Subclass of Multivalent Harmonic Functions with Missing Coefficients

A continuous function f u iv is a complex-valued harmonic function in a simply connected complex domain D ⊂ C if both u and v are real harmonic in D. It was shown by Clunie and Sheil-Small 1 that such harmonic function can be represented by f h g, where h and g are analytic in D. Also, a necessary and sufficient condition for f to be locally univalent and sense preserving in D is that |h′ z | > |g ′ z | see also, 2–4 . Denote by H the family of functions f h g, which are harmonic univalent and sense-preserving in the open-unit discU {z ∈ C : |z| < 1}with normalization f 0 h 0 f ′ z 0 − 1 0. For m ≥ 1, 0 ≤ β < 1, and γ ≥ 0, let R m, β, γ denote the class of all multivalent harmonic functions f h g with missing coefficients that are sense-preserving in U, and h, g are of the form