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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.

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