Let $G$ be a bounded domain, $f$ and $g$ holomorphic and continuous functions on whole $\overline{G}$ and let $|f(z)|=|g(z)|$ for all $z\in \partial G$. What can we say about $f$ and $g$?
Define: $h(z) = \frac{f(z)}{g(z)}$, if g has zero-points, then we cant say much because of the singularities. If g has no zero points, then with the maximum principle for bounded domains it follows that h takes its maximum on the edge. Because |f(z)|=|g(z)| for all $z\in \partial G$ it follows that the modulus is 1. If now f has zero points, then we cant say anything more about it. If f has no zero points, then we can use the minimum principle for bounded domains. With that it follows that also the modulus of the minimum must be 1.
From that we can conclude that : $|h(z)|=\frac{|f(z)|}{|g(z)|}=1$ for all $z\in G$. So if f and g have no zero points, $f(z)=\lambda g(z)$
Is this correct?