The growth and 1/4-theorems for starlike mappings in \(\mathbb{C}^ n\) (Q1177394)

\textit{Henri Cartan} [``Sur la possibilité d'étendre aux fonctions de plusieurs variables complexes la théorie des fonctions univalentes'', 129-155. Note added to \textit{P. Montel}, ``Leçons sur les fonctions univalentes ou multivalentes'', Paris (1933; Zbl 0006.35102)] suggested that geometric function theory of one complex variable (in particular, starlike and convex functions) should be extended to biholomorphic mappings of \(n\) complex variables. In noting some of the difficulties of the generalization, he pointed out that growth theorem and Koebe \({1\over 4}\)-theorem would extend neither to the polydisc nor to the ball. However, the authors show (in the note under review) that, for the class of the normalized one-to-one starlike mappings on the ball, versions of the theorem mentioned above hold. As a consequence they establish that ``the only balanced domain which is the image of the unit ball under a normalized biholomorphic mapping is the unit ball''.