Properties of Harmonic Functions Defined by Shear Construction
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Properties of Harmonic Functions Defined by Shear Construction Rasoul Aghalary1 · Mahdi Jahani Rad1
Received: 7 April 2017 / Accepted: 7 September 2017 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017
Abstract In this paper, we introduce a new subclass of harmonic functions defined by shear construction. Among other properties of this subclass, we study the convolution of its elements of it with some special subclasses of harmonic functions. Also, we provide coefficient conditions leading to harmonic mappings which are starlike of order α. Some interesting applications of the general results are also presented. Keywords Harmonic starlike and convex functions · Shear construction · Univalent harmonic function · Radii problem Mathematics Subject Classification (2010) Primary 30C45 · Secondary 30C80
1 Introduction Let A denote the class of all analytic functions in the unit disk U = {z ∈ C : |z| < 1}. Also let us denote by A0 the subclass of A consisting of all functions f with f (0) = f (0) − 1 = 0. Denote by H the class of all complex valued harmonic mappings f = h + g defined in the unit disk U = {z ∈ C : |z| < 1}. Such harmonic mappings are locally univalent and sense-preserving if and only if h (z) = 0 in U and the dilatation function ω, defined by ω = g / h , satisfies |ω(z)| < 1 for all z ∈ U (see [6]). The class of all univalent harmonic and sense-preserving mappings f = h + g in U , normalized by the conditions f (0) = 0
Rasoul Aghalary
[email protected]; [email protected] Mahdi Jahani Rad [email protected] 1
Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran
R. Aghalary, M. J. Rad
and fz (0) = 1, is denoted by SH . Therefore, a function f = h + g in the class SH has the representation f (z) = z +
∞
an z n +
n=2
∞
bn z n ,
z ∈ U.
n=1
If the co-analytic part g(z) ≡ 0 in U , then the class SH reduces to the usual class S of all 0 be the subclass of S whose members f normalized univalent analytic functions. Let SH H ∗ , K , and C , the subclasses satisfy the additional condition fz (0) = 0. We denote by SH H H of SH consisting of those functions which map U onto starlike, convex, and close-to-convex domains, respectively. S ∗ , K, and C will denote corresponding subclasses of S. A domain is said to be convex in the direction of eiγ , 0 ≤ γ < π , if every line parallel to the line through 0 and eiγ has either connected or empty intersection with . Let KH (γ ) and Kγ , 0 ≤ γ < π denote the subclass of SH and S, respectively, consisting of functions which map the unit disk U onto domains convex in the direction of eiγ . In particular, a domain convex in the horizontal direction will be denoted by CHD. The convolution or the product φ ∗ ψ of two analytic mappings φ(z) = ∞ ∞Hadamard n n n=0 an z and ψ(z) = n=0 bn z in U is defined as (φ ∗ ψ)(z) =
∞
an bn zn , z ∈ U.
n=0
In the harmonic case, with f (z) = h(z) + g(z) = z +
∞
an z n +
n=2
F (z) = H (z) + G(z) =
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