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Montel's theorem

Montel's theorem encompasses key results in complex analysis regarding the normality of families of holomorphic or meromorphic functions on a domain in the complex plane, introduced by the French mathematician Paul Montel in his foundational work on normal families around 1912.^[1] The classical version of Montel's theorem states that a family of holomorphic functions on an open set \Omega \subset \mathbb{C} is normal—meaning every sequence in the family has a subsequence that converges uniformly on compact subsets of \Omega to a holomorphic function—if and only if the family is locally uniformly bounded.^[2] This boundedness condition ensures equicontinuity via the Arzelà-Ascoli theorem, providing compactness in the space of holomorphic functions.^[2] A more advanced formulation, known as the Great Montel's Theorem, applies to meromorphic functions: a family of meromorphic functions on \Omega is normal if there are three distinct points w_1, w_2, w_3 in the extended complex plane \hat{\mathbb{C}} such that each function in the family omits these values, i.e., f(\Omega) \cap \{w_1, w_2, w_3\} = \emptyset for all f in the family.^[3] This result generalizes the basic theorem by replacing boundedness with the omission of three values, leveraging the spherical metric on \hat{\mathbb{C}} to control growth.^[3] These theorems are pivotal in complex function theory, underpinning proofs of major results such as the Riemann mapping theorem, which establishes biholomorphic equivalence between simply connected domains and the unit disk by applying Montel's theorem to families of normalized injective mappings to extract a limit function.^[4] They also facilitate extensions to Picard's theorems on value distribution and the study of iteration in dynamics, highlighting the compactness properties of normal families in understanding global behavior of analytic functions.^[3] Montel's original contributions, detailed in his 1916 memoir, laid the groundwork for these developments by adapting compactness ideas from real analysis to the holomorphic setting.^[5]

Background Concepts

Normal Families

A family \mathcal{F} of holomorphic functions on a domain \Omega \subset \mathbb{C} is normal if every sequence \{f_n\} \subset \mathcal{F} admits a subsequence that converges uniformly on every compact subset of \Omega either to a holomorphic function on \Omega or to infinity in the extended complex plane (meaning |f_{n_k}(z)| \to \infty uniformly on compact subsets).^[2] This notion of convergence, known as locally uniform or normal convergence, ensures that the limit function (or infinity) is well-behaved across the domain.^[6] The space \mathcal{H}(\Omega) of all holomorphic functions on \Omega, equipped with the compact-open topology, provides the natural framework for understanding normality. In this topology, a subbasis for the neighborhoods of a function f \in \mathcal{H}(\Omega) consists of sets \{g \in \mathcal{H}(\Omega) : \sup_{z \in K} |f(z) - g(z)| < \epsilon\}, where K \subset \Omega ranges over compact subsets and \epsilon > 0. A family \mathcal{F} \subset \mathcal{H}(\Omega) is normal if and only if it is relatively compact in this topology, meaning its closure is compact; this equivalence follows from the Arzelà-Ascoli theorem applied to restrictions on compact subsets.^[2] The concept of normal families originated with Paul Montel in his 1907 work on sequences of analytic functions, where he introduced ideas of compactness for such families without initially using the term "normal."^[6] Montel's definition focused on the existence of convergent subsequences for families admitting certain representations, laying the groundwork for modern criteria; over the subsequent decades, the notion evolved to encompass broader classes of functions, including meromorphic ones, while retaining the core emphasis on sequential compactness in the compact-open topology. For instance, the family of all holomorphic functions on the unit disk \mathbb{D} is not normal, as the sequence f_n(z) = n z diverges without any subsequence converging uniformly on compact subsets (e.g., on disks of radius $1/2). In contrast, families that are uniformly bounded on \mathbb{D} exhibit normality, highlighting how restrictive conditions can enforce the required compactness.^[2]

Holomorphic Functions and Domains

A holomorphic function is a complex-valued function f: D \to \mathbb{C}, where D is an open subset of the complex plane \mathbb{C}, that is complex differentiable at every point in D.^[7] Complex differentiability at a point z_0 \in D means the limit \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0} exists and is finite.^[7] Holomorphic functions satisfy key properties, such as the maximum modulus principle, which states that if f is holomorphic in a bounded domain D and continuous up to the boundary, then the maximum of |f(z)| on the closure of D is attained on the boundary.^[8] In complex analysis, a domain is defined as a nonempty open connected subset of the complex plane \mathbb{C}.^[9] Examples include the unit disk \{z \in \mathbb{C} : |z| < 1\}, the punctured plane \mathbb{C} \setminus \{0\}, and the upper half-plane \{z \in \mathbb{C} : \operatorname{Im}(z) > 0\}.^[9] These domains provide the natural settings for studying holomorphic functions, as connectivity ensures path integrals and other global properties behave consistently.^[9] A sequence of holomorphic functions \{f_n\} on a domain D converges uniformly on compact subsets of D if, for every compact K \subset D, the convergence is uniform on K.^[10] Such uniform convergence on compact subsets preserves holomorphy: the limit function f = \lim_{n \to \infty} f_n is holomorphic on D.^[10] This result follows from the Arzelà-Ascoli theorem applied to holomorphic functions, which leverages their equicontinuity on compact sets due to Cauchy's estimates.^[10] Local uniform boundedness for a holomorphic function f on a domain D means that for every compact subset K \subset D, there exists a constant M_K > 0 such that |f(z)| \leq M_K for all z \in K.^[11] This condition is equivalent to f being bounded on every compact subset of D, ensuring controlled growth locally.^[11]

Statements of the Theorem

Boundedness Criterion

The boundedness criterion, a foundational result in the theory of normal families, asserts that a family of holomorphic functions on a domain is normal provided it is locally uniformly bounded. Specifically, let G \subset \mathbb{C} be a domain and let \mathcal{F} be a family of holomorphic functions on G. If for every compact subset K \subset G there exists M_K > 0 such that |f(z)| \leq M_K for all z \in K and all f \in \mathcal{F}, then \mathcal{F} is normal on G.^[12] This criterion was established by Paul Montel in his 1916 paper "Sur les familles normales de fonctions analytiques," refining his earlier investigations into sequences of analytic functions from 1907 onward.^[13] A direct corollary is that local uniform boundedness is necessary for normality: if \mathcal{F} is not locally uniformly bounded, then \mathcal{F} is not normal. For instance, consider the family \{p_n(z) = z^n : n \in \mathbb{N}\} of holomorphic functions on the unit disk D = \{z \in \mathbb{C} : |z| < 1\}. This family is not locally bounded near any point on the boundary (approached radially), and sequences from it fail to have subsequences converging uniformly on compact subsets of D, confirming non-normality.^[14]^[15] Informally, the boundedness condition ensures that functions in \mathcal{F} cannot "escape to infinity" along any sequence of points in compact subsets of G, thereby guaranteeing equicontinuity (via Cauchy's estimates) and total boundedness in the compact-open topology. This setup enables the application of the Arzelà-Ascoli theorem to extract convergent subsequences, establishing normality.^[12]^[14]

Value Omission Criterion

The value omission criterion, also known as Montel's fundamental normality criterion, states that if F is a family of holomorphic functions on a domain G \subseteq \mathbb{C} such that each f \in F omits two distinct fixed values \alpha, \beta \in \mathbb{C} (i.e., f(G) \subseteq \mathbb{C} \setminus \{\alpha, \beta\}), then F is normal in the extended sense, meaning every sequence in F has a subsequence that converges uniformly on compact subsets of G to a holomorphic function or to the constant function \infty.^[16] This result was established by Paul Montel in 1907. The functions in such a family map the domain G into the twice-punctured complex plane \mathbb{C} \setminus \{\alpha, \beta\}, a hyperbolic Riemann surface whose restricted geometry ensures the normality of the family; in contrast, mapping into the full plane \mathbb{C} or the once-punctured plane \mathbb{C} \setminus \{\alpha\} does not guarantee normality.^[17] The universal covering space of the twice-punctured plane is the unit disk, and compositions with the covering map yield bounded lifts, which underpin the implicit local boundedness leading to normality.^[18] An illustrative example is the family of all holomorphic functions on the unit disk that map into the unit disk itself (e.g., by Schwarz lemma considerations or contractions), which omit any two fixed values outside the closed unit disk, such as 2 and 3, rendering the family normal by this criterion.^[16] The criterion is sharp in the sense that omitting only one value is insufficient to ensure normality. For instance, consider the family \{ f_n(z) = \exp(n z) \mid n = 1, 2, \dots \} of holomorphic functions on the unit disk \mathbb{D}; each f_n omits the value 0, but the family is not normal, as it fails to be locally bounded—for example, |f_n(1/2)| = \exp(n/2) \to \infty as n \to \infty.^[17]

Great Montel's Theorem

A more general version, known as the Great Montel's Theorem, applies to families of meromorphic functions: Let \mathcal{F} be a family of meromorphic functions on a domain \Omega \subset \mathbb{C}. If there exist three distinct points w_1, w_2, w_3 \in \hat{\mathbb{C}} such that f(\Omega) \cap \{w_1, w_2, w_3\} = \emptyset for all f \in \mathcal{F}, then \mathcal{F} is normal on \Omega. This extends the value omission criterion to meromorphic functions by using the spherical metric.^[3]

Proofs of the Criteria

Proof of Boundedness Implies Normality

To prove that a locally uniformly bounded family of holomorphic functions on a domain U \subset \mathbb{C} is normal, assume \mathcal{F} is such a family, meaning that for every compact subset K \subset U, there exists M_K > 0 such that |f(z)| \leq M_K for all f \in \mathcal{F} and z \in K.^[19] The proof proceeds by showing that the spherical derivatives of functions in \mathcal{F} are locally bounded, which implies equicontinuity in the chordal metric and hence relative compactness in the compact-open topology via Marty's theorem.^[10] Fix a compact subset K \subset U. Since \mathcal{F} is locally uniformly bounded, there exists an open neighborhood V of K contained in U such that |f(z)| \leq M for all f \in \mathcal{F} and z \in V, for some M > 0. Choose r > 0 such that the open r-neighborhood of K is contained in V. For any z \in K and f \in \mathcal{F}, Cauchy's integral formula gives

f'(z) = \frac{1}{2\pi i} \int_{|\zeta - z| = r} \frac{f(\zeta)}{(\zeta - z)^2} \, d\zeta,

|f'(z)| \leq \frac{1}{r} \max_{|\zeta - z| = r} |f(\zeta)| \leq \frac{M}{r}.

The spherical derivative is defined as

f^\#(z) = \frac{|f'(z)|}{1 + |f(z)|^2}.

Since |f(z)| \leq M on K, it follows that $1 + |f(z)|^2 \geq 1, yielding

f^\#(z) \leq |f'(z)| \leq \frac{M}{r}

for all z \in K and f \in \mathcal{F}. Thus, the spherical derivatives are uniformly bounded on K.^[19] By Marty's theorem, a family of meromorphic functions (here, holomorphic as a special case) is normal on U if and only if the family of its spherical derivatives is locally bounded on U.^[10] The local boundedness established above implies that \mathcal{F} is equicontinuous on every compact subset of U with respect to the chordal metric on the Riemann sphere. To derive equicontinuity explicitly, consider points z_1, z_2 \in K with |z_1 - z_2| < r/2. By the mean value theorem, there exists \xi between z_1 and z_2 such that |f(z_1) - f(z_2)| = |f'(\xi)| \cdot |z_1 - z_2| \leq (M/r) |z_1 - z_2|. Since |f(z)| \leq M, the chordal distance \chi(f(z_1), f(z_2)) \leq 2 |f(z_1) - f(z_2)| / (1 + |f(z_1)|^2) \leq 2 (M/r) |z_1 - z_2|, providing a uniform modulus of continuity on K. This holds for any covering of K by disks of radius r, confirming equicontinuity.^[19] The uniform boundedness and equicontinuity on compact sets imply, by the Arzelà–Ascoli theorem, that every sequence in \mathcal{F} has a subsequence converging uniformly on compact subsets of U to a holomorphic limit function (by the Weierstrass theorem on differentiation under uniform limits). Thus, \mathcal{F} is normal.^[19]

Proof of Omission Implies Normality

To prove that a family \mathcal{F} of meromorphic functions on a domain U \subset \mathbb{C}, each omitting three distinct fixed values a, b, c \in \hat{\mathbb{C}}, is normal, the argument leverages the Zalcman rescaling lemma and Picard's great theorem.^[6] Without loss of generality, by a suitable Möbius transformation (which preserves meromorphicity and normality), assume the omitted values are $0, 1, \infty. The thrice-punctured Riemann sphere \hat{\mathbb{C}} \setminus \{0, 1, \infty\} is hyperbolic, but the proof proceeds via contradiction using rescaling.^[3] Suppose \mathcal{F} is not normal. Then Zalcman's rescaling lemma implies the existence of a subsequence \{f_n\} \subset \mathcal{F}, points z_n \in U, radii \rho_n \to 0, such that the rescaled functions g_n(\zeta) = f_n(z_n + \rho_n \zeta) converge uniformly on compact subsets of \mathbb{C} to a meromorphic function g: \mathbb{C} \to \hat{\mathbb{C}} \setminus \{0, 1, \infty\}, with g nonconstant (since if constant, the original family would be normal by the boundedness criterion). Moreover, the spherical derivative satisfies g^\#(\zeta) \leq 1 for all \zeta \in \mathbb{C}.^[6] However, Picard's great theorem states that any nonconstant meromorphic function on \mathbb{C} omits at most two values in \hat{\mathbb{C}}. Thus, no such nonconstant g omitting three values exists, yielding a contradiction. Therefore, \mathcal{F} must be normal.^[3]

Necessity and Equivalence

Local Boundedness as Necessary Condition

To establish the necessity of local uniform boundedness in Montel's boundedness criterion, suppose a family \mathcal{F} of holomorphic functions on a domain \Omega \subset \mathbb{C} is normal but not locally uniformly bounded. Then there exists a compact set K \subset \Omega such that \mathcal{F} is not uniformly bounded on K, so there exists a sequence \{f_n\} \subset \mathcal{F} with \sup_{z \in K} |f_n(z)| \geq n for each n. By normality, there is a subsequence \{f_{n_k}\} converging uniformly on K to a holomorphic function f on \Omega. Since f is continuous on the compact set K, it is bounded: \sup_{z \in K} |f(z)| < \infty. Uniform convergence implies that for sufficiently large k, \sup_{z \in K} |f_{n_k}(z)| is also bounded, contradicting the choice of the sequence. Thus, normality implies local uniform boundedness.^[20] A concrete example illustrating non-normality due to lack of local boundedness is the family \{f_n(z) = n z : n \in \mathbb{N}\} on the unit disk \mathbb{D}. Here, |f_n(1)| = n \to \infty, so \mathcal{F} is not locally bounded on the compact set \{1/2 \leq |z| \leq 1\} \subset \mathbb{D}. Moreover, \mathcal{F} is not normal on \mathbb{D}, as any subsequence diverges to \infty uniformly on compact subsets avoiding 0 but converges to 0 near 0, precluding uniform convergence on the whole \mathbb{D}.^[11]

Equivalence in the Bounded Case

In the bounded case, Montel's theorem establishes a precise equivalence between normality and local uniform boundedness for families of holomorphic functions on a domain \Omega \subseteq \mathbb{C}. Specifically, a family \mathcal{F} of holomorphic functions on \Omega is normal if and only if it is locally uniformly bounded, meaning that for every compact subset K \subset \Omega, there exists a constant M_K > 0 such that |f(z)| \leq M_K for all z \in K and all f \in \mathcal{F}.^[20]^[21] This bidirectional formulation captures the full sharpness of the criterion, where local uniform boundedness ensures the existence of convergent subsequences uniformly on compact subsets, and conversely, normality implies such boundedness via properties of uniform limits of holomorphic functions.^[20] This equivalence was comprehensively developed in Paul Montel's 1927 treatise, where he synthesized earlier results on normal families and demonstrated the if-and-only-if relationship in the context of boundedness. Montel's work highlighted how this criterion provides a complete characterization for families that remain "well-behaved" under compactness in the space of holomorphic functions, influencing subsequent developments in complex analysis. In contrast, the value omission criterion—stating that a family omitting two fixed values in \mathbb{C} is normal—is not equivalent to normality, as it represents a stricter sufficient condition rather than a necessary one.^[20] While omitting two values implies local uniform boundedness (and thus normality) through growth estimates on the functions, the converse fails: there exist normal families that do not omit any two fixed values collectively. For instance, consider the family \mathcal{F} = \{f_n(z) = z/n : n = 1, 2, \dots \} of holomorphic functions on the entire complex plane \mathbb{C}. This family is locally uniformly bounded, hence normal, with subsequences converging uniformly on compact sets to the zero function; however, it omits no values commonly, as f_1(z) = z attains every complex number.^[21] Such examples underscore that boundedness provides the fundamental equivalence, while value omission serves as a specialized tool for ensuring normality without requiring explicit bounds.

Applications and Relations

Connections to Entire Function Theorems

Montel's theorem establishes deep connections to classical results in the theory of entire functions by specializing its normality criteria to the case of functions holomorphic on the entire complex plane \mathbb{C}. The boundedness criterion, which states that a locally bounded family of holomorphic functions on a domain is normal, directly generalizes Liouville's theorem when applied to entire functions. Specifically, a bounded family of entire functions is normal, meaning every sequence has a subsequence converging uniformly on compact subsets of \mathbb{C} to a holomorphic limit function. Since the family is globally bounded, the limit is a bounded entire function and thus constant by Liouville's theorem. Consequently, all functions in the family must be constant, underscoring that non-constant entire functions cannot belong to a non-trivial bounded family.^[22] The value omission criterion of Montel's theorem similarly links to Picard's little theorem. A family of holomorphic functions on a domain that collectively omit two fixed values in \mathbb{C} is normal. When restricted to entire functions, such a family admits a subsequence converging uniformly on compacta to an entire limit function that also omits the two values. By Picard's little theorem, this limit must be constant, implying that the original family consists solely of constant functions. This interplay demonstrates how Montel's theorem bridges local normality on domains to global rigidity for entire functions, where omitting two values enforces constancy.^[23] Bloch's principle offers a unifying heuristic for these connections, suggesting that a property compelling entire functions to be constant typically induces normality in families of holomorphic functions on more general domains. Boundedness exemplifies this: it yields constancy for entire functions via Liouville's theorem and normality for families via Montel's boundedness criterion. Omitting two values aligns similarly, producing constancy for entire functions by Picard's little theorem while ensuring normality for families under Montel's omission criterion. The principle highlights the global-to-local transition inherent in entire function theory, where properties like normality on a disk can imply broader structural constraints.^[24] A key illustration of the one-value omission limit arises with the family of scaled exponentials \{f_\lambda(z) = \lambda e^z : \lambda \in \mathbb{C}, |\lambda| \leq 1\}, consisting of entire functions that omit 0. This family is not normal, as the subsequence f_n(z) = e^{z - n} converges to 0 on \{z : \operatorname{Re} z < 0\} but diverges to \infty on \{z : \operatorname{Re} z > 0\}, precluding a uniformly convergent limit on \mathbb{C}. This example emphasizes that omitting a single value permits non-constant entire functions, such as the exponential itself, and fails to guarantee normality, contrasting sharply with the two-value case.^[25]

Role in Picard's Theorem and Bloch's Principle

Montel's theorem plays a pivotal role in establishing Picard's theorems by leveraging the normality of families of holomorphic functions to derive restrictions on value omission for entire or meromorphic functions. In the proof of Picard's little theorem, which states that a non-constant entire function omits at most one complex value, the argument proceeds by contradiction: assume a non-constant entire function f omits two values, say 0 and 1 (after a suitable Möbius transformation). Consider the family of rescaled and translated functions \{f(2^k \phi_a) \mid \phi_a \in \Aut(\Delta), k = 1, 2, \dots \}, where \Delta is the unit disk and \Aut(\Delta) denotes its automorphisms; this family consists of holomorphic functions on \Delta that omit 0 and 1. By Montel's theorem, since the family omits two fixed values, it is normal on \Delta.^[26] Applying Marty's theorem to bound the spherical derivatives and extracting a subsequence converging uniformly on compact subsets leads to f'(z) = 0 everywhere, implying f is constant, a contradiction.^[26] Thus, Montel's normality criterion forces the value omission bound. For Picard's great theorem, which asserts that near an essential singularity at z = a, a holomorphic function takes every complex value, with at most one exception, infinitely often, Montel's theorem similarly ensures normality in punctured neighborhoods. Assume f has an essential singularity at 0 and omits three distinct values a_1, a_2, a_3; by the Casorati-Weierstrass theorem, there exists a sequence c_n \to 0 such that the spherical derivative f^\#(c_n) \to \infty. Define the rescaled family f_n(z) = f(c_n + c_n z), holomorphic on the unit disk with f_n^\#(0) \to \infty, implying non-normality by Marty's criterion. However, each f_n omits a_1, a_2, a_3, so Montel's theorem declares the family normal, yielding a contradiction unless the singularity is not essential.^[27] This application highlights how local normality via Montel controls global behavior near singularities. Bloch's principle, a heuristic linking local normality to global restrictions, is formalized through Montel's theorem to imply Picard-type behavior on the plane. Attributed to André Bloch, the principle posits that a family of meromorphic functions on a domain is normal if no non-constant entire meromorphic function shares the same restrictive property (e.g., omitting values). Zalcman's formalization defines a "Bloch property" P such that if a single function on the plane satisfies P, it is constant, and families satisfying P locally are normal; Montel's value-omission criterion exemplifies this, as omitting three values forces constancy globally (Picard's little theorem) and normality locally.^[24] This duality ensures that Montel's local normality criterion extends to global Picard assertions, providing a unified framework for value distribution.^[3] Modern extensions of Montel's theorem to several complex variables preserve the normality criterion: a family of holomorphic maps from a domain in \mathbb{C}^m to \mathbb{P}^n(\mathbb{C}) omitting $2n+1 hyperplanes in general position is normal, generalizing the one-variable case (where n=1 corresponds to omitting 3 values in \hat{\mathbb{C}}) and facilitating analogs of Picard's theorems in higher dimensions.^[28] In non-archimedean fields, such as p-adic numbers, a version of Montel's theorem holds for analytic functions over complete valued fields, where families avoiding certain disks are normal, enabling studies of dynamics and value distribution in this setting; recent works further refine these for rational maps and periodic points.^[29]

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