Parametric representation and asymptotic starlikeness in $\mathbb {C}^n$

Authors:
Ian Graham, Hidetaka Hamada, Gabriela Kohr and Mirela Kohr

Journal:
Proc. Amer. Math. Soc. **136** (2008), 3963-3973

MSC (2000):
Primary 32H02; Secondary 30C45

DOI:
https://doi.org/10.1090/S0002-9939-08-09392-1

Published electronically:
June 9, 2008

MathSciNet review:
2425737

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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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Additional Information

**Ian Graham**

Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada

Email:
graham@math.toronto.edu

**Hidetaka Hamada**

Affiliation:
Faculty of Engineering, Kyushu Sangyo University, 3-1 Matsukadai 2-Chome, Higashi-ku Fukuoka 813-8503, Japan

Email:
h.hamada@ip.kyusan-u.ac.jp

**Gabriela Kohr**

Affiliation:
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., 400084 Cluj-Napoca, Romania

Email:
gkohr@math.ubbcluj.ro

**Mirela Kohr**

Affiliation:
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., 400084 Cluj-Napoca, Romania

Email:
mkohr@math.ubbcluj.ro

Keywords:
Asymptotic starlikeness,
biholomorphic mapping,
Loewner chain,
parametric representation,
spirallike mapping,
starlike mapping.

Received by editor(s):
December 6, 2006

Received by editor(s) in revised form:
October 15, 2007

Published electronically:
June 9, 2008

Additional Notes:
The first author was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A9221

The second author was partially supported by Grant-in-Aid for Scientific Research (C) no. 19540205 from the Japan Society for the Promotion of Science, 2007

The third and fourth authors were partially supported by the Romanian Ministry of Education and Research, CNCSIS Grant type A, 1472/2007

Communicated by:
Mei-Chi Shaw

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.