Let $f$ be a holomrphic non-constant function in $\Omega$, and a disc $C$, such that $\overline{C} \subset \Omega$.
I want to proove that if $|f|$ is constant at $\partial C$, then $f$ has at least a zero in $C$.
Really do not know where to start.
Appreciate any help.
(From here I edited it)
So, this is what I thought.
Lets suppose that $f$ has no zeros in $\overline{C}$, then $1/f$ is holomorphic at $\overline{C}$. Since $|f|=M$ is constant at $\partial C$ and we know that a holomorphic function in a close bounded set takes its maximum in the boundary, we know that for every $z\in \overline{C}$ we have $|f(z) \leq M$ and $|1/f(z)| \leq 1/M$ so we know that for every $z \in \overline{C}$ $f(z)=M$. Then $f$ its constant in $C$.
Using maximum modulus principle we have that f must be constant in $\Omega$ (absurd).
Then $f$ must have zeros in $\overline{C}$, if they are in the boundary then using that its maximum must be in the boundary, we have that $f=0$ in $C$, and then we have the same reasoning.
Finally, $f$ must have zeros at $C$.
