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 $k \ge 1$

Proof

$\ds \map f z$

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {\map {f^{\paren n} } 0} {n!} z^n$

Taylor Series of Holomorphic Function

$\ds $

$=$

$\ds \sum_{n \mathop = 1}^\infty \frac {\map {f^{\paren n} } 0} {n!} z^n$

as $\map f 0 = 0$

Lemma

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

$\Box$

$\ds \cmod {\map f z}$

$=$

$\ds \cmod {\sum_{n \mathop = 1}^\infty \frac {\map {f^{\paren n} } 0} {n!} z^n}$

$\ds $

$\le$

$\ds \sum_{n \mathop = 1}^\infty \frac { \cmod { \map {f^{\paren n} } 0 } } {n!} r^n$

$\ds $

$\le$

$\ds 2 M \sum_{n \mathop = 1}^\infty \paren {\frac r R}^n$

by Lemma

$\ds $

$=$

$\ds \frac {2 M r} {R - r}$

$\ds \cmod {\map {f^{\paren n} } z}$

$=$

$\ds \cmod {\sum_{n \mathop = 0}^\infty \frac {\map {f^{\paren {n + k} } } 0} {n!} z^n}$

Power Series is Termwise Differentiable within Radius of Convergence

$\ds $

$\le$

$\ds \sum_{n \mathop = 0}^\infty \frac { \cmod { \map {f^{\paren {n+k} } } 0} } {n!} r^n$

$\ds $

$\le$

$\ds 2 M \sum_{n \mathop = 0}^\infty \frac { \paren {n+k} ! } {R^{n+k} n! } r^n$

by Lemma

$\ds $

$=$

$\ds \frac {2M k!} {R^k} \sum_{n \mathop = 0}^\infty \frac { \paren {n+k} ! } {n! k!} \paren {\frac r R}^n$

$\ds $

$=$

$\ds \frac {2M k!} {R^k} \frac 1 {\paren {1 - \frac r R}^{k+1} }$

$\ds $

$=$

$\ds \frac {2 M R k!} {\paren {R - r}^{k+1} }$

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Source of Name

This entry was named for Émile Borel and Constantin Carathéodory.

Sources

1932: A.E. Ingham: The Distribution of Prime Numbers: Chapter $\text {III}$: Further Theory of $\map \zeta s$. Applications: $\S 6$: Theorem $\text E$
2006: Hugh L. Montgomery and Robert C. Vaughan: Multiplicative Number Theory: I. Classical Theory ... (previous) $6$: The Prime Number Theorem: $\S6.1$: A zero-free region: Lemma $6.2$