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# Monotonicity Properties of the Hurwitz Zeta Function

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 $$Q(x)=Q(x;\alpha,\beta;a,b)=\frac{(\zeta(\alpha,x))^a}{(\zeta(\beta,x))^b},$$ where $\alpha, \beta>1$ and $a,b>0$ are real numbers. We prove: (i) The function $Q$ is decreasing on $(0,\infty)$ if{}f $\alpha a-\beta b\geq \max(a-b,0)$. (ii) $Q$ is increasing on $(0,\infty)$ if{}f $\alpha a-\beta b\leq \min(a-b,0)$. An application of part (i) reveals that for all $x>0$ the function $s\mapsto [(s-1)\zeta(s,x)]^{1/(s-1)}$ is decreasing on $(1,\infty)$. This settles a conjecture of Bastien and Rogalski.