Bloch constants in several variables

Abstract

We give lower estimates for Bloch's constant for quasiregular holomorphic mappings. A holomorphic mapping of the unit ball Bn into Cn is K-quasiregular if it maps infinitesimal spheres to infinitesimal ellipsoids whose major axes are less than or equal to K times their minor axes. We show that if f is a K-quasiregular holomorphic mapping with the normalization det f'(O) = 1, then the image f(Bn) contains a schlicht ball of radius at least 1/12K1-1/n. This result is best possible in terms of powers of K. Also, we extend to several variables an analogous result of Landau for bounded holomorphic functions in the unit disk.