ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 30C Vasil’ev, Alexander, Universidad de Los Andes, Colombia Two point distortion theorems for quasiconformal mappings. Let U_{K} be a class of all K- quasiconformal homeomorphisms (q. c. h.) of the unit disk U with a motionless point z = 0 and H_{K} be a class of all K-q. c. h of the upper half-plane H with the normalization f(0) = 0, f(1) = 1, f(∞) = ∞. If we fix more than one point different from the normalization forming points in U or H there is known almost nothing about the mutual behaviour of values of q. c. h. from U_{K}, H_{K} at these poins. The upper boundary of the region of the functional I(f) = |f(z_{1})|+i|f(z_{2})|, 0 < z_{1} < z_{2} < 1, f ∈ U_{K} is determined in [1]. Here we determine the entire region for I(f), f ∈ H_{K}, z_{1}, z_{2} real, by the extremal length method for curve families. Symmetrization is applied for the analogous results for q. c. h of the plane. Reference:

1. A. Vasil’ev, Isoperimetric growth theorems for quasiconformal automorphisms of a disk, Izv. Vysshikh Uchebn. Zaved. Matematika 3, (1997). 14–21.