Growth, distortion and coefficient bounds for Carathéodory families in [formula omitted] and complex Banach spaces

Abstract

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