Differential inequalities and normality

Abstract

The well known Marty’s theorem asserts that a family $F$ of functions meromorphic in some domain $D$ is normal if and only if for every compact subset $K$ of $D$ there is a positive constant $C\left(K\right)$ such that

for every $f$ in $F$ and every $z$ in $D$. We reverse the sign of the inequality and discuss the connection between normality and differential inequalities of the type

where $h$ is some increasing and continuous function. As we shall see, some of these inequalities imply quasy-normality, a geometric extension of the notion of normality.