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

Let X be a complex Banach space with the unit ball B  . The family MM 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 MgMg of MM depending on a univalent function g  . We obtain growth theorems and coefficient bounds for holomorphic mappings in MgMg, including some sharp improvements of existing results. When g   is convex, we study the family RgRg consisting of holomorphic mappings f:B→Xf:B→X which have the property that the mapping Df(z)(z)Df(z)(z) belongs to MgMg. Further, we consider radius problems related to the family RgRg, when X is a complex Hilbert space. In particular, if X   is the Euclidean space CnCn, we obtain some quasiconformal extension results for mappings in RgRg. We also obtain some sufficient conditions for univalence and starlikeness in complex Banach spaces.