Abstract
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.
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