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WEAKENING OF THE HARDY PROPERTY FOR MEANS

Part of: Real functions: Miscellaneous topics Inequalities

Published online by Cambridge University Press:  09 July 2019

PAWEŁ PASTECZKA Open the ORCID record for PAWEŁ PASTECZKA [Opens in a new window]
PAWEŁ PASTECZKA*
Affiliation:
Institute of Mathematics, Pedagogical University of Cracow, Podchorążych str. 2, 30-084 Kraków, Poland email pawel.pasteczka@up.krakow.pl
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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

$$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }\mathscr{M}(a_{1},\ldots ,a_{n})<+\infty \quad \text{for all }a\in \ell _{1}(I).\end{eqnarray}$$
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.

Keywords

Hardy meanquasi-arithmetic meanselfmapping of ℓ1bounded operator

MSC classification

Primary: 26E60: Means
Secondary: 26D15: Inequalities for sums, series and integrals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 1 , February 2020 , pp. 118 - 129
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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