Symmetrization and the rate of convergence of semigroups of holomorphic functions

Abstract

Let $(\phi_t)$, $t\ge 0$, 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 $\Omega^\sharp$ be the Steiner symmetrization of $\Omega$ with respect to the real axis. Consider the semigroup $(\phi_t^\sharp)$ with Koenigs domain $\Omega^\sharp$ and let $\tau^\sharp$ be its Denjoy-Wolff point. We show that, up to a multiplicative constant, the rate of convergence of $(\phi_t^\sharp)$ is slower than that of $(\phi_t)$; that is, for every $t>0$, $|\phi_t(0)-\tau|\leq 4\pi\, |\phi_t^\sharp(0)-\tau^\sharp|$. The main tool for the proof is the harmonic measure.