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The main purpose of this note is to show a certain connection between quasiconformal and quasimobius maps. We prove that quasimobius maps are the natural boundary maps for quasiconformal maps from the upper half plane onto itself as well as for quasiconformal maps from the unit disc onto itself. In the first case we can remove the rather unpleasant condition that infinity is fixed. We think that the most interesting aspect of our observations is that the results concerning the unit disk follow by very simple arguments from the corresponding results on half planes by the use of quasimobius maps. Similar results concerning the disk are earlier proved by Krzyz (Quasicircles and harmonic measure. Ann Acad Sci Fenn. 12:19–24, 1987) and Douady and Earle (Conformally natural extension of homeomorphisms of the circle. Acta Math 157:23–48, 1986). However, using properties of quasimobius maps proved by Vaisala (Quasimobius maps. J Anal Math 44:218–234, 1984), Tukia and Vaisala (Quasisymmetric embeddings of metric spaces. Ann Acad Sci Fenn Ser A I Math 5:97–114, 1980) and Heinonen (Lectures on analysis on metric spaces. Berlin: Springer, 2001) we are able to prove some of these results in a much simpler way. In this respect our approach to these problems is more natural than the approach in Douady and Earle (Conformally natural extension of homeomorphisms of the circle. Acta Math 157:23–48, 1986) and Krzyz (Quasicircles and harmonic measure. Ann Acad Sci Fenn. 12:19–24, 1987). [See also Pommerenke (Boundary behaviour of conformal maps. Berlin: Springer, 1992)]. To the best of our knowledge the situation for the quasimobius maps is not considered in the literature. In a paper by Bonk et al. (Rigidity of Schottky sets. Am J Math 131(2):409–443, 2009) related results in higher dimensions are proved in Section 4.

We present a detailed and self-contained proof of the harmonic measure doubling characterization of bounded quasidisks due to Jerison and Kenig.

We give here a pair of characterizations for a euclidean disk D which are concerned with the hyperbolic geometry in D and in domains which contain D.

Introduction. This paper should be considered as a companion report to F.W. Gehring’s survey lectures ”Characterizations of quasidisks” given at this Summer School [7]. Notation, definitions and background results are given in that paper. In particular, D is a simply connected proper subdomain of R unless otherwise stated and D∗ denotes the exterior of D in R 2 . Many of the characterizations of quasidisks have been motivated by looking at properties of euclidean disks. It is therefore natural to go back and ask if any of the original properties in fact characterize euclidean disks. We follow the procedure in Gehring’s lectures and look at four different categories of properties

We study the geometry of plane domains and the uniform Holder continuity properties of analytic functions.

The purpose of this Round Table was to address the underrepresentation of women in mathematics. We wanted in particular to focus on how differences in culture and tradition affect this situation since the proportion of women in mathematics varies considerably from country to country.

We prove that a limitärperiodisch T-fraction, which corresponds to a rational function, converges (locally uniformly) to the original function in a certain domain.

We prove that a limitarperiodisch T-fraction, which corresponds to a rational function, has the property that — 1.