Montel's Theorem and Normal Families

Normal families provide the compactness framework for spaces of holomorphic functions, analogous to the Arzela-Ascoli theorem for continuous functions. Montel's theorem is the key tool for extracting convergent subsequences.

Normal Families

Definition7.5Normal family

A family $\mathcal{F}$ of holomorphic functions on a domain $D$ is normal if every sequence in $\mathcal{F}$ has a subsequence that converges uniformly on every compact subset of $D$ . The limit function is holomorphic by the theorem on uniform limits.

Theorem7.6Montel's theorem

A family $\mathcal{F}$ of holomorphic functions on a domain $D$ is normal if and only if it is locally bounded: for every compact $K \subset D$ , there exists $M_K$ such that $|f(z)| \leq M_K$ for all $f \in \mathcal{F}$ and $z \in K$ .

Proof of Montel's Theorem

Proof

Step 1: Equicontinuity from local boundedness. Fix a compact $K \subset D$ and let $\delta = \text{dist}(K, \partial D)/2$ . For $z_1, z_2 \in K$ with $|z_1 - z_2| < \delta$ , Cauchy's integral formula on $|w-z_1| = \delta$ gives:

$|f(z_1) - f(z_2)| = \left|\frac{1}{2\pi i}\oint_{|w-z_1|=\delta} f(w)\left(\frac{1}{w-z_1} - \frac{1}{w-z_2}\right)dw\right| \leq \frac{M |z_1-z_2|}{\delta^2} \cdot 2\pi\delta = \frac{2\pi M|z_1-z_2|}{\delta}.$

This gives uniform equicontinuity of $\mathcal{F}$ on $K$ .

Step 2: Arzela-Ascoli. Choose a countable dense subset $\{q_n\} \subset D$ . For any sequence $\{f_k\} \subset \mathcal{F}$ , use a diagonal argument: extract a subsequence converging at $q_1$ , then a sub-subsequence converging at $q_2$ , etc. The diagonal subsequence converges at all $q_n$ .

Step 3: Uniform convergence. Equicontinuity plus pointwise convergence on a dense set implies uniform convergence on compact subsets (standard argument via finite $\varepsilon$ -nets). $\blacksquare$

■

Montel's Great Theorem

Theorem7.7Montel's theorem (strong form)

A family $\mathcal{F}$ of holomorphic functions on a domain $D$ that all omit two fixed values $a, b \in \mathbb{C}$ ( $a \neq b$ ) is normal.

RemarkConnection to Picard

Montel's great theorem implies Picard's little theorem: if $f$ is entire and omits two values, consider $\mathcal{F} = \{f_n\}$ where $f_n(z) = f(nz)$ . Each $f_n$ omits the same two values, so by Montel, $\{f_n\}$ is normal. Analysis of the limit gives $f$ constant. The proof uses the theory of the modular function $\lambda$ .

Applications

ExampleFixed-point theorem for holomorphic self-maps

Let $f: \mathbb{D} \to \mathbb{D}$ be holomorphic. The iterates $f^n = f \circ \cdots \circ f$ form a normal family (bounded by $1$ ). If $f$ has no fixed point in $\mathbb{D}$ , the Denjoy-Wolff theorem says $f^n \to c$ uniformly on compacts for some $c \in \partial\mathbb{D}$ . If $f$ has a unique fixed point $z_0 \in \mathbb{D}$ and $f$ is not a rotation, then $f^n \to z_0$ uniformly on compacts.

ExampleVitali's convergence theorem

If $\{f_n\}$ is a normal family on $D$ and $f_n \to f$ pointwise on a set with an accumulation point in $D$ , then $f_n \to f$ uniformly on compact subsets. (Any convergent subsequence must have the same limit by the identity theorem.)

RemarkCompactness in function spaces

Montel's theorem is the complex-analytic analogue of the Arzela-Ascoli theorem. It provides the compactness needed for existence proofs throughout complex analysis: the Riemann mapping theorem, normal forms for conformal maps, and value distribution theory all rely on it fundamentally.