I'd like to know in which paper of H. Cartan I could find the following theorem :

Let $\Omega$ be a connected, open and bounded subset of $\mathbb{C}$. Let $a \in \Omega$ and $f \in H(\Omega,\Omega)$ with $f(a) = a$ and set $f_n = f o \ldots o f$ (n times).

Then

1) $\mid f'(a)\mid \le 1$

2) $\mid f'(a)\mid = 1 \Leftrightarrow f \in Aut (\Omega)$

3) $\mid f'(a)\mid \lt 1 \Rightarrow f_n \rightarrow^{uc} \phi_a $, where $\phi_a \in H(\Omega)$ is the constant function defined by $\phi_a(z) = a$ for every z in $\Omega$