The question is to find an analytic isomorphism from the open region between $x=1$ to $x=3$ to the upper half unit disk: $\{|z|<1, \text{Im}(z)>0\}$. I would know how to deal with this if it just asks to find an isomorphism from the $x>1$ to the full unit disk. In such a scenario where the region given is truncated, i.e not a full half plane or circle how do we go about approach this? Any hints would be appreciated!
I am thinking of finding an isomorphism such that it maps $x>1$ to the full unit circle $|z|<1$ and that it maps $x>3$ to the lower half circle. The restriction of this isomorphism to $1<x<3$ is the one we want. But how can I go about ensuring the latter condition is satisfied?