A remark on the maximal dilatation of a quasiconformal mapping

Vuorinen, Matti

doi:10.1090/S0002-9939-1984-0760934-6

A remark on the maximal dilatation of a quasiconformal mapping
HTML articles powered by AMS MathViewer

by Matti Vuorinen PDF

Proc. Amer. Math. Soc. 92 (1984), 505-508 Request permission

Abstract:

A lower bound for the maximal dilatation of a quasiconformal self-mapping of a uniform domain $G$ in ${R^n}$ with connected boundary is proved provided that the boundary is kept pointwise fixed by the mapping and that a prescribed point $a$ is mapped on another point $b$ in $G$. The lower bound is given in terms of the quasihyperbolic distance from $a$ to $b$.

References

G. D. Anderson and M. K. Vamanamurthy, An extremal displacement mapping in $n$-space, Complex analysis Joensuu 1978 (Proc. Colloq., Univ. Joensuu, Joensuu, 1978) Lecture Notes in Math., vol. 747, Springer, Berlin, 1979, pp. 1–9. MR 553026
F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Analyse Math. 36 (1979), 50–74 (1980). MR 581801, DOI 10.1007/BF02798768
F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172–199. MR 437753, DOI 10.1007/BF02786713
O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), no. 2, 383–401. MR 565886, DOI 10.5186/aasfm.1978-79.0413
Oswald Teichmüller, Ein Verschiebungssatz der quasikonformen Abbildung, Deutsche Math. 7 (1944), 336–343 (German). MR 18761

Lectures on

-dimensional quasiconformal mappings

Conformal invariants and quasiregular mappings