Lipschitz constants for a hyperbolic type metric under Möbius transformations

Yinping Wu; Gendi Wang; Gaili Jia; Xiaohui Zhang

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 2, page 445-460
  • ISSN: 0011-4642

Abstract

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Let D be a nonempty open set in a metric space ( X , d ) with ∂ D ≠ ∅ . Define h D , c ( x , y ) = log 1 + c d ( x , y ) d D ( x ) d D ( y ) , where d D ( x ) = d ( x , ∂ D ) is the distance from x to the boundary of D . For every c ≥ 2 , h D , c is a metric. We study the sharp Lipschitz constants for the metric h D , c under Möbius transformations of the unit ball, the upper half space, and the punctured unit ball.

How to cite

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Wu, Yinping, et al. "Lipschitz constants for a hyperbolic type metric under Möbius transformations." Czechoslovak Mathematical Journal 74.2 (2024): 445-460. <http://eudml.org/doc/299486>.

@article{Wu2024,
abstract = {Let $D$ be a nonempty open set in a metric space $(X,d)$ with $\partial D\ne \emptyset $. Define \[ h\_\{D,c\}(x,y)=\log \bigg (1+c\frac\{d(x,y)\}\{\sqrt\{d\_D(x)d\_D(y)\}\}\bigg ), \] where $d_D(x)=d(x,\partial D)$ is the distance from $x$ to the boundary of $D$. For every $c\ge 2$, $h_\{D,c\}$ is a metric. We study the sharp Lipschitz constants for the metric $h_\{D,c\}$ under Möbius transformations of the unit ball, the upper half space, and the punctured unit ball.},
author = {Wu, Yinping, Wang, Gendi, Jia, Gaili, Zhang, Xiaohui},
journal = {Czechoslovak Mathematical Journal},
keywords = {Lipschitz constant; hyperbolic type metric; Möbius transformation},
language = {eng},
number = {2},
pages = {445-460},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Lipschitz constants for a hyperbolic type metric under Möbius transformations},
url = {http://eudml.org/doc/299486},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Wu, Yinping
AU - Wang, Gendi
AU - Jia, Gaili
AU - Zhang, Xiaohui
TI - Lipschitz constants for a hyperbolic type metric under Möbius transformations
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 2
SP - 445
EP - 460
AB - Let $D$ be a nonempty open set in a metric space $(X,d)$ with $\partial D\ne \emptyset $. Define \[ h_{D,c}(x,y)=\log \bigg (1+c\frac{d(x,y)}{\sqrt{d_D(x)d_D(y)}}\bigg ), \] where $d_D(x)=d(x,\partial D)$ is the distance from $x$ to the boundary of $D$. For every $c\ge 2$, $h_{D,c}$ is a metric. We study the sharp Lipschitz constants for the metric $h_{D,c}$ under Möbius transformations of the unit ball, the upper half space, and the punctured unit ball.
LA - eng
KW - Lipschitz constant; hyperbolic type metric; Möbius transformation
UR - http://eudml.org/doc/299486
ER -

References

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