Symmetrization and the rate of convergence of semigroups of holomorphic functions

Abstract

Let $({\varphi }_t)$, $t\ge 0$, be a semigroup of holomorphic self-maps of the unit disk $\mathbb{D}$. Let $\Omega$ be its Koenigs domain and $\tau \in \partial \mathbb{D}$ be its Denjoy-Wolff point. Suppose that $0\in \Omega$ and let ${\Omega }^{♯}$ be the Steiner symmetrization of $\Omega$ with respect to the real axis. Consider the semigroup $({\varphi }_t^{♯})$ with Koenigs domain ${\Omega }^{♯}$ and let ${\tau }^{♯}$ be its Denjoy-Wolff point. We show that, up to a multiplicative constant, the rate of convergence of $({\varphi }_t^{♯})$ is slower than that of $({\varphi }_t)$; that is, for every $t>0$, $|{\varphi }_t(0)-\tau |\le 4\pi |{\varphi }_t^{♯}(0)-{\tau }^{♯}|$. The main tool for the proof is the harmonic measure.