Abstract view
Un lemme de Schwarz pour les boulesunités ouvertes


Published:19970301
Printed: Mar 1997
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.