DISTORTION-THEOREMS FOR BLOCH FUNCTIONS

We establish various distortion theorems for both normalized locally schlicht Bloch functions and normalized Bloch function with branch points. These distortion theorems give lower bounds on either \f'(z)\ or Re f'(z); most of our distortion theorems are sharp and all extremal functions identified. The main tools used in establishing these distortion theorems are the classical form of Julia's Lemma and a new version of Julia's Lemma that applies to certain multiple-valued analytic functions. As applications of these distortion theorems, we obtain known lower bounds for various Bloch constants and also establish improved lower bounds on a number of Marden constants for Bloch, normal and Yosida functions.