We show that a family F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} of meromorphic functions in a domain D\documentclass[12pt]{minimal}
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\begin{document}$$D$$\end{document} satisfying |f(k)|1+|f(j)|α(z)≥Cforallz∈Dandallf∈F\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{|f^{(k)}|}{1+|f^{(j)}|^\alpha }(z)\ge C \qquad \text{ for } \text{ all } z\in D \text{ and } \text{ all } f\in \mathcal {F}\end{aligned}$$\end{document}(where k\documentclass[12pt]{minimal}
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\begin{document}$$k$$\end{document} and j\documentclass[12pt]{minimal}
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\begin{document}$$j$$\end{document} are integers with k>j≥0\documentclass[12pt]{minimal}
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\begin{document}$$k>j\ge 0$$\end{document} and C>0\documentclass[12pt]{minimal}
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\begin{document}$$C>0$$\end{document}, α>1\documentclass[12pt]{minimal}
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\begin{document}$$\alpha >1$$\end{document} are real numbers) is quasi-normal. Furthermore, if all functions in F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} are holomorphic, the order of quasi-normality of F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} is at most j-1\documentclass[12pt]{minimal}
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\begin{document}$$j-1$$\end{document}. The proof relies on the Zalcman rescaling method and previous results on differential inequalities constituting normality.
