Entire one-to-one functions are linear Can we prove that every entire one-to-one function is linear?

I'll give the "usual" proof.
Note that by Little Picard, $f$ misses at most one point; but it is a homeomorphism onto its image, and the plane minus a point is not simply connected.  Thus $f$ is onto $\mathbb{C}$, and hence bijective.  Then $f$ has a holomorphic inverse, which is enough to imply $f$ is proper, that is, the pre-image of a compact set is compact.  This in turn implies
$$ \lim_{z\rightarrow\infty} f(z)=\infty,$$
and thus if we define $f(\infty)=\infty$, $f$ becomes a Möbius transformation of the Riemann sphere.  So $f$ has the form
$f(z) = \frac{az+b}{cz+d},$
and it is easy to see that if $f$ is entire on $\mathbb{C}$, then $c=0$.
