A first order proof of the improved 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 Lefèvre'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 bona fide Ingham type proof of the discrete Hardy inequality, and also provides the optimal constant.