A nonvanishing nonconstant holomorphic function cannot attain its minimum modulus in the interior.Suggest a slogan
Let \(\Omega\subseteq\CC\) be a connected open set and let \(f\) be holomorphic and nonvanishing on \(\Omega\).
If \(\abs{f}\) has a local minimum at some \(z_0\in\Omega\), then \(f\) is constant on \(\Omega\).
If moreover \(\Omega\) is bounded, \(f\) extends continuously to \(\overline{\Omega}\), and \(f\) is nonconstant, then \(\min_{\overline{\Omega}}\abs{f}\) is attained only at points of \(\partial\Omega\).