Convex mappings on the unit ball of \(\mathbb C^ n\)
The authors study convex mappings of the unit ball into \(\mathbb C^n\), i.e. mappings \(F : B \to \mathbb C^n\) such that \(F\) is univalent, \(F(0) = 0\), \(dF(0) = I\), and \(F(B)\) is convex. The main result of the paper is the following theorem: Theorem 1. Suppose \(f : \Delta \to \mathbb C\) is a univalent convex function on the unit disk \(\Delta \subset \mathbb C\) with \(f(0) = 0\), \(f'(0) = 1\). Form a new function \(F : B \to \mathbb C^2\), where \(B\) is the unit ball in \(\mathbb C^2\) and \(F(2,w) = (f(z), w \sqrt {f'(z)})\). Then \(f\) is a convex mapping. Theorem 1 can be generalized to any complex dimension.
scientific article; zbMATH DE number 807411