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

The proof relies on properties of the elliptic modular function and can be found here: J. L. Schiff (1993). Normal Families. Springer-Verlag. ISBN 0-387-97967-0. 

See also

• Montel's theorem