Differential inequalities and quasi-normal families

Abstract

We show that a family $\mathcal {F}$ of meromorphic functions in a domain $D$ satisfying

$$\begin{aligned} \frac{|f^{(k)}|}{1+|f^{(j)}|^\alpha }(z)\ge C \qquad \text{ for } \text{ all } z\in D \text{ and } \text{ all } f\in \mathcal {F}\end{aligned}$$

(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.