A note on convexity properties of functions related to the Hurwitz zeta and alternating Hurwitz zeta function

Abstract

Using the Hurwitz zeta and the alternating Hurwitz zeta function, ζ(s,a) and ζ⁎(s,a), it was shown through classical analysis and in a straightforward and unified manner that asζ(s,a) with a>0 and s>1 is strictly log-convex in s on (1,∞), whereas asζ⁎(s,a) for a,s>0 is strictly concave in s on (0,∞). As an immediate consequence, convexity properties of the Riemann zeta function as well as the Dirichlet beta, eta and lambda function were deduced. © 2020 Elsevier Inc.