If I have an analityc function $f(z)$ on a domain $B \subset \mathbb{C}$ and a simple and closed curve $C$ that encloses $B$, and if $|f|$ is constant over C, the $f$ is constant on $B$?
I haven’t found a counterexample but i dont know how to apply the maximum principle or apply some analityc continuation. Please some help with this.