Cerone, Pietro and Dragomir, Sever S (2005) Some Convexity Properties of Dirichlet Series with Positive Terms. Research report collection, 8 (4).

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 (1,∞) . 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.

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