Symmetrization and the rate of convergence of semigroups of holomorphic functions

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