Let $D$ be a bounded region and $f$ is an analytic function on $D$. Show that if there is a constant $c ≥ 0$ such that $|f(z)| = c$ for all $z$ in the boundary of $D$ then either $f$ is a constant function or $f$ has a zero in $D$.
Is this just a direct consequence of the Maximum-Modulus Theorem?