Borel-Carathéodory Lemma

Theorem
Let $D \subset \C$ be an open set with $0 \in D$.

Let $R > 0$ be such that the open disk $\map B {0, R} \subset D$.

Let $f: D \to \C$ be analytic with $\map f 0 = 0$.

Let $\map \Re {\map f z} \le M$ for $\cmod z \le R$.

Let $0 < r < R$.

Then for $\cmod z \le r$:
 * $(1): \quad \cmod {\map f z} \le \dfrac {2 M r} {R - r}$
 * $(2): \quad \cmod {\map {f^{\paren k} } z} \le \dfrac {2 M R k!} {\paren {R - r}^{k + 1} }$ for all $k \ge 1$