Montel’s fundamental normality test (published in 1912) provides a strong sufficient condition for normality: a family \(\mathscr {F}\) of functions meromorphic in a region \(\varOmega \) is normal there if there exist three distinct values a, b, c in the extended complex plane \({\mathbb {C}}_\infty \) such that each f in \(\mathscr {F}\) omits in \(\varOmega \) each of these values. In 1954 Montel published an extension of this which gives a necessary and sufficient condition for \(\mathscr {F}\) to be a normal family, and which contains his Fundamental Normality Test as a special case. This striking result does not seem to be well known and, unfortunately, there is a small error in the proof. Montel’s new condition for normality is the uniform separation of the pre-images \(f^{-1}(a_j)\), or fibers, of four distinct points \(a_1,a_2,a_3,a_4\) in \({\mathbb {C}}_\infty \). We point out the error in the proof, and establish an improved version of his result.

Montel的基本正态性检验（于1912年出版）为正态性提供了强大的充分条件：如果存在三个不同的值，则区域\（\ varOmega \）中亚纯函数的族\（\ mathscr {F} \）是正常的，b，ç在扩展复平面\（{\ mathbb {C}} _ \ infty \） ，使得每个˚F在\（\ mathscr {F} \）中省略了\（\ varOmega \）这些值的每。1954年，蒙特（ Montel）发布了对此的扩展，它为\（\ mathscr {F} \）提供了充要条件。成为一个正常的家庭，其中包含他的基本正常性测试作为特例。这个惊人的结果似乎并不为人所知，并且不幸的是，证明中存在一个很小的错误。Montel的正态性新条件是将\（f {{-1}（a_j）\）或纤维在\ {{中的四个不同点\（a_1，a_2，a_3，a_4 \）的均匀分离\ mathbb {C}} _​​ \ infty \）。我们指出了证明中的错误，并建立了他的结果的改进版本。