Moduli of quadrilaterals and extremal quasiconformal extensions of quasisymmetric functions

Abstract

Abstract. We establish a relationship between Strebel boundary dilatation of a quasisymmetric function of the unit circle and indicated by the change in the module of the quadrilaterals with vertices on the circle. By using general theory of universal Teichmüller space, we show that there are many quasisymmetric functions of the circle have the property that the smallest dilatation for a quasiconformal extension of a quasisymmetric function of the unit circle is larger than indicated by the change in the module of quadrilaterals with vertices on the circle. Mathematics Subject Classification (1991). Primary 32G15; Secondary 30C60; 30C75.Keywords. Boundary dilatation, quasisymmetric function, quasiconformal mapping, Strebel point. §1. IntroductionIn this paper, the following notation will be used. C = the finite complex plane; ∆ = {z ∈ C; |z| < 1}; Γ = ∂∆ (boundary of ∆);∆ = ∆∪Γ; ∆ r = {z; r < |z| < 1}, where 0 < r < 1; H = the upper half plane; R = the real line in C.

On Symmetric Quasicircles

“…The sufficiency has been proved in Theorem 4.1 (see the part (c) implies (a)). For the necessity, we observe that the argument for the degenerate case in the proof of [Wu,Theorem 1] shows that (4.12) j? H, mod(/i(0) w here H is the boundary dilatation of h and where the supremum is taken over all quadrilaterals Q in D with one of its sides having diameter less than S. Once again h is symmetric implies that H = 1.…”

Dilatations and exponents of quasisymmetric homeomorphisms

“…A quasisymmetric homeomorphism h of R onto itself is said to be induced by an affine mapping if it is the restriction to R of a map of the form φ 2 • A K • φ 1 , where A K (x + iy) = x + iKy is an affine map, while φ 1 and φ 2 are conformal mappings from a rectangle {x + iy : 0 < x < a, 0 < y < b} and its image {u + iv : 0 < u < a, 0 < v < Kb} under A K onto H, respectively. The necessary condition for M h = K h established by Wu (see [24]) and Yang (see [26]) can be stated as follows.…”

“…Later, Wu [24] and Yang [26] independently established a necessary condition such that M h = K h . In order to state their result, we need the following definitions.…”

“…The purpose of introducing the new names and new notation here is to give more intuitive terms and notation for these quantities. One should also note that the dilatation M h can be defined in terms of moduli of quadrilaterals with domain H and vertices on the real line (see [3] and [24]). …”

“…[24,26] Let h : R → R be a quasisymmetric homeomorphism. If M h = K h , then either h is induced by an affine mapping or H h = K h (that is, h has a substantial boundary point).…”