Suppose $f(z)$ and $g(z)$ are entire functions and that $f(z)$ is not constant. If $|f(z)| < |g(z)|$ for all $z \in \mathbb C, $ prove that $f(z)$ can not be a polynomial.
I was thinking what I could do was using the fact $|f(z)| < |g(z)|$ , I can argue $ \frac {|f(z)|} {|g(z)|}< 1$ if $g$ not equal $0$. And, I use the Louiville's Theorem to conclude $ \frac {|f(z)|} {|g(z)|}$ is constant. Then I don't know where to go with that. I think I am not going in the right direction. Please help.