Harmonic close-to-convex mappings
Let \(S_H\) denote the class of functions \(f= h+\overline g\) that are harmonic univalent in the unit disc \(\Delta= \{z:|z|< 1\}\) for which \(h\) and \(g\) have the forms \[ h(z)= z+\sum^\infty_{n=2} a_n z^n;\quad g(z)= \sum^\infty_{n=1} b_n z^n. \] In this paper the authors give some sufficient conditions for functions in \(S_H\) to be close-to-convex harmonic, or convex harmonic in \(\Delta\). Some convolution properties for harmonic functions are also investigated. For example it is proved: Theorem. If \(f= h+\overline g\) and \(\sum^\infty_{n=2} n|a_n|+ \sum^\infty_{n=1} n|b_n|\leq 1\), then \(f\) is convex harmonic in \(\Delta\).
scientific article; zbMATH DE number 1777330