Suppose I have a conformal mapping $f:D\to \Omega$ which takes a unit disc to a connected blob $\Omega\subset \mathbb{C}$. There should exist a conformal mapping $g:\mathbb{C}\setminus D\to \mathbb{C}\setminus\Omega$ (this is true right?)
My question is that, is there any theorem/formula can relate $g$ to $f$ other than they have to agree on the boundary $\partial \Omega$?
What about the mapping $h: D\to \mathbb{C}\setminus\Omega$?