Results 41 to 50 of about 74 (58)

The quasihyperbolic metric and associated estimates on the hyperbolic metric

Journal D'Analyse Mathematique, 1986
In this paper the authors present a nice contribution to the theory of the quasihyperbolic metric for domains in \({\mathbb{R}}^ n\). For a domain D in \({\mathbb{R}}^ n\) the quasihyperbolic distance \(k_ D(x,y)\) is obtained from the generalized Riemannian metric \(| dx| /d(x,\partial D)\), where d(x,\(\partial D)\) is the Euclidean distance from x ...
Gaven Martin, Martin Gaven J
exaly   +2 more sources

Old and New on the Quasihyperbolic Metric

1998
Let D be a proper subdomain of \( {\mathbb{R}^d}\). Following Gehring and Palka [GP] we define the quasihyperbolic distance between a pair x 1, x 2 of points in D as the infimum of \( {\smallint _\gamma }\frac{{ds}}{{D\left( {x,\partial D} \right)}}\) over all rectifiable curves γ joining x 1, x 2 in D.
Pekka Koskela, Koskela Pekka
exaly   +2 more sources

John domains and the quasihyperbolic metric

Complex Variables and Elliptic Equations, 1999
David Herron
exaly   +2 more sources

Quasihyperbolic metric and Quasisymmetric mappings in metric spaces

Transactions of the American Mathematical Society, 2015
J. Väisälä has developed the theory of quasiconformal maps in the context of Banach spaces in a series of papers. For an overview of this work see [\textit{J. Väisälä}, Banach Cent. Publ. 48, 55--118 (1999; Zbl 0934.30018)]. The paper under review builds on these ideas. For instance the authors prove that the quasihyperbolic metric is invariant under a
Huang, Xiaojun, Liu, Jinsong
openaire   +2 more sources

Conformal capacity and the quasihyperbolic metric

Indiana University Mathematics Journal, 1996
By examining certain capacity conditions we establish a connection between two classes of domains in \(\mathbb{R}^n\) introduced by Gehring and Martio. We present new characterizations, for uniform domains and for domains which satisfy a quasihyperbolic boundary condition, which are valid in the category of domains quasiconformally equivalent to ...
Herron, David A., Koskela, Pekka
openaire   +2 more sources

Quasiconformal removability and the quasihyperbolic metric

Indiana University Mathematics Journal, 2005
The authors establish an essentially sharp condition sufficient for the \(L^n\)-integrability of the quasihyperbolic metric in a domain \(\Omega\subset\mathbb{R}^n\). As a corollary, they prove a result concerning (quasi)conformal and \(W^{1,n}\)-removability of the boundary \(\partial\Omega\).
Koskela, Pekka, Nieminen, Tomi
openaire   +2 more sources

Domains with growth conditions for the quasihyperbolic metric

Journal d'Analyse Mathématique, 2000
The quasihyperbolic metric \(k_D(x,y)\) is a counterpart of the standard hyperbolic metric on a plane domain on an arbitrary proper subdomain \(D\) of \(\mathbb{R}^n\) [{F. W. Gehring}, \textit{B. P. Palka}, J. Anal. Math. 30, 172-199 (1976; Zbl 0349.30019)]. For \(x\in D\) let \(\delta_D(x)\) be the distance from \(x\) to \(\partial D\).
openaire   +2 more sources

An estimate for the quasiconformality coefficient of a domain via the curvature of its quasihyperbolic metric

Siberian Mathematical Journal, 1999
It is well known that, in the Poincaré model (that is, in the half-space \(x_n >0\)), the metric of the hyperbolic (or Lobachevskij) space is given by the formula \[ ds^2=\frac{dx_1^2 + dx_2^2 +\dots +dx_n^2}{x_n^{2}}. \] Similarly, the metric, defined by the formula \[ ds^2=\frac{dx_1^2 + dx_2^2 +\dots +dx_n^2}{\rho (x)^{2}} \] in a domain \(D\subset \
openaire   +2 more sources

The quasihyperbolic metric, growth, and John domains

1998
The Hardy-Littlewood (H-L) theorem states that if \(f\) is analytic in the unit disk \(B\) and if \[ | f'(z)|\leq m\text{ dist}(z,\partial B)^{\alpha- 1},\tag ...
openaire   +2 more sources

Quasihyperbolic Geodesics are Cone Arcs

Journal of Geometric Analysis, 2023
Saminathan Ponnusamy, Qingshan Zhou, Ponnusamy Saminathan   +2 more
exaly