Normality and uniqueness property of meromorphic function in terms of some differential polynomials

Abstract: In this paper, we will consider normality and uniqueness property of a family $\mathcal{F}$ of meromorphic functions when $[Q(f)]^{(k)}$ and $[Q(g)]^{(k)}$ share $\alpha$ ignoring multiplicities, for any $f,g\in \mathcal{F}$, where $Q$ is a polynomial and $\alpha $ is a small function. Our results do not need all of zeros of $Q$ have large order as other authors's results.