Maximum modulus on a bounded domain and a boundary-modulus consequence

Prove the following maximum modulus principle: suppose \(U\) is a bounded domain and \(f\) is a nonconstant continuous function on \(\overline U\) whose restriction to \(U\) is holomorphic. Show that for every \(z_0\in U\), \[ |f(z_0)|<\sup\{|f(z)|:z\in\partial U\}. \]

Furthermore, if \(|f(z)|\) is constant on \(\partial U\), show that \(f\) has a zero in \(U\).