The distortion of distances between points under maps is studied. We first prove a Schwarz-type lemma for quasiregular maps of the unit disk involving the visual angle metric. Then we investigate conversely the quasiconformality of a bilipschitz map
with respect to the visual angle metric on convex domains. For the unit ball or half space, we prove that a bilipschitz map with respect to the visual angle metric is also bilipschitz with respect to the hyperbolic metric. We also obtain various inequalities relating the visual angle metric to other metrics such as the distance ratio metric and the quasihyperbolic metric.
We prove Schwarz-Pick type estimates and coefficient estimates for a class of elliptic partial differential operators introduced by Olofsson. Then we apply these results to obtain a Landau type theorem.
We give study the Lipschitz continuity of Mobius transformations of a punctured disk onto another punctured disk with respect to the distance ratio metric.
Given a domain $G subsetneq Rn$ we study the quasihyperbolic and the distance ratio metrics of $G$ and their connection to the corresponding metrics of a subdomain $D subset G$. In each case, distances in the subdomain are always larger than in the o
riginal domain. Our goal is to show that, in several cases, one can prove a stronger domain monotonicity statement. We also show that under special hypotheses we have inequalities in the opposite direction.
Suppose that $E$ and $E$ denote real Banach spaces with dimension at least 2, that $Dsubset E$ and $Dsubset E$ are domains, and that $f: Dto D$ is a homeomorphism. In this paper, we prove the following subinvariance property for the class of uniform
domains: Suppose that $f$ is a freely quasiconformal mapping and that $D$ is uniform. Then the image $f(D_1)$ of every uniform subdomain $D_1$ in $D$ under $f$ is still uniform. This result answers an open problem of Vaisala in the affirmative.
Numpy and SciPy are program libraries for the Python scripting language, which apply to a large spectrum of numerical and scientific computing tasks. The Sage project provides a multiplatform software environment which enables one to use, in a unifie
d way, a large number of software components, including Numpy and Scipy, and which has Python as its command language. We review several examples, typical for scientific computation courses, and their solution using these tools in the Sage environment.
A new similarity invariant metric $v_G$ is introduced. The visual angle metric $v_G$ is defined on a domain $GsubsetneqRn$ whose boundary is not a proper subset of a line. We find sharp bounds for $v_G$ in terms of the hyperbolic metric in the partic
ular case when the domain is either the unit ball $Bn$ or the upper half space $Hn$. We also obtain the sharp Lipschitz constant for a Mobius transformation $f: Grightarrow G$ between domains $G$ and $G$ in $Rn$ with respect to the metrics $v_G$ and $v_{G}$. For instance, in the case $G=G=Bn$ the result is sharp.
We prove sharp bounds for the product and the sum of two hyperbolic distances between the opposite sides of hyperbolic Lambert quadrilaterals in the unit disk. Furthermore, we study the images of Lambert quadrilaterals under quasiconformal mappings f
rom the unit disk onto itself and obtain sharp results in this case, too.
We study expansion/contraction properties of some common classes of mappings of the Euclidean space ${mathbb R}^n, nge 2,,$ with respect to the distance ratio metric. The first main case is the behavior of Mobius 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.
