Let $f$ be an entire function such that $f$ maps the open unit ball $D$ around 0 to itself. Suppose further that $f ( 0 ) = 0$ and $f ( 1 ) = 1$. Show that $f ^ { \prime } ( 1 ) \in \mathbb { R }$ and that $\left| f ^ { \prime } ( 1 ) \right| \geq 1$.