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2010, Journal of Inequalities and Applications
The purpose of this paper is to explore conditions which guarantee Lipschitz-continuity of harmonic maps w.r.t. quasihyperbolic metrics. For instance, we prove that harmonic quasiconformal maps are Lipschitz w.r.t. quasihyperbolic metrics.
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.
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 .
Annales Academiae Scientiarum Fennicae Mathematica, 2010
We obtain a sharp estimate of the derivatives of harmonic quasiconformal extension u = P [φ] of a Lipschitz map φ : S n−1 → R n . We also consider additional conditions which provide that u is Lipschitz on the unit ball; in particular, we give characterizations of Lipschitz continuity of u in the planar case and in the upper half space setting. We also answer a question posed by Martio in [OM] and extend this to the case of several variables. + = {(x, y) : x ∈ R n , y > 0} in Theorem 3.
Journal of Inequalities and Applications, 2011
We prove that for harmonic quasiconformal mappings α-Hölder continuity on the boundary implies α-Hölder continuity of the map itself. Our result holds for the class of uniformly perfect bounded domains, in fact we can allow that a portion of the boundary is thin in the sense of capacity. The problem for general bounded domains remains open.
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.
2010
It is established a series of criteria for continuous and homeomorphic extension to the boundary of the so-called lower Q-homeomorphisms f between domains in R n = R n ∪ {∞}, n 2, under integral constraints of the type Φ(Q n−1 (x)) dm(x) < ∞ with a convex non-decreasing function Φ : [0, ∞] → [0, ∞]. It is shown that integral conditions on the function Φ found by us are not only sufficient but also necessary for a continuous extension of f to the boundary. It is given also applications of the obtained results to the mappings with finite area distortion and, in particular, to finitely bi-Lipschitz mappings that are a far reaching generalization of isometries as well as quasiisometries in R n. In particular, it is obtained a generalization and strengthening of the well-known theorem by Gehring-Martio on a homeomorphic extension to boundaries of quasiconformal mappings between QED (quasiextremal distance) domains.
We show two versions of the Koebe theorem: one for quasiregular harmonic functions and another for quasiconformal functions. We also give an elementary proof of a version of the Koebe one-quarter theorem for holomorphic functions. As an application, we show the harmonic analogue of the Koebe one-quarter theorem and that holomorphic functions (more generally, quasiregular harmonic functions) and their modulus have similar behaviour in a certain sense.
Journal d'Analyse Mathématique, 2011
It is established a series of criteria for continuous and homeomorphic extension to the boundary of the so-called lower Q-homeomorphisms f between domains in R n = R n ∪ {∞}, n 2, under integral constraints of the type Φ(Q n−1 (x)) dm(x) < ∞ with a convex non-decreasing function
Annales Academiae Scientiarum Fennicae Series A I Mathematica, 1969
Conformal Geometry and Dynamics of the American Mathematical Society, 2008
We prove that quasiconformal maps onto domains satisfying a suitable growth condition on the quasihyperbolic metric are uniformly continuous even when both domains are equipped with internal metric. The improvement over previous results is that the internal metric can be used also in the image domain. We also extend this result for conformal deformations of the euclidean metric on the unit ball of R n .
2011
An improved version of quasiinvariance property of the quasihyperbolic metric under Möbius transformations of the unit ball in R n , n ≥ 2, is given. Next, a quasiinvariance property, sharp in a local sense, of the quasihyperbolic metric under quasiconformal mappings is proved. Finally, several inequalities between the quasihyperbolic metric and other commonly used metrics such as the hyperbolic metric of the unit ball and the chordal metric are established.
2010
We prove in dimension n > 2 that a K-quasiconformal harmonic mapping u of the unit ball B n onto itself is Euclidean bi-Lipschitz if u(0) = 0 and K < 2 n−1. This is an extension of a similar result of Tam and Wan for hyperbolic harmonic mappings with respect to a hyperbolic metric. The proof uses Möbius transformations on the related space and a recent result of the first author, which states that harmonic quasiconformal self-mappings of the unit ball are Lipschitz continuous.
Journal d'Analyse Mathématique, 2006
We prove a version of "inner estimate" for quasi-conformal diffeomorphisms, which is satisfying certain estimate concerning their laplacian. As an application of this estimate, we show that quasiconformal harmonic mappings between smooth domains (with respect to the approximately analytic metric), have bounded partial derivatives; in particular, these mappings are Lipschitz.
Annales Academiae Scientiarum Fennicae Mathematica, 2013
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.
Journal d'Analyse Mathématique, 1986
In this paper we investigate the geometry of the quasihyperbolic metric of domains in R". This metric arises from the conformally flat generalized Riemannian metric d(x, aD)-'ldx I. Due to the fact that the density cl(x, aD) -t is not necessarily differentiable, the classical theories of Riemannian geometry do not apply to this metric. The quasihyperbolic metric has been found to have many interesting and varied applications in geometric function theory. In particular, quasiconformal mappings are quasi-isometries of this metric for sufficiently far-lying points, also bounds on the quasihyperbolic metric in terms of other metrics imply that a domain is uniform which then implies certain injectivity criteria for locally-Lipschitz mappings, amongst others. In fact there is quite a strong relationship between uniform domains and the quasihyperbolic metric. Most of these basic results on the quasihyperbolic metric can be found in [3], [2] and . We note here that the quasihyperbolic metric is complete and generates the usual topology on a proper subdomain of R". Further, geodesics (length minimizing curves) always exist for this metric and these geodesics have Lipschitz continuous first derivatives, which is in fact best possible.
Acta Mathematica, 1998
Annales Academiae Scientiarum Fennicae Mathematica, 2015
The triangular ratio metric is studied in subdomains of the complex plane and Euclidean n-space. Various inequalities are proven for it. The main results deal with the behavior of this metric under quasiconformal maps. We also study the smoothness of metric disks with small radii.
Journal of the Australian Mathematical Society, 2014
Suppose that E and E ′ denote real Banach spaces with dimension at least 2 and that D ⊂ E and D ′ ⊂ E ′ are domains. In this paper, we establish, in terms of the j D metric, a necessary and sufficient condition for the homeomorphism f : E → E ′ to be FQC. Moreover, we give, in terms of the j D metric, a sufficient condition for the homeomorphism f : D → D ′ to be FQC. On the other hand, we show that this condition is not necessary.
Journal d'Analyse Mathématique, 2009
We study the local growth of quasiconformal mappings in the plane. Estimates are given in terms of integral means of the pointwise angular dilatations. New sufficient conditions for a quasiconformal mapping f to be either Lipschitz or weakly Lipschitz continuous at a point are given.
2016
The triangular ratio metric is studied in subdomains of the complex plane and Euclidean n-space. Various inequalities are proven for this metric. The main results deal with the behavior of this metric under quasiconformal maps. We also study the smoothness of metric disks with small radii.
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