Fundamental normality test

In complex analysis, a mathematical discipline, the fundamental normality test gives sufficient conditions to test the normality of a family of analytic functions. It is another name for the stronger version of Montel's theorem.

Statement

Let $\mathcal{F}$ be a family of analytic functions defined on a domain $\Omega$ . If there are two fixed complex numbers a and b such that for all ƒ ∈ $\mathcal{F}$ and all x ∊ $\Omega$ , f(x) ∉ {a, b}, then $\mathcal{F}$ is a normal family on $\Omega$ .

The proof relies on properties of the elliptic modular function and can be found here:

{{cite book | author = J. L. Schiff | title = Normal Families | publisher = Springer-Verlag | year = 1993 | isbn=0-387-97967-0 }}

Fundamental normality test

Statement

See also