Application of Riemann Mapping Theorem to conformal automorphism Let $\Omega=G -\{a,b\},$ where $G$ is non-empty simply connected bounded domain in $\mathbb{C}$ and $a \neq b.$ Could anyone advise me how to find all conformal automorphism of $\Omega \ ?$ I tried to invoke the use of Riemann mapping theorem to no avail. 
Thank you/ 
 A: Since $G$ is bounded, every holomorphic map $f:\Omega\to\Omega$ extends to a holomorphic map on $G$. (Singularities are removable when the function is bounded nearby.) The extended map (call it $F$) still takes values in $\Omega$, since its image is open. And it is still injective, because when injectivity of a holomorphic map fails, there is an infinite set of points with multiple preimages. (More precisely: if $w$ has at least two preimages $z_1,z_2$, so does every point sufficiently close to $w$; to prove this, consider the images of  the neighborhoods of $z_1,z_2$.)  
So, $F$ is an injective holomorphic map, hence a  homeomorphism onto its image. Since $G$ is simply connected, so is $F(G)$. We conclude that $F(G)=G$. The problem reduces to identifying conformal automorphisms of $G$ that map the set $\{a,b\}$ onto itself. 
This is where the Riemann mapping theorem helps by reducing to the case $G=\mathbb D$ and $a=0$. There are two Möbius transformations $\phi$ that satisfy $\phi(\{0,b\})=\{0,b\}$. One is the identity map, the other is an involution exchanging $0$ and $b$.
