# 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

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)
2. 2.
Clunie, J., Sheil-Small, T.: Harmonic univalent functions. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 9, 3–25 (1984)
3. 3.
Duren, P.: Harmonic Mappings in the Plane. Cambridge University press, Cambridge (2004)
4. 4.
Janowski, W.: Some extremal problems for certain families of analytic functions I. Ann. Pol. Math. XXVII, 297–326 (1973)
5. 5.
Kaplan, W.: Close-to-convex functions. Mich. Math. J. 1(2), 169–184 (1952)
6. 6.
Umezawa, T.: Multivalently close-to-convex functions. Proc. Am. Math. Soc. 8(5), 869–874 (1957)

## 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