Concavity of weighted arithmetic means with applications

Arkady Berenstein; Alek Vainshtein

doi:10.1007/s000130050101

Concavity of weighted arithmetic means with applications

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Abstract

We prove that the following three conditions together imply the concavity of the sequence {∑ⁿ_i=0 α_iβ_i/ ⁿ∑_i=0 α_i}: concavity of {β_n}, log-concavity of {α_n} and nonincreasing of {(β_n - β_n-1)/(α_n-1/α_n-α _n-2/α_n-1)}. As a consequence we get necessary and sufficient conditions for the concavity of the sequences {S_n-1(x)/S_n(x)} and {S^l_n(x)/S_n(x)} for any nonnegative x, where S_n(x) is the nth partial sum of a power series with arbitrary positive coefficients {α_n}.

Original language	English
Pages (from-to)	120-126
Number of pages	7
Journal	Archiv der Mathematik
Volume	69
Issue number	2
DOIs	https://doi.org/10.1007/s000130050101
State	Published - 1 Aug 1997

Bibliographical note

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

ASJC Scopus subject areas

Mathematics (all)

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10.1007/s000130050101

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title = "Concavity of weighted arithmetic means with applications",

abstract = "We prove that the following three conditions together imply the concavity of the sequence {∑ni=0 αiβi/ n∑i=0 αi}: concavity of {βn}, log-concavity of {αn} and nonincreasing of {(βn - βn-1)/(αn-1/αn-α n-2/αn-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}.",

author = "Arkady Berenstein and Alek Vainshtein",

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

year = "1997",

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doi = "10.1007/s000130050101",

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T1 - Concavity of weighted arithmetic means with applications

AU - Berenstein, Arkady

AU - Vainshtein, Alek

N1 - Funding Information: *) The research of this author is supported by the Rashi Foundation.

PY - 1997/8/1

Y1 - 1997/8/1

N2 - We prove that the following three conditions together imply the concavity of the sequence {∑ni=0 αiβi/ n∑i=0 αi}: concavity of {βn}, log-concavity of {αn} and nonincreasing of {(βn - βn-1)/(αn-1/αn-α n-2/αn-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}.

AB - We prove that the following three conditions together imply the concavity of the sequence {∑ni=0 αiβi/ n∑i=0 αi}: concavity of {βn}, log-concavity of {αn} and nonincreasing of {(βn - βn-1)/(αn-1/αn-α n-2/αn-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}.

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DO - 10.1007/s000130050101

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SN - 0003-889X

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SP - 120

EP - 126

JO - Archiv der Mathematik

JF - Archiv der Mathematik

IS - 2

ER -

Concavity of weighted arithmetic means with applications

Abstract

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