Complex Analysis - Rational Functions

[ilikegroupreps]

Homework Statement

I'm studying for my final exam and came across this problem:
Let f and g be entire analytic functions and |f(z)|<|g(z)| when |z|>1. Show that f/g is a rational function.

The Attempt at a Solution

I really have no clue where to go :(

 

[christoff]

Well, you know that since |f(z)|<|g(z)| on |z|>1, f(z)/g(z) is bounded on |z|>1. Does this tell you anything?