Maximum modulus principle

Let \(\Omega\subseteq\CC\) be open and let \(f\) be holomorphic on \(\Omega\).

1. If \(\abs f\) has a local maximum at \(z_0\in\Omega\), then \(f\) is constant on a neighborhood of \(z_0\).

2. If \(\Omega\) is bounded and connected and \(f\) extends continuously to \(\overline\Omega\), then either \(f\) is constant or \(\abs f\) attains its maximum \(M\coloneqq\max_{z\in\overline\Omega}\abs{f(z)}\) only at points of \(\bd\Omega\).