Montel's theorem

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. 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. 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 F {displaystyle {mathcal {F}}} is a family ofmeromorphic functions on an open set D {displaystyle D} . If z 0 ∈ D {displaystyle z_{0}in D} is such that F {displaystyle {mathcal {F}}} is not normal at z 0 {displaystyle z_{0}} , and U ⊂ D {displaystyle Usubset D} is a neighborhood of z 0 {displaystyle z_{0}} , then ⋃ f ∈ F f ( U ) {displaystyle igcup _{fin {mathcal {F}}}f(U)} is densein the complex plane. 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 a , b ∈ C , {displaystyle a,bin mathbb {C} ,} is normal. The conditions in the above theorems are sufficient, but not necessary for normality. Indeed, the family { z ↦ z } {displaystyle {zmapsto z}} is normal, but does not omit any complex value. The first version of Montel's theorem is a direct consequence of Marty's Theorem (whichstates that a family is normal if and only if the spherical derivatives are locally bounded)and Cauchy's integral formula. 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 F {displaystyle {mathcal {F}}} omit the same neighborhood of the point z 1 {displaystyle z_{1}} . By postcomposing with the map z ↦ 1 z − z 1 {displaystyle zmapsto {frac {1}{z-z_{1}}}} 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 C ∖ { a , b } {displaystyle mathbb {C} setminus {a,b}} . (Such a covering is given by the elliptic modular function).