# 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 $\Omega$

• This statement is known as the Schwarz lemma in the case of the disc, and Pick's Theorem or the Schwarz-Pick Lemma when transferred to the hyperbolic metric of general domains. (Here there isn't the assumption that a is fixed, and the conclusion in 2) being that f is a covering map. Your statement follows from this version.) Unfortunately I can't say that I really know the history of the statement. On what basis do you think it should have been first stated in this form by Cartan? May 12, 2014 at 8:40
• Do you mean that $f(\Omega)\subseteq \Omega$ implies that $|f'(a)|\leq 1$ ? May 12, 2014 at 10:02
• Thanks for these answers and sorry for the delay. @LasseRempe-Gillen : I saw it it a textbook of H. Quéffellec & C. Zuily entitled "Eléments d'Analyse" (p. 153 of the 2nd edition). Since I wanted to use similar arguments in my next paper, I wanted to know if there was not a more general statement. Indeed, it could generalize it, but it would be unnecessary if it's already done. May 13, 2014 at 21:15
• @DuchampGérardH.E. : in the proof, this inequality is indeed a consequence of the f-stability of $\Omega$ by applying Cauchy inequalities and also using the fact that a is a fixed point. May 13, 2014 at 21:15
• If you are looking for a generalization in complex dimension 1, then this is given by the (hyperbolic version of) the Schwarz lemma. May 25, 2014 at 22:58

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.

• Thanks for this precise answer :-) That's what I was seeking. May 13, 2014 at 21:19
• I've quickly read the article, but it seems that the theorem I stated is not mentioned. Of course, there are some similarities with the Schwartz lemma, but it is quite different. May 13, 2014 at 22:50

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.)

• OK, thanks for these useful additional details :-) May 13, 2014 at 21:22
• I've quickly read the article, but it seems that the theorem I stated is not mentioned. Of course, there are some similarities with the Schwartz lemma, but it is quite different. May 13, 2014 at 22:51
• As stated in my answer, if you are looking for the theorem that you state, this is the Schwarz lemma, via covering theory. May 25, 2014 at 22:57