Maximum modulus principle

Theorem 4.1 (Maximum modulus principle). Let \(D\) be a bounded domain, and let \(f\) be continuous on the closure of \(D\) and analytic in \(D\). Then \(\abs {f}\) attains its maximum over the closure on the boundary of \(D\). If \(\abs {f}\) also attains it at a point of \(D\), then \(f\) is constant.

Source: [1, ch. 4]