On negative results concerning weak-Hardy means

Abstract

. We establish the test which allows to show that a mean does not admit 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 mean on R + . More precisely, for every mean M : S ∞ n =1 R n + → R as above, the inequality M ( a 1 ) + M a 1 , a 2 ) + · · · < ∞ holds for all a ∈ ℓ 1 ( R + ) if and only if there exists a positive, real constant C (depending only on M ) such that ) for every sequence a ∈ ℓ 1 ( R + ) .