Find a certain analytic function Let $D_1$ and $D_2$ be two disjoint disks.Find an analytic function $f$ defined on the upper half plane such that $f$ takes every value in $D_1$ exactly once and every value of $D_2$ exactly twice.
I don't have any clue about this problem and any advice is welcome.

Consider the upper half plane $\mathbf{H}$ minus two rays as divided into three wedges $A = \{ e^{i\theta}\,|\,0<\theta<\pi/3\}, B = \{ e^{i\theta}\,|\,\pi/3<\theta<2\pi/3\}, C = \{ e^{i\theta}\,|\,2\pi/3<\theta<\pi\}$. Now the map $z\mapsto z^3$ takes $A$ biholomorphically to $\mathbf{H}$, $B$ biholomorphically to $-\mathbf{H}$, and $C$ biholomorphically to $\mathbf{H}$. This means that the cubing map is two-to-one from $\mathbf{H}$ onto $\mathbf{H}$  and one-to-one from $\mathbf{H}$ onto $-\mathbf{H}$. Now you can just move your disks (using, say, a Möbius transformation) so that one lies in $\mathbf{H}$ and one lies in $-\mathbf{H}$.
