A bounded complex-valued harmonic function that attains its maximum modulus inside is constant.Suggest a slogan
Let \(D\subseteq\CC\) be a connected open set and let \(h=u+iv\) be a complex-valued function on \(D\) whose real and imaginary parts are harmonic. If \(\abs{h(z)}\le M\) for all \(z\in D\) and \(\abs{h(z_0)}=M\) for some \(z_0\in D\), then \(h\) is constant on \(D\).
Write \(h(z_0)=Me^{i\varphi}\) and put \(U\coloneqq\Re\big(e^{-i\varphi}h\big)=u\cos\varphi+v\sin\varphi\), a real harmonic function on \(D\). Then \(U\le\abs{h}\le M\) on \(D\) and \(U(z_0)=M\), so by the strict maximum principle for real harmonic functions on a connected open set, \(U\equiv M\). Since \(M=U\le\abs{e^{-i\varphi}h}\le M\), the imaginary part of \(e^{-i\varphi}h\) vanishes, so \(e^{-i\varphi}h\equiv M\) and \(h\equiv Me^{i\varphi}\).