Parametric representation and asymptotic starlikeness in ℂⁿ

Graham, Ian; Hamada, Hidetaka; Kohr, Gabriela; Kohr, Mirela

doi:10.1090/S0002-9939-08-09392-1

Parametric representation and asymptotic starlikeness in $\mathbb {C}^n$
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by Ian Graham, Hidetaka Hamada, Gabriela Kohr and Mirela Kohr PDF

Proc. Amer. Math. Soc. 136 (2008), 3963-3973 Request permission

Abstract:

In this paper we consider the notion of asymptotic starlikeness in the Euclidean space $\mathbb {C}^n$. 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 $\alpha \in (-\pi /2,\pi /2)$ 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 $S$ is asymptotically starlike; however, in dimension $n\geq 2$ this is no longer true.

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