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On negative results concerning weak-Hardy means

Volume 171 / 2023

Paweł Pasteczka Colloquium Mathematicum 171 (2023), 153-158 MSC: Primary 26D15; Secondary 26E60. DOI: 10.4064/cm8749-4-2022 Published online: 2 August 2022


We establish a test which allows one to show that a mean does not have 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 means on $\mathbb {R}_+$.

More precisely, for every mean $\mathscr{M} \colon \bigcup _{n=1}^\infty {\mathbb R}_+^n \to {\mathbb R}$ as above, the inequality $${\mathscr M}(a_1)+{\mathscr M}(a_1,a_2)+\dots \lt \infty $$ holds for all $a \in \ell _1(\mathbb R _+)$ if and only if there exists a positive real constant $C$ (depending only on $\mathscr{M} $) such that $${\mathscr M}(a_1)+{\mathscr M}(a_1,a_2)+\dots \lt C \cdot (a_1+a_2+\cdots )$$ for every sequence $a \in \ell _1({\mathbb R}_+)$.


  • Paweł PasteczkaInstitute of Mathematics
    Pedagogical University of Kraków
    Podchorążych 2
    30-084 Kraków, Poland

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