The classical spherical type criterion of F. Marty states that a family

$\mathcal{F}$ of functions meromorphic in a domain

$D$ is a normal family iff a spherical estimate

$|{f}^{\text{'}}\left(z\right)|\le {c}_{K}{(1+|f\left(z\right)|}^{2})$ holds uniformly for

$f\in \mathcal{F}$ and points

$z$ in any compact subset

$K\subset D$. A Euclidean type criterion of an entirely different nature is originally due to

*A. J. Lohwater* and

*C. Pommerenke* [Ann. Acad. Sci. Fenn., Ser. A I 550, 12 p. (1973;

Zbl 0275.30027)] with a later treatment given by

*L. Zalcman* [Am. Math. Mon. 82, 813-817 (1975;

Zbl 0315.30036)]. The present paper considers similar criterion for normality of families of functions whose zeros are of degree at least

$k$, where

$k$ is a positive integer. As an example of how these results can be applied, the authors prove that a family

$\mathcal{F}$ of meromorphic functions is normal if each function

$f\in \mathcal{F}$ has only poles of degree at least

$k+2$ and satisfies

${f}^{\left(k\right)}-a{f}^{3}\ne b$ everywhere, where

$a$ and

$b$ are fixed complex numbers. This result was established by D. Drasin for holomorphic functions where

$k=1$.