The aim of this paper is to find a broad family of means defined on a subinterval of
$I\subset [0,+\infty )$
such that
$$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }\mathscr{M}(a_{1},\ldots ,a_{n})<+\infty \quad \text{for all }a\in \ell _{1}(I).\end{eqnarray}$$
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.