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Some convexity properties of Dirichlet series with positive terms
Author(s) -
Cerone Pietro,
Dragomir Sever S.
Publication year - 2009
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200610783
Subject(s) - mathematics , convexity , dirichlet series , riemann zeta function , series (stratigraphy) , pure mathematics , function (biology) , bounded function , lambda , dirichlet eta function , dirichlet distribution , mathematical analysis , combinatorics , paleontology , physics , evolutionary biology , financial economics , optics , economics , biology , boundary value problem
Some basic results for Dirichlet series ψ with positive terms via log‐convexity properties are pointed out. Applications for Zeta, Lambda and Eta functions are considered. The concavity of the function 1/ ψ is explored and, as a main result, it is proved that the function 1/ ζ is concave on ( ζ –1 ( e ), ∞). As a consequence of this fundamental result it is noted that Zeta at the odd positive integers is bounded above by the harmonic mean of its immediate even Zeta values which are known explicitly (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)