On Hardy type inequalities for weighted means

TL;DR

This paper establishes weighted Hardy inequalities for a broad class of means, identifying conditions under which the smallest constant C exists and characterizing the extremal weights for symmetric, monotone means.

Contribution

The paper provides a comprehensive analysis of weighted Hardy inequalities for monotone and symmetric means, extending classical results and identifying optimal weights.

Findings

Weighted Hardy inequality holds for monotone, symmetric means satisfying Kedlaya's inequality.

The optimal Hardy constant is achieved with constant weights.

Conditions on weights ensure the inequality's validity for a broad family of means.

Abstract

The aim of this paper is to establish weighted Hardy type inequality in a broad family of means. In other words, for a fixed vector of weights $(\lambda_n)_{n=1}^\infty$ and a weighted mean $\mathscr{M}$, we search for the smallest number $C$ such that $$\sum_{n=1}^{\infty} \lambda_n \mathscr{M} \big((x_1,\dots,x_n),(\lambda_1,\dots,\lambda_n)\big) \le C \sum_{n=1}^{\infty} \lambda_nx_n \text{ for all admissible }x.$$ The main results provide a definite answer in the case when $\mathcal{M}$ is monotone and satisfies the weighted counterpart of the Kedlaya inequality. In particular, if $\mathcal{M}$ is symmetric, Jensen-concave, and the sequence $\big(\tfrac{\lambda_n}{\lambda_1+\cdots+\lambda_n}\big)$ is nonincreasing. In addition, it is proved that if $\mathcal{M}$ is a symmetric and monotone mean, then the biggest possible weighted Hardy constant is achieved if $\lambda$ is the constant vector.