Let $\mathcal{F}$ be a family of holomorphic functions in a domain
$D$; let $k$ be a positive integer; let $h$ be a positive number; and let
$a$ be a
function holomorphic in $D$ such that $a(z)\not= 0$ for $z\in D$. For
$k\not= 2$ we show that
if, for every $f\in \mathcal{F}$, all zeros of $f$ have
multiplicity at
least
$
k$,
$f(z)=0$ $\Longrightarrow $ $f^{(k)}(z)=a(z)$, and $f^{(k)}(z)=a(z)$
$\Longrightarrow $ $|f^{(k+1)}(z)|\le h$, then $\mathcal{F}$ is normal
in
$D$.
For $k=2$ we prove the following result:
Let $s\ge 4$ be an even integer. If, for
every $f\in \mathcal{F}$, all zeros of $f$ have multiplicity at
least $
2$,
$f(z)=0$
$\Longrightarrow $ $f''(z)=a(z)$, and $f''(z)=a(z)$ $\Longrightarrow $
$|f'''(z)|+|f^{(s)}(z)|\le h$, then $\mathcal{F}$ is normal in $D$. This
improves the well-known normality criterion of Miranda.