Monotonicity Properties of the Hurwitz Zeta Function

[摘要] Let$$zeta(s,x)=sum_{n=0}^{infty}frac{1}{(n+x)^s} quad{(s>1,, x>0)}$$be the Hurwitz zeta function and let$$Q(x)=Q(x;alpha,eta;a,b)=frac{(zeta(alpha,x))^a}{(zeta(eta,x))^b},$$ where $alpha, eta>1$ and $a,b>0$ are real numbers. We prove: (i) The function $Q$ is decreasing on $(0,infty)$ if{}f $alpha a-eta bgeq max(a-b,0)$.(ii) $Q$ is increasing on $(0,infty)$ if{}f $alpha a-eta bleqmin(a-b,0)$.An application of part (i) reveals that for all $x>0$ the function $smapsto [(s-1)zeta(s,x)]^{1/(s-1)}$ is decreasing on $(1,infty)$. This settlesa conjecture of Bastien and Rogalski.

[学科分类] 数学（综合）

[关键词] subdifferential;directionally regular function;approximate convex function;subdifferentially and directionally stable function