Existence of biholomorphic function for two fixed points with $f(z_1) = z_2$ Let $U \subset C$ be a simply connected open subspace and $z_1, z_2 \in U$. I want to show that then there exists a biholomorphic function $f: U \rightarrow U$ with $f(z_1) = z_2$, but I have no idea how to do so.
If $f$ is biholomorphic, $f$ is injective. Hence, from $f(z_1) = f(z_2)$ we get $z_1=z_2$.
If $f$ is biholomorphic, $f$ is injective. Hence, from $f(z_1) = f(z_2)$ we get $z_1=z_2$.