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

30C45 30C55