Change of variable and discrete Hardy inequality

TL;DR

This paper presents a simplified and more direct proof of the discrete Hardy inequality by explicitly formulating a change of variable estimate, removing unnecessary conditions, and achieving the optimal constant.

Contribution

It introduces a clear change of variable approach that simplifies the proof of the discrete Hardy inequality and removes superfluous conditions from previous methods.

Findings

Provides an explicit one-sided summation estimate for absolutely convergent series.

Simplifies the proof of the discrete Hardy inequality.

Achieves the optimal constant in the 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.