I have found in several books the following affirmation :
Let $f: \Delta \rightarrow \Delta$ be a non constant holomorphic function that extends continuously to $\overline{\Delta}$, $\Delta$ being the open unit disk. Then $f$ is a finite Blaschke product, i.e. of the form $$B(z) = e^{i \theta} \prod_{k=1}^d \frac{z- a_i}{1-\overline{a_i} z}$$
Now it is not hard to check that those rational fractions do indeed send $\Delta$ onto $\Delta$, but I do not know where to find a proof that they are the only ones.
Where can I find that proof in the literature ? I am not interested in the infinite product case, only finite. (Of course, if anyone is kind enough to post a proof, that would do fine as well =)
I think you have to assume that $f$ is non constant and that $f$ maps the unit circle into itself. In this case, the result follows by the maximum modulus principle.
Indeed, first note that $f$ has a finite number of zeros in $\mathbb{D}$, otherwise since $f$ is not identically zero the zeros would accumulate on the circle, which contradicts the fact that $|f|=1$ there.
Then, consider $B$ a finite blaschke product that has exactly the same zeros than $f$. Then $B/f$ and $f/B$ are holomorphic in $\mathbb{D}$, continuous in $\overline{\mathbb{D}}$. Furthermor, $|B/f|=1$ and $|f/B|=1$ on the unit circle. By the maximum modulus principle, we get that $|B/f| \equiv 1$ in $\mathbb{D}$ and thus $B/f$ is a unimodular constant. This gives the result.
Note that infinite blaschke products do not extend continuously to the unit circle : they have (essential) singularities where the zeros accumulate.