On Hardy type inequalities for weighted quasideviation means

TL;DR

This paper establishes sharp Hardy inequalities for weighted quasideviation means, revealing how the Hardy constant depends on weight sequences and the mean’s behavior near zero.

Contribution

It provides explicit Hardy constants for a broad class of weighted quasideviation means, extending previous results with new sharp bounds.

Findings

Determines the smallest Hardy constant for weighted quasideviation means.

Shows the Hardy constant depends on the limit of weight ratios and mean behavior near zero.

Provides a unified approach to Hardy inequalities for these means.

Abstract

Using recent results concerning the homogenization and the Hardy property of weighted means, we establish sharp Hardy constants for concave and monotone weighted quasideviation means and for a few particular subclasses of this broad family. More precisely, for a mean $\mathscr{D}$ like above and a sequence $(\lambda_n)$ of positive weights such that $\lambda_n/(\lambda_1+\dots+\lambda_n)$ is nondecreasing, we determine the smallest number $H \in (1,+\infty]$ such that $$ \sum_{n=1}^\infty \lambda_n \mathscr{D}\big((x_1,\dots,x_n),(\lambda_1,\dots,\lambda_n)\big) \le H \cdot \sum_{n=1}^\infty \lambda_n x_n \text{ for all }x \in \ell_1(\lambda). $$ It turns out that $H$ depends only on the limit of the sequence $(\lambda_n/(\lambda_1+\dots+\lambda_n))$ and the behaviour of the mean $\mathscr{D}$ near zero.