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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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