A Normal Criterion Concerning Sequence of Functions and their Differential Polynomials

In this paper, a normality criterion concerning a sequence of meromorphic functions and their differential polynomials is obtained. Precisely, we have proved: Let $\left\{f_j\right\}$ be a sequence of meromorphic functions in the open unit disk $\mathbb{D}$ such that, for each $j,$ $f_j$ has poles of multiplicity at least $m,m\in \mathbb{N}\mathrm{.}$ Let $\left\{h_j\right\}$ be a sequence of holomorphic functions in $\mathbb{D}$ such that $h_j\to h$ locally uniformly in $\mathbb{D},$ where $h$ is holomorphic in $\mathbb{D}$ and $h\equiv ̸0.$ Let $Q[f_j]$ be a differential polynomial of $f_j$ having degree ${\lambda }_Q$ and weight ${\mu }_Q\mathrm{.}$ If, for each $j,$ $f_j(z)\mathrm{\ne }0$ and $Q[f_j]-h_j$ has at most ${\mu }_Q+{\lambda }_Q(m-1)-1$ zeros, ignoring multiplicities, in $\mathbb{D},$ then $\left\{f_j\right\}$ is normal in $\mathbb{D}\mathrm{.}$