On the boundary correspondence of quasiconformal mappings of domains bounded by quasicircles

Terence John Reed

Vol. 28 (1969), No. 3, 653–661

DOI: 10.2140/pjm.1969.28.653

Abstract

A Jordan curve ℒ is a quasicircle if there exists a constant C(1 ≦ C < ∞) such that the cross ratio (z₁,z₂,z₃,z₄) of any four points z₁,⋯,z₄ in order on ℒ satisfies |(z₁,z₂,z₃,z₄)|≦ C. It is shown that the boundary correspondence f induced by a quasiconformal homeomorphism of two Jordan domains bounded by quasicircles satisfies |(w₁,w₂,w₃,w₄)|≦ A|(z₁,z₂,z₃,z₄)|^α, (A ≧ 1,0 < α ≦ 1) where w_k = f(z_k) and the points are in order on the boundary. A converse to this result is proved and estimates are computed in each direction.

Mathematical Subject Classification

Primary: 30.47

Milestones

Received: 18 April 1968

Revised: 5 July 1968

Published: 1 March 1969