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Harmonic univalent functions with negative coefficients. (English) Zbl 0908.30013
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 \(\sum^\infty_{n= 2} n^2(a_n+ b_n)\leq 1\).

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
Full Text: DOI
[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
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