I had to show what a one-to-one analytic function from the plane to itself could possibly be.
So, I studied the behavior of such a function at infinity:
Case 1: Such a function cannot have no singularity (or a removable singularity) at infinity, since then the function would be entire and bounded, and by Liouville's Theorem, is constant, which contradicts the one-to-one assumption of f(z).
Case 2: Such a function cannot have an essential singularity at infinity, since then the function would attain every complex value (except for maybe one value) infinitely often in a (every) neighborhood of infinity, which again contradicts the one-to-one assumption of f(z). This comes from Picard's theorem.
Cases 1 & 2 imply that the function must have a pole at infinity.
We conclude that f(z) must be a polynomial.
Is this ok?
My concern is that I feel that I've made a jump from saying that a 1-1 entire function with a pole at infinity is a polynomial -- as if it were a definition. Can I treat this property of a polynomial as a definition? (Which would then make my proof complete.)