Multivalent harmonic starlike functions. (English) Zbl 1019.30010

Ahuja, Om P.; Jahangiri, Jay M.

Ann. Univ. Mariae Curie-Skłodowska, Sect. A 55, 1-13 (2001).

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.

Reviewer: O.Fekete (Freiburg)

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

Keywords:

harmonic multivalent function; starlike functions; extreme points; multivalent harmonic functions