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 French mathematician
Paul Montel
Paul Antoine Aristide Montel (29 April 1876 – 22 January 1975) was a French mathematician. He was born in Nice, France and died in Paris, France. He researched mostly on holomorphic functions in complex analysis.
Montel was a student of Émil ...
, and give conditions under which a family of holomorphic functions is
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 subset of the
complex numbers is
normal if and only if it is locally uniformly bounded.
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
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
.
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
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
In topology, puncturing a manifold is removing a finite set of points from that manifold. The set of points can be small as a single point. In this case, the manifold is known as once-punctured. With the removal of a second point, it becomes twic ...
plane
. (Such a covering is given by the
elliptic modular function
In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is ho ...
).
This version of Montel's theorem can be also derived from
Picard's theorem
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.
The theorems
Little Picard Theorem: If a function f: \mathbb \to\mathbb i ...
,
by using
Zalcman's lemma.
Relationship to theorems for entire functions
A heuristic principle known as
Bloch's Principle Bloch's Principle is a philosophical principle in mathematics
stated by André Bloch.
Bloch states the principle in Latin as: ''Nihil est in infinito quod non prius fuerit in finito,'' and explains this as follows: Every proposition in whose sta ...
(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
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.
The theorems
Little Picard Theorem: If a function f: \mathbb \to\mathbb i ...
.
See also
*
Montel space
*
Fundamental normality test
Notes
References
*
*
*
{{PlanetMath attribution, title=Montel's theorem, id=5754
Compactness theorems
Theorems in complex analysis