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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.

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Here's a plan. For all of this, note that as $f$ and $g$ do not vanish at any point of $\overline G$ we can make a lot of arguments using $f/g$ and its reciprocal.

We have a statement about absolute values on a boundary, which reminds us of the maximum modulus principle. Use this theorem to prove that $|f| = |g|$ on all of $\overline{G}$. What does the open mapping theorem tell us about a holomorphic function from $G$ to $S^1$?

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    Merci……………………..2011-12-06