Let $f$ and $g$ be two entire functions and suppose that there is $r > 0$ such that $|f(z)| \le |g(z)|$ for all $z$ with $|z| \ge r$. What can be said about $f$ and $g$?
I know that outside $D(0,r)$, $f/g$ is holomorphic (argument just like is usually done in the case where the inequality holds on the whole $\mathbb C$). But for all we know, nothing can be said about $f/g$ inside $D(0,r)$. Can someone give a hint or point me in the right direction? Thanks.
Hint: Special case: Assume $r=1$ and $g$ is nonzero on the closed unit disc. Then $f/g$ is an entire function that is bounded in absolute value by $1.$ Hence by Liouville, $f=cg$ for some constant $c$ with $|c|\le 1.$
More general: Again assume $r=1$ but only assume $g$ is nonzero on the unit circle. Consider $fB_g/g,$ where $B_g$ is the finite Blaschke product whose zeros match those of $g.$