On improvements of the Hardy, Copson and Rellich inequalities

TL;DR

This paper introduces new improved versions of classical Hardy, Copson, and Rellich inequalities in one dimension using a factorization method and a generalized discrete Laplacian, establishing their optimality and interrelations.

Contribution

It develops extended improved discrete Hardy, Rellich, and Copson inequalities, proving their optimality and linking improvements across related inequalities.

Findings

Established extended improved discrete Hardy and Rellich inequalities.

Proved the Copson inequality admits an optimal improvement.

Linked improvements of Knopp inequalities to Rellich inequalities.

Abstract

Using a method of factorization and by introducing a generalized discrete Dirichlet's Laplacian matrix $(-\Delta_{\Lambda})$, we establish an extended improved discrete Hardy's inequality and Rellich inequality in one dimension. We prove that the discrete Copson inequality (E.T. Copson, \emph{Notes on a series of positive terms}, J. London Math. Soc., 2 (1927), 9-12.) in one-dimension admits an improvement. We also prove that the improved Copson's weights are optimal (in fact \emph{critical}). It is shown that improvement of the Knopp inequalities (Knopp in J. London Math. Soc. 3(1928), 205-211 and 5(1930), 13-21) lies on improvement of the Rellich inequalities. Further, an improvement of the generalized Hardy's inequality (Hardy in Messanger of Math. 54(1925), 150-156) in a special case is obtained.