Multivariate Dynamic Sneak-Out Inequalities on Time Scales

Bennett and Grosse-Erdmann [1] introduce the “sneak-out” principle concerned with the equivalence of two series. Bohner and Saker [2] extended the sneak-out principle on time scales and proved some new dynamic sneak-out inequalities and their converses on time scales which, as special cases, with T � N, contain the discrete inequalities obtained by Bennett and Grosse-Erdmann (Section 6 in [1]). However, the sneak-out principle on time scales can be applied to formulate the corresponding integral inequalities by choosing T � R. (e paper aims to extend the work given by Bohner and Saker in [2] for functions depending on more than one parameter. Some other inequalities, such as Hardytype, Hardy-Copson, and Copson-Leindler-type inequalities, are also studied for functions of more than one parameter [3–5] via time scales’ calculus. Some literature concerning with time scale can be seen in [6–13]. (e paper is organized as follows. Section 2 provides some basics from time scales’ calculus. Section 3 features two dynamic inequalities of the Copson type, which are needed to prove further results. In Section 4, we present sneak-out inequalities on time scales for functions depending on more than one parameter. 2. Preliminaries


Introduction
Bennett and Grosse-Erdmann [1] introduce the "sneak-out" principle concerned with the equivalence of two series. Bohner and Saker [2] extended the sneak-out principle on time scales and proved some new dynamic sneak-out inequalities and their converses on time scales which, as special cases, with T � N, contain the discrete inequalities obtained by Bennett and Grosse-Erdmann (Section 6 in [1]). However, the sneak-out principle on time scales can be applied to formulate the corresponding integral inequalities by choosing T � R. e paper aims to extend the work given by Bohner and Saker in [2] for functions depending on more than one parameter. Some other inequalities, such as Hardytype, Hardy-Copson, and Copson-Leindler-type inequalities, are also studied for functions of more than one parameter [3][4][5] via time scales' calculus. Some literature concerning with time scale can be seen in [6][7][8][9][10][11][12][13]. e paper is organized as follows. Section 2 provides some basics from time scales' calculus. Section 3 features two dynamic inequalities of the Copson type, which are needed to prove further results. In Section 4, we present sneak-out inequalities on time scales for functions depending on more than one parameter.

Preliminaries
A time scale T as well as close set in R are nonempty [14,15]. Some examples of time scales are Z, R, and Cantor set. Assume that infT � ϕ, where ϕ is empty set and sup T � ∞. A time-scale interval is denoted by [t 0 , ∞) T ≔ [t 0 , ∞) ∩ T, for t 0 ∈ T. e operators σ: } are forward as well as backward jump operators, respectively, for l ∈ T. e point l ∈ T is right-scattered if it satisfies σ(l) > l, and left-scattered if ρ(l) < l. e points which are at the same time left-scattered as well as right-scattered are called isolated. Furthermore, the point l ∈ T is right-dense if it satisfies l < Sup T and σ(l) � l, and left-dense if it satisfies l > infT and σ(l) � l; furthermore, the point is called dense if it is left-dense as well as right-dense at the same time. A function μ: T ⟶ [0, ∞), defined by μ(l) ≔ σ(l) − l, is called the graininess function.
If a function g: T ⟶ R is continuous at all right-dense points, the left-hand limits exist and are finite at left-dense points in T; then, it is right-dense continuous (rd-continuous) on T. e set denoted by C r d (T) contain all rdcontinuous functions on T.

Dynamic Copson-Type Inequalities for Finite Numbers of Parameters
We assume throughout that all the functions are nonnegative and the integrals considered exist. For h ∈ N, ι ∈ 1, 2, . . . , h { }, let T ι be time scales.

Journal of Mathematics
Suppose (16) is true for 1 ≤ h ≤ p. To prove for h � p + 1, by using H 2 , we have defined as Use the right-hand side part of inequality (8) with Use (5) "p times" in second term of (22):  (23) and (25) to obtain By applying (5) "p times" on (25) and making simplification, we obtain
In this case, (29) takes the form Note that (41) is extension of Example 4.3 in [2].
(54) us, by mathematical induction, (54) holds for all h ∈ N, which completes the proof. (55) Proof. We prove the result by using mathematical induction. For h � 1, statement is true by eorems 4.6 in [2]. Assume for 1 ≤ h ≤ p, (55) holds. To prove the result for h � p + 1, take L.H.S of (55) with h � p + 1 in the following form: where