A sharp form of the discrete Hardy inequality and the Keller-Pinchover-Pogorzelski inequality

TL;DR

This paper presents a concise proof of a recent Hardy-type inequality, establishes its optimality, and identifies a remainder term that transforms it into an identity, enhancing understanding of the inequality's structure.

Contribution

The paper provides a simplified proof of the Keller-Pinchover-Pogorzelski inequality, confirms its optimality, and introduces a remainder term to convert it into an exact identity.

Findings

Short proof of the Hardy-type inequality

Proof of the inequality's optimality

Identification of a remainder term that makes the inequality an identity

Abstract

We give a short proof of a recently established Hardy-type inequality due to Keller, Pinchover, and Pogorzelski together with its optimality. Moreover, we identify the remainder term which makes it into an identity.