Linear dilatation of quasiconformal maps in space
Suppose that \(f\) is a homeomorphism from a domain \(D\) in \(\mathbb{R}^n\) into \(\mathbb{R}^n\). Then \(H(x,f)= \limsup_{t\to 0} L(x,f,r)/ l(x,f,r)\) defines the linear dilatation of \(f\) at \(x\in D\). Here \(L(x,f,r)= \max_z \{|f(z)- f(x)|: |z-r|=r\}\) and \(l(x,f,r)\) is defined similarly with max replaced by min. Let \(H(K,D)\) be \(\sup H(x,f)\) over all \(x\in D\) and all \(K\)-quasiconformal homeomorphisms \(f:D\to \mathbb{R}^n\). The study of \(H(K,D)\) has a long history: In 1957 Mori proved \(H(K,D)\leq e^{\pi K}\) for \(D\subset \mathbb{R}^n\) and in [\textit{O. Lehto, K. J. Virtanen} and \textit{J. Väisälä}, Ann. Acad. Sci. Fenn., Ser. A I 273, 1-14 (1959; Zbl 0090.05102)] the exact value of \(H(K,R^2)\) was found in terms of the 2-capacity of the Teichmüller ring. In [\textit{F. W. Gehring}, Trans. Am. Math. Soc. 103, 353-393 (1962; Zbl 0113.05805)] Gehring generalized the Mori result to \(\mathbb{R}^n\), \(n\geq 2\). In this paper a new explicit upper bound for \(H(K,D)\leq h_1(K)\), \(D\subset \mathbb{R}^n\), is obtained. Here \(h_1(K)\to 1\) as \(K\to 1\). The method is based on the theory developed in [\textit{G. D. Anderson, M. K. Vamanamurthy} and \textit{M. Vuorinen}, Conformal invariants, inequalities and quasiconformal maps (1997; Zbl 0885.30012)].
scientific article; zbMATH DE number 1425066
scientific article