The $p$-Hardy-Rellich-Birman inequalities on the half-line

TL;DR

This paper extends classical discrete $p$-Hardy inequalities to higher-order derivatives on the half-line, introduces new inequalities with optimal constants, and connects discrete and continuous $p$-Birman inequalities.

Contribution

It generalizes the $p$-Hardy inequality to higher-order derivatives, introduces discrete $p$-Rellich and $p$-Birman inequalities, and provides an alternative proof for the continuous case.

Findings

Derived discrete $p$-Rellich and $p$-Birman inequalities with optimal constants.

Established a variant of the Copson inequality with negative exponent.

Connected discrete inequalities to their continuous counterparts.

Abstract

The classical discrete $p$-Hardy inequality establishes a sharp relationship between the $\ell^{p}$-norms of a sequence and its discrete derivative. In this paper, we generalize this inequality to discrete derivatives of arbitrary integer order $\ell \geq 1$, yielding discrete $p$-Rellich ($\ell=2$) and general $p$-Birman ($\ell \geq 3$) inequalities. As a key step in the proof, we deduce a variant of the Copson inequality with a negative exponent, which may be of independent interest. Furthermore, we demonstrate how the continuous $p$-Birman inequality can be recovered from our discrete version, providing an alternative proof of this classical result. All constants in the obtained inequalities are shown to be optimal.