A class of analytic functions defined by Ruscheweyh derivative

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