  We show that the family of all holomorphic functions $f$ in a domain $D$
satisfying $$\frac{|f^{(k)}|}{1+|f|}(z)\le C \qquad \mbox{ for all } z\in D$$
(where $k$ is a natural number and $C>0$) is quasi-normal. Furthermore, we give
a general counterexample to show that for $\alpha>1$ and $k\ge2$ the condition
$$\frac{|f^{(k)}|}{1+|f|^\alpha}(z)\le C \qquad \mbox{ for all } z\in D$$ does
not imply quasi-normality.
