Boundary behavior of a conformal mapping

John Marafino

Vol. 153 (1992), No. 1, 147–155

DOI: 10.2140/pjm.1992.153.147

Abstract

Let D be a simply connected plane domain, not the whole plane. Let R^∗ denote those accessible boundary points of D such that D twists violently about them; that is, if α ∈ R^∗ and w(α) denotes its complex coordinate, then

where arg(w − w(α)) is defined and continuous in D. We show that if a certain geometric condition holds at each point of a set W^∗ ⊂ R^∗, then W^∗ is a D-conformal null set. Let L_ν denote the ray with terminal point w(α), α ∈ R^∗, having inclination ν, 0 ≤ ν < 2π. Let m denote Lebesgue measure on L_ν and set

Let W^∗ = {α ∈ R^∗ : there exists L_νi, i = 1,2,3, at w(α) such that |ν_i − ν_j| = (2∕3)π, 1 ≤ i < j ≤ 3, and u(ν_i) < 1 for i = 1,2,3}.

Theorem W^∗ is a D-conformal null set.

Mathematical Subject Classification 2000

Primary: 30C35

Secondary: 30D40

Milestones

Received: 14 August 1989

Revised: 5 February 1991

Published: 1 March 1992