It is well known that if $f(z)$ and $g(z)$ are both holomorphic on a (path-)connected open set $C$ and $|f(z)|=|g(z)|$ on $C$ then $f(z)$=$cg(z)$ on $C$ for some constant $c$. Do we have a robust version of this theorem like the following?
Suppose for simplicity $C$ is the unit disk and ${\rm dist}$ is some distance measure such as L2-distance. If ${\rm dist}_C(|f|,|g|)$ is small then ${\rm dist}_{C}(f,cg)$ is also small for some $c$.
Any relevant information will be greatly appreciated!