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Parametric Representation of Univalent Mappings in Several Complex Variables

Published online by Cambridge University Press:  20 November 2018

Ian Graham ,
Hidetaka Hamada  and
Gabriela Kohr
Ian Graham
Affiliation:
Department of Mathematics University of Toronto Toronto, Ontario M5S 3G3, e-mail: graham@math.toronto.edu
Hidetaka Hamada
Affiliation:
Faculty of Engineering Kyushu Kyoritsu University 1-8 Jiyugaoka, Yahatanishi-ku Kitakyushu 807-8585 Japan, email: hamada@kyukyo-u.ac.jp
Gabriela Kohr
Affiliation:
Faculty of Mathematics Babeş-Bolyai University 1 M. Kogălniceanu Str. 3400 Cluj-Napoca Romania, email: gkohr@math.ubbcluj.ro
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Abstract

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Let $B$ be the unit ball of ${{\mathbb{C}}^{n}}$ with respect to an arbitrary norm. We prove that the analog of the Carathéodory set, i.e. the set of normalized holomorphic mappings from $B$ into ${{\mathbb{C}}^{n}}$ of “positive real part”, is compact. This leads to improvements in the existence theorems for the Loewner differential equation in several complex variables. We investigate a subset of the normalized biholomorphic mappings of $B$ which arises in the study of the Loewner equation, namely the set ${{S}^{0}}\left( B \right)$ of mappings which have parametric representation. For the case of the unit polydisc these mappings were studied by Poreda, and on the Euclidean unit ball they were studied by Kohr. As in Kohr’s work, we consider subsets of ${{S}^{0}}\left( B \right)$ obtained by placing restrictions on the mapping from the Carathéodory set which occurs in the Loewner equation. We obtain growth and covering theorems for these subsets of ${{S}^{0}}\left( B \right)$ as well as coefficient estimates, and consider various examples. Also we shall see that in higher dimensions there exist mappings in $S(B)$ which can be imbedded in Loewner chains, but which do not have parametric representation.

Keywords

32H0230C45
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 2 , 01 April 2002 , pp. 324 - 351
Copyright
Copyright © Canadian Mathematical Society 2002

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