Bloch constants in several variables (Q2701661)

A holomorphic mapping of the unit ball \(B^n\) into \(\mathbb C^n\) is called \(K\)-quasiregular if it maps infinitesimal spheres to infinitesimal ellipsoids whose major axes are less than or equal to \(K\) times their minor axes. A holomorphic mapping \(f\) from \(B^n\) into \(\mathbb C^n\) is said to be a Wu \(K\)-mapping if \(|f'|\leq K|\det f'|^{1/n}.\) NEWLINENEWLINENEWLINEThe authors give two new lower bounds for Bloch's constant \(\beta(K,n)\) for Wu \(K\)-mapping. It is shown 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 \(K.\) NEWLINENEWLINENEWLINEThe known theorem of Landau for bounded holomorphic functions in the unit disk is extended to several variables.