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
- 1978: John B. Conway: Functions of One Complex Variable (2nd ed.): $\S 7.2$