Univalent harmonic mappings and linearly connected domains

We investigate the relationship between the univalence of f and of h in the decomposition f = h + g of a sense-preserving harmonic mapping defined in the unit disk D ⊂ C. Among other results, we determine the holomorphic univalent maps h for which there exists c > 0 such that every harmonic mapping of the form f = h + g with |g | < c|h | is univalent. The notion of a linearly connected domain appears in our study in a relevant way. © 2006 Elsevier Inc. All rights reserved