Let \(f: \Omega\to \CC\) be holomorphic and suppose there is a \(z_0 \in \Omega\) with \(\abs{f(z_0)}\leq \abs{f(z)}\) for all \(z\in \Omega\). Show that either \(f(a) = 0\) or \(f\) is constant.
Solution
Suppose \(f(z_0)\neq 0\), then the inequality forces there to be no zeros in \(\Omega\). So \(g(z) \da 1/f(z)\) is nonzero and holomorphic on \(\Omega\) and \(\abs{g(z)}\leq \abs{1\over f(z_0)} \da \abs{g(z_0)}\). Since \(z_0\in \Omega\), the MMP forces \(g\) to be constant, and thus so is \(f\).