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 $x\in \Omega$, $f(x)\not \in \{a,b\}$, then ${\mathcal {F}}$ is a normal family on $\Omega$.

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

See also

• Montel's theorem