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

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