Generalized dynamic inequalities similar to Hardy’s inequality involving a convex function on time scales

Abstract

In this paper, we establish some new generalizations of dynamic inequalities similar to Hardy's inequality on a time scale $\mathbb{T}$, by applying Jensen's inequality, integration by parts and chain rule on time scales. In particular, when $\mathbb{T}=\mathbb{R}$, we get the classical inequalities known from the literature, while in the discrete case $\mathbb{T}=\mathbb{N}$, the obtained inequalities are essentially new. In addition, we show that our results are  more accurate than some recent dynamic inequalities known from the literature.  Finally, we establish the corresponding relations in  quantum calculus, when $\mathbb{T}=q^{\mathbb{N}_{0}}$,  $q>1$.