Bohr's phenomenon on a regular condensator in the complex plane

Abstract

We prove the following generalisation of Bohr theorem : let $K\subset \mathbb{C}$ a continuum, $(F_n{)}_n$ its Faber polynomials, ${\Omega }_R=\{{\Phi }_K<R\},(R>1)$ the levels sets of the Green function; then there exists $R_0>1$ such that for any $f={\sum }_n{a}_n{F}_n\in \mathcal{O}({\Omega }_{R_0})$ : $f({\Omega }_{R_0})\subset D(0,1)$ implies ${\sum }_n|a_n|\cdot |F_n{|}_K<1$.

12 pages - V3 - English version, to appear in Computional Methods and Function Theory