A real harmonic function attaining an interior maximum on a connected domain is constant.Suggest a slogan
Let \(D\subseteq\CC\) be a connected open set, let \(u\colon D\to\RR\) be harmonic, and let \(M\in\RR\) satisfy \(u(z) \leq M\) for all \(z \in D\). If \(u(z_0)=M\) for some \(z_0 \in D\), then \(u(z)=M\) for all \(z \in D\).