Analytic function with 3 or less repeats

[Grothard]

Homework Statement

A function f that is analytic in the whole plane does not take any value more than 3 times. What can it be?

Relevant equations

Cauchy-Riemann equations

The attempt at a solution

I am aware that any non-constant analytic function is open, but I'm not sure where to go from there.
The function cannot be a polynomial of degree 4 or higher since it would have 4 or more roots. It could be a polynomial of degree 3 or less I think.
I'm not sure how I'd go about proving that nothing other than a polynomial can work (or if that's even true).

 

[Gib Z]

This should help: http://en.wikipedia.org/wiki/Weierstrass_factorization_theorem