Schwarz's Lemma

Theorem
Let $f$ be a function holomorphic on the unit disk, $D$.

Let $\map f 0 = 0$ and $\cmod {\map f z} \le 1$ for all $z \in D$.

Then $\cmod {\map {f'} 0} \le 1$, and $\cmod {\map f z} \le \cmod z$ for all $z \in D$.