Planar harmonic and quasiconformal mappings

Complex Analysis and Operator Theory, 2011

The aim of this paper is to investigate some properties of planar harmonic and biharmonic mappings. First, we use the Schwarz lemma and the improved estimates for the coefficients of planar harmonic mappings to generalize earlier results related to Landau's constants for harmonic and biharmonic mappings. Second, we obtain a new Landau's Theorem for a certain class of biharmonic mappings. At the end, we derive a relationship between the images of the linear connectivity of the unit disk D under the planar harmonic mappings f = h + g and under their corresponding analytic counterparts F = h − g.

Nonlinear Analysis: Theory, Methods & Applications, 2015

In this paper, we establish a three circles type theorem, involving the harmonic area function, for harmonic mappings. Also, we give bounds for length and area distortion for harmonic quasiconformal mappings. Finally, we will study certain Lipschitz-type spaces on harmonic mappings.

2012

We prove that a harmonic diffeomorphism between two Jordan domains with C 2 boundaries is a (K, K) quasiconformal mapping for some constants K ≥ 1 and K ≥ 0 if and only if it is Lipschitz continuous. In this setting, if the domain is the unit disk and the mapping is normalized by three boundary points condition we give an explicit Lipschitz constant in terms of simple geometric quantities of the Jordan curve which surrounds the codomain and (K, K). The results in this paper generalize and extend several recently obtained results.

Filomat, 2015

Recently G. Alessandrini-V. Nesi and Kalaj generalized a classical result of H. Kneser (RKC-Theorem). Using a new approach we get some new results related to RKC-Theorem and harmonic quasiconformal (HQC) mappings. We also review some results concerning bi-Lipschitz property for HQC-mappings between Lyapunov domains and related results in planar case using some novelty. Following the notation in [1], denote by D f the derivative of f and note that det Df is Jacobian of f .

Computational Methods and Function Theory, 2004

A geometric interpretation of the Schwarzian of a harmonic mapping is given in terms of geodesic curvature on the associated minimal surface, generalizing a classical formula for analytic functions. A formula for curvature of image arcs under harmonic mappings is applied to derive a known result on concavity of the boundary. It is also used to characterize fully convex mappings, which are related to fully starlike mappings through a harmonic analogue of Alexander's theorem.

Journal of Mathematical Analysis and Applications, 2007

We prove versions of the Ahlfors-Schwarz lemma for quasiconformal euclidean harmonic functions and harmonic mappings with respect to the Poincaré metric.

Indagationes Mathematicae

In this article, we determine the radius of univalence of sections of normalized univalent harmonic mappings for which the range is convex (resp. starlike, close-to-convex, convex in one direction). Our result on the radius of univalence of section s n,n (f) is sharp especially when the corresponding mappings have convex range. In this case, each section s n,n (f) is univalent in the disk of radius 1/4 for all n ≥ 2, which may be compared with classical result of Szegö on conformal mappings.

Archiv der Mathematik, 2014

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.

Journal d'Analyse Mathématique, 2003

Computational Methods and Function Theory, 2005

A necessary and sufficient condition is given for a planar harmonic mapping f to be locally decomposable as a univalent harmonic mapping F of an analytic function ϕ.

2009

Page 1. Mapping problems and harmonic univalent mappings Antti Rasila Helsinki University of Technology [email protected] (Mainly based on P. Duren's book Harmonic mappings in the plane) Helsinki Analysis Seminar, 2009-11-16 FILE: plharm3-1.tex, 2009-11-04, printed: 2009-11-16, 17.52 Antti Rasila () Mapping problems and harmonic mappings 2009-11-16 1 / 36 Page 2.

Filomat, 2015

We analyze the properties of harmonic quasiconformal mappings and by comparing some suitably chosen conformal metrics defined in the unit disc we obtain some geometrically motivated inequalities for those mappings (see for instance ). In particular, we obtain the answers to many questions concerning these classes of functions which are related to the determination of different properties that are of essential importance for validity of the results such as those that generalize famous inequalities of the Schwarz-Pick type. The approach used is geometrical in nature, via analyzing the properties of the Gaussian curvature of the conformal metrics we are dealing with. As a consequence of this approach we give a note to the co-Lipschicity of harmonic quasiconformal self mappings of the unit disc at the origin.

The Journal of Analysis, 2020

In this paper, we study a family of sense-preserving harmonic mappings whose analytic part is convex in one direction. We first establish the bounds on the pre-Schwarzian norm. Next, we obtain radius of fully starlike and radius of fully convex for this family of harmonic mappings.

arXiv: Complex Variables, 2019

In this article, we introduce a new family of sense preserving harmonic mappings f in the open unit disk and prove that functions in this family are close-to-convex. We give some basic properties such as coefficient bounds, growth estimates, convolution and determine the radius of convexity for the functions belonging to this family. In addition, we construct certain harmonic univalent polynomials belonging to this family.

Mathematische Zeitschrift, 2008

We present some recent results on the topic of quasiconformal harmonic maps. The main result is that every quasiconformal harmonic mapping w of C 1,µ Jordan domain Ω 1 onto C 1,µ Jordan domain Ω is Lipschitz continuous, which is the property shared with conformal mappings. In addition, if Ω has C 2,µ boundary, then w is bi-Lipschitz continuous. These results have been considered by the authors in various ways.