This is an exercise from a book called "theory of complex functions" I am trying to solve:
Let $G$ be a bounded region, $f,g$ continuous and zero-free in $\overline{G}$ and holomorphic in $G$. With $|f(z)|=|g(z)| \ \ \ \ \ \forall z \in \partial G$ It follows that there exists a $\lambda \in S^{1}$ such that $f(z)= \lambda g(z) \ \ \ \forall z \in \overline{G}$
I dont know how to even begin.
Does anybody see how to begin? Please, do tell.