Let $D,U \subseteq \mathbb{C}$ be non-empty domains s.t. $\overline{D} \subseteq U$ with compact $\overline{D}$. Assume $f,g$ are holomorphic on $U$. If $f,g$ have no zeros in $\overline{D}$ and $|f(z)| = |g(z)|$ for $z \in \partial{D}$ then there exists a $\lambda \in \mathbb{C}$ s.t. $|\lambda| = 1$ and $f = \lambda g$.
I started like this: Consider $h := \frac{f|_{\overline{D}}}{g|_{\overline{D}}}$. Because $g$ has no zeros $h$ is holomorphic on $\overline{D}$. Since $\overline{D}$ is compact and $|h(z)| = 1$ for $z \in \partial{D}$ by the maximum principle, we get $h(\overline{D}) \subseteq U_1(0)$. How should I continue?