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En análisis complejo, un área de las matemáticas, teorema de Montel refiere a uno de dos teoremas sobre familias de funciones holomorfas. Estos reciben su nombre en honor a Paul Montel, y brindan condiciones bajo las cuales una familia de funciones holomorfas resulta ser normal.

Toda familia uniformemente acotada es normal

La primera, y la más simple, versión del teorema afirma que una familia de funciones holomorfas uniformemente acotada definida en un subconjunto abierto del plano complejo es normal.

Este teorema nos lleva al siguiente corolario. This theorem has the following formally stronger corollary. Supongamos que ${\mathcal {F}}$ es una familia de funciones meromorfas en un conjunto abierto $D$. Si $z_{0}\in D$ es tal que ${\mathcal {F}}$ no es normal en $z_{0}$, y $U\subset D$ es un entorno de $z_{0}$, entonces $\bigcup _{f\in {\mathcal {F}}}f(U)$ es denso en el plano complejo.

Funciones que omiten dos valores

La versión más fuerte del teorema de Montel (a la cual ocasionalmete se llama Fundamental Normality Test) afirma que una familia de funciones holomorfas, todas las cuales omiten los mismos dos valores $a,b\in \mathbb {C} ,$ es normal.

Necessity

Las condiciones en los anteriores teorwmas son suficientes para garantizar normalidad, pero no necesarias. Indeed, the family $\{z\mapsto z\}$ is normal, but does not omit any complex value.

Proofs

The first version of Montel's theorem is a direct consequence of Marty's Theorem (which states that a family is normal if and only if the spherical derivatives are locally bounded) and Cauchy's integral formula.

This theorem has also been called the Stieltjes–Osgood theorem, after Thomas Joannes Stieltjes and William Fogg Osgood.

The Corollary stated above is deduced as follows. Suppose that all the functions in ${\mathcal {F}}$ omit the same neighborhood of the point $z_{1}$. By postcomposing with the map $z\mapsto {\frac {1}{z-z_{1}}}$ we obtain a uniformly bounded family, which is normal by the first version of the theorem.

The second version of Montel's theorem can be deduced from the first by using the fact that there exists a holomorphic universal covering from the unit disk to the twice punctured plane $\mathbb {C} \setminus \{a,b\}$. (Such a covering is given by the elliptic modular function).

This version of Montel's theorem can be also derived from Picard's theorem, by using Zalcman's lemma.

Relationship to theorems for entire functions

A heuristic principle known as Bloch's Principle (made precise by Zalcman's lemma) states that properties that imply that an entire function is constant correspond to properties that ensure that a family of holomorphic functions is normal.

For example, the first version of Montel's theorem stated above is the analog of Liouville's theorem, while the second version corresponds to Picard's theorem.

See also

• Montel space
• Fundamental normality test

References

• John B. Conway (1978). Functions of One Complex Variable I. Springer-Verlag. ISBN 0-387-90328-3.
• "Montel theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• J. L. Schiff (1993). Normal Families. Springer-Verlag. ISBN 0-387-97967-0.

This article incorporates material from Montel's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Category:Compactness theorems Category:Theorems in complex analysis