Quasi-normality induced by differential inequalities

We show that the family Fk of all meromorphic functions f in a domain D satisfying (where k is a natural number and C>0) is quasi-normal. The proof relies mainly on the Zalcman-Pang rescaling lemma which we employ to get estimates for the behaviour of certain quotients of higher derivatives of the functions in Fk at certain points close to a point of non-normality.