Characterization of the Hardy property of means and the best Hardy constants

TL;DR

This paper characterizes Hardy means within broad classes of symmetric, increasing, Jensen concave, and repetition invariant means, providing a formula for the optimal Hardy constant that bounds the sum of means by the sum of sequence elements.

Contribution

It offers a broad characterization of Hardy means and derives a formula for the best Hardy constant within specific classes of means.

Findings

Characterization of Hardy means in broad classes of means.

Derivation of a formula for the optimal Hardy constant.

Identification of conditions under which means satisfy Hardy inequalities.

Abstract

The aim of this paper is to characterize in broad classes of means the so-called Hardy means, i.e., those means $M\colon\bigcup_{n=1}^\infty \mathbb{R}_+^n\to\mathbb{R}_+$ that satisfy the inequality $$ \sum_{n=1}^\infty M(x_1,\dots,x_n) \le C\sum_{n=1}^\infty x_n $$ for all positive sequences $(x_n)$ with some finite positive constant $C$. One of the main results offers a characterization of Hardy means in the class of symmetric, increasing, Jensen concave and repetition invariant means and also a formula for the best constant $C$ satisfying the above inequality.