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Some convexity properties of Dirichlet series with positive terms

✍ Scribed by Pietro Cerone; Sever S. Dragomir

Publisher: John Wiley and Sons
Year: 2009
Tongue: English
Weight: 151 KB
Volume: 282
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.200610783

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✦ Synopsis

Abstract

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)

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