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Multivalent harmonic starlike functions. (English) Zbl 1019.30010

Denote by \(H(m)\) the class of multivalent harmonic functions \(f=h+ \overline g\) that are sense-preserving in the unit disk \(D=\{z: |z |<1\}\) and \(h\) and \(g\) are of the form \[ h(z)=z^m+ \sum^\infty_{n=2} a_{n+ m-1} z^{n+m-1},\;g(z)=\sum^\infty_{n=1}b_{n+m-1}z^{n+m-1},\;|b_m |<1. \tag{1} \] For \(m\geq 1\) let \(SH(m)\) denote the subclass of \(H(m)\) consisting of harmonic starlike functions and let \(TH(m)\) denote the subclass of \(SH(m)\) so that \(h\) and \(g\) are of the form \[ h(z)=z^m- \sum^\infty_{n=2} |a_{n+m-1} |z^{n+m-1},\;g(z)=\sum^\infty_{n=1} |b_{n+m-1} |z^{n+m-1}.\tag{2} \] Sufficient coefficient bounds for functions of the form (1) to be in \(SH(m)\) are given. This bounds are necessary if \(f\in TH(m)\). Extreme points, distortion and covering theorems, convolutions and convex combination conditions for these classes of functions are also determined.

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
30C55 General theory of univalent and multivalent functions of one complex variable
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