If $f$ and $g$ are entire functions such that $|f(z)|\lt|g(z)|$ for $|z|\gt1$, then show that $f(z)/g(z)$ is a rational function.
I am thinking that if I can just prove that $f/g$ is a meromorphic function on the extended plane, and then maybe show that $g(z)$ can't be zero at some point in $|z|\gt1$ so that $f/g$ is rational?
Someone dropped a hint on this website before that suppose ${z_1,...z_n}$ are the zeros of $g(z)$, then let $p(z)=(z-z_1)...(z-z_n)$, we have $p(z)f(z)/g(z)$ is entire and $f$ is entire, then according to a theorem, $f$ is a polynomial. And we can prove that $g$ has no zero in the range, so $f(z)/g(z)$ is rational. Is this correct? Thank you very much.