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.