I'm doing some exercises for an exam and I've come across this one from a past comprehensive that I can't solve, can anyone give me any tips/hints?
Suppose that $f$ is analytic in a domain $G$ in the complex plane and not constant. Let $D$ be a disc whose closure is contained in $G$. Suppose $|f|$ is constant on $\delta D$. Show that $f$ has at least one zero in $D$.
I've tried showing that since $|f|$ is constant on $\delta D$ $f$ is constant on $\delta D$ (via CR equations), but I am not sure that is right (can anyone tell me if it is or not?).