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On a subclass of harmonic univalent functions. (English) Zbl 1173.30011
Let $$h(z)=z+ \sum_{k=2}^\infty a_k z^k$$, $$b(z)=\sum_{k=1}^\infty b_k z^k$$ be holomorphic functions in the unit disc and let $$f(z)=h(z) + \bar{g}(z)$$. The authors study the class $$HS(m, n, \alpha)$$ of complex valued harmonic functions $$f$$ such that $\sum_{k=1}^\infty (k^m -\alpha k^n)(|a_k| +|b_k|) \leq (1-\alpha) (1-|b_1|),$ where $$m$$ and $$n$$ are integers, $$m\geq 1, n\geq 0, m>n,$$ and $$\alpha \in (0;1).$$
Among other things the authors prove that the class $$HS(m, n, \alpha)$$ consists of univalent sense preserving harmonic mappings and that the following estimates are true: $|f(z)| \leq |z|(1+|b_1|) + \frac{1-\alpha}{2^m -\alpha 2^n}(1-|b_1|)|z|^2 ,$ $|f(z)| \geq (1-|b_1|) (|z| -\frac{1-\alpha}{2^m -\alpha 2^n} |z|^2).$

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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