The Bohr inequality for ordinary Dirichlet series
Volume 175 / 2006
Studia Mathematica 175 (2006), 285-304
MSC: 30B10, 30B50, 40A05, 42A45, 42B30, 11M41.
DOI: 10.4064/sm175-3-7
Abstract
We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if $f(s) = \sum_{n=1}^{\infty}a_nn^{-s}$ with $\| f \|_{\infty} := \sup_{\Re s > 0} |f(s)| < \infty$, then $\sum_{n=1}^{\infty}|a_n|n^{-2} \leq \| f \|_{\infty}$ and even slightly better, and $\sum_{n=1}^{\infty}|a_n|n^{-1/2} \leq C\| f \|_{\infty}$, $C$ being an absolute constant.
