C V ] 3 N ov 2 01 1 DIFFERENTIAL INEQUALITIES , NORMALITY AND QUASI-NORMALITY

We prove that if D is a domain in C, α > 1 and C > 0, then the family F of functions f meromorphic in D such that |f ′(z)| 1 + |f(z)|α > C for every z ∈ D is normal in D. For α = 1, the same assumptions imply quasi-normality but not necessarily normality.