Bloch constants in several variables

We give lower estimates for Bloch's constant for quasiregular holomorphic mappings. A holomorphic mapping of the unit ball $B^n$ into $\mathbf{C}^n$ 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 $\det f'(0) =1,$ then the image $f(B^n)$contains a schlicht ball of radius at least $1/12K^{1-1/n}.$ 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.