How do I show that if $f$ and $g$ are entire and $|f|\ge |g|$, then there is some $\beta$ such that $f(z) = \beta g(z)$ for all $z$?
Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$.