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 RoperSuffridge 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.

Complete Quasiconvex Mappings in Polydisc

“…Recently, the authors [3] extended the above two kinds of mappings to the unit ball of a general complex Banach space and proved that the class of the quasiconvex mappings coincides with the class of quasiconvex mappings of type A in all Banach spaces. As a result, they call these two classes of mappings unified quasiconvex mappings, which are shown to have the same growth and covering theorems as those of the convex mappings.…”

“…Using the analytic characterization of the convex mappings of one complex variable, we introduce the following definition in [3]:…”

“…It is shown in [3] that the concept of the quasiconvex mapping of type A is equivalent to that of the quasiconvex mapping.…”

The decomposition theorem for the family of complete quasi-convex mappings

“…Taishun Liu and Wenjun Zhang [8] extended the definitions of Q A (Ω) and Q C (Ω) to the open unit ball B in complex Banach space, and proved that…”

The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains

“…We also generalized the Roper-Suffridge operator to the Banach spaces in [12][13][14][15]. All of these papers studied the properties of the generalized Roper-Suffridge operator on a domain D p with some p j = 2.…”