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

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 ${\mathbb{T}\mathbb{=}\mathbb{N}}_0,$ we derived discrete inequalities with negative parameters for monotone sequences, offering fundamentally new results. When $\mathbb{T}\mathbb{=}\mathbb{R},$ 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.