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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 $$\sum^\infty_{n= 2} n^2(a_n+ b_n)\leq 1$$.

##### 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
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##### References:
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