Bohr’s Phenomenon on a Regular Condenser in the Complex Plane

Abstract

We prove the following generalization of Bohr’s Theorem: let $K\subset{\rm C}$ be a continuum, $(F_{K,n})_{n\geq 0}$ its Faber polynomials, ω_nr the level sets of the Green function of ℂ K with singularity at infinity. Then there exists a radius $R_{0}>0$ such that for any $f=\sum_{n}a_{n}F_{K,n}\in {O}(\Omega_{R_0})$ with $f(\Omega_{R_0})\subset D(0,1) {\rm \ we \ have} \sum_{n}|a_{n}|.||F_{K,n}||_K<1$.