Here is my problem:
Let $f(z)$ be an entire function such that $|f '(z)| < |f(z)|$ for all $z \in \mathbb{C}$, Show that there exists a constant A such that $|f(z)| < A*e^{|z|}$ for all $z \in \mathbb{C}$. I am trying to use Liouville's theorem to prove and try to set $g(z)=|f '(z)|/|f(z)| < 1$ and then g(z) is constant. I am not sure if my thinking is right and how to prove this problem? Thanks