de Branges spaces with bi-Lipschitz phase for large distances

Lipschitz continuity of quasiconformal mappings and of the solutions to second order elliptic PDE with respect to the distance ratio metric

The main aim of this paper is to study the Lipschitz continuity of certain $(K, K')$-quasiconformal mappings with respect to the distance ratio metric, and the Lipschitz continuity of the solution of a quasilinear differential equation with respect to the distance ratio metric.

Sharp Lipschitz constants for the distance ratio metric

We study expansion/contraction properties of some common classes of mappings of the Euclidean space ${\mathbb R}^n, n\ge 2\,,$ with respect to the distance ratio metric. The first main case is the behavior of Möbius transformations of the unit ball in ${\mathbb R}^n$ onto itself. In the second main case we study the polynomials of the unit disk onto a subdomain of the complex plane. In both cases sharp Lipschitz constants are obtained.

Lipschitz conditions and the distance ratio metric

We give study the Lipschitz continuity of Möbius transformations of a punctured disk onto another punctured disk with respect to the distance ratio metric.

度量空间之间映射的距离函数与Lipschitz类

The distance function and Lipschitz classes of mappings between metric spaces

本文研究度量空间之间映射的局部Lipschitz性质与全局Lipschitz性质的关系，利用距离函数和Hardy-Littlewood定理的度量空间版本，将Dyakonov关于单位圆盘解析函数的Lipschitz类刻画推广到赋范空间的一致域上。

On the hyperbolic distance of $n$-times punctured spheres

The length of the shortest closed geodesic in a hyperbolic surface $X$ is called the systole of $X.$ When $X$ is an $n$-times punctured sphere $\hat{ \mathbb{C}} \setminus A$ where $A \subset \hat{\mathbb{C}}$ is a finite set of cardinality $n\ge4,$ we define a quantity $Q(A)$ in terms of cross ratios of quadruples in $A$ so that $Q(A)$ is quantitatively comparable with the systole of $X.$ We next propose a method to construct a distance function $d_X$ on a punctured sphere $X$ which is Lipschitz equivalent to the hyperbolic distance $h_X$ on $X.$ In particular, when the construction is based …

Hyperbolic Distance versus Quasihyperbolic Distance in Plane Domains

We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct an example where the spaces are not quasiisometrically equivalent.