Conformal maps from upper-half plane to two disjoint disks

Question

Given two disjoint disks $D_1$ and $D_2$. How to find a analytic function $f$ on upper half plane so that $f$ takes every value in $D_1$ exactly once and every value in $D_2$ exactly twice?

Answer 1

Let us do it for SOME two disks first. The function $f(z)=z^3$ in the upper half-plane covers the upper half-plane twice and lower half-plane once. To see this, draw the images of the first quadrant and of the second quadrant separately, and then paste them together. So every disk $D_1$ in the lower half-plane is covered exactly once and every disk $D_2$ in the upper half-plane is covered exactly twice. Now you can arrange this for any given $D_1$ and $D_2$ by replacing your $f$ by $af+b$ with appropriate constants $a,b$.