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 $f=\sum_n a_n F_n\in\mathscr O(\Omega_{R_0})$ : $f(\Omega_{R_0})\subset D(0,1)$ implies $\sum_n|a_n|\cdot|F_n|_K<1$.