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