Let f(x) be a function which is defined in the open unit disk (|z| < 1) and is analytic there. f(z) maps the unit disk onto itself k times, meaning |f(z)| < 1 for all |z| < 1 and every point in the unit disk has k preimages under f(z). Prove that f(z) must be a rational function. Furthermore, show that the degree of its denominator cannot exceed k. If this was limited to k=1 I think we could use the Riemann mapping theorem, but for larger k I am quite lost. How does one go about proving that an arbitrary analytic function with these givens must be rational? I've seen a form of this question in several places, but I still can't grasp how one would tackle such a problem.