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

TL;DR

This paper generalizes Bohr's theorem to complex plane condensers using Faber polynomials and Green function level sets, establishing conditions under which holomorphic functions map certain domains into the unit disk.

Contribution

It introduces a new Bohr-type inequality involving Faber polynomials and Green functions for continua in the complex plane, extending classical results.

Findings

Established a Bohr phenomenon for functions on level sets of Green functions.

Proved the existence of a threshold R_0 for the domain levels.

Derived a bound involving Faber polynomial coefficients and the continuum.

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 $\sum_n|a_n|\cdot|F_n|_K<1$.