On a Subclass of Harmonic Convex Functions of Complex Order

A continuous function f u iv is a complex-valued harmonic function in a complex domain Ω if both u and v are real and harmonic inΩ. In any simply connected domainD ⊂ Ω, we can write f h g, where h and g are analytic inD. We call h the analytic part and g the coanalytic part of f . A necessary and sufficient condition for f to be locally univalent and orientation preserving in D is that |h′ z | > |g ′ z | in D see 1 . Denote by SH the family of functions f h g, which are harmonic, univalent, and orientation preserving in the open unit discU {z : |z| < 1} so that f is normalized by f 0 h 0 fz 0 − 1 0. Thus, for f h g ∈ SH, the functions h and g analytic in U can be expressed in the following forms: