The extension operator in banach spaces for locally biholomorphic mappings

In this article, the generalized Roper—Suffridge extension operator in Banach spaces for locally biholomorphic mappings is introduced. It is proved that this operator preserves the starlikeness on some domains in Banach spaces but does not preserves convexity for some cases. Moreover, the growth theorem, covering theorem, and the radius of starlikeness are discussed. Some results of Roper and Suffridge, Gong and Liu, Graham et al in Cn are extended to Hilbert spaces or Banach spaces.