Raimo Näkki

Analytic functions and quasiconformal mappings in stolz angles and cones

Let f be analytic and bounded in the unit disk B 2 or quasiconformal in the unit ball Bn of , and let z 0 be a boundary point of B 2 or Bn . We assume that for a suitable majornat μ and for all z on ∂ B 2 when f is analytic, or for all z on an from 0 to z 0 when f is quasiconforml. We then obtain estimates for when z 1 and z 2 lie in a Stolz angle or cone with vertex at z 0.

Boundary angles, cusps and conformal mappings

Let f be a conformal mapping of a bounded Jordan domain D in the complex plane onto the unit disk . This paper examines the consequences for the local geometry of D near a boundary point z 0 of the mapping f-or, to be more precise, of the homeomorphic extension of this mapping to the closure of D—satisfying a Holder condition at z 0 or, alternatively, of its inverse satisfying a Holder condition at the point f(z 0). In particular, the compatibility of Holder conditions with the presence of cusps in the boundary of D is investigated.

Cluster sets and quasiconformal mappings

Certain classical results on cluster sets and boundary cluster sets of analytic functions, due to Iversen, Lindelöf, Noshiro, Tsuji, Ohtsuka, Pommerenke, Carmona, Cufi and others, are extended to n-dimensional quasiconformal mappings. Unlike what is usually the case in the context of analytic functions, our considerations are not restricted to mappings of a disk or ball only. It is shown, for instance, that quasiconformal cluster sets and boundary cluster sets, taken at a non-isolated boundary point of an arbitrary domain, coincide. More refined versions are established in the special case where the domain is the open unit ball. These include cluster set considerations of the induced radial boundary extension and results where certain exceptional sets on the boundary are allowed for.

Cone conditions and quasiconformal mappings

Let f be a quasiconformal mapping of the open unit ball B n = {x ∈ R n : | x | < l× in euclidean n-space R n onto a bounded domain D in that space. For dimension n= 2 the literature of geometric function theory abounds in results that correlate distinctive geometric properties of the domain D with special behavior, be it qualitative or quantitative, on the part of f or its inverse. There is a more modest, albeit growing, body of work that attempts to duplicate in dimensions three and above, where far fewer analytical tools are at a researcher’s disposal, some of the successes achieved in the plane along such lines. In this paper we contribute to that higher dimensional theory some observations relating the behavior of f and f -1 to one of the venerable geometric conditions in analysis, the cone condition prominent in potential theory, geometric measure theory, and elsewhere. We first demonstrate that, when D obeys a specific interior cone condition along its boundary, f must satisfy a uniform Holder condition in B n . With regard to f -1, the dual result one might anticipate — that an exterior cone condition satisfied by D at its boundary would lead to a uniform Holder estimate for f -l in D — is not, in general, true. We show, however, that in the presence of a certain auxiliary condition on D, one which is implied by an exterior cone condition when D is a Jordan domain in the plane, such a cone condition does exert a definite influence on the modulus of continuity of f -1.