Harmonic Quasiconformal Mappings Research Articles

Quasiconformal solutions to elliptic partial differential equations

In this paper, we assume that \(G\) and \(\Omega\) are two Jordan domains in \(\mathbb{R}^n\) with \(\mathcal{C}^2\) boundaries, where \(n\ge 2\), and prove that every quasiconformal mapping \(f\in\mathcal{W}^{2,1+\epsilon}_{\mathrm{loc}}\) of \(G\) onto \(\Omega\), satisfying the elliptic partial differential inequality \(|L_ A[f]|\lesssim (\|Df\|^2+|g|)\), with \(g\in\mathcal{L}^p(G)\), where \(p>n\), is Lipschitz continuous. The result is sharp since for \(p=n\), the mapping \(f\) is not necessarily Lipschitz continuous. This extends several results for harmonic quasiconformal mappings.

Hölder continuity of quasiconformal harmonic mappings from the unit ball to a spatial domain with [formula omitted] boundary

We prove that every quasiconformal mapping from the harmonic β-Bloch space between the unit ball and a spatial domain with C1 boundary is globally α-Hölder continuous for α<1−β, with the Hölder coefficient that does not depend neither on the mapping nor on β. An analogous result also holds for Lipschitz continuous, quasiconformal harmonic mappings for α<1. This is an approach towards the extension of some results from the complex plane obtained by Warschawski (1951) for conformal mappings and Kalaj (2022) for quasiconformal harmonic mappings.

Erratum to “Harmonic quasiconformal mappings between $\mathcal{C}^1$ smooth Jordan domains”

Erratum to “Harmonic quasiconformal mappings between $\mathcal{C}^1$ smooth Jordan domains”

Hyperbolically Lipschitz Continuity, Area Distortion, and Coefficient Estimates for (K, K′)-Quasiconformal Harmonic Mappings of the Unit Disk

Hyperbolically Lipschitz Continuity, Area Distortion, and Coefficient Estimates for (K, K′)-Quasiconformal Harmonic Mappings of the Unit Disk

Quasiconformal Harmonic Mappings Between the Unit Ball and a Spatial Domain with C1,α Boundary

We prove the following. If f is a harmonic quasiconformal mapping between the unit ball in $\mathbb {R}^{n}$ and a spatial domain with C1,α boundary, then f is Lipschitz continuous in B. This generalizes some known results for n = 2 and improves some others in higher dimensional case.

Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for -quasiconformal harmonic mappings of unit disk

UDC 517.51 We study the hyperbolically Lipschitz continuity, Euclidean and hyperbolic area distortion theorem, and coefficient estimate for the classes of -quasiconformal harmonic mappings from the unit disk onto itself.

Bohr Phenomenon for Locally Univalent Functions and Logarithmic Power Series

In this article we prove Bohr inequalities for sense-preserving $K$-quasiconformal harmonic mappings defined in $\mathbb{D}$ and obtain the corresponding results for sense-preserving harmonic mappings by letting $K\to\infty$. One of the results includes the sharpened version of a theorem by Kayumov $\textit{et. al.}$ ($\textit{Math. Nachr.}$, 291 (2018), no. 11--12, 1757--1768). In addition Bohr inequalities have been established for uniformly locally univalent holomorphic functions, and for $\log(f(z)/z)$ where $f$ is univalent or inverse of a univalent function.

Erratum to "Quasiconformal harmonic mappings with the convex holomorphic part"

Erratum to "Quasiconformal harmonic mappings with the convex holomorphic part"

Quasiconformal harmonic mappings with the convex holomorphic part

Quasiconformal harmonic mappings with the convex holomorphic part

Quasiconformal harmonic mappings with unbounded gradient

Let $f=h+\bar{g}$ be a harmonic mapping of the unit disk $\mathbf{U}$ ontoa convex domain, where $h$ and $g$ are analytic in $\mathbf{U}$ and $g(0)=0$. In this paper,we first find the equivalent description for $\Lambda_f$, the gradient of $f$,which has the control growth function $\frac{1}{(1-|z|)^\alpha}$, where $z\in\mathbf{U}$ and $0\leq\alpha\leq1$.Then, we obtain the sufficient conditions for a $v$-Bloch harmonic mapping to be quasiconformal.Moreover, if $f(\mathbf{U})$ is a general domain (which may not be convex) and $h$ is convex, then we give the necessary andsufficient conditions for $f$ to be quasiconformal.

Bi-Lipschitzity of quasiconformal harmonic mappings in n-dimensional space with respect to k-metric

We explore conditions which guarantee bi-Lipschitzity of harmonic quasiconformal maps with respect to k-metric. We prove that harmonic k-quasiconformal maps with nonzero Jacobian between any two domains in Rn are bi-Lipschitz with respect to k-metric, and prove the converse too.

Quasiconformal Harmonic mappings and the curvature of the boundary

We estimate the Jacobian of harmonic mapping of the unit disk onto a smooth and convex Jordan domain via the boundary function and the boundary curvature of the image domain. By using this result we make some asymptotically sharp estimates of constant of quasiconformality for harmonic diffeomorphisms between the unit disk and the convex domains via their boundary mappings.

Linear combinations of harmonic quasiconformal mappings convex in one direction

In this paper, we introduce a new class $\mathscr{S}_{H} (k, γ; \phi)$ of harmonic quasiconformal mappings, where $k \in [0,1), γ \in [0,π)$ and $\phi$ is an analytic function. Sufficient conditions for the linear combinations of mappings in such classes to be in a similar class, and convex in a given direction, are established. In particular, we prove that the images of linear combinations in this class, for special choices of $γ$ and $\phi$, are convex.

Quasiconformal harmonic mappings between Dini-smooth Jordan domains

Quasiconformal harmonic mappings between Dini-smooth Jordan domains

Muckenhoupt weights and Lindelöf theorem for harmonic mappings

We extend the result of Lavrentiev which asserts that the harmonic measure and the arc-length measure are A∞ equivalent in a chord-arc Jordan domain. By using this result we extend the classical result of Lindelöf to the class of quasiconformal (q.c.) harmonic mappings by proving the following assertion. Assume that f is a quasiconformal harmonic mapping of the unit disk U onto a Jordan domain. Then the function A(z)=arg⁡(∂φ(f(z))/z) where z=reiφ, is well-defined and smooth in U⁎={z:0<|z|<1} and has a continuous extension to the boundary of the unit disk if and only if the image domain has C1 boundary.

A note on the harmonic quasiconformal diffeomorphisms of the unit disc

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 [15, 17, 20]). 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.

Invertible harmonic and harmonic quasiconformal mappings

Recently G. Alessandrini - V. Nesi and Kalaj generalized a classical result of H. Kneser (RKCTheorem). 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.

Estimation of hyperbolically partial derivatives of ρ-harmonic quasiconformal mappings and its applications

In this paper, for a general metric density , we build a differential inequality to study the hyperbolically partial derivative of -harmonic quasiconformal mappings and generalize a result given by Kne z̆ević and Mateljević. As one application, we obtain the hyperbolically ()-biLipschitz continuity of such class of mappings. As another application, we generalize the classical Koebe Theorem to this class of mappings and use this result to study its quasihyperbolically biLipschitz continuity.

Harmonic quasiconformal mappings on the upper half-plane

We study the analytic characteristic property for the sense-preserving univalent harmonic mappings on the upper half-plane onto itself. Using the representing formula for positive harmonic mappings on the upper half-plane, one necessary and sufficient condition for sense-preserving univalent harmonic mappings on the upper half-plane onto itself to be harmonic quasiconformal mappings is obtained. As an application, by the property of the two-sided univalent harmonic mappings on the upper half-plane onto itself, we prove that the set of univalent harmonic quasiconformal mappings on the upper half-plane onto itself with respect to composition is not a group.

A class of harmonic quasiconformal mappings with strongly hyperbolically convex ranges

In this article we study a class of harmonic quasiconformal mappings with strongly hyperbolically convex ranges. After establishing a differential inequality for strongly hyperbolically convex domains, we show that their hyperbolically partial derivatives have explicit bounds determined by the constant of quasiconformality. As an application we show that they are hyperbolically Lipschitz with explicit coefficients given by the constant of quasiconformality.