Montel's Theorem

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Theorem

Let $U \subseteq \C$ be an open subset of the complex numbers.

Let $\map \HH U$ be the space of holomorphic mappings on $U$.


Then a family of mappings $\FF \subseteq \map \HH U$ is normal if and only if $\FF$ is locally bounded.


Proof

Normal implies locally bounded

By the Arzelà-Ascoli Theorem, every normal family is locally bounded.


Locally bounded implies normal

By the Arzelà-Ascoli Theorem, any locally bounded and locally uniformly equicontinuous family is normal.

Hence it suffices to show that $\FF$ is locally uniformly equicontinuous.

As a Family of Lipschitz Continuous Functions with same Lipschitz Constant is Uniformly Equicontinuous, it suffices to show that every point $z_0 \in U$ has a neighbourhood $N$ such that all functions from $\FF$ have the same Lipschitz constant on $N$.

To this end, take any $z_0 \in U$.

By the assumption that $\FF$ is locally bounded, we may choose $R > 0$ such that for all $z \in B := \map B {x; R}$ (the closed disk of radius $R$ around $x$), we have $\size {\map f z} < M$.

Now take any $r \in \tuple {0, R}$ and let $B' = \map B {x; r}$.

We then have for any $z, z' \in B'$ and $w \in B$:

$\size {\paren {w - z} \paren {w - z'} } \ge \paren {R - r}^2$

This allows us to derive:

\(\ds \size {\map f z - \map f {z'} }\) \(=\) \(\ds \size {\frac 1 {2 \pi i} \oint_{\partial B} \frac {\map f w} {w - z} \rd w - \frac 1 {2 \pi i} \oint_{\partial B} \frac {\map f w} {w - z'} \rd w}\) Cauchy's Integral Formula
\(\ds \) \(=\) \(\ds \size {\frac 1 {2 \pi i} \oint_{\partial B} \paren {\frac {\map f w} {w - z} - \frac {\map f w} {w - z'} } \rd w}\) Linear Combination of Contour Integrals
\(\ds \) \(=\) \(\ds \size {\frac 1 {2 \pi i} \oint_{\partial B} \frac {\map f w \paren {w - z'} - \map f w \paren {w - z} } {\paren {w - z} \paren {w - z'} } \rd w}\)
\(\ds \) \(=\) \(\ds \size {\frac {\paren {z - z'} }{2 \pi i} \oint_{\partial B} \frac {\map f w} {\paren {w - z} \paren {w - z'} } \rd w}\)
\(\ds \) \(\le\) \(\ds \size {\frac 1 {2 \pi} } \size {z - z'} 2 \pi \frac M {\paren {R - r}^2}\) Triangle Inequality for Contour Integrals
\(\ds \) \(=\) \(\ds \frac M {\paren {R - r}^2} \size {z - z'}\)

Therefore, $M \paren {R - r}^{-2}$ is a Lipschitz constant for all functions in $\FF$ on $B'$.

$\blacksquare$


Source of Name

This entry was named for Paul Antoine Aristide Montel.


Sources