Editing Montel's theorem

{{short description|Two theorems about families of holomorphic functions}}
In [[complex analysis]], an area of [[mathematics]], '''Montel's theorem''' refers to one of two [[theorem]]s about  families of [[holomorphic function]]s. These are named after French mathematician [[Paul Montel]], and give conditions under which a family of holomorphic functions is [[normal family|normal]].

==Locally uniformly bounded families are normal==
The first, and simpler, version of the theorem states that a family of holomorphic functions defined on an [[open set|open]] [[subset]] of the [[complex number]]s is [[normal family|normal]] if and only if it is locally uniformly bounded.

This theorem has the following formally stronger corollary. Suppose that
<math>\mathcal{F}</math> is a family of
meromorphic functions on an open set <math>D</math>. If <math>z_0\in D</math> is such that
<math>\mathcal{F}</math> is not normal at <math>z_0</math>, and <math>U\subset D</math> is a neighborhood of <math>z_0</math>, then <math>\bigcup_{f\in\mathcal{F}}f(U)</math> 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 <math>a,b\in\mathbb{C},</math> is normal.

==Necessity==
The conditions in the above theorems are sufficient, but not necessary for normality. Indeed, 
the family <math>\{z\mapsto z\}</math> 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]].<ref>{{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
| isbn = 978-0-582-32388-9
| accessdate = 2009-03-01
}}</ref>

This theorem has also been called the Stieltjes–Osgood theorem, after [[Thomas Joannes Stieltjes]] and [[William Fogg Osgood]].<ref>{{cite book |author=Reinhold Remmert, [[Leslie M. Kay]] |url=https://books.google.com/books?id=BHc2b0iCoy8C |title=Classical Topics in Complex Function Theory |publisher=Springer |year=1998 |pages=154 | isbn=978-0-387-98221-2 |accessdate=2009-03-01}}</ref>
 
The Corollary stated above is deduced as follows. Suppose that all the functions in <math>\mathcal{F}</math> omit the same neighborhood of the point <math>z_1</math>. By postcomposing with the map <math>z\mapsto \frac{1}{z-z_1}</math> 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 <math>\mathbb{C}\setminus\{a,b\}</math>. (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 [[Bloch's Principle|Zalcman's lemma]].

==Relationship to theorems for entire functions==
A heuristic principle known as [[Bloch's principle]] (made precise by [[Bloch's Principle#Zalcman's lemma|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 (complex analysis)|Liouville's theorem]], while the second version corresponds to [[Picard's theorem]].

==See also==
*[[Montel space]]
*[[Fundamental normality test]]
*[[Riemann mapping theorem]]

==Notes==
{{reflist}}

==References==
*{{cite book | author = John B. Conway | title = Functions of One Complex Variable I | publisher = Springer-Verlag | year = 1978 | isbn=0-387-90328-3 }}
*{{SpringerEOM|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 }}

{{PlanetMath attribution|title=Montel's theorem|id=5754}}

[[Category:Compactness theorems]]
[[Category:Theorems in complex analysis]]