During the early 1900s, Hilbert made the discovery of this inequality (refer to [10]) Here, ${\left\{a_s\right\}}_{s=1}^{\infty }$ and ${\left\{c_n\right\}}_{n=1}^{\infty }$ are real sequences satisfying $0<{\sum }_{s=1}^{\infty }{a}_s^2<\infty$ and $0<{\sum }_{n=1}^{\infty }{c}_n^2<\infty .$ This particular expression is known as Hilbert’s double series inequality.

In [17], Schur demonstrated that $\pi$ in (1) is the most optimal constant achievable. Additionally, he unveiled the integral counterpart of (1), which later became recognized as the Hilbert integral inequality, taking the form where $f,g$ are measurable functions satisfying$0<{\int }_0^{\infty }{f}^2\left(\eta \right)d\eta <\infty$ and $0<{\int }_0^{\infty }{g}^2\left(\tau \right)d\tau <\infty .$

In [8], an extension of (1) is presented as follows: suppose $l,r>1$ with $1/l+1/r=1,$ ${\left\{a_s\right\}}_{s=1}^{\infty },$ ${\left\{c_n\right\}}_{n=1}^{\infty }$ are real sequences satisfying $0<{\sum }_{s=1}^{\infty }{a}_s^r<\infty$ and $0<{\sum }_{n=1}^{\infty }{c}_n^l<\infty ,$ then Here, $\pi /sin(\pi /r)$ is the optimal constant.

In [9], the authors derived the integral counterpart of (3) as Here, $f,g\ge 0$ are measurable functions satisfying $0<{\int }_0^{\infty }{f}^r\left(\eta \right)d\eta <\infty$ and $0<{\int }_0^{\infty }{g}^l\left(\tau \right)d\tau <\infty .$

In [14], new inequalities akin to the ones presented in (3) and (4) were established as follows: let $l,r>1$ with $1/l+1/r=1.$ Consider sequences $a_w:\left\{0,1,2,...,s\right\}\subset \mathbb{N}\to \mathbb{R}$ and $c_{\vartheta }:\left\{0,1,2,...,n\right\}\subset \mathbb{N}\to \mathbb{R}$where $a\left(0\right)=c\left(0\right)=0.$ Then Here, $\nabla a_w=a_w-a_{w-1},$ $\nabla c_{\vartheta }=c_{\vartheta }-c_{\vartheta -1}$ and Moreover, if $l,r>1$ with $1/l+1/r=1,$ $f\left(w\right)$ and $g\left(\vartheta \right)$ are real-valued continuous functions with $f\left(0\right)=g\left(0\right)=0,$ then Here, In [7], Chang-Jian et al. established a set of new inequalities that are similar to extensions of Hilbert’s double-series inequality and derived their integral analogues. These inequalities are outlined as follows: let $r_j>1$ such that $1/l_j+1/r_j=1$ and $a_j\left(w_j\right)$ are real sequences defined for $w_j=0,1,2,...,s_j$ where $s_j\in \mathbb{N}$ and $a_j\left(0\right)=0;$ $j=1,2,...,n.$ Define the operator ∇ as $\nabla a_j\left(w_j\right)=a_j\left(w_j\right)-a_j(w_j-1).$ Then Here, Also, they proved that if $h_j\ge 1,$ $l_j,r_j>1$ are constants with $1/r_j+1/l_j=1,$ $f_j\left(w_j\right)$ are real valued differentiable functions defined on $[0,{\eta }_j)$ where ${\eta }_j\in \left(0,\infty \right)$ and $f_j\left(0\right)=0;$ $j=1,2,...,n,$ then where Furthermore, they established that if $l_j,r_j>1$ such that $1/r_j+1/l_j=1,$ $a_j(w_j,z_j)$ be real sequences defined for $\left(w_j,z_j\right)$ where $w_j=0,1,2,...,s_j,$ $z_j=0,1,2,...,n_j;$ $s_j,n_j\in \mathbb{N}$ and $a_j(0,z_j)=a_j(w_j,0)=0$ $\forall j=1,2,...,n.$ Define the operators ${\nabla }_1$ and ${\nabla }_2$ by Then Here,

In recent decades, a novel theory known as time scale theory has emerged, aimed at unifying continuous calculus and discrete calculus. The results presented in this paper encompass classical continuous and discrete inequalities as special cases when $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{N},$ respectively. Moreover, these inequalities can be extended to analogous inequalities on various time scales, such as $\mathbb{T}=q^{\stackrel{-}{\mathbb{Z}}}$ for $q>1$. Many researchers have delved into dynamic inequalities on time scales, and for a more comprehensive understanding of these dynamic inequalities on time scales, readers are referred to the following papers: [3,4,12,15,16,19,20,21].

The primary objective of this paper is to establish analogous formulas for Hilbert-type inequalities (7), (9) and (8) within the framework of time scales in delta calculus. It’s important to note that these formulas are derived under specific conditions, which are $a_j\left(s_j\right)=0$ and$a_j(s_j,z_j)=a_j(w_j,n_j)=0$ $\forall j=1,2,...,n.$ These conditions differ from those utilized in a previous work [7]. The outcomes of our research provide novel insights and estimations for these specific categories of inequalities. In particular, we have introduced multivariate summation inequalities for extensions of the Hilbert inequality, which were previously unproven. Additionally, we have obtained their corresponding integral expressions. The proofs of these results are based on the application of Hölder’s inequality on time scales and the mean inequality.

The paper is structured as follows: After this introductory section, the subsequent section offers an overview of fundamental concepts in time scale calculus, which serve as the basis for our proofs. The final section is dedicated to presenting our main findings.

In what follows, the time scale $\mathbb{T}$ is a nonempty closed subset of $\mathbb{R}$, and it could be an interval, a union of intervals, or even a set of isolated points. The real numbers (continuous case), integers (discrete case), and various amalgamations of the two constitute the most prevalent instances of time scales. Given $\nu \in \mathbb{T}$, we establish $\sigma :\mathbb{T}\to \mathbb{T}$ and $\mu :\mathbb{T}\to \mathbb{R}$ as $\sigma \left(\nu \right):=inf\{\alpha \in \mathbb{T}:\alpha >\nu \}$ and $\mu \left(\nu \right):=\sigma \left(\nu \right)-\nu \ge 0.$ These components are referred to as the forward jump operator and the forward graininess function, correspondingly. Considering a function $\Im :\mathbb{T}\to \mathbb{R}$, we introduce the notation: Additionally, we establish the interval ℓ within the context of $\mathbb{T}$ as: Now, let’s examine a function $\Im :\mathbb{T}\to \mathbb{R}$ and we introduce the concept of the "delta derivative" or "$\Delta$ differentiable" of ℑ at a point $v\in \mathbb{T}$ as follows.

Following is a description of the concept of an integral on time scales.

It’s widely acknowledged that any rd-continuous function possesses an antiderivative. As a result, we can deduce the following outcomes.

Next, we present the Hölder’s inequality in two dimensions on time scales.

Let ${\mathbb{T}}_1,$ ${\mathbb{T}}_2$ be time scales, $C{C}_{rd}$ denote the set of functions $\Im \left(\tau ,\xi \right)$ on ${\mathbb{T}}_1\times {\mathbb{T}}_2,$ where ℑ is $rd-$continuous in $\tau ,$ $\xi$ and $C{C}_{rd}^{\prime }$ denote the set of all functions $C{C}_{rd}$ for which both the ${\Delta }_1$ partial derivative with respect to $\tau$ and ${\Delta }_2$ partial derivative with respect to $\xi$ exist and are in $C{C}_{rd}.$

Throughout this paper, we will operate under the assumption that the functions are rd-continuous and we will also consider the existence of the integrals. To substantiate our results, it is necessary to prove the following lemma.