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Zbl 0959.30003
Silverman, H.; Silvia, E.M.
Subclasses of harmonic univalent functions. (English)
[J] N. Z. J. Math. 28, No.2, 275-284 (1999). ISSN 1171-6096/e

Denote by ${\cal S}_H$ the class of functions $f$ of the form $f=h+\overline g$ that are harmonic univalent and sense-preserving in the disk $\Delta= \{z:|z|<1\}$ with $f(0)=f_z(0) -1=0$; ${\cal S}^0_H$ -- the subclass of ${\cal S}_H$ for which $f_{\overline z}(0)=0$; ${\cal S}^*_H$ and ${\cal K}_H$ -- the subclasses of $S_H$ consisting of functions $f$ that map $\Delta$ onto starlike and convex domains, respectively. The authors consider the classes ${\cal T}_H^*$ and ${\cal T}{\cal K}_H$ that are subclasses of ${\cal S}^*_H$ and ${\cal K}_H$, respectively, for which the coefficients of functions $f=h+\overline g$ take the form $$h(z)=z -\sum^\infty_{n=2} a_nz^n,\ g(z)= -\sum^\infty_{n=1} b_nz^n,\ 0\le b_1<1;\ a_n,b_n\ge 0,\ n=2,3,\dots.$$ In [{\it H. Silverman}, J. Math. Anal. Appl. 220, No. 1, 283-289 (1998; Zbl 0908.30013)], the families ${\cal T}_H^{*0} ={\cal T}^*_H \cap{\cal S}^0_H$ and ${\cal T}{\cal K}^0_H ={\cal T}{\cal K}^0_H \cap{\cal S}^0_H$ where investigated. In this note the authors generalize results for ${\cal T}^{*0}_H$ to the classes ${\cal T}^*_H$ and ${\cal T}{\cal K}_H$ and also consider the family ${\cal T}_H^*(b)$, $0\le b<1$, consisting of functions $f\in{\cal T}^*_H$ for which $b_1=b$ is fixed.
[Zbigniew J.Jakubowski (Łódź)]

MSC 2000:: *30C45 Special classes of univalent and multivalent functions
31A05 Harmonic functions, etc. (two-dimensional)

Keywords: complex-valued harmonic; univalent functions; harmonic starlike functions; harmonic convex functions

Citations: Zbl 0908.30013

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