1. Introduction
During the early 1900s, Hilbert made the discovery of this inequality (refer to [
1])
Here, and are real sequences satisfying and . This particular expression is known as Hilbert’s double series inequality.
In [
2], Schur demonstrated that
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
are measurable functions satisfying
and
.
In [
3], an extension of (
1) is presented as follows: suppose
with
,
,
are real sequences satisfying
and
, then
Here, is the optimal constant.
In [
4], the authors derived the integral counterpart of (
3) as
Here, are measurable functions satisfying and .
In [
5], new inequalities akin to the ones presented in (
3) and (
4) were established as follows: let
with
. Consider sequences
and
where
. Then
Here,
,
and
Moreover, if
with
,
and
are real-valued continuous functions with
, then
In [
6], Chang-Jian et al. proved some new inequalities of Hilbert type in the difference calculus with “n-dimension” and derived their integral analogues. These inequalities are outlined as follows: let
such that
and
are real sequences defined for
, where
and
. Define the operator ∇ as
. Then
Also, they proved that if
,
are constants with
,
are real valued differentiable functions defined on
, where
and
, then
where
Furthermore, they established that if
such that
,
are real sequences defined for
where
,
and
. Define the operators
and
by
For more details about Hilbert type inequalities, see the papers [
5,
6,
7,
8]. As applications of our work, we refer to the papers [
9,
10]. 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
and
, respectively. Moreover, these inequalities can be extended to analogous inequalities on various time scales, such as
for
. 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 papers [
11,
12,
13,
14,
15,
16,
17].
The primary objective of this paper is to establish analogous formulas for Hilbert-type inequalities (
7) and (
8) within the framework of time scales in delta calculus. It is important to note that these formulas are derived under specific conditions, which are
and
. These conditions differ from those utilized in a previous work [
6]. 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.
2. Basic Principles
In what follows, the time scale
is a nonempty closed subset of
, 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
, we establish
and
as
and
. These components are referred to as the forward jump operator and the forward graininess function, correspondingly. Considering a function
, we introduce the notation:
Additionally, we establish the interval
ℓ within the context of
as:
Below, we present the concept of the delta derivative along with its properties. We also delve into the chain rule, integration by parts, Fubini’s theorem, and the mean inequality, which are discussed and analyzed in the references [
4,
18,
19,
20,
21] and others.
Definition 1 ([
20])
. We use the term “Δ differentiable" to describe a function ℑ being differentiable at , if , there is a neighborhood W of v such that for some β the inequalityis true and, in this case, we write .
Theorem 1 (Properties of delta-derivatives [
20])
. Assume ℑ is a function and let , then Theorem 2 (Chain Rule [
20])
. Given that is a continuous and Δ differentiable and is continuously differentiable, then Definition 2 ([
20])
. A function ℑ is characterized as continuous when it exhibits continuity at every right-dense point within and possesses finite left-sided limits at left-dense points in . We use the symbol to represent the sets of all rd-continuous functions, and the symbol to represent the set of all continuous functions. The following is a description of the concept of an integral on time scales.
Definition 3 ([
20])
. ℜ is Δ antiderivative of ℑ ifAs a result, for , we deduce the integral of ℑ as It is widely acknowledged that any rd-continuous function possesses an antiderivative. As a result, we can deduce the following outcomes.
Theorem 3 ([
20])
. If , then Theorem 4 ([
20])
. If , and , , thenIf , then .
.
Lemma 1 (Integration by parts [
19])
. If and , , then Theorem 5 ([
19])
. Let and . Then Lemma 2 (Hölder’s Inequality [
19])
. If and , , thenwhere and .
Let , be time scales, denote the set of functions on , where ℑ is continuous in , and denote the set of all functions , for which both the partial derivative with respect to and partial derivative with respect to exist, and are in .
Lemma 3 ([
18], Theorem 3.3)
. Let with , , and such that . Then, Lemma 4 (Fubini’s theorem [
21])
. If and is integrable, then Lemma 5 (Mean inequality [
4])
. If for , then 3. Main Results
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.
Lemma 6. Let with and , where . Then Proof. By utilizing Lemma 5 with
and
, we deduce that
Since
, then (
17) becomes
which is (
16). □
Theorem 6. Let , such that and with . Then Proof. By utilizing the property (5) of Theorem 4, we deduce that
Since
, then
and then
Substituting (
21) into (
20), we observe that
therefore
Applying (
13) on
with
,
and
, we have
and then
By substituting (
23) into (
22) and applying (
16) on
with
, we acquire
Dividing (
24) on
and integrating over
from
to
,
, we conclude that
Again, using (
13) on
with
,
and
, we obtain
and then
Substituting (
26) into (
25), we obtain
Now, using (
12) on
with
and
we find that
where
. Combining (
28) with (
27), we obtain
Corollary 1. Let in Theorem 6, , such that and be real sequences with . Then, and Here, Δ is the forward difference operator and A is specified as in (19). Corollary 2. Let in Theorem 6, , such that and with . Then, and where A is given by (19). Corollary 3. Let for , such that and be real sequences with . Then, and where A is given by (19) and In the following, we generalize the last theorem for two variables.
Theorem 7. Let , such that , , with for and . Then Here, the derivative of is the derivative with respect to the first variable τ and the derivative of is the derivative with respect to the second variable ξ.
Proof. Applying the property (5) of Theorem 4, Fubini’s theorem and using the hypothesis
, we obtain
and then
Applying (
14) on
with
,
and
, we see that
and then
Substituting (
32) into (
31) and applying (
16) on
, we obtain
Dividing (
33) on
, integrating over
from
to
and over
from
to
for
and using (
34), we conclude that
Again, using (
14) on
with exponents
and
,
we observe that
and then
Substituting (
36) into (
35) and applying the Fubini theorem, we see that
Now, by applying (
12) on
with
we find that
where
. By integrating (
38) over
from
to
and using the Fubini theorem, we have
Again, using (
12) on the term
with
we see that
where
. Substituting (
40) into (
39) and applying Fubini’s theorem, we obtain
Substituting (
41) into (
37), we obtain
Corollary 4. Let in Theorem 7, , such that and be real sequences with for and , where . Then, , and where B is given by (30). Corollary 5. Let in Theorem 7, , such that and with for and , where . Then, , and where B is given by (30). Corollary 6. Let for , , such that and are real sequences with for and , where . Then, , and Here, B is given by (30) and the derivative of is the derivative with respect to the first variable τ and the derivative of is the derivative with respect to the second variable ξ.