Monotonicity Properties of the Hurwitz Zeta Function

Alzer, Horst

Canadian mathematical bulletin, Tome 48 (2005) no. 3, pp. 333-339

Résumé

Let $$\zeta \left( s,x \right)=\sum\limits_{n=0}^{\infty }{\frac{1}{{{\left( n+x \right)}^{s}}}}\left( s>1,x>0 \right)$$ be the Hurwitz zeta function and let $$Q\left( x \right)=Q\left( x;\alpha ,\beta ;a,b \right)=\frac{{{\left( \zeta \left( \alpha ,x \right) \right)}^{a}}}{{{\left( \zeta \left( \beta ,x \right) \right)}^{{{b}'}}}}$$ where $\alpha ,\beta >1$ and $a,b>0$ are real numbers. We prove: (i) The function $Q$ is decreasing on $\left( 0,\infty\right)$ iff $\alpha a-\beta b\ge \max \left( a-b,0 \right)$ . (ii) $Q$ is increasing on $\left( 0,\infty\right)$ iff $\alpha a-\beta b\le \min \left( a-b,0 \right)$ . An application of part (i) reveals that for all $x>0$ the function $s\mapsto {{\left[ \left( s-1 \right)\zeta \left( s,x \right) \right]}^{1/\left( s-1 \right)}}$ is decreasing on $\left( 1,\infty\right)$ . This settles a conjecture of Bastien and Rogalski.

Détail

DOI : 10.4153/CMB-2005-031-3

Mots-clés : 11M35, 26D15