Given $|f(z)|<|g(z)|$ on some domain and both are analytic, prove that f(z)/g(z) is analytic.

Question

This seems to be obvious because whenever $g(z)=0$, $f(z)$ is also 0. I think this can be proved using Caucy Estimates. But I'd like to see if there is some other ways/easier way to do this.

Answer 1

Let $D$ be the domain of $f$ and $g$ and let $Z=g^{-1}(0)$. I will assume that $g$ is not the null function. The set$Z$ is a discrete subset of $D$ and $\frac fg\colon D\setminus Z\longrightarrow\mathbb C$ is analytic, since it is the quotient of two analytic functions. Besides, $\frac fg$ is bounded near each point of $Z$ and therefore, by Riemann's theorem, you can extend it to an analytic function from $D$ into $\mathbb C$.

Answer 2

Let $D$ be the domain. Since $g(z)$ can never be $0$ in $D,$ (otherwise we would have $0\le |f(z)| <0$ for some $z\in D,$ contradiction), it is elementary that $f(z)/g(z)$ is analytic in $D.$