The boundary of a simply connected domain at an inner tangent point

John Marafino

Vol. 154 (1992), No. 1, 165–178

DOI: 10.2140/pjm.1992.154.165

Abstract

Let T^∗ be the set of accessible boundary points at which the inner tangent to ∂D exists. That is, if a ∈ T^∗ and w(a) represents its complex coordinate, then there exists a unique ν(a), 0 ≤ ν(a) < 2π, such that for each 𝜖 > 0 (𝜖 < ) there exists a δ > 0 such that

Let γ(a,r) represent the unique component of D ⊂{|w − w(a)| = r} that intersects the inner normal {w(a) + ρe^iν(a) : ρ > 0}, L(a,r) denote the length of γ(a,r) and set A(a,r) = ∫ ₀^rL(a,r′)dr′. Finally let, AD^∗ be those points of T^∗ at which a non-zero angular derivative exists.

Our main result is a purely geometrical proof of a theorem that describes the boundary of D near a ∈ T^∗. As a consequence we have

1. a geometric description of the boundary of D near almost every a ∈ AD^∗ that is a generalization of the geometric behavior of a smooth curve,
2. an answer on T^∗ and hence on AD^∗ of the two open questions and conjectures made by McMillan in [3, p. 739] concerning the length and area ratios

Mathematical Subject Classification 2000

Primary: 30C99

Milestones

Received: 11 September 1989

Published: 1 May 1992