Advanced Hardy-type inequalities with negative parameters involving monotone functions in delta calculus on time scales

1.   Introduction

In ^[1], Hardy established a foundational result in the theory of inequalities with positive parameters, demonstrating the discrete inequality:

where $\gamma > 1,$ $\mathcal{E}(s)\geq 0$ for $s\geq 1\ $, and $0 < \sum_{s = 1}^{\infty }\mathcal{E}^{\gamma }(s) < \infty.$ Hardy [2, Theorem A] also derived the corresponding integral inequality of (1.1):

where $\gamma > 1$, and ${\rm{℧}}(\mathfrak{r})\geq 0$ such that $0 < \int_{0}^{\infty }{\rm{℧}}^{\gamma }(\mathfrak{r})d\mathfrak{r} < \infty.$ The constant $(\gamma /(\gamma -1))^{\gamma }$ is optimal in both inequalities.

Since the emergence of these two inequalities (1.1) and (1.2), they have garnered significant attention from scientists and researchers, with many of them working on improving and generalizing them using various methods (see ^[3,4,5]). In parallel to advancements in positive parameter inequalities, there has been a growing interest in Hardy-type inequalities with negative parameters. For example, Bicheng ^[6] demonstrated that if $\gamma < 0,$ $\varrho \in \mathbb{R },$ $\varrho \neq 1,$ ${\rm{℧}}(\mathfrak{r})\geq 0$, and $0 < \int_{0}^{\infty }\mathfrak{r}^{-\varrho }\left(\mathfrak{r}{\rm{℧}}(\mathfrak{r})\right) ^{\gamma }d\mathfrak{r} < \infty,$ then

He also established that if $\varrho < 1,$ then

where $\left(\gamma /\left(1-\varrho \right) \right) ^{\gamma }$ and $\left(\gamma /\left(\varrho -1\right) \right) ^{\gamma }$ is optimal in both inequalities (1.3) and (1.4).

Further advancements are presented in ^[7], where the authors extended these results. They showed that if $\ \gamma < 0,$ $\varrho > 1$, and ${\rm{℧}}(\mathfrak{r}), \mathbb{\Im }(\mathfrak{r}) > 0$ such that $\mathfrak{r}/\mathbb{\Im }(\mathfrak{r})$ is a nondecreasing function, then

Additionally, if $0\leq \varrho < 1$, and ${\rm{℧}}(\mathfrak{r}), \mathbb{ \Im }(\mathfrak{r}) > 0$ such that $\mathfrak{r}/\mathbb{\Im }(\mathfrak{r})$ is a nonincreasing function, then

Moreover, if $\gamma < 0,$ $\varrho < 0$, and ${\rm{℧}}(\mathfrak{r}), \mathbb{\Im }(\mathfrak{r}) > 0$ such that $\mathfrak{r}/\mathbb{\Im }(\mathfrak{r})$ is a nondecreasing function, then

The transition from Hardy's inequalities with positive parameters to those involving negative parameters illustrates a rich field of study, revealing a deeper structure and broader applicability of these mathematical tools. These developments highlight the ongoing evolution in the theory of Hardy-type inequalities, encompassing both positive and negative parameter cases and their various generalizations.

More recently, many scientists have used the famous theory known as time scale theory to study various classical inequalities, especially the famous Hardy inequality.

Among these scientists was P. Řehák ^[8], who was able to obtain the time scale form of Hardy's inequality, making discrete inequality (1.1) and integral version (1.2) a special case of it. He proved that if $\mathbb{T}$ is a time scale, $\varrho > 1,$ and $\Omega (z) = \int_{d}^{z}\xi (\delta)\Delta \delta,$ for $z\in \lbrack d, \infty)_{ \mathbb{T}}$, then

unless $\xi \equiv 0$. If, in addition, $\mu (z)/z\rightarrow 0$ as $z\rightarrow \infty,$ then $\left(\frac{\varrho }{\varrho -1}\right) ^{\varrho }$ is sharp. Refer to Section 2 for the notations used here and for the calculus applied in proving the main results of this paper.

In ^[9], the authors presented a time scale form of (1.5)–(1.7) by using nabla calculus, respectively, as follows: Let $b\in \mathbb{T},$ $\varepsilon < 0,$ $\varepsilon ^{\ast } = \varepsilon /\left(\varepsilon -1\right),$ $\varrho > 1$ thinspace and ${\rm{℧}}, \mathbb{\Im }\in C_{ld}\left([b, \infty)_{\mathbb{T}}, \mathbb{R}^{+}\right)$ with $\left(\zeta -b\right) /\mathbb{\Im }\left(\zeta \right)$ is nondecreasing. If

then

where $\mathcal{G}(\zeta) = \int_{\zeta }^{\infty }{\rm{℧}}(\mathfrak{z})\nabla \mathfrak{z}$ and

Additionally, if $b\in \mathbb{T},$ $\varepsilon < 0,$ $0\leq \varrho < 1$, and ${\rm{℧}}, \mathbb{\Im }\in C_{ld}\left([b, \infty)_{\mathbb{T}}, \mathbb{R}^{+}\right)$ such that $\left(\zeta -b\right) /\mathbb{\Im }\left(\zeta \right)$ is a nonincreasing function. If (1.9) holds, then

where $\mathcal{M}(\zeta) = \int_{b}^{\zeta }{\rm{℧}}(\mathfrak{z})\nabla \mathfrak{z}$ and

Moreover, if $b\in \mathbb{T},$ $\varepsilon < 0,$ $\varrho < 0$, and ${\rm{℧}}, \mathbb{\Im }\in C_{ld}\left([b, \infty)_{\mathbb{T}}, \mathbb{R} ^{+}\right)$ such that $\left(\zeta -b\right) /\mathbb{\Im }\left(\zeta \right)$ is a nondecreasing function. If (1.9) holds, then

where $\mathcal{M}(\zeta) = \int_{b}^{\zeta }{\rm{℧}}(\mathfrak{z})\nabla \mathfrak{z}$ and

Recently, new results have emerged regarding the Hardy inequality through various types of time scale calculus, such as the time scale delta integral (see ^{[10,11,12,13]}), which broadens the applications of dynamic inequalities in studying the qualitative behavior of dynamic equations, as referenced in ^[14,15,16].

In fact, the study of Hardy's inequality with a negative parameter using the idea of time scale $\mathbb{T}$ has not been exposed to many researchers. Therefore, in this paper, we will attempt to obtain some new results in this area through time scale calculus. Specifically, we will prove a time scale version of (1.5)–(1.7) and also obtain the discrete analogues of these inequalities.

The organization of the paper is as follows. In Section 2, we present some lemmas on time scales. In Section 3, we state and prove our results.

2.   Definitions and basic lemmas

In 2001, Martin and Allan ^[17,18] introduced the concept of a time scale $\mathbb{T}$, which is defined as a nonempty closed subset of $\mathbb{R}$. For any $\varkappa, \varrho \in \mathbb{T},$ the forward jump operator is defined by $\sigma (\varkappa): = \inf \{s\in \mathbb{T} :s > \varkappa \}$ and the backward jump operator by $\rho (\varrho): = \sup \{s\in \mathbb{T}:s > \varrho \}.$ The graininess function $\mu$ for a time scale $\mathbb{T}$ is defined by $\mu (\tau): = \sigma (\tau)-\tau \geq 0.$ A point $\zeta \in \mathbb{T}$ is called:

$\bullet$ Right-dense if $\sigma (\zeta) = \zeta \mathfrak{; }$

$\bullet$ Left-dense if $\rho (\zeta) = \zeta \mathfrak{; }$

$\bullet$ Right-scattered if $\sigma (\zeta) > \zeta;$

$\bullet$ Left-scattered if $\rho (\zeta) < \zeta \mathfrak{.}$

If $\mathbb{T}$ has a left-scattered maximum $\eta$, then $\mathbb{T}^{k} = \mathbb{T}-\{\eta \};$ otherwise, $\mathbb{T}^{k} = \mathbb{T}$.

In the following, for a function ${\rm{℧}}:\mathbb{T}\rightarrow \mathbb{R},$ we denote ${\rm{℧}}(\sigma (\tau))$ as ${\rm{℧}}^{\sigma }(\tau).$ The notation $[\tau, \varrho ]\cap \mathbb{T}$ is denoted as $[\tau, \varrho ]_{\mathbb{T}}.$

Definitions:

$\bullet$ Rd-continuous function ^[17]:A function ${\rm{℧}} :\mathbb{T}\rightarrow \mathbb{R}$ is $rd$-continuous if it is continuous at right-dense points and has finite left-sided limits at left-dense points. The set of rd-continuous functions is denoted by $\mathrm{C}_{rd}({\mathbb{T }}, {\mathbb{R)}}$.

$\bullet$ Delta derivative ^[17]:For ${\rm{℧}}:\mathbb{T} \rightarrow \mathbb{R}$ and $\mathfrak{z}\in \mathbb{T},$ the delta derivative ${\rm{℧}}^{\Delta }(\mathfrak{z})$ exists if, for any $\varepsilon > 0$, there is a neighborhood $W = (\mathfrak{z}-\delta,$ $\mathfrak{z}+\delta)\cap \mathbb{T}$ of $\mathfrak{z}$ for some $\delta > 0$, such that

$\bullet$ Antiderivative and delta integral ^[17]:A function $\mathcal{G}:\mathbb{T}\rightarrow \mathbb{R}$ is a delta antiderivative of ${\rm{℧}}$ if $\mathcal{G}^{\Delta }(\mathfrak{z}) = {\rm{℧}}(\mathfrak{z}),$ $\forall \mathfrak{z}\in \mathbb{T}^{k}.$ The delta integral of ${\rm{℧}}$ is given by

It is noted that every rd-continuous function ${\rm{℧}}$ has an antiderivative. In particular, if $\mathfrak{z}_{0}\in \mathbb{T}$, then

We now present the main lemmas on $\mathbb{T}$ that will be utilized to support our conclusions.

Main lemmas:

$\bullet$ Chain rule [17, Theorem 1.90]: If $\mathbb{\Im }: \mathbb{R}\rightarrow \mathbb{R}$ is continuous, $\mathbb{\Im }: \mathbb{T}\rightarrow \mathbb{R}$ is $\Delta -$differentiable $,$ and ${\rm{℧}}:\mathbb{R}\rightarrow \mathbb{R}$ is continuously differentiable, then

$\bullet$ Integration by Parts ^[19]:For $\varrho, \tau \in \mathbb{T}$ and $\phi, \varphi \in \mathrm{C} _{rd}(\mathbb{[}\varrho, \tau \mathbb{]}_{\mathbb{T}}, \mathbb{R}),$

$\bullet$ Reversed Hölder's inequality ^[19]: For $\varrho, \tau \in \mathbb{T}$ and $\phi, \omega \in \mathrm{C}_{rd}(\mathbb{[}\varrho, \tau \mathbb{ ]}_{\mathbb{T}}, \mathbb{R}^{+}),$

where $\gamma < 0,$ and $1/\gamma +1/\nu = 1.$

3.   Main results

In this section, we present our key results. Prior to stating the upcoming theorem, we establish a few preliminary assumptions: all integrals considered throughout the paper are assumed to exist. Additionally, we assume the presence of a positive constant $\aleph \geq 1,$ such that

In the following theorem, we will present the time scale version of inequality (1.5).

Theorem 3.1. Consider $b\in \mathbb{T},$ $\gamma < 0,$ $\gamma ^{\ast } = \gamma /\left(\gamma -1\right),$ $\varrho > 1$, and ${\rm{℧}}, \mathbb{\Im }\in C_{rd}\left([b, \infty)_{\mathbb{T}}, \mathbb{R}^{+}\right)$ such that $\left(\mathfrak{r}-b\right) /\mathbb{\Im }\left(\mathfrak{r}\right)$ is nondecreasing. If (3.1) is satisfied, then

where $\mathcal{G}(\mathfrak{r}) = \int_{\mathfrak{r}}^{\infty }{\rm{℧}}(\mathfrak{z})\Delta \mathfrak{z}$ and

Proof. Start with

Applying reversed Hölder's inequality (2.3), we obtain

Applying (2.1) on $\left(\mathfrak{z}-b\right) ^{\frac{\varrho -1}{ \gamma }},$ we have

Since $\varrho > 1,$ $\gamma < 0$, and $d\geq \mathfrak{z},$ then $\left(\varrho -1-\gamma \right) /\gamma < 0,$ so

Substituting (3.6) into (3.5), we find

Integrating (3.7) over $\mathfrak{z}$ from $\sigma \left(\mathfrak{ r}\right)$ to $\infty,$ we observe

Since $\gamma ^{\ast } > 0,$ we obtain

Substituting (3.8) into (3.4), we get

From (3.3) and (3.9), we have for $\gamma < 0$ that

Multiplying (3.10) by $\left[ \mathbb{\Im }(\mathfrak{r})\right] ^{-\varrho }$ and then integrating over $\mathfrak{r}$ from $b$ to $\infty,$ we get

Applying (2.2) on $\int_{b}^{\infty }\left(\sigma \left(\mathfrak{r} \right) -b\right) ^{\frac{\varrho -1}{\gamma ^{\ast }}}\left[ \mathbb{\Im }(\mathfrak{r})\right] ^{-\varrho }\left(\int_{\sigma \left(\mathfrak{r} \right) }^{\infty }\left(\mathfrak{z}-b\right) ^{\frac{1+\gamma -\varrho }{ \gamma ^{\ast }}}\left[ {\rm{℧}}(\mathfrak{z})\right] ^{\gamma }\Delta \mathfrak{z}\right) \Delta \mathfrak{r}$, we conclude

where $u_{1}(\mathfrak{r}) = \int_{b}^{\mathfrak{r}}\left(\sigma \left(\mathfrak{z}\right) -b\right) ^{\frac{\varrho -1}{\gamma ^{\ast }}}\left[ \mathbb{\Im }(\mathfrak{z})\right] ^{-\varrho }\Delta \mathfrak{z}.$ Using (3.1) $,$ we have

Since $\left(\mathfrak{z}-b\right) /\mathbb{\Im }\left(\mathfrak{z}\right)$ is nondecreasing and $\varrho > 1$, we have for $\mathfrak{z}\leq \mathfrak{r}$ that

Substituting (3.13) into (3.12), we observe that

From (3.11) and (3.14), we get

Applying (2.1) on $\left(\mathfrak{z}-b\right) ^{\frac{1-\varrho }{ \gamma }},$ we observe

Now, we consider two cases:

Case 1: For $1-\varrho \leq \gamma,$ we have $\frac{1-\varrho }{\gamma }-1\geq 0.$ From (3.16), we have $\left(d-b\right) ^{\frac{1-\varrho }{ \gamma }-1}\geq \left(\mathfrak{z}-b\right) ^{\frac{1-\varrho }{\gamma }-1},$ and

Integrating (3.17) over $\mathfrak{z}$ from $b$ to $\mathfrak{r},$ we get

Using (3.1), (3.15), and (3.18), we have

which matches (3.2) with $\mathcal{Q} = \left(\gamma /\left(1-\varrho \right) \right) ^{\gamma }\aleph ^{\frac{\varrho -1}{\gamma ^{\ast }}}.$

Case 2: For $1-\varrho \geq \gamma,$ we have $\frac{1-\varrho }{\gamma }-1\leq 0.$ From (3.16), we see that $\left(d-b\right) ^{\frac{1-\varrho }{ \gamma }-1}\geq \left(\sigma \left(\mathfrak{z}\right) -b\right) ^{\frac{ 1-\varrho }{\gamma }-1},$and

Integrating (3.19) over $\mathfrak{z}$ from $b$ to $\mathfrak{r},$ we have

Substituting (3.20) into (3.15), we observe

which matches (3.2) with $\mathcal{Q} = \left(\gamma /\left(1-\varrho \right) \right) ^{\gamma }\aleph ^{\varrho }.$ □

Remark 3.1. In Theorem 3.1, when $\mathbb{T = R}$ and $b = 0,$ we have $\sigma \left(\mathfrak{r}\right) = \mathfrak{r}.$ Consequently, we see that (3.1) holds with $\aleph = 1.$ As a result, (3.2) simplifies to (1.5), and for $\mathbb{\Im }(\mathfrak{r}) = \mathfrak{r},$ we obtain (1.3).

Corollary 3.1. If $\mathbb{T = N}_{0},$ $b = 0,$ $\varrho > 1, \gamma < 0\ $, and $\left\{ s_{n}\right\} _{n = 0}^{\infty },$ $\left\{ t_{n}\right\} _{n = 0}^{\infty }$ are positive sequences such that $n/t_{n}$ is nondecreasing, then

where

Here,

Since $-1/\left(n+1\right) \geq -1/2,$ then $\left(n-b\right) /\left(\sigma \left(n\right) -b\right) \geq 1/2,$ and (3.1) holds with $\aleph = 2.$

Corollary 3.2. Let $\mathbb{T} = q^{\mathbb{N}_{0}}$ for $q > 1,$ $b\in \mathbb{T},$ $\gamma < 0,$ $\gamma ^{\ast } = \gamma /\left(\gamma -1\right),$ $\varrho > 1$, and ${\rm{℧}}, \mathbb{\Im }\ $be posituve sequences on $[b, \infty)$ such that $\left(\mathfrak{r}-b\right) /\mathbb{\Im }\left(\mathfrak{r}\right)$ is nondecreasing. If

then

where $\mathcal{G}(\mathfrak{r}) = \sum_{\mathfrak{z = r}}^{\infty }(q-1) \mathfrak{z}{\rm{℧}}(\mathfrak{z})$ and

In the following theorem, we will present the time scale version of inequality (1.6).

Theorem 3.2. Assume $b\in \mathbb{T},$ $\gamma < 0,$ $\gamma ^{\ast } = \gamma /\left(\gamma -1\right),$ $0\leq \varrho < 1$, and ${\rm{℧}}, \mathbb{\Im } \in C_{rd}\left([b, \infty)_{\mathbb{T}}, \mathbb{R}^{+}\right)$ such that $\left(\mathfrak{r}-b\right) /\mathbb{\Im }(\mathfrak{r})$ is nonincreasing. If (3.1) holds, then

where $\Omega (\mathfrak{r}) = \int_{b}^{\mathfrak{r}}{\rm{℧}}(\mathfrak{z})\Delta \mathfrak{z}$ and

Proof. To prove this theorem, we consider two cases:

Case 1: For $\left(\varrho -1\right) /\gamma \geq 1.$ Start with

Applying (2.3) on (3.22), we get

From this and the previous inequality, we have

Applying (2.1) on $\left(\mathfrak{z}-b\right) ^{\frac{\varrho -1}{ \gamma }},$ we observe that

Since $\left(\varrho -1\right) /\gamma \geq 1,$ then $\left(d-b\right) ^{ \frac{\varrho -\gamma -1}{\gamma }}\leq \left(\sigma \left(\mathfrak{z} \right) -b\right) ^{\frac{\varrho -\gamma -1}{\gamma }},$ and (3.24) becomes

By integrating (3.25) over $\mathfrak{z}$ from $b$ to $\sigma \left(\mathfrak{r}\right),$ we get

Substituting (3.26) into (3.23), since $\gamma ^{\ast } > 0$, we observe

For $\gamma < 0,$ this yields

Multiplying (3.27) by $\left[ \mathbb{\Im }(\mathfrak{r})\right] ^{-\varrho }$ and then integrating over $\mathfrak{r}$ from $b$ to $\infty,$ we find

Applying (2.2) on

we obtain

where

Using (3.1) $,$ we have

Since $\left(\mathfrak{z}-b\right) /\mathbb{\Im }(\mathfrak{z})$ is nonincreasing and $0\leq \varrho < 1,$ we have for $\mathfrak{z}\geq \mathfrak{r}$ that $\left(\left(\mathfrak{z}-b\right) /\mathbb{\Im }(\mathfrak{z})\right) ^{\varrho }\leq \left(\left(\mathfrak{r}-b\right) / \mathbb{\Im }(\mathfrak{r})\right) ^{\varrho }$, and then (3.29) becomes

From (3.16), since $\left(1-\varrho \right) /\gamma < 0,$ we have $\left(d-b\right) ^{\frac{1-\varrho }{\gamma }-1}\geq \left(\sigma \left(\mathfrak{z}\right) -b\right) ^{\frac{1-\varrho }{\gamma }-1}$ and

thus,

Substituting (3.31) into (3.30) and using (3.1), since $\left(1-\varrho \right) /\gamma ^{\ast }-1 > 0$, we observe

Substituting (3.32) into (3.28), we have

which matches (3.21) with $\mathcal{J} = \aleph ^{\frac{\varrho -1}{\gamma }}\left(\gamma /\left(\varrho -1\right) \right) ^{\gamma }.$

Case 2: For $\left(\varrho -1\right) /\gamma \leq 1.$ We have

Applying (2.3) on $\int_{b}^{\sigma \left(\mathfrak{r}\right) }\left(\mathfrak{z}-b\right) ^{\frac{\varrho -\gamma -1}{\gamma \gamma ^{\ast }}} \left[ \left(\mathfrak{z}-b\right) ^{\frac{1+\gamma -\varrho }{\gamma \gamma ^{\ast }}}{\rm{℧}}(\mathfrak{z})\right] \Delta \mathfrak{z},$we find

From (3.33) and (3.34), we have

Using (3.5), since $0 < \left(\varrho -1\right) /\gamma \leq 1,$ we have $\left(d-b\right) ^{\frac{\varrho -\gamma -1}{\gamma }}\leq \left(\mathfrak{ z}-b\right) ^{\frac{\varrho -\gamma -1}{\gamma }}$ and

and then

Substituting (3.36) into (3.35), since $\gamma ^{\ast } > 0$, we conclude

For $\gamma < 0,$ this yields

Multiplying the last inequality by $\left[ \mathbb{\Im }(\mathfrak{r})\right] ^{-\varrho }$ and then integrating over $\mathfrak{r}$ from $b$ to $\infty,$ we observe

Applying (2.2) on $\int_{b}^{\infty }\left(\sigma \left(\mathfrak{r} \right) -b\right) ^{\frac{\varrho -1}{\gamma ^{\ast }}}\left[ \mathbb{\Im }(\mathfrak{r})\right] ^{-\varrho }\left(\int_{b}^{\sigma \left(\mathfrak{r} \right) }\left(\mathfrak{z}-b\right) ^{\frac{1+\gamma -\varrho }{\gamma ^{\ast }}}\left[ {\rm{℧}}(\mathfrak{z})\right] ^{\gamma }\Delta \mathfrak{z }\right) \Delta \mathfrak{r}$, we obtain

where

With (3.1), $\left(\mathfrak{z}-b\right) /\mathbb{\Im }(\mathfrak{z})$ is nonincreasing, and $\varrho > 0$, we have for $\mathfrak{z}\geq \mathfrak{r}$ that

Since $\left(1-\varrho \right) /\gamma < 0,$ then by using (3.16), we have

therefore,

Substituting (3.39) into (3.38), we obtain

Substituting (3.40) into (3.37), we get

which matches (3.21) with $\mathcal{J} = \aleph ^{\varrho }\left(\gamma /\left(\varrho -1\right) \right) ^{\gamma }.$ □

Remark 3.2. In Theorem 3.2, when $\mathbb{T = R}$ and $b = 0,$ we have $\sigma \left(\mathfrak{r}\right) = \mathfrak{r}.$ Consequently, we see that (3.1) holds with $\aleph = 1.$ As a result, (3.21) simplifies to (1.6), and for $\mathbb{\Im }\left(\mathfrak{r}\right) = \mathfrak{r},$ we get (1.4).

Corollary 3.3. If $\mathbb{T = N}_{0}$, $b = 0$ and $\left\{ s_{n}\right\} _{n = 0}^{\infty },$ $\left\{ t_{n}\right\} _{n = 0}^{\infty }\ $ are positive sequences with the property that $n/t_{n}$ is nonincreasing, then

where

Here, the inequality (3.1) holds with $\aleph = 2.$

Corollary 3.4. Let $\mathbb{T} = q^{\mathbb{N}_{0}}$ for $q > 1,$ $b\in \mathbb{T},$ $\gamma < 0,$ $\gamma ^{\ast } = \gamma /\left(\gamma -1\right),$ $0\leq \varrho < 1$, and ${\rm{℧}}, \mathbb{\Im }$ be posituve sequences on $[b, \infty)$ such that $\left(\mathfrak{r}-b\right) /\mathbb{\Im }(\mathfrak{r})$ is nonincreasing. If

holds, then

where $\Omega (\mathfrak{r}) = \sum_{\mathfrak{z = }b}^{\mathfrak{r/}q}(q-1) \mathfrak{z}{\rm{℧}}(\mathfrak{z})$ and

In the following theorem, we will present the time scale version of inequality (1.7).

Theorem 3.3. Assume $b\in \mathbb{T},$ $\gamma < 0,$ $\gamma ^{\ast } = \gamma /\left(\gamma -1\right),$ $\varrho < 0\ $, and ${\rm{℧}}, \mathbb{\Im }\in C_{rd}\left([b, \infty)_{\mathbb{T}}, \mathbb{R}^{+}\right)$ such that $\left(\mathfrak{r}-b\right) /\mathbb{\Im }(\mathfrak{r})$ is nondecreasing. If (3.1) holds, then

where $\Omega (\mathfrak{r}) = \int_{b}^{\mathfrak{r}}{\rm{℧}}(\mathfrak{z})\Delta \mathfrak{z}$ and

Proof. We consider the following two cases to prove this theorem.

Case 1: For $\left(\varrho -1\right) /\gamma \leq 1.$ Start with

Applying (2.3) on $\int_{b}^{\sigma \left(\mathfrak{r}\right) }\left(\mathfrak{z}-b\right) ^{\frac{\varrho -\gamma -1}{\gamma \gamma ^{\ast }}} \left[ \left(\mathfrak{z}-b\right) ^{\frac{1+\gamma -\varrho }{\gamma \gamma ^{\ast }}}{\rm{℧}}(\mathfrak{z})\right] \Delta \mathfrak{z},$ we get

From (3.43) and (3.44), we have

Since $0 < \left(\varrho -1\right) /\gamma \leq 1,$ and by using (3.5), we find

thus,

Substituting (3.46) into (3.45), we conclude

For $\gamma < 0,$ we have

Multiplying the last inequality by $\left[ \mathbb{\Im }(\mathfrak{r}) \right] ^{-\varrho }$ and then integrating over $\mathfrak{r}$ from $b$ to $\infty,$ we observe

Applying (2.2) on

we observe that

where

Since $\sigma \left(\mathfrak{z}\right) \geq \mathfrak{z}$ and $\varrho < 0,$ we get

Since $\left(\mathfrak{z}-b\right) /\mathbb{\Im }(\mathfrak{z})$ is nondecreasing and $\varrho < 0,$ (3.48) becomes

Since $\varrho < 0\ $and $\gamma < 0,$ by using (3.16), we get

and then

Substituting (3.50) into (3.49), we obtain

Substituting (3.51) into (3.47), we observe

which is (3.42) with $\mathcal{M} = \left(\gamma /\left(\varrho -1\right) \right) ^{\gamma }.$

Case 2: For $\left(\varrho -1\right) /\gamma \geq 1.$ We have

Applying (2.3), we get

Substituting (3.53) into (3.52), we obtain

Since $\left(\varrho -1\right) /\gamma \geq 1,$ then by using (3.5), we have

By integrating (3.55) over $\mathfrak{z}$ from $b$ to $\sigma \left(\mathfrak{r}\right),$ we get

Substituting (3.56) into (3.54), since $\gamma ^{\ast } > 0$, we observe

For $\gamma < 0,$ this yields

Multiplying (3.57) with $\left[ \mathbb{\Im }(\mathfrak{r})\right] ^{-\varrho }$ and then integrating over $\mathfrak{r}$ from $b$ to $\infty,$ we see

Applying (2.2) on $\int_{b}^{\infty }\left(\sigma \left(\mathfrak{r} \right) -b\right) ^{\frac{\varrho -1}{\gamma ^{\ast }}}\left[ \mathbb{\Im }(\mathfrak{r})\right] ^{-\varrho }\left(\int_{b}^{\sigma \left(\mathfrak{r} \right) }\left(\sigma \left(\mathfrak{z}\right) -b\right) ^{\frac{1+\gamma -\varrho }{\gamma ^{\ast }}}\left[ {\rm{℧}}(\mathfrak{z})\right] ^{\gamma }\Delta \mathfrak{z}\right) \Delta \mathfrak{r},$ we get

where

Since $\sigma \left(\mathfrak{z}\right) \geq \mathfrak{z}$, and $\varrho < 0,$ we find

Since $\left(\mathfrak{z}-b\right) /\mathbb{\Im }(\mathfrak{z})$ is nondecreasing, $\varrho < 0$ and $\mathfrak{z}\geq \mathfrak{r}$, (3.59) becomes

Since $\varrho < 0$ and $\gamma < 0,$ then by using (3.16), we have

and then

Substituting (3.61) into (3.60) and using (3.1) (note that $\frac{\varrho -1}{\gamma }-\varrho \geq 0$), we observe that

Substituting (3.62) into (3.58), we obtain

which is (3.42) with $\mathcal{M} = \left(\gamma /\left(\varrho -1\right) \right) ^{\gamma }\aleph ^{\frac{\varrho -1}{\gamma }-\varrho }.$ □

Remark 3.3. In Theorem 3.3, if $\mathbb{T = R}$, and $b = 0,$ then (3.1) holds with $\aleph = 1$, and (3.42) reduces to (1.7). In addition, for $\mathbb{\Im }\left(\mathfrak{r}\right) = \mathfrak{r},$ we get (1.4).

Corollary 3.5. If $\mathbb{T = N}_{0},$ $b = 0,$ $\varrho,$ $\gamma < 0$, and $\left\{ s_{n}\right\} _{n = 0}^{\infty }$, $\left\{ t_{n}\right\} _{n = 0}^{\infty }\ $ are positive sequences with the property $n/t_{n}$ bing nondecreasing $,$ then (3.1) holds with $\aleph = 2.$ Consequently, the following inequality holds:

where

Corollary 3.6. Let $\mathbb{T} = q^{\mathbb{N}_{0}}$ for $q > 1,$ $b\in \mathbb{T},$ $\gamma,$ $\gamma ^{\ast } = \gamma /\left(\gamma -1\right),$ $\varrho < 0\ $, and ${\rm{℧}}, \mathbb{\Im }$ be positive sequences on $[b, \infty)$ such that $\left(\mathfrak{r}-b\right) /\mathbb{\Im }(\mathfrak{r})$ is nondecreasing. If

holds, then

where $\Omega (\mathfrak{r}) = \sum_{\mathfrak{z = }b}^{\mathfrak{r/}q}(q-1) \mathfrak{z}{\rm{℧}}(\mathfrak{z})$ and

4.   Conclusions

This work extends Hardy's foundational inequalities by exploring their generalizations with negative parameters within the framework of time scale theory. We have derived new results by providing time scale versions of previously established inequalities, along with their discrete analogues. These contributions offer a more comprehensive perspective on Hardy-type inequalities, demonstrating their flexibility and potential for further research. Our findings underscore the importance of integrating time scale calculus into classical inequality theory, unveiling promising directions for future investigations.

Looking ahead, we plan to expand on these results by applying alpha-conformable fractional derivatives on time scales, facilitating a deeper exploration of fractional calculus in this setting. Additionally, we aim to broaden our findings by examining their application within the framework of diamond-alpha calculus, which we believe will offer fresh insights into this developing field.

Author contributions

A. M. Ahmed, A. I. Saied and H. M. Rezk: Investigation, Software, Supervision, Writing-original draft; M. Zakarya, A. A. I Al-Thaqfan and M. Ali: Writing-review editing, Funding. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through large Research Project under grant number RGP 2/190/45.

Conflict of interest

The authors declare that there are no conflict of interests in this paper.