An improved discrete Hardy inequality

TL;DR

This paper presents an improved version of the classical discrete Hardy inequality, providing tighter bounds for sums involving non-negative sequences, which enhances the understanding of inequalities in analysis.

Contribution

The authors derive a sharper inequality than the classical discrete Hardy inequality, advancing the theoretical framework of inequalities in mathematical analysis.

Findings

Established a new, improved discrete Hardy inequality

Provided tighter bounds for non-negative sequences

Enhanced theoretical understanding of inequality bounds

Abstract

We improve the classical discrete Hardy inequality \begin{equation*}\label{1} \sum _{{n=1}}^{\infty }a_{n}^{2}\geq \left({\frac {1}{2}}\right)^{2} \sum _{{n=1}}^{\infty }\left({\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right)^{2}, \end{equation*} where $\{a_n\}_{n=1}^\infty$ is any sequence of non-negative real numbers.