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# Un lemme de Schwarz pour les boules-unités ouvertes

Published:1997-03-01
Printed: Mar 1997
• Jean-Pierre Vigué
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## Abstract

Let $B_1$ and $B_2$ be the open unit balls of ${\bbd C}^{n_1}$ and ${\bbd C}^{n_2}$ for the norms $\Vert\,{.}\,\Vert_1$ and $\Vert\,{.}\, \Vert_2$. Let $f \colon B_1 \rightarrow B_2$ be a holomorphic mapping such that $f(0)=0$. It is well known that, for every $z \in B_1$, $\Vert f(z)\Vert_2 \leq \Vert z \Vert_1$, and $\Vert f'(0)\Vert \leq 1$. In this paper, I prove the converse of this result. Let $f \colon B_1 \rightarrow B_2$ be a holomorphic mapping such that $f'(0)$ is an isometry. If $B_2$ is strictly convex, I prove that $f(0) =0$ and that $f$ is linear. I also define the rank of a point $x$ belonging to the boundary of $B_1$ or $B_2$. Under some hypotheses on the ranks, I prove that a holomorphic mapping such that $f(0) = 0$ and that $f'(0)$ is an isometry is linear.
 MSC Classifications: 32H15 - unknown classification 32H1532H02 - Holomorphic mappings, (holomorphic) embeddings and related questions

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