Univalent harmonic mappings and linearly connected domains

We investigate the relationship between the univalence of f and of h in the decomposition f = h + over(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 + over(g, -) with | g^′ | < c | h^′ | is univalent. The notion of a linearly connected domain appears in our study in a relevant way.