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Zbl 0908.30013
Silverman, Herb
Harmonic univalent functions with negative coefficients.
(English)
[J] J. Math. Anal. Appl. 220, No.1, 283-289 (1998). ISSN 0022-247X

Denote by $S_H$ the class of functions $f$ of the form: (1) $f= h+\overline g$ that are harmonic univalent and sense-preserving in the unit disc $\Delta= \{z:| z|< 1\}$ for which $f(0)= f_z(0)- 1=0$ and by $S^0_H$ the subclass of $S_H$ for which $f_{\overline z}(0)= 0$.\par Let: (2) $h(z)= z+\sum^\infty_{n= 2} a_nz^n$, $g(z)= \sum^\infty_{n= 2} b_nz^n$, $z\in \Delta$. Denote by $S^{*0}_H$ and $K^0_H$ the subclasses of $S^0_H$ consisting of functions $f$ that map $\Delta$ onto starlike and convex domains, respectively. Let $T^{*0}_H$ and $TK^0_H$ be the subclasses of $S^{*0}_H$ and $K^0_H$, respectively, whose coefficients $f= h+\overline g$ take the form: (3) $h(z)= z- \sum^\infty_{n= 2} a_nz^n$, $a_n\ge 0$; $g(z)= -\sum^\infty_{n=2} b_nz^n$, $b_n\ge 0$, $z\in\Delta$.\par In the present paper mentioned above classes of harmonic functions are considered. The author proves among others: Theorem 1. If $f$ of the form (1-2) satisfies $\sum^\infty_{n= 2}n(| a_n|+| b_n|)\le 1$, then $f\in S^{*0}_H$. Corollary 1. If $f$ of the form (1-2) satisfies $\sum^\infty_{n= 2} n^2(| a_n|+ | b_n|)\le 1$, then $f\in K^0_H$. Theorem 2. For $f$ of the form (1), (3), $f\in T^{*0}_H$ if and only if $\sum^\infty_{n= 2} n(a_n+ b_n)\le 1$. Theorem 3. For $f$ of the form (1), (3), $f\in TK^0_H$ if and only if $\sum^\infty_{n= 2} n^2(a_n+ b_n)\le 1$.
[Z.J.Jakubowski (Łódź)]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
31A05 Harmonic functions, etc. (two-dimensional)

Keywords: harmonic univalent functions; harmonic convex functions; harmonic starlike functions; functions with negative coefficients

Cited in: Zbl 0959.30003

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