Weakening of Hardy property for means

Abstract

The aim of this paper is to find a broad family of means defined on a subinterval of $I\subset [0,+\infty )$ such that $$\sum _{n=1}^{\infty }\mathcal{M}(a_1,\dots ,a_n)<+\infty \text{ for all }a\in {\ell }_1(I).$$ Equivalently, the averaging operator $$(a_1,a_2,a_3,\dots )\mapsto (a_1,\mathcal{M}(a_1,a_2),\mathcal{M}(a_1,a_2,a_3),\dots )$$ is a selfmapping of ${\ell }_1(I)$. This property is closely related to so-called Hardy inequality for means (which additionally requires boundedness of this operator). In fact we prove that these two properties are equivalent in a family of Gini means and Gaussian product of Power means. Moreover it is shown that this is not the case for quasi-arithmetic means.

References in corpus (2)

• Characterization of the Hardy property of means and the best Hardy constants
• On the best Hardy constant for quasi-arithmetic means and homogeneous deviation means

Cited by in corpus (1)

• On properties of weighted Hardy constant for means