Discrete Hardy-type Inequalities

TL;DR

This paper investigates discrete Hardy-type inequalities, deriving variational formulas for optimal constants, proposing improved bounds, and comparing constants across different intervals to enhance understanding of these inequalities.

Contribution

It introduces a new improved factor for upper estimates of optimal constants, which is smaller and best possible, advancing the theoretical understanding of discrete Hardy inequalities.

Findings

Derived variational formulas for optimal constants

Proposed an improved, best possible upper estimate factor

Compared optimal constants across different discrete intervals

Abstract

This paper studies the Hardy-type inequalities on the discrete intervals. The first result is the variational formulas of the optimal constants. Using these formulas, one may obtain an approximating procedure and the known basic estimates of the optimal constants. The second result, which is the main innovation of this paper, is about the factor of basic upper estimates. An improved factor is presented, which is smaller than the known one and is best possible. Some comparison results are included for comparing the optimal constants on different intervals.