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Zbl 1019.30010
Ahuja, Om P.; Jahangiri, Jay M.
Multivalent harmonic starlike functions.
(English)
[J] Ann. Univ. Mariae Curie-Skłodowska, Sect. A 55, 1-13 (2001). ISSN 0365-1029

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 $m\ge 1$ 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.
[O.Fekete (Freiburg)]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
30C50 Coefficient problems for univalent and multivalent functions
30C55 General theory of univalent and multivalent functions

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

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