Change of variable and discrete Hardy inequality
Abstract
For absolutely convergent series we state explicitly a one-sided summation estimate that can be viewed as the discrete analogue of the change of variable formula on the half line. This estimate is implicit in Pascal Lef\`evre's recent elegant proof of the classical discrete Hardy inequality. Here we remove a superfluous irrationality condition therein and point out the change of variable character of his approach. This leads to a simpler, shorter and \textit{bona fide} Ingham type proof of the discrete Hardy inequality, and also provides the optimal constant.
Keywords
Cite
@article{arxiv.2307.04971,
title = {Change of variable and discrete Hardy inequality},
author = {Yi C. Huang},
journal= {arXiv preprint arXiv:2307.04971},
year = {2023}
}
Comments
This is a companion note of ''A first order proof of the improved discrete Hardy inequality. Arch. Math. 117, 671-674 (2021)'' and was written in 08/2021. In light of the supersolution language of Pinchover et al., our motivation is to unify, at least conceptually, the proofs for both the discrete and the continuous Hardy inequalities