On Refinements of Multidimensional Inequalities of Hardy‐Type via Superquadratic and Subquadratic Functions

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In this article, we discuss several novel generalized Ostrowski-type inequalities for functions whose derivative module is relatively convex in time scales calculus. Our core findings are proved by using the integration by parts technique, Hölder’s inequality, and the chain rule on time scales. These derived inequalities expand the existing literature, enriching specific integral inequalities within this domain.

Section: Introductionmentioning

confidence: 99%

Inequalities of Ostrowski Type for Functions Whose Derivative Module Is Relatively Convex on Time Scales

Rezk,

Saied,

Ali

et al. 2024

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“…Since the first Hardy-type inequality on time scales was proposed, many researchers have further generalized and refined this inequality (see, [21][22][23][24]). Recently, there are more new results about the Hardy inequality via other kinds of time scale calculus, such as time scale delta integral [25] and time scale nabla integral [26,27].…”

mentioning

confidence: 99%

Novel Hardy-Type Inequalities with Submultiplicative Functions on Time Scales Using Delta Calculus

Rezk

Saied

Ali

et al. 2023

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In this study, we apply Hölder’s inequality, Jensen’s inequality, chain rule and the properties of convex functions and submultiplicative functions to develop an innovative category of dynamic Hardy-type inequalities on time scales delta calculus. A time scale, denoted by T, is any closed nonempty subset of R. In time scale calculus, results are unified and extended. As particular cases of our findings (when T=R), we have the continuous analogues of inequalities established in some the literature. Furthermore, we can find other inequalities in different time scales, such as T=N, which, to the best of the authors’ knowledge, is a largely novel conclusion.

Some New Generalized Inequalities of Hardy Type Involving Several Functions on Time Scale Nabla Calculus

Saied

AlNemer

Zakarya

et al. 2022

In this article, we establish several new generalized Hardy-type inequalities involving several functions on time-scale nabla calculus. Furthermore, we derive some new multidimensional Hardy-type inequalities on time scales nabla calculus. The main results are proved by applying Minkowski’s inequality, Jensen’s inequality and Arithmetic Mean–Geometric Mean inequality. As a special case of our results, when T=R, we obtain refinements of some well-known continuous inequalities and when T=N, the results which are essentially new.