On negative results concerning weak-Hardy means

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 $\mathcal{M}:{\bigcup }_{n=1}^{\infty }{\mathbb{R}}_{+}^n\to \mathbb{R}$ as above, the inequality $$\mathcal{M}(a_1)+\mathcal{M}(a_1,a_2)+\cdots <\infty$$ holds for all $a\in {\ell }^1({\mathbb{R}}_{+})$ if and only if there exists a positive, real constant $C$ (depending only on $\mathcal{M}$) such that $$\mathcal{M}(a_1)+\mathcal{M}(a_1,a_2)+\cdots <C\cdot (a_1+a_2+\cdots )$$ for every sequence $a\in {\ell }^1({\mathbb{R}}_{+})$.