Maximum Principle for holomorphic functions:
let $f \colon G\to {\mathbb C}\ mathbb{C}$ $ be holomorphic and $z_0$ some interior point of $G$. By the Poisson-Formula, $f(z_0)$ is essentially the mean of values of $f$ on a circle around $z_0$. Thus, $|f(z_0)|$ is not maximal.

