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

Part of the Springer Optimization and Its Applications book series (SOIA, volume 68)

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)=zp+cnp+1znp+1+cnp+2znp+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)=znψ(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)=znψ(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 

References

  1. 1.
    Bernardi, S.D.: New distortion theorems for functions of positive real part and applications to the univalent convex functions. Proc. Am. Math. Soc. 45, 113–118 (1974) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Clunie, J., Sheil-Small, T.: Harmonic univalent functions. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 9, 3–25 (1984) MathSciNetMATHGoogle Scholar
  3. 3.
    Duren, P.: Harmonic Mappings in the Plane. Cambridge University press, Cambridge (2004) MATHCrossRefGoogle Scholar
  4. 4.
    Janowski, W.: Some extremal problems for certain families of analytic functions I. Ann. Pol. Math. XXVII, 297–326 (1973) MathSciNetGoogle Scholar
  5. 5.
    Kaplan, W.: Close-to-convex functions. Mich. Math. J. 1(2), 169–184 (1952) MATHCrossRefGoogle Scholar
  6. 6.
    Umezawa, T.: Multivalently close-to-convex functions. Proc. Am. Math. Soc. 8(5), 869–874 (1957) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Daniel Breaz
    • 1
  • Yasar Polatog̃lu
    • 2
  • Nicoleta Breaz
    • 3
  1. 1.Department of Mathematics“1 Decembrie 1918” University of Alba IuliaAlba IuliaRomania
  2. 2.Department of Mathematics and Computer ScienceKültür UniversityIstanbulTurkey
  3. 3.“1 Decembrie 1918” University of Alba IuliaAlba IuliaRomania

Personalised recommendations