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.

Dedicated to Professor Themistocles M. Rassias on the occasion of his 60th birthday.