On the rate of convergence of parabolic semigroups of holomorphic functions

Abstract

Let \(\{\phi _t\}_{t\ge 0}\) be a semigroup of holomorphic self-maps of the unit disk \({\mathbb D}\) with Denjoy–Wolff point 1 and associated planar domain \(\Omega \). We further assume that for all \(t>0\), \(\phi _t^\prime (1)=1\) (angular derivative), namely the semigroup is parabolic. We study the rate of convergence of the semigroup to 1. We prove then that for every \(z\in {\mathbb D}\), there exists a positive constant \(C\) such that \(|\phi _t(z)-1|\le C\;t^{-1/2}\). If, in addition, \(\Omega \) is contained in a half-plane, then we prove a stronger estimate: \(|\phi _t(z)-1|\le C\;t^{-1}\).