On Mori’s theorem for quasiconformal maps in the 𝑛-space

Bhayo, B.; Vuorinen, M.

doi:10.1090/S0002-9947-2011-05281-5

On Mori’s theorem for quasiconformal maps in the $n$-space
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by B. A. Bhayo and M. Vuorinen PDF

Trans. Amer. Math. Soc. 363 (2011), 5703-5719 Request permission

Abstract:

R. Fehlmann and M. Vuorinen proved in 1988 that Mori’s constant $M(n,K)$ for $K$-quasiconformal maps of the unit ball in $\mathbf {R}^n$ onto itself keeping the origin fixed satisfies $M(n,K) \to 1$ when $K\to 1.$ Here we give an alternative proof of this fact, with a quantitative upper bound for the constant in terms of elementary functions. Our proof is based on a refinement of a method due to G.D. Anderson and M. K. Vamanamurthy. We also give an explicit version of the Schwarz lemma for quasiconformal self-maps of the unit disk. Some experimental results are provided to compare the various bounds for the Mori constant when $n=2.$

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