On ε quasi-convex mappings in the unit ball of a complex Banach space ✩

In this paper, a new class of biholomorphic mappings named “ε quasi-convex mapping” is introduced in the unit ball of a complex Banach space. Meanwhile, the definition of ε-starlike mapping is generalized from ε ∈ [0, 1] to ε ∈ [−1, 1]. It is proved that the class of ε quasi-convex mappings is a proper subset of the class of starlike mappings and contains the class of ε starlike mappings properly for some ε ∈ [−1, 0)∪(0, 1]. We give a geometric explanation for ε-starlike mapping with ε ∈ [−1, 1] and prove that the generalized Roper– Suffridge extension operator preserves the biholomorphic ε starlikeness on some domains in Banach spaces for ε ∈ [−1, 1]. We also give some concrete examples of ε quasi-convex mappings or ε starlike mappings for ε ∈ [−1, 1] in Banach spaces or Cn. Furthermore, some other properties of ε quasi-convex mapping or ε-starlike mapping are obtained. These results generalize the related works of some authors. © 2005 Elsevier Inc. All rights reserved