Chapter

Nonlinear Analysis

Volume 68 of the series Springer Optimization and Its Applications pp 79-89

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

  • Daniel BreazAffiliated withDepartment of Mathematics, “1 Decembrie 1918” University of Alba Iulia Email author 
  • , Yasar Polatog̃luAffiliated withDepartment of Mathematics and Computer Science, Kültür University
  • , Nicoleta BreazAffiliated with“1 Decembrie 1918” University of Alba Iulia

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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.

Key words

Generalized p-valent Janowski convex function Generalized p-valent Janowski close-to-convex function Radius of convexity

Mathematics Subject Classification

30C45 30C55