  Let $f$ be a transcendental meromorphic function defined in the complex plane
$\mathbb{C}$, and $\varphi(\not\equiv 0,\infty)$ be a small function of $f$. In
this paper, We give a quantitative estimation of the characteristic function
$T(r, f)$ in terms of $N\left(r,\frac{1}{M[f]-\varphi(z)}\right)$ as well as
$\ol{N}\left(r,\frac{1}{M[f]-\varphi(z)}\right)$, where $M[f]$ is the
differential monomial, generated by $f$.\par Moreover, we prove one normality
criterion: Let $\mathscr{F}$ be a family of analytic functions on a domain $D$
and let $k(\geq1)$, $q_{0}(\geq 3)$, $q_{i}(\geq0)$ $(i=1,2,\ldots,k-1)$,
$q_{k}(\geq1)$ be positive integers. If for each $f\in \mathscr{F}$, $f$ has
only zeros of multiplicity at least $k$, and
$f^{q_{0}}(f')^{q_{1}}...(f^{(k)})^{q_{k}}\not=1$, then $\mathscr{F}$ is normal
on domain $D$.
