A nonconstant holomorphic function reaches its largest modulus only on the boundary.Suggest a slogan
Let \(\Omega\subseteq\CC\) be a connected open set and let \(f\) be holomorphic on \(\Omega\).
If \(\abs{f}\) attains a local maximum at some \(z_0\in\Omega\), then \(f\) is constant on \(\Omega\).
If \(\Omega\) is bounded and \(f\) extends continuously to \(\overline\Omega\), then \(\max_{\overline\Omega}\abs{f}=\max_{\bd\Omega}\abs{f}\).