Let \(\Omega\subseteq\CC\) be a bounded connected open set, and let \(f\) be holomorphic on \(\Omega\), continuous on \(\overline\Omega\), and nowhere zero on \(\Omega\). Then \[ \min_{z\in\overline\Omega}\abs{f(z)}=\min_{z\in\bd\Omega}\abs{f(z)}. \]
If \(f\) vanishes at a point of \(\bd\Omega\), both minima are \(0\). Otherwise \(f\) is nowhere zero on \(\overline\Omega\), and the function \(1/f\) is holomorphic on \(\Omega\) and continuous on \(\overline\Omega\), so by the maximum modulus principle, \(\max_{\overline\Omega}\abs{1/f}=\max_{\bd\Omega}\abs{1/f}\).
The hypothesis that \(f\) has no zeros is needed: \(f(z)=z\) on \(\DD\) has \(\min_{\overline\DD}\abs{f}=0\), attained only at the interior point \(0\), while \(\abs{f}=1\) on \(\bd\DD\).
See [@SS03, Chapter 3, Theorem 4.5 and Corollary 4.6].