Multivalent harmonic starlike functions (Q2769250)

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 NEWLINE\[NEWLINEh(z)=z^m+ \sum^\infty_{n=2} a_{n+ m-1} z^{n+m-1},\;g(z)=\sum^\infty_{n=1}b_{n+m-1}z^{n+m-1},\;|b_m |<1. \tag{1}NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINEh(z)=z^m- \sum^\infty_{n=2} |a_{n+m-1} |z^{n+m-1},\;g(z)=\sum^\infty_{n=1} |b_{n+m-1} |z^{n+m-1}.\tag{2}NEWLINE\]NEWLINE 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.