On certain subclasses of multivalently meromorphic close-to-convex maps

Let Mₚ denote the class of functions f of the form ${\displaystyle f\left(z\right)=\frac{1}{z^p}+{\sum }_{k=0}^{\infty }aₖz^k}$, p a positive integer, in the unit disk E = {|z| < 1}, f being regular in 0 < |z| < 1. Let ${\displaystyle L_{n,p}(\alpha )=\{f:f\in Mₚ,Re\{-\left(\frac{z^{p+1}}{p}\right)(Dⁿf)\prime \}>\alpha \}}$, α < 1, where ${\displaystyle Dⁿf=\frac{{\left(z^{n+p}f\left(z\right)\right)}^{\left(n\right)}}{z^{p}n!}}$. Results on ${\displaystyle L_{n,p}(\alpha )}$ are derived by proving more general results on differential subordination. These results reduce, by putting p =1, to the recent results of Al-Amiri and Mocanu.