I was trying to solve the following problem:


Let $f: D \longrightarrow D$ be proper holomorphic (so that means it is a Blaschke product with finitely many factors). Suppose $\{ a\_1, ..., a\_n \} \subset D$ be the set of branch points of $f$ and the ramification degree of $a\_j$ is $m\_j \in [2, \infty) \cap \mathbb{N}$ (by this I mean there exists $\alpha\_j \in D$ such that $f( \alpha\_j) = a\_j$ and $\alpha\_j$ is a ramification point of degree $m\_j$).


The only progress I have made so far is that I can extend this map to be a holomorphic map from the Riemann sphere to itself. Then I can get a lower bound on the degree of the rational map using the Riemann-Hurwitz formula.


I would like to know how to construct such a function and if this problem has connections to any other phenomenon in (algebraic) geometry. Thanks in advance.



Constructing proper holomorphic self-mappings of the unit disk with a given set of branch points and corresponding ramification degrees
This (existence and uniqueness) is proved in the paper in much more general setting (in fact, the result is due to E. Picard):


M. Heins, ‘On a class of conformal metrics’, Nagoya Math. J. 21 (1962) 1–50.


For a more recent and simpler proof of the special case that you ask, see


MR1479037 (99d:30009)
Zakeri, Saeed
On critical points of proper holomorphic maps on the unit disk.
Bull. London Math. Soc. 30 (1998), no. 1, 62–66.


