On the improvements of Hardy and Copson inequalities

TL;DR

This paper revisits and extends discrete Hardy and Copson inequalities, establishing improved versions with optimality and analyzing related sequence space structures.

Contribution

It introduces extended improved discrete Hardy and Copson inequalities with proven optimality and explores fundamental properties of associated sequence spaces.

Findings

Established an extended improved discrete Hardy inequality with optimality.

Achieved an improvement of the discrete Copson inequality in a specific case.

Analyzed the structure of sequence spaces related to the inequalities.

Abstract

In this current work, we revisit the recent improvement of the discrete Hardy's inequality in one dimension and establish an extended improved discrete Hardy's inequality with its optimality. We also study one-dimensional discrete Copson's inequality (E.T. Copson, \emph{Notes on a series of positive terms}, J. London Math. Soc., 2 (1927), 9-12.), and achieve an improvement of the same in a particular case. Further, we study some fundamental structures such as completeness, K\"{o}the-Toeplitz duality, separability, etc. of the sequence spaces which originated from the improved discrete Hardy and Copson inequalities in one dimension.