Some Convexity Properties of Dirichlet Series with Positive Terms

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


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.

Item type Article
Subjects Historical > FOR Classification > 0101 Pure Mathematics
Current > Collections > Research Group in Mathematical Inequalities and Applications (RGMIA)
Keywords Dirichlet series, zeta function, lambda function, logarithmic convexity, log-convex
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