Symmetrization and the rate of convergence of semigroups of holomorphic functions

Let , , be a semigroup of holomorphic self‐maps of the unit disk . Let be its Koenigs domain and be its Denjoy–Wolff point. Suppose that and let be the Steiner symmetrization of with respect to the real axis. Consider the semigroup with Koenigs domain and let be its Denjoy–Wolff point. We show that, up to a multiplicative constant, the rate of convergence of is slower than that of ; that is, for every , . The main tool for the proof is the harmonic measure.