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Let Mₚ denote the class of functions f of the form $f\left(z\right)=1/z^p+{\sum }_{k=0}^{\infty }aₖz^k$, p a positive integer, in the unit disk E = |z| < 1, f being regular in 0 < |z| < 1. Let $L_{n,p}\left(\alpha \right)=f:f\in Mₚ,Re-(z^{p+1}/p){\left(Dⁿf\right)}^{\text{'}}>\alpha$, α < 1, where $Dⁿf={\left(z^{n+p}f\left(z\right)\right)}^{\left(n\right)}/(z^{p}n!)$. Results on $L_{n,p}\left(\alpha \right)$ 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.

The function $f\left(z\right)=z^p+{\sum }_{k=1}^{\infty }{a}_{p+k}{z}^{p+k}$ (p ∈ ℕ = 1,2,3,...) analytic in the unit disk E is said to be in the class $K_{n,p}\left(h\right)$ if ($D^{n+p}f)/\left(D^{n+p-1}f\right)\prec h$, where $D^{n+p-1}f=\left(z^p\right)/\left({(1-z)}^{p+n}\right)*f$ and h is convex univalent in E with h(0) = 1. We study the class $K_{n,p}\left(h\right)$ and investigate whether the inclusion relation $K_{n+1,p}\left(h\right)\subseteq K_{n,p}\left(h\right)$ holds for p > 1. Some coefficient estimates for the class are also obtained. The class $A_{n,p}(a,h)$ of functions satisfying the condition $a*\left(D^{n+p}f\right)/\left(D^{n+p-1}f\right)+(1-a)*\left(D^{n+p+1}f\right)/\left(D^{n+p}f\right)\prec h$ is also studied.