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, and give conditions under which a family of holomorphic functions is 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.
is a family of meromorphic functions on an open set
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
The conditions in the above theorems are sufficient, but not necessary for normality.
is normal, but does not omit any complex value.
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.
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).
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.
This article incorporates material from Montel's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.