Existence of certain analytic functions

Question

Let $D_1,D_2\subset\mathbb{C}$ be two disjoint disks. Is there an analytic function defined on the upper half plane $f:\mathbb{H}\rightarrow\mathbb{C}$ such that $f$ takes each value in $D_1$ exactly once and eaach value in $D_2$ exactly twice? By argument principle, one can consider an analytic function $f$ such that the argument of its image $f(z)$ along $\partial\mathbb{H}$ has winding number 1 about $D_1$ and winding number 2 about $D_2$. Since $D_1$ and $D_2$ are disjoint, such a function always exists. Without loss of generality, we can use rotation and translation to put the centers of $D_1$ and $D_2$ on the real axis, but I don't konw how to write down $f$ explicitly.

Answer 1

Define $f(z)=z^{5/2}$, with $f(i)=\mathrm{e}^{5\pi i/4}$. Then the open first quartile (i.e., $\{(x,y): x,y>0\}$) is taken twice, while the second, third and fourth exactly once.

So, how to modify this example in order to get an example for your question. Let $C$ be a straight line of the plane with the property that the two disk lie in the opposite side the $C$, and $C'$ a perpendicular one with the property that both disks lie on the same side of $C'$. Assume that their intersection is $z_0=x_0+iy_0$ and $C$, $C'$ are $$ C: (t-x_0)\cos\vartheta=(t-y_0)\sin\vartheta,\quad t\in\mathbb R, $$ $$ C': (t-x_0)\sin\vartheta +(t-y_0)\cos\vartheta,\quad t\in\mathbb R. $$ Then $g(z)=\mathrm{e}^{i\vartheta}(z-z_0)^{5/2}$ is one such analytic function.