On a subclass of harmonic convex functions of complex order (Q436530)

Assume that NEWLINE\[NEWLINEh(z)= z+\sum^\infty_{n=2} a_n z^n,\quad g(z)= \sum^\infty_{n=1} b_n z^n,\;|b_1|< 1,\;|z|< 1.NEWLINE\]NEWLINE Functions of the form \(f(z)= h(z)+ \overline{g(z)}\), univalent and orientation preserving on the unit disk \(|z|< 1\), are called harmonic mappings and their class is denoted by \(S_H\). The authors impose an additional condition on the functions of the form \(\text{Re}\{F(z,f(z),b,\lambda)\}>\gamma\), being a generalization of the inequality \(\text{Re}\{1+ b^{-1}{zh''(z)\over h'(z)}\}> \gamma\), and define a class \(SC_H(b,\gamma, \lambda)\subset S_H\) of convex functions of complex order. For special choices of the parameters their class reduces to some earlier ones considered by other authors. They give, in terms of the coefficients \(a_n\), \(b_n\), a sufficient condition for \(f\in SC_H(b,\gamma,\lambda)\) (which in the case \(a_n\leq 0\), \(b_n\geq 0\) is also necessary) and investigate various properties of the functions.