }_a^c{\Im }^r\left(\delta \right)\Delta \delta \right]}^{{\displaystyle \frac{1}{r}}}{\left[{\int }_a^c{\mathrm{{\rm Y}}}^{\lambda }\left(\delta \right)\Delta \delta \right]}^{{\displaystyle \frac{1}{\lambda }}},$$

(17)

where $r>1$ and $1/r+1/\lambda =1.$

3. Main Results

This section is devoted to presenting and proving the main results. During the work, we will make the assumption that the functions are positive and that the integrals under consideration exist. The proofs are based on the following fundamental inequality (see [17], p. 51)

$$C^{\lambda }+D^{\lambda }\le {\left(C+D\right)}^{\lambda }\le 2^{\lambda -1}\left(C^{\lambda }+D^{\lambda }\right),\mathrm{if}C,D\ge 0,\lambda \ge 1;$$

(18)

Theorem 4.

Let $𝚤,m\in \mathbb{T}$, $\eta \in C_{rd}({[𝚤,m]}_{\mathbb{T}},{\mathbb{R}}^{+})$ be a non-increasing function on ${[𝚤,m]}_{\mathbb{T}}$, $p\ge 0$ and $𝓁\ge 1$. If $\chi :{[𝚤,m]}_{\mathbb{T}}⟶\mathbb{R}$ is Δ-differentiable with $\chi \left(𝚤\right)\ne 0$, then

$$\begin{array}{ccc}\hfill {\int }_{𝚤}^m{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p{\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta & \le & 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{(m-𝚤)}^p{\int }_{𝚤}^m\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{p+𝓁}\Delta \vartheta \hfill \\ & & +2^{p-1}{\left|\chi \left(𝚤\right)\right|}^p{\int }_{𝚤}^m\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta .\hfill \end{array}$$

(19)

Proof. 

Define the function ${\mathrm{{\rm Y}}}_1\left(\vartheta \right)$ by

$${\mathrm{{\rm Y}}}_1\left(\vartheta \right)={\int }_{𝚤}^{\vartheta }{\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\epsilon \right){\left|{\chi }^{\Delta }\left(\epsilon \right)\right|}^{𝓁}\Delta \epsilon ,$$

(20)

so

$${\mathrm{{\rm Y}}}_1\left(𝚤\right)=0,and{\mathrm{{\rm Y}}}_1^{\Delta }\left(\vartheta \right)={\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\ge 0.$$

(21)

Since

$$\left|\chi \left(\vartheta \right)\right|-\left|\chi \left(𝚤\right)\right|\le |\chi \left(\vartheta \right)-\chi \left(𝚤\right)|\le |{\int }_{𝚤}^{\vartheta }{\chi }^{\Delta }\left(\epsilon \right)\Delta \epsilon |\le {\int }_{𝚤}^{\vartheta }\left|{\chi }^{\Delta }\left(\epsilon \right)\right|\Delta \epsilon .$$

Then

$$\left|\chi \left(\vartheta \right)\right|\le {\int }_{𝚤}^{\vartheta }|{\chi }^{\Delta }\left(\epsilon \right)|\Delta \epsilon +|\chi \left(𝚤\right)|={\int }_{𝚤}^{\vartheta }{\eta }^{{\displaystyle \frac{-1}{p+𝓁}}}\left(\epsilon \right){\eta }^{{\displaystyle \frac{1}{p+𝓁}}}\left(\epsilon \right)|{\chi }^{\Delta }\left(\epsilon \right)|\Delta \epsilon +|\chi \left(𝚤\right)|.$$

Since the weight function $\eta$ is positive and integrable on ${[𝚤,m]}_{\mathbb{T}}$, Hölder’s inequality (17) can be applied with indices ℓ and ${\displaystyle \frac{𝓁}{𝓁-1}}$, when $𝓁>1$, we have

$$\begin{array}{ccc}\hfill \left|\chi \right(\vartheta \left)\right|& \le & {\left({\int }_{𝚤}^{\vartheta }{\left({\eta }^{{\displaystyle \frac{-1}{p+𝓁}}}\left(\epsilon \right)\right)}^{{\displaystyle \frac{𝓁}{𝓁-1}}}\Delta \epsilon \right)}^{{\displaystyle \frac{𝓁-1}{𝓁}}}{\left({\int }_{𝚤}^{\vartheta }{\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\epsilon \right){\left|{\chi }^{\Delta }\left(\epsilon \right)\right|}^{𝓁}\Delta \epsilon \right)}^{{\displaystyle \frac{1}{𝓁}}}+\left|\chi \left(𝚤\right)\right|\hfill \\ & \le & {\eta }^{{\displaystyle \frac{-1}{p+𝓁}}}\left(\vartheta \right){(\vartheta -𝚤)}^{{\displaystyle \frac{𝓁-1}{𝓁}}}{\mathrm{{\rm Y}}}_1^{{\displaystyle \frac{1}{𝓁}}}\left(\vartheta \right)+\left|\chi \left(𝚤\right)\right|.\hfill \end{array}$$

(22)

By taking the power p of (22), we get

$${\left|\chi \left(\vartheta \right)\right|}^p\le {\left[{\eta }^{{\displaystyle \frac{-1}{p+𝓁}}}\left(\vartheta \right){(\vartheta -𝚤)}^{{\displaystyle \frac{𝓁-1}{𝓁}}}{\mathrm{{\rm Y}}}_1^{{\displaystyle \frac{1}{𝓁}}}\left(\vartheta \right)+\left|\chi \left(𝚤\right)\right|\right]}^p.$$

(23)

Now, by applying (18) on (23), it follows that

$${\left|\chi \left(\vartheta \right)\right|}^p\le 2^{p-1}{\eta }^{{\displaystyle \frac{-p}{p+𝓁}}}\left(\vartheta \right){(\vartheta -𝚤)}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}{\mathrm{{\rm Y}}}_1^{{\displaystyle \frac{p}{𝓁}}}\left(\vartheta \right)+2^{p-1}{\left|\chi \left(𝚤\right)\right|}^p,$$

(24)

which yields that

$${\eta }^{{\displaystyle \frac{p}{p+𝓁}}}{\left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p\le 2^{p-1}{(\vartheta -𝚤)}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}{\mathrm{{\rm Y}}}_1^{{\displaystyle \frac{𝓁}{𝓁}}}\left(\vartheta \right)+2^{p-1}{\eta }^{{\displaystyle \frac{p}{p+𝓁}}}\left(\vartheta \right){\left|\chi \left(𝚤\right)\right|}^p.$$

(25)

Thus, clearly in the case when $𝓁=1$, we get

$$\left|\chi \left(\vartheta \right)\right|\le {\eta }^{{\displaystyle \frac{-1}{p+1}}}\left(\vartheta \right){\mathrm{{\rm Y}}}_1\left(\vartheta \right)+\left|\chi \left(𝚤\right)\right|.$$

(26)

Now, from (25) and (21), we obtain

$$\begin{array}{ccc}& & {\int }_{𝚤}^m{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p|{\chi }^{\Delta }{\left(\vartheta \right)|}^{𝓁}\Delta \vartheta ={\int }_{𝚤}^m{\eta }^{{\displaystyle \frac{p}{p+𝓁}}}{\left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p{\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & \le & {\int }_{𝚤}^m\left[2^{p-1}{(\vartheta -𝚤)}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}{\mathrm{{\rm Y}}}_1^{{\displaystyle \frac{p}{𝓁}}}\left(\vartheta \right)+2^{p-1}{\eta }^{{\displaystyle \frac{p}{p+𝓁}}}\left(\vartheta \right){\left|\chi \left(𝚤\right)\right|}^p\right]{\mathrm{{\rm Y}}}_1^{\Delta }\left(\vartheta \right)\Delta \vartheta \hfill \\ & =& {\int }_{𝚤}^m{2}^{p-1}{(\vartheta -𝚤)}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}{\mathrm{{\rm Y}}}_1^{{\displaystyle \frac{p}{𝓁}}}\left(\vartheta \right){\mathrm{{\rm Y}}}_1^{\Delta }\left(\vartheta \right)\Delta \vartheta +{\int }_{𝚤}^m{2}^{p-1}{\eta }^{{\displaystyle \frac{p}{p+𝓁}}}\left(\vartheta \right){\left|\chi \left(𝚤\right)\right|}^p{\mathrm{{\rm Y}}}_1^{\Delta }\left(\vartheta \right)\Delta \vartheta .\hfill \end{array}$$

Moreover, it follows by (21), that

$$\begin{array}{ccc}& & {\int }_{𝚤}^m{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p{\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & \le & 2^{p-1}{(m-𝚤)}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}{\int }_{𝚤}^m{\mathrm{{\rm Y}}}_1^{{\displaystyle \frac{p}{𝓁}}}\left(\vartheta \right){\mathrm{{\rm Y}}}_1^{\Delta }\left(\vartheta \right)\Delta \vartheta +2^{p-1}{\int }_{𝚤}^m{\eta }^{{\displaystyle \frac{p}{p+𝓁}}}{\left(\vartheta \right)\left|\chi \left(𝚤\right)\right|}^p{\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & \le & 2^{p-1}{(m-𝚤)}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}{\int }_{𝚤}^m{\mathrm{{\rm Y}}}_1^{{\displaystyle \frac{p}{𝓁}}}\left(\vartheta \right){\mathrm{{\rm Y}}}_1^{\Delta }\left(\vartheta \right)\Delta \vartheta +2^{p-1}{\left|\chi \left(𝚤\right)\right|}^p{\int }_{𝚤}^m\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta .\hfill \end{array}$$

(27)

From (21), and (13), we obtain

$${\left[{\mathrm{{\rm Y}}}_1^{{\displaystyle \frac{p+𝓁}{𝓁}}}\left(\vartheta \right)\right]}^{\Delta }={\displaystyle \frac{p+𝓁}{𝓁}}{\mathrm{{\rm Y}}}_1^{{\displaystyle \frac{p}{𝓁}}}\left(c\right){\mathrm{{\rm Y}}}_1^{\Delta }\left(\vartheta \right),c\in [\vartheta ,\sigma \left(\vartheta \right)].$$

Since ${\mathrm{{\rm Y}}}_1^{\Delta }\left(\vartheta \right)\ge 0$ and $\vartheta <c$, we get

$${\left[{\mathrm{{\rm Y}}}_1^{{\displaystyle \frac{p+𝓁}{𝓁}}}\left(\vartheta \right)\right]}^{\Delta }={\displaystyle \frac{p+𝓁}{𝓁}}{\mathrm{{\rm Y}}}_1^{{\displaystyle \frac{p}{𝓁}}}\left(c\right){\mathrm{{\rm Y}}}_1^{\Delta }\left(\vartheta \right)\ge {\displaystyle \frac{p+𝓁}{𝓁}}{\mathrm{{\rm Y}}}_1^{{\displaystyle \frac{p}{𝓁}}}\left(\vartheta \right){\mathrm{{\rm Y}}}_1^{\Delta }\left(\vartheta \right).$$

(28)

Substituting from (28) into (27) and using ${\mathrm{{\rm Y}}}_1\left(𝚤\right)=0$, we have

$$\begin{array}{ccc}& & {\int }_{𝚤}^m{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p{\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & \le & 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{(m-𝚤)}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}{\int }_{𝚤}^m{\left[{\mathrm{{\rm Y}}}_1^{{\displaystyle \frac{p+𝓁}{𝓁}}}\left(\vartheta \right)\right]}^{\Delta }+2^{p-1}{\left|\chi \left(𝚤\right)\right|}^p{\int }_{𝚤}^m\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & =& 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{(m-𝚤)}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}\left[{\mathrm{{\rm Y}}}_1^{{\displaystyle \frac{p+𝓁}{𝓁}}}\left(m\right)\right]+2^{p-1}{\left|\chi \left(𝚤\right)\right|}^p{\int }_{𝚤}^m\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta .\hfill \end{array}$$

(29)

Now, by applying (17) with indices ${\displaystyle \frac{p+𝓁}{𝓁}}$ and ${\displaystyle \frac{𝓁}{p+𝓁}}$ on (20), we have

$$\begin{array}{ccc}\hfill {\mathrm{{\rm Y}}}_1\left(m\right)& =& {\int }_{𝚤}^m{\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & \le & {\left({\int }_{𝚤}^{m}1\Delta \vartheta \right)}^{{\displaystyle \frac{p}{p+𝓁}}}{\left({\int }_{𝚤}^m{\left({\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\right)}^{{\displaystyle \frac{p+𝓁}{𝓁}}}\Delta \vartheta \right)}^{{\displaystyle \frac{𝓁}{p+𝓁}}}\hfill \\ & =& {(m-𝚤)}^{{\displaystyle \frac{p}{p+𝓁}}}{\left({\int }_{𝚤}^m{\left({\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\right)}^{{\displaystyle \frac{p+𝓁}{𝓁}}}\Delta \vartheta \right)}^{{\displaystyle \frac{𝓁}{p+𝓁}}}.\hfill \end{array}$$

(30)

Substituting (30) into (29), we have

$$\begin{array}{ccc}\hfill {\int }_{𝚤}^m{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p{\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta & \le & 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{(m-𝚤)}^p{\int }_{𝚤}^m\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{p+𝓁}\Delta \vartheta \hfill \\ & & +2^{p-1}{\left|\chi \left(𝚤\right)\right|}^p{\int }_{𝚤}^m\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta ,\hfill \end{array}$$

which is (19). □

The following result can be regarded as a dual version of Theorem 4, established under analogous assumptions, where the boundary condition is taken at the right endpoint of the interval.

Theorem 5.

Let $m,𝚥\in \mathbb{T}$ and $\eta \in C_{rd}({[m,𝚥]}_{\mathbb{T}},{\mathbb{R}}^{+})$ be a non-increasing function on ${[m,𝚥]}_{\mathbb{T}},$$p\ge 0$ and $𝓁\ge 1$. Suppose that $\chi :{[m,𝚥]}_{\mathbb{T}}⟶\mathbb{R}$ is Δ-differentiable with $\chi \left(𝚥\right)\ne 0$. Then

$$\begin{array}{ccc}\hfill {\int }_m^{𝚥}\eta \left(\vartheta \right)|{\chi }^{\sigma }{\left(\vartheta \right)|}^p{\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta & \le & 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{(𝚥-m)}^p{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{p+𝓁}\Delta \vartheta \hfill \\ & & +2^{p-1}{\left|\chi \left(𝚥\right)\right|}^p{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta .\hfill \end{array}$$

(31)

Proof. 

Define the function ${\mathrm{{\rm Y}}}_2\left(\vartheta \right)$ by

$${\mathrm{{\rm Y}}}_2\left(\vartheta \right)={\int }_{\vartheta }^{𝚥}{\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\epsilon \right){\left|{\chi }^{\Delta }\left(\epsilon \right)\right|}^{𝓁}\Delta \epsilon ,$$

(32)

so

$${\mathrm{{\rm Y}}}_2\left(𝚥\right)=0,\mathrm{and}{\mathrm{{\rm Y}}}_2^{\Delta }\left(\vartheta \right)=-{\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\ge 0.$$

(33)

Since

$$|{\chi }^{\sigma }\left(\vartheta \right)|-|\chi \left(𝚥\right)|\le |{\chi }^{\sigma }\left(\vartheta \right)-\chi \left(𝚥\right)|=|\chi \left(𝚥\right)-{\chi }^{\sigma }\left(\vartheta \right)|\le |{\int }_{\sigma \left(\vartheta \right)}^{𝚥}{\chi }^{\Delta }\left(\epsilon \right)\Delta \epsilon |\le {\int }_{\sigma \left(\vartheta \right)}^{𝚥}\left|{\chi }^{\Delta }\left(\epsilon \right)\right|\Delta \epsilon .$$

Then

$$|{\chi }^{\sigma }\left(\vartheta \right)|\le {\int }_{\sigma \left(\vartheta \right)}^{𝚥}|{\chi }^{\Delta }\left(\epsilon \right)|\Delta \epsilon +|\chi \left(𝚥\right)|={\int }_{\sigma \left(\vartheta \right)}^{𝚥}{\eta }^{{\displaystyle \frac{-1}{p+𝓁}}}\left(\epsilon \right){\eta }^{{\displaystyle \frac{1}{p+𝓁}}}\left(\epsilon \right)|{\chi }^{\Delta }\left(\epsilon \right)|\Delta \epsilon +|\chi \left(𝚥\right)|.$$

Since the weight function $\eta$ is positive and integrable on ${[𝚤,m]}_{\mathbb{T}}$, Hölder’s inequality (17) can be applied with indices ℓ and ${\displaystyle \frac{𝓁}{𝓁-1}}$, when $𝓁>1$, we have

$$\begin{array}{ccc}\hfill |{\chi }^{\sigma }\left(\vartheta \right)|& \le & {\left({\int }_{\sigma \left(\vartheta \right)}^{𝚥}{\left({\eta }^{{\displaystyle \frac{-1}{p+𝓁}}}\left(\epsilon \right)\right)}^{{\displaystyle \frac{𝓁}{𝓁-1}}}\Delta \epsilon \right)}^{{\displaystyle \frac{𝓁-1}{𝓁}}}{\left({\int }_{\sigma \left(\vartheta \right)}^{𝚥}{\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\epsilon \right){\left|{\chi }^{\Delta }\left(\epsilon \right)\right|}^{𝓁}\Delta \epsilon \right)}^{{\displaystyle \frac{1}{𝓁}}}+\left|\chi \left(𝚥\right)\right|\hfill \\ & \le & {\eta }^{{\displaystyle \frac{-1}{p+𝓁}}}\left(\vartheta \right){(𝚥-\sigma \left(\vartheta \right))}^{{\displaystyle \frac{𝓁-1}{𝓁}}}{\left({\mathrm{{\rm Y}}}_2^{\sigma }\right)}^{{\displaystyle \frac{1}{𝓁}}}\left(\vartheta \right)+\left|\chi \left(𝚥\right)\right|.\hfill \end{array}$$

(34)

Taking the power p of (34), we get

$$|{\chi }^{\sigma }{\left(\vartheta \right)|}^p\le {\left[{\eta }^{{\displaystyle \frac{-1}{p+𝓁}}}\left(\vartheta \right){(𝚥-\sigma \left(\vartheta \right))}^{{\displaystyle \frac{𝓁-1}{𝓁}}}{\left({\mathrm{{\rm Y}}}_2^{\sigma }\right)}^{{\displaystyle \frac{1}{𝓁}}}\left(\vartheta \right)+\left|\chi \left(𝚥\right)\right|\right]}^p.$$

(35)

Now, by applying (18) on (35), we deduce

$$|{\chi }^{\sigma }{\left(\vartheta \right)|}^p\le 2^{p-1}{\eta }^{{\displaystyle \frac{-p}{p+𝓁}}}\left(\vartheta \right){(𝚥-\sigma \left(\vartheta \right))}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}{\left({\mathrm{{\rm Y}}}_2^{\sigma }\right)}^{{\displaystyle \frac{p}{𝓁}}}\left(\vartheta \right)+2^{p-1}{\left|\chi \left(𝚥\right)\right|}^p,$$

(36)

which yields that

$${\eta }^{{\displaystyle \frac{p}{p+𝓁}}}\left(\vartheta \right)|{\chi }^{\sigma }{\left(\vartheta \right)|}^p\le 2^{p-1}{(𝚥-\sigma \left(\vartheta \right))}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}{\left({\mathrm{{\rm Y}}}_2^{\sigma }\right)}^{{\displaystyle \frac{p}{𝓁}}}\left(\vartheta \right)+2^{p-1}{\eta }^{{\displaystyle \frac{p}{p+𝓁}}}\left(\vartheta \right){\left|\chi \left(𝚥\right)\right|}^p.$$

(37)

In the special case when $𝓁=1$, we get

$$|{\chi }^{\sigma }\left(\vartheta \right)|\le {\eta }^{{\displaystyle \frac{-1}{p+1}}}\left(\vartheta \right){\mathrm{{\rm Y}}}_2^{\sigma }\left(\vartheta \right)+\left|\chi \left(𝚥\right)\right|.$$

(38)

From (37) and (33), we have

$$\begin{array}{ccc}& & {\int }_m^{𝚥}\eta \left(\vartheta \right)|{\chi }^{\sigma }{\left(\vartheta \right)|}^p|{\chi }^{\Delta }{\left(\vartheta \right)|}^{𝓁}\Delta \vartheta ={\int }_m^{𝚥}{\eta }^{{\displaystyle \frac{p}{p+𝓁}}}\left(\vartheta \right)|{\chi }^{\sigma }{\left(\vartheta \right)|}^p{\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & \le & {\int }_m^{𝚥}\left[2^{p-1}{(𝚥-\sigma \left(\vartheta \right))}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}{\left({\mathrm{{\rm Y}}}_2^{\sigma }\right)}^{{\displaystyle \frac{p}{𝓁}}}\left(\vartheta \right)+2^{p-1}{\eta }^{{\displaystyle \frac{p}{p+𝓁}}}\left(\vartheta \right){\left|\chi \left(𝚥\right)\right|}^p\right][-{\mathrm{{\rm Y}}}_2^{\Delta }\left(\vartheta \right)]\Delta \vartheta \hfill \\ & =& {\int }_m^{𝚥}{2}^{p-1}{(𝚥-\sigma \left(\vartheta \right))}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}{\left({\mathrm{{\rm Y}}}_2^{\sigma }\right)}^{{\displaystyle \frac{p}{𝓁}}}\left(\vartheta \right)(-{\mathrm{{\rm Y}}}_2^{\Delta }\left(\vartheta \right))\Delta \vartheta +{\int }_m^{𝚥}{2}^{p-1}{\eta }^{{\displaystyle \frac{p}{p+𝓁}}}\left(\vartheta \right){\left|\chi \left(𝚥\right)\right|}^p(-{\mathrm{{\rm Y}}}_2^{\Delta }\left(\vartheta \right))\Delta \vartheta ,\hfill \end{array}$$

from (33), we get

$$\begin{array}{ccc}& & {\int }_m^{𝚥}\eta \left(\vartheta \right)|{\chi }^{\sigma }{\left(\vartheta \right)|}^p{\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & \le & 2^{p-1}{(𝚥-m)}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}{\int }_m^{𝚥}-{\left({\mathrm{{\rm Y}}}_2^{\sigma }\right)}^{{\displaystyle \frac{p}{𝓁}}}\left(\vartheta \right){\mathrm{{\rm Y}}}_2^{\Delta }\left(\vartheta \right)\Delta \vartheta +2^{p-1}{\int }_m^{𝚥}{\eta }^{{\displaystyle \frac{p}{p+𝓁}}}\left(\vartheta \right)|\chi \left(𝚥\right){|}^p{\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & \le & 2^{p-1}{(𝚥-m)}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}{\int }_m^{𝚥}-{\left({\mathrm{{\rm Y}}}_2^{\sigma }\right)}^{{\displaystyle \frac{p}{𝓁}}}\left(\vartheta \right){\mathrm{{\rm Y}}}_2^{\Delta }\left(\vartheta \right)\Delta \vartheta +2^{p-1}|\chi \left(𝚥\right){|}^p{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta .\hfill \end{array}$$

(39)

Using (13), and (33), we get

$${[-{\left({\mathrm{{\rm Y}}}_2\right)}^{{\displaystyle \frac{p+𝓁}{𝓁}}}\left(\vartheta \right)]}^{\Delta }=-{\displaystyle \frac{p+𝓁}{𝓁}}{\left({\mathrm{{\rm Y}}}_2\right)}^{{\displaystyle \frac{p}{𝓁}}}\left(c\right){\mathrm{{\rm Y}}}_2^{\Delta }\left(\vartheta \right),c\in [\vartheta ,\sigma \left(\vartheta \right)].$$

Since ${\mathrm{{\rm Y}}}_2^{\Delta }\left(\vartheta \right)\ge 0$ and $c<\sigma \left(\vartheta \right)$, we obtain

$${[-{\left({\mathrm{{\rm Y}}}_2\right)}^{{\displaystyle \frac{p+𝓁}{𝓁}}}\left(\vartheta \right)]}^{\Delta }=-{\displaystyle \frac{p+𝓁}{𝓁}}{\left({\mathrm{{\rm Y}}}_2\right)}^{{\displaystyle \frac{p}{𝓁}}}\left(c\right){\mathrm{{\rm Y}}}_2^{\Delta }\left(\vartheta \right)\ge -{\displaystyle \frac{p+𝓁}{𝓁}}{\left({\mathrm{{\rm Y}}}_2^{\sigma }\right)}^{{\displaystyle \frac{p}{𝓁}}}\left(\vartheta \right){\mathrm{{\rm Y}}}_2^{\Delta }\left(\vartheta \right).$$

(40)

Substituting (40) into (39) and using the fact that ${\mathrm{{\rm Y}}}_2\left(𝚥\right)=0$, we have

$$\begin{array}{ccc}& & {\int }_m^{𝚥}\eta \left(\vartheta \right)|{\chi }^{\sigma }{\left(\vartheta \right)|}^p{\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & \le & 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{(𝚥-m)}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}{\int }_m^{𝚥}{[-\left({\mathrm{{\rm Y}}}_2^{{\displaystyle \frac{p+𝓁}{𝓁}}}\left(\vartheta \right)\right)]}^{\Delta }+2^{p-1}{\left|\chi \left(𝚥\right)\right|}^p{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & =& 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{(𝚥-m)}^{{\displaystyle \frac{p(𝓁-1)}{𝓁}}}\left[{\mathrm{{\rm Y}}}_2^{{\displaystyle \frac{p+𝓁}{𝓁}}}\left(m\right)\right]+2^{p-1}{\left|\chi \left(𝚥\right)\right|}^p{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta .\hfill \end{array}$$

(41)

Using (17) in (32) with indices ${\displaystyle \frac{p+𝓁}{𝓁}}$ and ${\displaystyle \frac{𝓁}{p+𝓁}}$, we have

$$\begin{array}{ccc}\hfill {\mathrm{{\rm Y}}}_2\left(m\right)& =& {\int }_m^{𝚥}{\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & \le & {\left({\int }_m^{𝚥}1\Delta \vartheta \right)}^{{\displaystyle \frac{p}{p+𝓁}}}{\left({\int }_m^{𝚥}{\left({\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\right)}^{{\displaystyle \frac{p+𝓁}{𝓁}}}\Delta \vartheta \right)}^{{\displaystyle \frac{𝓁}{p+𝓁}}}\hfill \\ & =& {(𝚥-m)}^{{\displaystyle \frac{p}{p+𝓁}}}{\left({\int }_m^{𝚥}{\left({\eta }^{{\displaystyle \frac{𝓁}{p+𝓁}}}\left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\right)}^{{\displaystyle \frac{p+𝓁}{𝓁}}}\Delta \vartheta \right)}^{{\displaystyle \frac{𝓁}{p+𝓁}}}.\hfill \end{array}$$

(42)

Substituting (42) into (41), we have

$$\begin{array}{ccc}\hfill {\int }_m^{𝚥}\eta \left(\vartheta \right)|{\chi }^{\sigma }{\left(\vartheta \right)|}^p{\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta & \le & 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{(𝚥-m)}^p{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{p+𝓁}\Delta \vartheta \hfill \\ & & +2^{p-1}{\left|\chi \left(𝚥\right)\right|}^p{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta .\hfill \end{array}$$

The proof is completed. □

Combining Theorems 4 and 5 by choosing $m={\displaystyle \frac{𝚤+𝚥}{2}},$ we obtain the following result.

Theorem 6.

Let $𝚤,𝚥\in \mathbb{T}$ and $\eta \in C_{rd}({[𝚤,𝚥]}_{\mathbb{T}},{\mathbb{R}}^{+})$ be a non-increasing function on ${[𝚤,𝚥]}_{\mathbb{T}},$ $p\ge 0$ and $𝓁\ge 1$. Suppose that $\chi :{[𝚤,𝚥]}_{\mathbb{T}}⟶\mathbb{R}$ is Δ-differentiable with $\chi \left(𝚤\right)\ne 0$ and $\chi \left(𝚥\right)\ne 0$. Then

$$\begin{array}{ccc}\hfill {\int }_{𝚤}^{𝚥}{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p{\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta & \le & 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{\left({\displaystyle \frac{𝚥-𝚤}{2}}\right)}^p{\int }_{𝚤}^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{p+𝓁}\Delta \vartheta \hfill \\ & +& 2^{p-1}{\left(\right|\chi \left(𝚤\right)|}^p+{\left|\chi \left(𝚥\right)\right|}^p){\int }_{𝚤}^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta .\hfill \end{array}$$

(43)

Proof. 

$$\begin{array}{ccc}\hfill {\int }_{𝚤}^{𝚥}{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p{\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta & =& {\int }_{𝚤}^m{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p|{\chi }^{\Delta }{\left(\vartheta \right)|}^{𝓁}\Delta \vartheta +{\int }_m^{𝚥}{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p{\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & \le & {\int }_{𝚤}^m{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p|{\chi }^{\Delta }{\left(\vartheta \right)|}^{𝓁}\Delta \vartheta +{\int }_m^{𝚥}\eta \left(\vartheta \right)|{\chi }^{\sigma }{\left(\vartheta \right)|}^p{\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & \le & 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{(m-𝚤)}^p{\int }_{𝚤}^m\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{p+𝓁}\Delta \vartheta \hfill \\ & & +2^{p-1}{\left|\chi \left(𝚤\right)\right|}^p{\int }_{𝚤}^m\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & & +2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{(𝚥-m)}^p{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{p+𝓁}\Delta \vartheta \hfill \\ & & +2^{p-1}{\left|\chi \left(𝚥\right)\right|}^p{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \hfill \\ & =& 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{\left({\displaystyle \frac{𝚥-𝚤}{2}}\right)}^p\hfill \\ & & \times \left[{\int }_{𝚤}^m\eta \left(\vartheta \right)|{\chi }^{\Delta }{\left(\vartheta \right)|}^{p+𝓁}\Delta \vartheta +{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{p+𝓁}\Delta \vartheta \right]\hfill \\ & & +2^{p-1}{\left(\right|\chi \left(𝚤\right)|}^p+{\left|\chi \left(𝚥\right)\right|}^p)\hfill \\ & & \times \left[{\int }_{𝚤}^m\eta \left(\vartheta \right)|{\chi }^{\Delta }{\left(\vartheta \right)|}^{𝓁}\Delta \vartheta +{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta \right]\hfill \\ & =& 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{\left({\displaystyle \frac{𝚥-𝚤}{2}}\right)}^p{\int }_{𝚤}^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{p+𝓁}\Delta \vartheta \hfill \\ & +& 2^{p-1}{\left(\right|\chi \left(𝚤\right)|}^p+{\left|\chi \left(𝚥\right)\right|}^p){\int }_{𝚤}^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\Delta }\left(\vartheta \right)\right|}^{𝓁}\Delta \vartheta .\hfill \end{array}$$

□

In the following, we present several corollaries that illustrate how the obtained dynamic inequalities reduce to their continuous and discrete counterparts. By taking particular choices of the time scale $\mathbb{T}$, namely $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{N}$, Theorems 4–6 yield a family of integral and summation inequalities that extend and unify many known results in the classical setting.

Corollary 1.

Putting $\mathbb{T}=\mathbb{R}$, in Theorem 4, then (19), takes the form

$$\begin{array}{ccc}\hfill {\int }_{𝚤}^m{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p{\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{𝓁}d\vartheta & \le & 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{(m-𝚤)}^p{\int }_{𝚤}^m\eta \left(\vartheta \right){\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{p+𝓁}d\vartheta \hfill \\ & & +2^{p-1}{\left|\chi \left(𝚤\right)\right|}^p{\int }_{𝚤}^m\eta \left(\vartheta \right){\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{𝓁}d\vartheta .\hfill \end{array}$$

(44)

Corollary 2.

If $\mathbb{T}=\mathbb{R}$, in Theorem 5, then (31), become

$$\begin{array}{ccc}\hfill {\int }_m^{𝚥}{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p{\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{𝓁}d\vartheta & \le & 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{(𝚥-m)}^p{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{p+𝓁}d\vartheta \hfill \\ & & +2^{p-1}{\left|\chi \left(𝚥\right)\right|}^p{\int }_m^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{𝓁}d\vartheta .\hfill \end{array}$$

(45)

Corollary 3.

If we consider the case $\mathbb{T}=\mathbb{R}$, in Theorem 6, the inequality (43), will be

$$\begin{array}{ccc}\hfill {\int }_{𝚤}^{𝚥}{\eta \left(\vartheta \right)\left|\chi \left(\vartheta \right)\right|}^p{\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{𝓁}d\vartheta & \le & 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{\left({\displaystyle \frac{𝚥-𝚤}{2}}\right)}^p{\int }_{𝚤}^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{p+𝓁}d\vartheta \hfill \\ & +& 2^{p-1}{\left(\right|\chi \left(𝚤\right)|}^p+{\left|\chi \left(𝚥\right)\right|}^p){\int }_{𝚤}^{𝚥}\eta \left(\vartheta \right){\left|{\chi }^{\prime }\left(\vartheta \right)\right|}^{𝓁}d\vartheta .\hfill \end{array}$$

(46)

Corollary 4.

In Theorem 4, if $\mathbb{T}=\mathbb{N}$ and $p,𝓁\in {\mathbb{R}}^{+}$ s.t. $p\ge 0$ and $𝓁\ge 1,$${\left\{{\eta }_{𝚤}\right\}}_{0\le 𝚤\le N}$ is nonnegative real sequence, such that ${\left\{{\chi }_{𝚤}\right\}}_{0\le 𝚤\le N}$ is a sequence of real numbers with $\chi \left(𝚤\right)\ne 0$, then

$$\begin{array}{ccc}\hfill \sum _{n=𝚤}^{N-1}{\eta \left(n\right)\left|\chi \left(n\right)\right|}^p{|\Delta \chi \left(n\right)|}^{𝓁}& \le & 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{(m-𝚤)}^p\sum _{n=𝚤}^{N-1}\eta \left(n\right){|\Delta \chi \left(n\right)|}^{p+𝓁}\hfill \\ & & +2^{p-1}{\left|\chi \left(𝚤\right)\right|}^p\sum _{n=𝚤}^{N-1}\eta \left(n\right){|\Delta \chi \left(n\right)|}^{𝓁}.\hfill \end{array}$$

(47)

Corollary 5.

In Theorem 5, if $\mathbb{T}=\mathbb{N}$, and $p,𝓁\in {\mathbb{R}}^{+}$ s.t. $p\ge 0$ and $𝓁\ge 1$, ${\left\{{\eta }_{𝚤}\right\}}_{0\le 𝚤\le N}$ is nonnegative real sequence, such that ${\left\{{\chi }_{𝚤}\right\}}_{0\le 𝚤\le N}$ is a sequence of real numbers with $\chi \left(𝚥\right)\ne 0$, then

$$\begin{array}{ccc}\hfill \sum _{n=N}^{𝚥-1}{\eta \left(n\right)\left|\chi \left(n\right)\right|}^p{|\Delta \chi \left(n\right)|}^{𝓁}& \le & 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{(𝚥-m)}^p\sum _{n=N}^{𝚥-1}\eta \left(n\right){|\Delta \chi \left(n\right)|}^{p+𝓁}\hfill \\ & & +2^{p-1}{\left|\chi \left(𝚥\right)\right|}^p\sum _{n=N}^{𝚥-1}\eta \left(n\right){|\Delta \chi \left(n\right)|}^{𝓁}.\hfill \end{array}$$

(48)

Corollary 6.

In Theorem 6, if $\mathbb{T}=\mathbb{N}$, and $p,𝓁\in {\mathbb{R}}^{+}$ s.t. $p\ge 0$ and $𝓁\ge 1$, ${\left\{{\eta }_{𝚤}\right\}}_{0\le 𝚤\le N}$ is nonnegative real sequence, such that ${\left\{{\chi }_{𝚤}\right\}}_{0\le 𝚤\le N}$ is a sequence of real numbers with $\chi \left(𝚤\right)\ne 0$ and $\chi \left(𝚥\right)\ne 0$, then

$$\begin{array}{ccc}\hfill \sum _{n=𝚤}^{𝚥-1}{\eta \left(n\right)\left|\chi \left(n\right)\right|}^p{|\Delta \chi \left(n\right)|}^{𝓁}& \le & 2^{p-1}{\displaystyle \frac{𝓁}{p+𝓁}}{\left({\displaystyle \frac{𝚥-𝚤}{2}}\right)}^p\sum _{n=𝚤}^{𝚥-1}\eta \left(n\right){|\Delta \chi \left(n\right)|}^{p+𝓁}\hfill \\ & & +2^{p-1}{\left(\right|\chi \left(𝚤\right)|}^p+{\left|\chi \left(𝚥\right)\right|}^p){\int }_{n=𝚤}^{𝚥-1}\eta \left(n\right){|\Delta \chi \left(n\right)|}^{𝓁}.\hfill \end{array}$$

(49)

Remark 1.

Putting $\chi \left(𝚤\right)=0$, in Corollary 1, the inequality (44), reduces to the inequality (6).

Remark 2.

When $\chi \left(𝚥\right)=0$, in Corollary 2, (45) will be reduced to the inequality (7).

Remark 3.

When $\chi \left(𝚤\right)=0$ and $\chi \left(𝚥\right)=0$, in Corollary 3, (46) reduces to the inequality (8).

Remark 4.

When $\chi \left(𝚤\right)=0$ and $\eta \left(\vartheta \right)=1$, in Corollary 1, (44), reduces to the inequality (3).

Remark 5.

When $\chi \left(𝚤\right)=0$, $\eta \left(\vartheta \right)=1$ and $𝓁=1$, in Corollary 1, (44) reduces to the inequality (2).

Remark 6.

When $\chi \left(𝚤\right)=0$, in Theorem 4, (19) reduces to the inequality (10).

Remark 7.

When $\chi \left(𝚥\right)=0$, in Theorem 5, (31) reduces to the inequality (11).

Remark 8.

When $\chi \left(𝚤\right)=0$ and $\chi \left(𝚥\right)=0$, in Theorem 6, the inequality (43) reduces to the inequality (12).

Remark 9.

When $\chi \left(𝚤\right)=0$, in Corollary 4, the inequality (47) reduces to the following inequality

$$\sum _{n=𝚤}^{N-1}{\eta \left(n\right)\left|\chi \left(n\right)\right|}^p{|\Delta \chi \left(n\right)|}^{𝓁}\le {\displaystyle \frac{𝓁}{p+𝓁}}{(m-𝚤)}^p\sum _{n=𝚤}^{N-1}\eta \left(n\right){|\Delta \chi \left(n\right)|}^{p+𝓁}.$$

Remark 10.

When $\chi \left(𝚥\right)=0$, in Corollary 5, the inequality (48), reduces to the following inequality

$$\sum _{n=N}^{𝚥-1}{\eta \left(n\right)\left|\chi \left(n\right)\right|}^p{|\Delta \chi \left(n\right)|}^{𝓁}\le {\displaystyle \frac{𝓁}{p+𝓁}}{(𝚥-m)}^p\sum _{n=N}^{𝚥-1}\eta \left(n\right){|\Delta \chi \left(n\right)|}^{p+𝓁}.$$

Remark 11.

When $\chi \left(𝚤\right)=0$ and $\chi \left(𝚥\right)=0$, in Corollary 6, (49), reduces to the following inequality

$$\begin{array}{c}\hfill \sum _{n=𝚤}^{𝚥-1}{\eta \left(n\right)\left|\chi \left(n\right)\right|}^p{|\Delta \chi \left(n\right)|}^{𝓁}\le {\displaystyle \frac{𝓁}{p+𝓁}}{(𝚥-𝚤)}^p\sum _{n=𝚤}^{𝚥-1}\eta \left(n\right){|\Delta \chi \left(n\right)|}^{p+𝓁}.\end{array}$$

4. Conclusions

In this article, several weighted dynamic Opial-type inequalities involving integrals of the product of a function and its delta derivative on time scales have been established. The presented results extend and refine existing inequalities in the literature, with the continuous and discrete cases obtained as natural corollaries of the general dynamic forms. As a future direction, these inequalities may be further generalized to the setting of conformable fractional and diamond–alpha calculi on time scales, providing a unified framework for studying fractional dynamic systems and their qualitative properties.