×

Salagean-type harmonic univalent functions. (English) Zbl 1021.30013

Assume the following notations: \[ f(z)= z+ \sum_{m=2}^\infty a_mz^m,\;g(z)= \sum_{m=1}^\infty b_mz^m, \]
\[ f(z)= h(z)+ \overline{g(z)};\;D^nh(z)= z+ \sum_{m=2}^\infty m^n a_mz^m, \]
\[ D^ng(z)= \sum_{m=1}^\infty m^n b_mz^m,\quad D^nf(z)= D^nh(z)+ (-1)^n \overline{D^ng(z)},\;|z|< 1. \] The authors define a new class \(H(n,\alpha)\) of harmonic mappings of the unit disk by imposing on \(f\) the following condition: Re\(\{ \frac {D^{n+1}f(z)} {D^nf(z)}\}> \alpha\), \(0\leq \alpha< 1\). They give a sufficient condition for \(f\) to be in \(H(n,\alpha)\), which is also necessary provided the coefficients \(a_m\), \(b_m\) are all negative, and investigate properties of such functions. The case \(\alpha=0\) was considered earlier by A. Ganczar [Demonstr. Math. 34, 549-558 (2001; Zbl 0988.30009)].

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable

Citations:

Zbl 0988.30009
PDFBibTeX XMLCite
Full Text: EuDML EMIS