Concavity of weighted arithmetic means with applications

Arkady Berenstein, Alek Vainshtein

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Abstract

We prove that the following three conditions together imply the concavity of the sequence {∑ni=0 αiβi/ ni=0 αi}: concavity of {βn}, log-concavity of {αn} and nonincreasing of {(βn - βn-1)/(αn-1nn-2n-1)}. As a consequence we get necessary and sufficient conditions for the concavity of the sequences {Sn-1(x)/Sn(x)} and {Sln(x)/Sn(x)} for any nonnegative x, where Sn(x) is the nth partial sum of a power series with arbitrary positive coefficients {αn}.

Original languageEnglish
Pages (from-to)120-126
Number of pages7
JournalArchiv der Mathematik
Volume69
Issue number2
DOIs
StatePublished - 1 Aug 1997

Bibliographical note

Funding Information:
*) The research of this author is supported by the Rashi Foundation.

ASJC Scopus subject areas

  • General Mathematics

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