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.