Original article about a theorem of Cartan on iterations of analytic functions 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 $\Omega$
 A: Cartan's theorem (1931) is the extension of Schwarz's lemma to multivariable functions. See The Schwarz lemma at the boundary, page 9.
H. Cartan, Les fonctions de deux variables complexes et le problème de la représentation analytique, J. Math. Pures Appl. 19 (1931) 1-114.
A: As mentioned by Carlo, the result of Cartan that you may be looking for is actually a generalization of the Schwarz lemma to several variables. I think the paper in question may be
H. Cartan, Sur les fonctions de deux variables complexes. Les transformations d’un domaine borné $D$ en un domaine intérieur à $D$, Bulletin de la Société Mathématique de France (1930)
Volume: 58, page 199-219. https://eudml.org/doc/86572
Assuming that you are interested in the one-dimensional version as stated above, it follows from the original Schwarz lemma together with the uniformization theorem (proved by Poincaré and Koebe). I would suggest looking at the sources cited in the Encyclopedia of Mathematics entry, which refers in particular to "The Schwarz lemma" by Dineen and pp. 191-192 of "An introduction to classical complex analysis" by Burchel. (I do not have these books to hand.)
