Differential inequalities and quasi-normal families

Abstract

We show that a family $\mathcal{F}$ of meromorphic functions in a domain $D$ satisfying $$\frac{|f^{(k)}|}{1+|f^{(j)}|^α}(z)\ge C \qquad \mbox{for all} z\in D \mbox{and all} f\in {\cal F}$$ (where $k$ and $j$ are integers with $k>j\ge 0$ and $C>0$, $\alpha >1$ are real numbers) is quasi-normal. Furthermore, if all functions in $\mathcal{F}$ are holomorphic, the order of quasi-normality of $\mathcal{F}$ is at most $j-1$. The proof relies on the Zalcman rescaling method and previous results on differential inequalities constituting normality.