Parametric representation and asymptotic starlikeness in ℂⁿ

Abstract

In this paper we consider the notion of asymptotic starlikeness in the Euclidean space C n \mathbb {C}^n . In the case of the maximum norm, asymptotic starlikeness was introduced by Poreda. We have modified his definition slightly, adding a boundedness condition. We prove that the notion of parametric representation which arises in Loewner theory can be characterized in terms of asymptotic starlikeness; i.e. they are equivalent notions. (A regularity assumption of Poreda is not needed.) In particular, starlike mappings and spirallike mappings of type α ∈ ( − π / 2 , π / 2 ) \alpha \in (-\pi /2,\pi /2) are asymptotically starlike. Therefore this notion is a natural generalization of starlikeness. However, we give an example of a spirallike mapping with respect to a linear operator which is not asymptotically starlike. In the case of one complex variable, any function in the class S S is asymptotically starlike; however, in dimension n ≥ 2 n\geq 2 this is no longer true.