Moduli of quadrilaterals and extremal quasiconformal extensions of quasisymmetric functions
Let \(f\) be a sense-preserving quasisymmetric homeomorphism of the unit circle onto itself, \(K_1(f)\) be the maximal dilatation of the extremal extension of \(f\) to the unit disk, and \(H(f)\) be the boundary dilatation of \(f\). Define \(K_0(f)=\sup_Q\{{M(f(Q))\over M(Q)}\}\) where sup is taken over all quadrilaterals \(Q\) with the unit disk as its domain, and \(M(Q)\) is the conformal modulus of \(Q\). It is well known that \(K_0(f)\leq K_1(f)\) and \(H(f)\leq K_1(f)\). If there is a quadrilateral \(Q\) such that \(K_0(f)={M(f(Q))\over M(Q)}\), then we denote \(K_0(f)\) by \(K^q_0(f)\); if \(K_0(f)>{M(f(Q))\over M(Q)}\) for every \(Q\), then we denote \(K_0(f)\) by \(K^d_0(f)\). In this paper, the author discusses the relationship among \(K_0(f)\), \(K^q_0(f)\), \(K_0^d(f)\), \(K_1(f)\) and \(H(f)\), and obtain some interesting results such as: Theorem 1. Either \(K_0(f)= K^q_0 (f)\), or \(K^d_0(f)\leq H(f)\). Theorem 2. If \(K^q_0(f)=K_1(f)\), then \([f]\) is a Strebel point in the universal Teichmüller space. Theorem 4. If \(H(f)=1\) and \(f\neq id\), then \(K_0(f)<K_1(f)\).
scientific article; zbMATH DE number 1129400