Schwarz's Lemma

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

Let $f\left({0}\right) = 0$ and $\left\vert f\left({z}\right) \right\vert \le 1$ for all $z \in D$.

Then $\left\vert f'\left({0}\right) \right\vert \le 1$, and $\left\vert f\left({z}\right) \right\vert \le \left\vert z \right\vert$ for all $z \in D$.