Borel-Carathéodory Lemma

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

Let $R > 0$ be such that $B \left({0, R}\right) \subset D$.

Let $f: D \to \C$ be analytic with $f \left({0}\right) = 0$.

Let $\Re \left({f \left({z}\right)}\right) \le M$ for $\left\lvert{z}\right\rvert \le R$.

Let $0 < r < R$.

Then for $\left\lvert{z}\right\rvert \le r$:
 * $(1): \quad \left\lvert{f \left({z}\right)}\right\rvert \le \dfrac {2 M r} {R - r}$
 * $(2): \quad \left\lvert{f^{\left({k}\right)} \left({z}\right)}\right\rvert \le \dfrac {2 M R k!} {\left({R - r}\right)^{k + 1} }$ for all $k \ge 1$