First-order univalence criteria, interior chord-arc conditions and quasidisks

Extensions of the Laurent Decomposition and the spaces $A^p(Ω)$

“…There is a C > 0 such that for any two points z, w ∈ Ω there is a curve in Ω connecting them with length less than C|z − w|. This is the so called "interior chord arc condition" ( [1]). 6.…”

“…The spaces C p (∂Ω), 0 ≤ p ≤ +∞, are defined via the parametrization of ∂Ω induced by any Riemann mapping of D onto Ω. In order to extend the results of the preceding section, we place additional hypotheses on the geometry of Ω, such as the interior chord arc condition ( [1]) and the boundedness of the geodesics ( [11]), or on the Riemann mapping φ : D → Ω itself, such as φ ∈ A p (D) and φ ′ (z) 0 for all z ∈ D. Under some of those conditions, we also prove that the spaces A p (Ω) and A p (D) are isomorphic as Banach spaces, for p < +∞, and that A ∞ (Ω) has no norm inducing its natural topology.…”

Stable geometric properties of pluriharmonic and biholomorphic mappings, and Landau–Bloch’s theorem

“…Let f = h + g be a pluriharmonic mapping from B n into C n , where h and g are holomorphic in B n . Then (1) f is called stable pluriharmonic univalent in B n if for every A ∈ L(C n , C n ) with A = 1, the mappings f A = h + gA are univalent in B n .…”

“…It is not difficult to verify that a 1-linearly connected domain is convex. For extensive discussions on linearly connected domains, see [1,5,9,10,19]. In [10], the authors discussed the relationship between linear connectivity of the images of D under the planar harmonic mappings f = h + g and under their corresponding holomorphic counterparts h, where h and g are holomorphic in D. In [11,Theorem 5.3], Clunie and Sheil-Small established an effective and beautiful method of constructing sense-preserving univalent harmonic mappings defined on the unit disk, which is popularly called shear construction.…”

A Class of Weierstrass–Enneper Lifts of Harmonic Mappings

“…In order to prove Theorem 4 we will make use of the following theorem from [1]. See also [7] for a similar application.…”