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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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