On the best Hardy constant for quasi-arithmetic means and homogeneous deviation means

TL;DR

This paper characterizes Hardy means, which satisfy a specific inequality involving sums, and determines their Hardy constants for concave quasi-arithmetic and homogeneous deviation means.

Contribution

It provides the first explicit determination of Hardy constants for concave quasi-arithmetic and homogeneous deviation means.

Findings

Explicit Hardy constants are derived for the specified classes of means.

The results unify and extend previous partial findings on Hardy inequalities for means.

The characterization helps identify means with optimal Hardy bounds.

Abstract

The aim of this paper is to characterize 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 $C$. The smallest constant $C$ satisfying this property is called the Hardy constant of the mean $M$. In this paper we determine the Hardy constant in the cases when the mean $M$ is either a concave quasi-arithmetic or a concave and homogeneous deviation mean.