# A class of analytic functions defined by Ruscheweyh derivative

• Volume: 54, Issue: 2, page 167-178
• ISSN: 0066-2216

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## Abstract

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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 (${D}^{n+p}f\right)/\left({D}^{n+p-1}f\right)\prec h$, where ${D}^{n+p-1}f=\left({z}^{p}\right)/\left({\left(1-z\right)}^{p+n}\right)*f$ 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 ${A}_{n,p}\left(a,h\right)$ of functions satisfying the condition $a*\left({D}^{n+p}f\right)/\left({D}^{n+p-1}f\right)+\left(1-a\right)*\left({D}^{n+p+1}f\right)/\left({D}^{n+p}f\right)\prec h$ is also studied.

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