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# Convex Subordination Chains in Several Complex Variables

Published:2009-06-01
Printed: Jun 2009
• Ian Graham
• Gabriela Kohr
• John A. Pfaltzgraff
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## Abstract

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, subordination
 MSC Classifications: 32H02 - Holomorphic mappings, (holomorphic) embeddings and related questions 30C45 - Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)

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