“Montel's theorem”的意思、由来-开放百科全书

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Montel's theorem

释义

Uniformly bounded families are normal
Functions omitting two values
Necessity
Proofs
Relationship to theorems for entire functions
See also
Notes
References

In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after Paul Montel, and give conditions under which a family of holomorphic functions is normal.

Uniformly bounded families are normal

The first, and simpler, version of the theorem states that a uniformly bounded family of holomorphic functions defined on an open subset of the complex numbers is normal.

This theorem has the following formally stronger corollary. Suppose that

is a family of

meromorphic functions on an open set . If is such that

is not normal at , and is a neighborhood of , then is dense

in the complex plane.

Functions omitting two values

The stronger version of Montel's Theorem (occasionally referred to as the Fundamental Normality Test) states that a family of holomorphic functions, all of which omit the same two values is normal.

Necessity

The conditions in the above theorems are sufficient, but not necessary for normality. Indeed,

the family 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.^[1]

This theorem has also been called the Stieltjes–Osgood theorem, after Thomas Joannes Stieltjes and William Fogg Osgood.^[2]

The Corollary stated above is deduced as follows. Suppose that all the functions in omit the same neighborhood of the point . By postcomposing with the map 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 . (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.

Notes

1. ^{{cite book| url = https://books.google.com/books?id=HwqjxJOLLOoC| title = Progress in Holomorphic Dynamics| author = Hartje Kriete| publisher = CRC Press| year = 1998| pages = 164| accessdate = 2009-03-01}}
2. ^{{cite book| url = https://books.google.com/books?id=BHc2b0iCoy8C| title = Classical Topics in Complex Function Theory| author = Reinhold Remmert, Leslie Kay| publisher = Springer| year = 1998| pages = 154| accessdate = 2009-03-01}}

References

{{cite book | author = John B. Conway | title = Functions of One Complex Variable I | publisher = Springer-Verlag | year = 1978 | isbn=0-387-90328-3 }}
{{springer|title=Montel theorem|id=p/m064890}}
{{cite book | author = J. L. Schiff | title = Normal Families | publisher = Springer-Verlag | year = 1993 | isbn=0-387-97967-0 }}

2 : Compactness theorems|Theorems in complex analysis

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