Reversed Weighted Hardy-type Inequalities with Negative Indices

Abstract

This research paper presents a comprehensive investigation of novel Hardy-type dynamic inequalities that incorporate two independent weight functions, denoted as u and v. A distinctive feature of this work is its focus on time scales calculus with negative parameters, a generalization that unifies and extends discrete and continuous analysis. The basic methodology involves the application of the reverse H¨older’s inequality and the Minkowski integral inequality to rigorously deduce all essential results. To illustrate the adaptability of our results, we provide explicit examples of the corresponding discrete and integral analogues for various time scales: when T = N (the natural numbers, indicating discrete sequences), T = l N0 for l > 1 (a quantum time scale), and T = R (the real numbers, signifying the classical continuous case). This paper situates its findings within a wider mathematical framework by demonstrating how they contain and extend certain cases of reverse Hardy-type dynamic inequalities previously formulated by distinguished scholars including Prokhorov, Kufner, Yang, Nguyen, and Benaissa. Consequently, this work presents a cohesive framework that broadens the theoretical terrain of Hardy-type inequalities.