Given $f$ entire function on $\mathbb C$ and $f$ one-one. Is it true that $f$ is linear?
At least among polynomials the only such functions are linear!
Given $f$ entire function on $\mathbb C$ and $f$ one-one. Is it true that $f$ is linear?
At least among polynomials the only such functions are linear!
The link given by @Patience leads to a proof, but one can avoid the heavier things like Picard, Casorati-Weierstrass and the very notion of essential singularity. Liouville's theorem is enough.
Pick a point $a$ such that $f\,'(a)\ne 0$. (I don't even want to argue that $f\,'$ never vanishes). Normalize so that $a=0$, $f(0)=0$, and $f\,'(0)=1$. Since $f$ is an open map, there exists $r>0$ such that $\{w:|w|