The well known Marty’s theorem asserts that a family of functions meromorphic in some domain is normal if and only if for every compact subset of there is a positive constant such that
for every in and every in . We reverse the sign of the inequality and discuss the connection between normality and differential inequalities of the type
where 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.