A new formulation of Hardy-type dynamic inequalities on time scales

In this paper, we introduce a novel formulation of dynamic Hardy-type inequalities on a time scale, motivated by a recently-established convexity approach in the Haar measure.The classical Hardy inequality is refined so that the classical Lebesgue-measure constant is replaced by the sharp constant $1$. We obtain time-scale analogues on finite intervals with best constants, and, for nonincreasing and nondecreasing functions, reversed inequalities with explicit weights described by incomplete $\beta$-functions. To establish our results, we employ two distinct time scales and apply the chain rule, together with the substitution rule, the derivative of inverse functions, and Fubini's theorem for delta integration. Our approach generalizes classical integral inequalities in the continuous setting, while yielding fundamentally new inequalities in the discrete setting. Furthermore, we explore the application of our results in the quantum case, demonstrating their broader relevance.