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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: June 9, 2008
MathSciNet review: 2425737
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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

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09392-1
PII: S 0002-9939(08)09392-1
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.