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My problem is to prove that every non-constant proper holomorphic map of unit disc into itself is product of finitely many disc automorphisms. As a hint I have: If $f$ is proper and $f=Mg$ where $M$ is automorphism than $|g|\leq 1$ on disc.

I proved that preimage of every singleton is a set of finitely many points, and I suspect that this gives that we will have only finitely many automorphisms. I know the clasification of unit disc automorphisms and my idea is to prove that automorphisms in product will be $\varphi_a$ where $f(a)=0$, with corresponding multiplicities. For this I would use the fact in hint and some kind of induction.

So, can you help me with proof of hint and to give me some idea about the rest: am I on right way or no?

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