The generalized Roper–Suffridge extension operator in Banach spaces (II)

Abstract

In this paper, we generalize the Roper–Suffridge extension operator from C n to Banach spaces. It is proved that this operator preserves the biholomorphic ɛ starlikeness on some domains in Banach spaces. From these, we may construct a lots of concrete examples about biholomorphic ɛ starlike mappings on some domains Ω in C n , or Hilbert spaces, or Banach spaces from univalent ɛ starlike functions on the unit disc U in C. Meanwhile, the growth theorems of the corresponding mappings are given. Some results of Gong and Liu, Roper and Suffridge, Graham et al. in C n are extended to Hilbert spaces or Banach spaces.