Distortion theorems for biholomorphic convex mappings in \({\mathbb C}^n\) (Q1856868)

Let \({\mathbf K}^{\mathbf n}\) be the family of all biholomorphic mappings of the open unit ball \(B^n=\{z\in \mathbb{C}^n\mid \|z\|< 1\}\) to \(\mathbb{C}^n\) such that \(f(0)=0\), \(Df(0)=I\) and \(f(B^n)\) is a convex domain in \(\mathbb{C}^n\). The authors consider the known inequality \[ \bigl(1+\|z\|\bigr)^{-2} \leq\bigl\|Df(z) \bigr\|\leq \bigl(1-\|z\|\bigr)^{-2} \] and show that the lower bound is not sharp.