Time-Scale Integral Inequalities of Copson with Steklov Operator in High Dimension

The paper derives some new time-scale (TS) dynamic inequalities for multiple integrals. The obtained inequalities are special cases of Copson integral using Steklov operator in (TS) version with high dimension. We prove the inequalities with several formulas for the operator and in different cases m > μ + 1 and m < μ + 1 for every μ ≥ 1, using time-scales (TSs) setting for integral properties, chain rules, Fubini’s theorem, and H¨older’s inequality.


Introduction
Equations and inequalities are the core of scienti c study and have a great in uence on a huge number of applications. A large number of physical phenomena and engineering studies have been analyzed and explained through equations and inequalities. For this reason, the study in this eld developed rapidly and many types of inequalities and equations appeared. Dynamic inequalities on (TS) are some of the important inequalities that were extended by a lot of researchers and have interesting applications. Furthermore, dynamic inequalities are used to study the behaviour of dynamic equations.
Mathematical analysis has been the most important study in mathematics for the past three decades. Integral inequalities are one of the main studies and the core of mathematical analysis. In the 20th century, a signi cant part of science was numerical inequalities as the rst composition to be released in 1934, through the published study by P'olya et al. [1]. is framework of inequalities played a vital role in the improvement processes and various applications of mathematics.
A large number of essential studies of integral inequalities appeared in the twentieth century, including pure and applied mathematics study. In 1920, Hardy produced the discrete Hardy inequality [2]. is inequality was also proved by himself in [3] (see also [4]), using the variations calculus to obtain the following inequality that is very valuable across both technological sciences and mathematics. If p > 1 and h ≥ 0 in (0, ∞) and where Λ p(p − 1) − 1 is the best possible constant (BPC). Several important assessments and their implementation are done by inequality (1). Furthermore, the inequality is true in where 0 < b 0 h p (x)dx < ∞. e classical inequality of Hardy declares that if p > 1 and h is nonnegative and measurable on (a, b), then (2) is valid except h ≡ 0 a.e. in (a, b), considering the (BPC).
Let h and v be functions such that they are nonnegative measurable on (0, ∞); Many papers included new extensions and generalizations for the inequalities above in more general settings. For instance, in 1979, some generalizations of Hardy-type inequality were proved by Chan [7]. en, in 1992, Pachpatte [8] generalized the inequalities that were produced by Chan [7]. In 2005, P. Rehak used (TS) setting to extend Hardy's inequalities [9]. In 2015, Pachpatte's inequalities [8] were extended by Saker and O'Regan [10], with setting of (TSs). Later, some extensions of (TSs) Hardy inequalities were done for functions with high dimensions (see, for example, [11][12][13][14]).
In 2021, Albalawi and Khan generalized the main integral of Hardy and Copson inequalities, using the Steklov operator. e operator is defined in the following formulas with considering conditions in two cases (for more details, see [15]). e aim of this paper is extending the study in [16] that was used for some new Hardy-type inequalities to obtain new special Copson inequalities with the Steklov operator (see [15]) in (TS) versions with high dimension. e results below are proved in two cases m > μ + 1 and m < μ + 1 by considering some general conditions that can be applied for any variable in the integral. To achieve this paper, we use (TSs) settings in integrals properties, chain rules, Hölder's inequality, and Fubini's theorem. e paper takes the following structure: After introduction, the main concepts of (TSs) are presented in Section 2.
en, in Section 3, we generalized a class of Copson inequalities pertaining the Steklov operator with (TS) in high dimension. Lastly, conclusion of our results is presented.

Preliminaries and Lemmas on Time Scales
We state the main concepts of (TSs) that are used in this paper (for more details about (TS) calculus, see [17,18]).
(TS) calculus in continuous case and discrete analysis was introduced by Hilger [19] in 1988. We denote to a subset (TS) of the real numbers R by T. Hence, the sets of numbers R, Z, and N can be considered as (TSs).
Let σ: T ⟶ T be a forward jump operator, such that In the case if points are right-scattered and leftscattered at the same time, then they will be isolated.
Let g: T ⟶ R be a continuous function and if it satisfied the continuity at all right-dense points in T and the limits of the left-sided exist (finite) at all left-dense points in T, then g is known rd-continuous. We use C r (T, R) to denote the space of all rd-continuous.
A function g: e Δ-derivative of a function g in high order n ∈ N is given by If the Δ-derivative of g Δ n− 1 (t) exists, the following examples show that the delta derivative for every number set of (TSs).
If T � R, then , for all t ∈ T.
Let g: T ⟶ R; if g is continuous at right-scattered t, then it is delta-derivative of the function g, given by In the case of t is not right-scattered, then the derivative of g is given by 2 Journal of Mathematics Here, the limit exists. Note that if T � R, we have .
e Cauchy integral of a delta-differential function of e time-scale integration by parts formula is given by Lemma 3 (dynamic Hölder inequality). Let a, d ∈ T and Theorem 4 (Fubini's theorem [20]). Let (Υ, N, μ Δ ) and (Σ, L, c Δ ) be (TS) measure spaces with finite dimension.
To be more accurate, if ξ:

Main Results
A new (TS) version of Copson-type inequality with Steklov operator for multiple integrals is obtained in this section. We consider the nonnegative rd-continuous functions w l , f l , g l , and v l are Δ-integrable and defined integrals. roughout this paper, we set K(t 1 , . . . , t k ) as the Copson-Steklov-type operator considering the existence of the integral and also finite. Define the operator where p ≥ 1 and m > μ + 1.
Proof. We write the left side of (10) as follows: where Γ k is the k-term Using formula (6) for integration by parts to compute Γ k , we have . . . , t k )) p , and hence, Assume λ ≥ 1 such that where F Δ k � (zF/zt k ), and since c l ∈ [t l , σ(t l )], we have Substituting the previous quantities in (12) and since V l (∞) � ∞ and w l (a) � 0, then we have Assume μ ≥ 1 such that 4 Journal of Mathematics en, Hölder's inequality (8) with indices p and p/(p − 1) can be applied: Substituting Γ k in (11) and applying Fubini's eorem 4, then we obtain the inequality Corollary 7. If T � R in Corollary 6, we obtain Remark 8. Assume μ � 0 and β > λ in Corollary 7; then, we have Corollary 3 in Theorem 9. Let T l be (TS) and a ∈ [0, ∞) T l , for 1 ≤ l ≤ k with l, k ∈ N. In addition, let w l , f l , g l , and v l be nonnegative and rd-continuous functions on [a, ∞) T l . Furthermore, assume there exist λ, μ ≥ 1 such that where F Δ l � (zF/zt l ); for every l,  en, where p ≥ 1 and 0 ≤ m < μ + 1.
Proof. We write the left side of (17) as follows: Use formula (6) to calculate the following k− term: where Using (7) and the product rule (5), there exists c k ∈ [s k , σ(s k )] such that Assume μ ≥ 1 such that Since V Δ k k (s k ) � v k (s k ) ≥ 0, s k ≤ c k ≤ σ(s k ), and 0 ≤ m < 1, then 6 Journal of Mathematics By integration, we have Now, we calculate (− K p (t 1 , . . . , t k )) Δ k and we obtain Assume λ ≥ 1 such that g Δ k k t k g σ k t k ≥ λ F Δ k t 1 , . . . , t k F t 1 , . . . , t k , where F t 1 , . . . , t k ≔ g k t k ∞ t k 1 g k s k v k s k V k s k f s 1 , . . . , s k Δs k , and since V k (∞) � ∞ and c k ≥ s k , then we have K p t 1 , . . . , t k Δ k ≥ pK p− 1 t 1 , . . . , t k (λ + 1)g σ k t k F Δ k t 1 , . . . , t k ≥ p(λ + 1)K p− 1 t 1 , . . . , t k − g σ k t k g k t k v k t k V k t k f t 1 , . . . , t k . en, Hence, we have operators and solve the singularity that appeared in eorem 12 with case m > μ + 1.

Data Availability
All data that support the findings of this study are included within the article.

Conflicts of Interest
e author declares no conflicts of interest.