Precise Coefficient Estimates for Close-to-Convex Harmonic Univalent Mappings

Abstract

The class SH consists of harmonic, univalent, and sense-preserving functions f in the open unit disk U={z:|z|<1}, such that f=h+g, where h(z)=z+∑n=2∞anzn and g(z)=∑n=1∞a−nzn. Let S0H, CH, and C0H denote the subclass of SH with a−1=0, the subclass of SH with f being a close-to-convex mapping, and the intersection of S0H and CH, respectively. In this paper, for f∈C0H and f∈CH, we prove that the harmonic analogue of the Bieberbach conjecture and the generalization of the Bieberbach conjecture are true.