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A subclass of harmonic univalent functions with positive coefficients. (English) Zbl 1206.30016
Summary: Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disc \(U\) can be written in the form \(f=h+\overline g\), where \(h\) and \(g\) are analytic in \(U\). In this paper, the authors introduce the class \(R_H(\beta)\), \(1<\beta\leq 2\), which consists of the harmonic univalent functions \(f=h+\overline g\) where \(h\) and \(g\) are of the form
\[ h(z)=z+\sum_{k=2}^\infty | a_k| z^k \quad\text{and}\quad g(z)=\sum_{k=2}^\infty | b_k | z^k \] and \(\text{Re\,}\big\{h'(z)-g'(z)\big\}<\beta\). We obtain distortion bounds, extreme points, and radii of convexity for the functions in this class, and discuss a class preserving integral operator. We also show that the class studied in this paper is closed under convolution and convex combinations.

MSC:
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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