Generalized p-Valent Janowski Close-to-Convex Functions and Their Applications to the Harmonic Mappings

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Abstract

Let A(p,n), n≥1, p≥1 be the class of all analytic functions in the open unit disc $\mathbb{D}= \{z||z|<1 \}$ of the form s(z)=z p +c np+1 z np+1+c np+2 z np+2+⋯ and let s(z) be an element of A(p,n), if s(z) satisfies the condition $(1+z\frac {s''(z)}{s'(z)})=\frac{1+A\varphi(z)}{1+B\varphi(z)}$ , then s(z) is a called generalized p-valent Janowski convex function, where A,B are arbitrary fixed real numbers such that −1≤B<A≤1, and φ(z)=z n ψ(z) with ψ(z) being analytic in $\mathbb {D}$ and satisfying the condition |ψ(z)|<1 for every $z\in\mathbb{D}$ . The class of generalized p-valent Janowski convex functions is denoted by C(p,n,A,B). Let s(z) be an element of A(p,n), then s(z) is a generalized p-valent Janowski close-to-convex function for $z\in\mathbb{D}$ , if there exists a function ϕ(z)∈C(p,n,A,B) such that $\frac{s'(z)}{\phi'(z)}=\frac{1+A\varphi(z)}{1+B\varphi(z)}$ . (−1≤BA≤1,φ(z)=z n ψ(z), ψ(z) is analytic and |ψ(z)|<1 for every $z\in\mathbb{D}$ ). The class of such functions is denoted by K(p,n,A,B).

The aim of this paper is to give an investigation of the class K(p,n,A,B) and its application to the harmonic mappings.