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

Abstract

In this research, we aim to explore generalizations of Hardy-type inequalities using nabla Hölder’s inequality, nabla Jensen’s inequality, chain rule on nabla calculus and leveraging the properties of convex and submultiplicative functions. Nabla calculus on time scales provides a unified framework that unifies continuous and discrete calculus, making it a powerful tool for studying various mathematical problems on time scales. By utilizing this approach, we seek to extend Hardy-type inequalities beyond their classical continuous or discrete settings to a more general time scale domain. As specific instances of our discoveries, we have the integral inequalities previously established in the existing literature.

Generalized Dynamic Inequalities of Copson Type on Time Scales

“…Since Z(ξ) is a non-increasing function and σ(ξ) ≥ ξ, we observe that Z σ (ξ) ≤ Z(ξ). Moreover, we then have from (13) that…”

“…The concept of symmetry in this context may be connected to exploring patterns, invariances or relationships within the established dynamic inequalities on time scales. For more results about the dynamic inequalities, see the book [11] and the papers [12,13].…”