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Subclasses of harmonic univalent functions. (English) Zbl 0959.30003
Denote by $${\mathcal S}_H$$ 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 $$f(0)=f_z(0) -1=0$$; $${\mathcal S}^0_H$$ – the subclass of $${\mathcal S}_H$$ for which $$f_{\overline z}(0)=0$$; $${\mathcal S}^*_H$$ and $${\mathcal K}_H$$ – the subclasses of $$S_H$$ consisting of functions $$f$$ that map $$\Delta$$ onto starlike and convex domains, respectively. The authors consider the classes $${\mathcal T}_H^*$$ and $${\mathcal T}{\mathcal K}_H$$ that are subclasses of $${\mathcal S}^*_H$$ and $${\mathcal K}_H$$, respectively, for which the coefficients of functions $$f=h+\overline g$$ take the form $h(z)=z -\sum^\infty_{n=2} a_nz^n,\;g(z)= -\sum^\infty_{n=1} b_nz^n,\;0\leq b_1<1;\;a_n,b_n\geq 0,\;n=2,3,\dots.$ In [H. Silverman, J. Math. Anal. Appl. 220, No. 1, 283-289 (1998; Zbl 0908.30013)], the families $${\mathcal T}_H^{*0} ={\mathcal T}^*_H \cap{\mathcal S}^0_H$$ and $${\mathcal T}{\mathcal K}^0_H ={\mathcal T}{\mathcal K}^0_H \cap{\mathcal S}^0_H$$ where investigated. In this note the authors generalize results for $${\mathcal T}^{*0}_H$$ to the classes $${\mathcal T}^*_H$$ and $${\mathcal T}{\mathcal K}_H$$ and also consider the family $${\mathcal T}_H^*(b)$$, $$0\leq b<1$$, consisting of functions $$f\in{\mathcal T}^*_H$$ for which $$b_1=b$$ is fixed.

##### 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