Improved Results on Weighted Hardy-Type Inequalities Based on the Principles of Time Scale Calculus

Abstract

Over the past decades, the study of Hardy-type inequalities has led to significant developments in both theory and applications, giving rise to a wide range of refinements and generalizations in weighted and unweighted settings. Building upon the weighted Hardy-type inequalities established by Sulaiman and the generalized forms proposed by Sroysang, this paper develops new sharp inequalities within the framework of time scale calculus. The approach unifies continuous and discrete settings by employing delta integrals and accommodating general weight functions under the assumption monotonicity. Our main theorems extend the continuous inequalities of Sulaiman and Sroysang to arbitrary time scales, and yield new discrete analogues as direct consequences. Proofs rely on Hölder's inequality, the chain rule, and integration by parts adapted to the time scale context. Several corollaries and illustrative examples are included, highlighting that various known continuous and discrete inequalities appear as exceptional cases of the presented framework.