A condition for univalence in the polydisk (Q1045723)

The authors study the Schwarzian derivative operator. This operator has been defined by R.~Hernandez. This definition is based on analogs of the Schwarzian derivative for locally biholomorphic mappings in \(\mathbb{C}^n\), introduced by T.~Oda. The authors prove that, if \(F\) is a locally biholomorphic mapping and the norm of its Schwarzian derivative operator is bounded by \(1/3\sqrt{2},\) then \(F\) is univalent. Using the example of a mapping \(F(z)=(f(z_1),z_2,\dots,z_n)\) where \(f\) is locally univalent in the disk \(\{z_1:|z|<1\}\), the authors demonstrate the effect of the obtained sufficient condition of univalence.