On the Rates of Convergence of Orbits in Semigroups of Holomorphic Functions

Let (t) be a continuous semigroup of holomorphic self-maps of the unit disk D with Denjoy-Wolff point D. We study the rate of convergence of the forward orbits of (t) to the Denjoy-Wolff point by finding explicit bounds for the quantity |t(z) - |, z D, t > 0. We further discuss the corresponding rate of convergence for the backward orbits of (t).