Silverman, Herb Harmonic univalent functions with negative coefficients. (English) Zbl 0908.30013 J. Math. Anal. Appl. 220, No. 1, 283-289 (1998). 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 \(f_{\overline z}(0)= 0\). 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) \(h(z)= z- \sum^\infty_{n= 2} a_nz^n\), \(a_n\geq 0\); \(g(z)= -\sum^\infty_{n=2} b_nz^n\), \(b_n\geq 0\), \(z\in\Delta\). 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 \(\sum^\infty_{n= 2}n(| a_n|+| b_n|)\leq 1\), then \(f\in S^{*0}_H\). Corollary 1. If \(f\) of the form (1-2) satisfies \(\sum^\infty_{n= 2} n^2(| a_n|+ | b_n|)\leq 1\), then \(f\in K^0_H\). Theorem 2. For \(f\) of the form (1), (3), \(f\in T^{*0}_H\) if and only if \(\sum^\infty_{n= 2} n(a_n+ b_n)\leq 1\). Theorem 3. For \(f\) of the form (1), (3), \(f\in TK^0_H\) if and only if \(\sum^\infty_{n= 2} n^2(a_n+ b_n)\leq 1\). Reviewer: Z.J.Jakubowski (Łódź) Cited in 1 ReviewCited in 40 Documents MSC: 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions Keywords:harmonic univalent functions; harmonic convex functions; harmonic starlike functions; functions with negative coefficients PDF BibTeX XML Cite \textit{H. Silverman}, J. Math. Anal. Appl. 220, No. 1, 283--289 (1998; Zbl 0908.30013) Full Text: DOI References: [1] Cima, J.A.; Livingston, A.E., Integral smoothness properties of some harmonic mappings, Complex variables, 11, 95-110, (1989) · Zbl 0724.30011 [2] Clunie, J.; Sheil-Small, T., Harmonic univalent functions, Ann. acad. sci. fenn. ser. A I math., 9, 3-25, (1984) · Zbl 0506.30007 [3] Sheil-Small, T., Constants for planar harmonic mappings, J. London math. soc. (2), 42, 237-248, (1990) · Zbl 0731.30012 [4] Silverman, H., Univalent functions with negative coefficients, Proc. amer. math. soc., 51, 109-116, (1975) · Zbl 0311.30007 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.