Differential inequalities and a Marty-type criterion for quasi-normality

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.