Math 6210 • Real Analysis

Let \(\Omega \) be an unbounded domain in the plane with the property that there exists a sequence \(\{h_n\}\) of functions holomorphic on a neighborhood of \(\overline {\Omega }\) such that each \(h_n\) vanishes at infinity on \(\Omega \), \(|h_n|\le 1\) on \(\overline {\Omega }\) for each \(n\), and \(\{h_n\}\) converges uniformly to \(1\) on each compact subset of \(\overline {\Omega }\). Prove that if \(f\) is a function which is bounded and holomorphic in a neighborhood of \(\overline {\Omega }\) and if \(|f|\le 1\) on \(\partial \Omega \), then \(|f|\le 1\) on all of \(\overline {\Omega }\).