Precise coefficient estimates for close-to-convex harmonic univalent mappings

The class S-H consists of harmonic, univalent, and sense-preserving functio ns f in the open unit disk U = {z : \z \ < 1}, such that f = h + (g) over b ar, where h(z) = z + Sigma (infinity)(n=2) a(n)z(n) and Sigma (infinity)(n= 1) = a(-n)z(n). Let S-H(0), CH, and C-H(0) denote the subclass of S-H with a(-1) = 0, the subclass of S-H with f being a close-to-convex mapping, and the intersection of S-H(0), and C-H, respectively. In this paper, for f is an element of C-H(0) and f is an element of C-H, we prove that the harmonic analogue of the Bieberbach conjecture and the generalization of the Bieber bach conjecture are true. (C) 2001 Academic Press.