On negative results concerning weak-Hardy means

TL;DR

This paper establishes a test to determine when a mean does not have the weak-Hardy property and proves that, under certain conditions, the Hardy and weak-Hardy properties are equivalent for specific classes of means.

Contribution

The paper introduces a test for non-weak-Hardy means and shows the equivalence of Hardy and weak-Hardy properties for homogeneous, symmetric, repetition invariant, Jensen concave means.

Findings

Hardy and weak-Hardy properties are equivalent under specified conditions.

A criterion to identify non-weak-Hardy means is established.

The equivalence is proven for means on bR_+ with certain invariance and concavity properties.

Abstract

We establish the test which allows to show that a mean does not admit a weak-Hardy property. As a result we prove that Hardy and weak-Hardy properties are equivalent in the class of homogeneous, symmetric, repetition invariant, and Jensen concave mean on $\mathbb{R}_+$. More precisely, for every mean $\mathscr{M} \colon \bigcup_{n=1}^\infty \mathbb{R}_+^n \to \mathbb{R}$ as above, the inequality $$\mathscr{M}(a_1)+\mathscr{M}(a_1,a_2)+\dots<\infty$$ holds for all $a \in \ell^1(\mathbb{R}_+)$ if and only if there exists a positive, real constant $C$ (depending only on $\mathscr{M}$) such that $$\mathscr{M}(a_1)+\mathscr{M}(a_1,a_2)+\dots<C \cdot (a_1+a_2+\cdots)$$ for every sequence $a \in \ell^1(\mathbb{R}_+)$.