Prove that:
if $f$ is a meromorphic function on $\mathbb{C}$, such that $\mathbb{C}_\infty-f(\mathbb C)$ has at least three points then $f$ is a constant.
I think that this should use transformations to take the function be entire, and then use Little Picard theorem, however I do not know how to do it..
any advice will be appreciated!
Perhaps a bit less abstract:
Choosing well $a,b,c,d,ad-bc=1$ then $g(z)=\frac{af(z)+b}{cf(z)+d}$ is entire and it misses $0$ and $1$.
Then $\lambda^{-1}(g(z))$ is entire with image in the upper half-plane so that $\frac1{i+\lambda^{-1}(g(z))}$ is entire and bounded thus constant.
The difficulty is constructing $\lambda$, showing that $\lambda^{-1}$ is locally analytic on $\Bbb{C-\{0,1\}}$ and that it implies that (the correct branch of) $\lambda^{-1}(g(z))$ is entire.