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