A boundary bound on the modulus of a harmonic function controls its modulus throughout the domain.Suggest a slogan
Let \(D\subseteq\CC\) be a bounded domain, and let \(h\colon D\to\CC\) be harmonic and extend continuously to \(\overline D\). If \(\abs{h(z)}\le M\) for all \(z\in\bd D\), then \(\abs{h(z)}\le M\) for all \(z\in D\).