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

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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 $6.2$