Let \(f\) be analytic in a domain \(D\) and suppose there exists a point \(z_0 \in D\) such that \(\left|\,f(z)\,\right| \le \left|\,f(z_0)\,\right| \, \forall z \in D\). Then \(f\) is constant in \(D\) !

As a consequence, if \(D \subset \mathbb{C}\) is a bounded domain, and if \(f: \overline D \mapsto \mathbb{C}\) is continuous on \(\overline D\) and analytic in \(D\), then \(\left|\, f \,\right|\) reaches it's maximum on \(\partial D\).

(If I understand the notation correctly, \(\overline D\) refers to the union of both the open domain \(D\) and it's boundary, whereas \(\partial D\) refers to only the boundary.)

More at Analysis of a Complex Kind.

# posted by rot13(Unafba Pune) @ 3:49 PM