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

Abstract

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 C n . Furthermore, some other properties of ε quasi-convex mapping or ε-starlike mapping are obtained. These results generalize the related works of some authors.