Let $G$ be an open connected set and $f, g$ analytic functions on $G$. If $|f|\le |g|$ then there exists an analytic function $h$ such that $f(z)=h(z)g(z)$.
We know $|f/g|\le 1$ everywhere in $G$, and so $f/g$ is bounded near any singularities. (The singularities must be isolated, for otherwise they have a limit and the function is identically zero.) So we can extend $f/g$ to be analytic in all of $G$.
But where do I go from here? Presumably the connectedness of $G$ is important, but I don't see how it helps in manipulating $f$ and $g$ to make an $h$ appear
EDIT: As pointed out in the comments, I am already done. Take $h= f/g$.