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 $m,~m\in\mathbb{ N}.$ Let $\left\{ h_j\right\}$ be a sequence of holomorphic functions in $\mathbb{ D}$ such that $h_j\rightarrow h$ locally uniformly in $\mathbb{ D},$ where $h$ is holomorphic in $\mathbb{ D}$ and $h\not\equiv 0.$ Let $Q[f_j]$ be a differential polynomial of $f_j$ having degree $\lambda_Q$ and weight $\mu_Q.$ If, for each $j,$ $f_j(z)\neq 0$ 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 $\mathbb{ D}.$
Categories: math.CV