Parametric representation and asymptotic starlikeness in

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 . 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 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 is asymptotically starlike; however, in dimension this is no longer true.

**1.**H. Cartan,*Sur la possibilité d'étendre aux fonctions de plusieurs variables complexes la théorie des fonctions univalentes*, 129-155, Note added to P. Montel,*Leçons sur les fonctions univalentes ou multivalentes*, Gauthier-Villars, Paris, 1933.**2.**Mark Elin, Simeon Reich, and David Shoikhet,*Complex dynamical systems and the geometry of domains in Banach spaces*, Dissertationes Math. (Rozprawy Mat.)**427**(2004), 62. MR**2071666**, 10.4064/dm427-0-1**3.**Sheng Gong,*Convex and starlike mappings in several complex variables*, Mathematics and its Applications (China Series), vol. 435, Kluwer Academic Publishers, Dordrecht; Science Press, Beijing, 1998. With a preface by David Minda. MR**1689825****4.**Ian Graham, Hidetaka Hamada, and Gabriela Kohr,*Parametric representation of univalent mappings in several complex variables*, Canad. J. Math.**54**(2002), no. 2, 324–351. MR**1892999**, 10.4153/CJM-2002-011-2**5.**I. Graham, H. Hamada, G. Kohr and M. Kohr,*Asymptotically spirallike mappings in several complex variables*, J. Anal. Math., to appear.**6.**Ian Graham, Hidetaka Hamada, Gabriela Kohr, and Ted J. Suffridge,*Extension operators for locally univalent mappings*, Michigan Math. J.**50**(2002), no. 1, 37–55. MR**1897032**, 10.1307/mmj/1022636749**7.**Ian Graham and Gabriela Kohr,*Geometric function theory in one and higher dimensions*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 255, Marcel Dekker, Inc., New York, 2003. MR**2017933****8.**Ian Graham, Gabriela Kohr, and Mirela Kohr,*Loewner chains and parametric representation in several complex variables*, J. Math. Anal. Appl.**281**(2003), no. 2, 425–438. MR**1982664**, 10.1016/S0022-247X(03)00127-6**9.**Ian Graham, Gabriela Kohr, and Mirela Kohr,*Basic properties of Loewner chains in several complex variables*, Geometric function theory in several complex variables, World Sci. Publ., River Edge, NJ, 2004, pp. 165–181. MR**2115789****10.**Kenneth R. Gurganus,*Φ-like holomorphic functions in 𝐶ⁿ and Banach spaces*, Trans. Amer. Math. Soc.**205**(1975), 389–406. MR**0374470**, 10.1090/S0002-9947-1975-0374470-1**11.**Hidetaka Hamada and Gabriela Kohr,*Subordination chains and the growth theorem of spirallike mappings*, Mathematica**42(65)**(2000), no. 2, 153–161 (2001). MR**1988620****12.**T. Liu,*The growth theorems and covering theorems for biholomorphic mappings on classical domains*, Doctoral Thesis, Univ. Sci. Tech. China, 1989.**13.**J. A. Pfaltzgraff,*Subordination chains and univalence of holomorphic mappings in 𝐶ⁿ*, Math. Ann.**210**(1974), 55–68. MR**0352510****14.**Christian Pommerenke,*Univalent functions*, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen; Studia Mathematica/Mathematische Lehrbücher, Band XXV. MR**0507768****15.**T. Poreda,*On the univalent holomorphic maps of the unit polydisc in 𝐶ⁿ which have the parametric representation. I. The geometrical properties*, Ann. Univ. Mariae Curie-Skłodowska Sect. A**41**(1987), 105–113 (1989) (English, with Polish summary). MR**1049182****16.**T. Poreda,*On the univalent holomorphic maps of the unit polydisc in 𝐶ⁿ which have the parametric representation. II. The necessary conditions and the sufficient conditions*, Ann. Univ. Mariae Curie-Skłodowska Sect. A**41**(1987), 115–121 (1989) (English, with Polish summary). MR**1049183****17.**Tadeusz Poreda,*On generalized differential equations in Banach spaces*, Dissertationes Math. (Rozprawy Mat.)**310**(1991), 50. MR**1104523****18.**Simeon Reich and David Shoikhet,*Nonlinear semigroups, fixed points, and geometry of domains in Banach spaces*, Imperial College Press, London, 2005. MR**2022955****19.**T. J. Suffridge,*The principle of subordination applied to functions of several variables*, Pacific J. Math.**33**(1970), 241–248. MR**0261040****20.**T. J. Suffridge,*Starlike and convex maps in Banach spaces*, Pacific J. Math.**46**(1973), 575–589. MR**0374914****21.**T. J. Suffridge,*Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions*, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), Springer, Berlin, 1977, pp. 146–159. Lecture Notes in Math., Vol. 599. MR**0450601**

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

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.