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
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.
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.
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.
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).
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.
- John B. Conway (1978). Functions of One Complex Variable I. Springer-Verlag. ISBN 0-387-90328-3.
- Hazewinkel, Michiel, ed. (2001), "Montel theorem", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- J. L. Schiff (1993). Normal Families. Springer-Verlag. ISBN 0-387-97967-0.