Borel-Carathéodory Lemma/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:

$\ds \forall n \in \Z_{\ge 1} : \quad \frac {\cmod {\map {f^{\paren n} } 0} }{ n! } \le \frac {2 M} {R^n}$

Proof

By Cauchy Integral Theorem:

$\ds \forall k \in \Z_{\ge 0} : \quad \oint_{\partial D} z^{k-1} \map f z \rd z = 0$

Parametrizing $\partial D$ by $R e^{2 \pi i t}$:

$\ds \forall k \in \Z_{\ge 0} : \quad \int _0^1 e^{2\pi i k t} \map f {R e^{2 \pi ikt} } \rd t = 0$

On the other hand:

Let $\theta_n := \map \arg {\map {f^{\paren n} } 0}$.

Then:

Since:

$\ds \int_0^1 \map f {R e^{2 \pi i t} } \paren {2 + e^{\paren {2 \pi n t + \theta_n} i } } \rd t = 0$

we have:

$\blacksquare$