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A new class of Salagean-type harmonic univalent functions. (English) Zbl 1085.30018

Let harmonic functions \(f\) in the unit disk \(U\) be given by \[ f(z)=h(z)+\overline{g(z)}=z+\sum_{k=2}^{\infty}a_kz^k+\overline{\sum_{k=1}^{\infty}b_kz^k}, \] \(| b_1| <1\), \(z\in U\). For \(0\leq\alpha<1\), \(m\in\mathbb N\), \(n\in\mathbb N_0\), \(m>n\), let \(S_H(m,n;\alpha)\) denote the family of harmonic functions \(f\) such that \[ \text{Re}\frac{D^mf(z)}{D^nf(z)}>\alpha,\;\;\;z\in U, \] where \[ D^mf(z)=z+\sum_{k=2}^{\infty}k^ma_kz^k+(-1)^m\overline{\sum_{k=1}^{\infty}k^mb_kz^k}. \] The subclass \(\overline S_H(m,n;\alpha)\) consists of \(f=h+\overline g\in S_H(m,n;\alpha)\) so that \(a_k\leq0\) for \(k\geq2\) and \((-1)^mb_k\leq0\) for \(k\geq1\). The author gives a coefficient inequality which guarantees that \(f\) is sense preserving, harmonic univalent in \(U\) and \(f\in S_H(m,n;\alpha)\). This inequality is necessary for \(f\in\overline S_H(m,n;\alpha).\) Also, the author describes the extreme points of closed convex hulls of \(\overline S_H(m,n;\alpha)\), finds the distortion bounds and the convexity radius in \(\overline S_H(m,n;\alpha)\) and shows that \(\overline S_H(m,n;\alpha)\) is invariant under convolution and convex combination.

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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References:

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