Some results on a subclass of harmonic mappings of order alpha

Abstract

Let SH be the class of harmonic mappings defined by SH = { f = h(z) + g(z)| h(z) = z + ∑∞ n=2 anzⁿ, g(z) = b1z + ∑∞ n=2 bnzⁿ, b1 < 1 } where h(z) and g(z) are analytic. Additionally f(z) ∈ SH(α) ⇔ | zh′ (z) − zg′(z) h(z) + g(z) − 1 − b1 1 + b1| < | 1 − b1 1 + b1| − α, z ∈ U, 0 ≤ α < 1 − b1 1 + b1 In the present work, by considering the analyticity of the functions defined by R. M. Robinson [7], we discuss the applications to the harmonic mappings.