Growth, distortion and coefficient bounds for Carathéodory families in and complex Banach spaces

Let X be a complex Banach space with the unit ball B. The family M is a natural generalization to complex Banach spaces of the well-known Carathéodory family of functions with positive real part on the unit disc. We consider subfamilies M g of M depending on a univalent function g. We obtain growth theorems and coefficient bounds for holomorphic mappings in M g , including some sharp improvements of existing results. When g is convex, we study the family R g consisting of holomorphic mappings f : B → X which have the property that the mapping D f ( z ) ( z ) belongs to M g . Further, we consider radius problems related to the family R g , when X is a complex Hilbert space. In particular, if X is the Euclidean space C n , we obtain some quasiconformal extension results for mappings in R g . We also obtain some sufficient conditions for univalence and starlikeness in complex Banach spaces.