WEAKENING OF THE HARDY PROPERTY FOR MEANS.

The aim of this paper is to find a broad family of means defined on a subinterval of $I\subset [0,+\infty)$ such that $$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }\mathscr{M}(a_{1},\ldots ,a_{n}) Equivalently, the averaging operator $(a_{1},\,a_{2},a_{3}\,,\ldots)\mapsto (a_{1},\,\mathscr{M}(a_{1},a_{2}),\,\mathscr{M}(a_{1},a_{2},a_{3}),\ldots)$ is a selfmapping of $\ell _{1}(I)$. This property is closely related to the so-called Hardy inequality for means (which additionally requires boundedness of this operator). We prove that these two properties are equivalent in a broad family of so-called Gini means. Moreover, we show that this is not the case for quasi-arithmetic means, that is functions $f^{-1}(\sum f(a_{i})/n)$ , where $f:I\rightarrow \mathbb{R}$ is continuous and strictly monotone, $n\in \mathbb{N}$ and $a\in I^{n}$. However, the weak Hardy property is localisable for this family. [ABSTRACT FROM AUTHOR]