In this paper we introduce a definition of the pre-Schwarzian and the Schwarzian derivatives of any locally univalent harmonic mapping f in the complex plane without assuming any additional condition on the (second complex) dilatation ω f of f . Using the new definition for the Schwarzian derivative of harmonic mappings, we prove analogous theorems to those by Chuaqui, Duren, and Osgood. Also, we obtain a Becker-type criterion for the univalence of harmonic mappings.

Given any sense preserving harmonic mapping f=h+ḡ in the unit disk, we prove that for all |λ|=1 the functions fλ=h+λḡ are univalent (resp. close-to-convex, starlike, or convex) if and only if the analytic functions Fλ=h+λg are univalent (resp. close-to-convex, starlike, or convex) for all such λ. We also obtain certain necessary geometric conditions on h in order that the functions fλ belong to the families mentioned above. In particular, we see that if fλ are univalent for all λ on the unit circle, then h is univalent.

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.A planar harmonic mapping is a complex-valued harmonic function f (z), z = x + iy, defined on some domain Ω ⊂ C. When Ω is simply connected, the mapping has a canonical decomposition f = h + g, where h and g are analytic in Ω. Since the Jacobian of f is given by |h | 2 − |g | 2 , it is locally univalent and orientation-preserving if and only if |g | < |h |, or equivalently if h (z) = 0 and the dilatation ω = g /h has the property |ω(z)| < 1 in Ω.Fundamental questions regarding univalent harmonic mappings are still to be resolved, including important coefficient estimates, and the exact nature of the analogue of the Riemann mapping theorem. There are beautiful results such as the shear construction of Clunie and Sheil-Small [2],

Abstract. We prove that if the Schwarzian norm of a given complex-valued locally univalent harmonic mapping f in the unit disk is small enough, then f is, indeed, globally univalent and can be extended to a quasiconformal mapping in the extended complex plane.

Abstract. Let f be a sense-preserving harmonic mapping in the unit disk. We give a sufficient condition in terms of the pre-Schwarzian derivative of f to ensure that it can be extended to a quasiconformal map in the complex plane.