Univalent harmonic mappings and linearly connected domains

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.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],

Quasiconformal harmonic mappings and close-to-convex domains

“…In this short note, by using some results of Clunie and Sheil-Small ([4]) we improve a result by Chuaqui and Hernández ( [2]) and answer a question posed there. In addition, for a given harmonic diffeomorphism (quasiconformal harmonic mapping) we produce a large class of harmonic diffeomorphisms (quasiconformal harmonic mappings).…”

A simple deformation of quasiconformal harmonic mappings in the unit disk

“…3]. We would like to express our sincere thanks to the referee who pointed out the paper [2] and gave us the suggestion to improve the original Corollary 2.2 into the present one.…”

“…It is worth noting here the paper [2] by Chuaqui and Hernández. They studied the relationship between the injectivity of the mappings F and H under the assumption that Ω is a linearly connected domain.…”

Properties of Some Classes of Planar Harmonic and Planar Biharmonic Mappings

“…In a recent paper, Chuaqui and Hernández [4] investigated the relationship between the univalency of planar harmonic mappings f and the linear connectivity of f (D), see for example [4,Theorem 1]. We now state our next result whose proof will be given in Sect.…”