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Search: MSC category 32H02 ( Holomorphic mappings, (holomorphic) embeddings and related questions )

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1. CJM 2009 (vol 61 pp. 566)

Graham, Ian; Hamada, Hidetaka; Kohr, Gabriela; Pfaltzgraff, John A.
 Convex Subordination Chains in Several Complex Variables In this paper we study the notion of a convex subordination chain in several complex variables. We obtain certain necessary and sufficient conditions for a mapping to be a convex subordination chain, and we give various examples of convex subordination chains on the Euclidean unit ball in $\mathbb{C}^n$. We also obtain a sufficient condition for injectivity of $f(z/\|z\|,\|z\|)$ on $B^n\setminus\{0\}$, where $f(z,t)$ is a convex subordination chain over $(0,1)$. Keywords:biholomorphic mapping, convex mapping, convex subordination chain, Loewner chain, subordinationCategories:32H02, 30C45

2. CJM 2002 (vol 54 pp. 324)

Graham, Ian; Hamada, Hidetaka; Kohr, Gabriela
 Parametric Representation of Univalent Mappings in Several Complex Variables Let $B$ be the unit ball of $\bb{C}^n$ with respect to an arbitrary norm. We prove that the analog of the Carath\'eodory set, {\it i.e.} the set of normalized holomorphic mappings from $B$ into $\bb{C}^n$ 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 $S^0(B)$ 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 $S^0(B)$ obtained by placing restrictions on the mapping from the Carath\'eodory set which occurs in the Loewner equation. We obtain growth and covering theorems for these subsets of $S^0(B)$ 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. Categories:32H02, 30C45