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

Abstract

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.

Exploring Generalized Hardy-Type Inequalities via Nabla Calculus on Time Scales

“…As a result, the primary goal of this study is to expand ( 9)- (11) for nabla time scale calculus. The foundational principles of these principal theorems draw inspiration from the work presented in the paper [20], wherein analogous findings were delineated within the domain of time scale delta calculus.…”

“…Recently, the authors [20] presented a time scale form of ( 9)-( 11) by using delta calculus, respectively, as follows: If κ, ψ ≥ 0 are nondecreasing functions, then…”