A normality criterion for a family of meromorphic functions

We consider a family $\mathscr{F}$ 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 $\mathscr{F}$ under certain conditions such as $f\neq 0$, $P[f]\neq 0$ and all the zeros of the function $P[f] - \psi^d$ have multipicity at least $\displaystyle{\frac{w+1}{w-1}}$, for each $f \in \mathscr{F}$.