Approximation by biholomorphisms Assuming two domains $\Omega_1 \subset \Omega_2 \subset \mathbb{C}$ satisfy the criteria in Runge's theorem. So we know that any holomorphic function $f: \Omega_1\to \Omega_1$ can be approximated uniformly on compacts by (the restriction to $\Omega_1$ of) a sequence of holomorphic functions $f_i:\Omega_2 \to \Omega_2$. 
I have seen that if the function $f$ is non-zero, one may assume the approximating sequence $\{f_i\}$ consists solely of non-zero functions.
My question is if $f$ is biholomorphic  of $\Omega_1$, (i.e. has a holomorphic inverse), can we approximate it with a sequence of biholomorphisms on $\Omega_2$?

No. Counterexamples abound, obtained by choosing things so that $\Omega_2$ has a small automorphism group.
For example, let $\Omega_1$ be the right half plane, $\Omega_2=\Bbb C$, and $f(z)=1/z$. Recall that every biholomorphic map from $\Bbb C$ to itself has the form $z\mapsto az+b$; it's clear that such things cannot aproximate $f$ uniformly on $[1,2]$.
