A normality criterion for a family of meromorphic functions

We consider a family $\mathcal{F}$ of meromorphic functions defined in a domain $D$, a holomorphic function $\psi$ and a homogeneous differential polynomial $P[f]$ of degree $d$ with weight $w$. In this paper, we prove the normality of $\mathcal{F}$ under certain conditions such as $f\mathrm{\ne }0$, $P[f]\mathrm{\ne }0$ and all the zeros of the function $P[f]-{\psi }^d$ have multipicity at least ${\displaystyle \frac{w+1}{w-1}}$, for each $f\in \mathcal{F}$.