Advanced Hardy-type inequalities with negative parameters involving monotone functions in delta calculus on time scales

Abstract

In this study, we introduced several novel Hardy-type inequalities with negative parameters for monotone functions within the framework of delta calculus on time scales $ \mathbb{T} $. As an application, when $ \mathbb{T = N}_{0}, $ we derived discrete inequalities with negative parameters for monotone sequences, offering fundamentally new results. When $ \mathbb{T = R}, $ we established continuous analogues of inequalities that have appeared in previous literature. Additionally, we presented inequalities for other time scales, such as $ \mathbb{T} = q^{\mathbb{N}_{0}} $ for $ q > 1, $ which, to the best of the authors' knowledge, represented largely novel contributions.

Keywords

• hardy's inequality
• negative parameter
• monotone functions
• hölder's inequality
• weighted inequalities
• delta calculus
• time scales