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The Gehring–Hayman inequality for quasihyperbolic geodesics

Published online by Cambridge University Press:  24 October 2008

Juha Heinonen  and
Steffen Rohde
Juha Heinonen
Affiliation:
University of Michigan, Department of Mathematics, Ann Arbor, MI 48109, USA
Steffen Rohde
Affiliation:
University of Michigan, Department of Mathematics, Ann Arbor, MI 48109, USA
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Extract

The quasihyperbolic metric in a proper subdomain D of Rn is defined by

where the infimum is taken over all rectifiable arcs γ in D joining x and y. There always exists an arc, called a quasihyperbolic geodesic in D, for which the infimum above is attained. We refer to [3], [4], [16], and [17] for the motivation and basic properties of the quasihyperbolic metric.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 3 , November 1993 , pp. 393 - 405
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Gehring, F. W. and Hayman, W. K.. An inequality in the theory of conformal mapping. J. Math. Pure Appl. (9) 41 (1962), 353–361.Google Scholar
[2]Gehring, F. W. and Martio, O.. Quasiextremal distance domains and extension of quasi-conformal mappings. J. Analyse Math. 45 (1985), 181–206.CrossRefGoogle Scholar
[3]Gehring, F. W. and Osgood, B. C.. Uniform domains and the quasi-hyperbolic metric. J. Analyse Math. 36 (1979), 50–74.CrossRefGoogle Scholar
[4]Gehring, F. W. and Palka, B. P.. Quasiconformally homogeneous domains. J. Analyse Math. 30 (1976), 172–199.CrossRefGoogle Scholar
[5]Gehring, F. W. and Väisälä, J.. Hausdorff dimension and quasiconformal mappings. J. London Math. Soc. 6 (1973), 504–512.CrossRefGoogle Scholar
[6]Heinonen, J. and Näkki, R.. Quasiconformal distortion on arcs. J. Analyse Math. (to appear).Google Scholar
[7]Pommerenke, Ch.. Uniformly perfect sets and the Poincaré metric. Arch. Math. 32 (1979), 192–199.CrossRefGoogle Scholar
[8]Pommerenre, Ch.. Boundary Behaviour of Conformal Maps (Springer-Verlag, 1992).CrossRefGoogle Scholar
[9]Stein, E. M.. Singular Integrals and Differentiability Properties of Functions (Princeton University Press, 1970).Google Scholar
[10]Tukia, P.. A quasiconformal group not isomorphic to a Möbius group. Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 149–160.CrossRefGoogle Scholar
[11]Tukia, P. and Väisälä, J.. Lipschitz and quasiconformal approximation and extension. Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 303–342.CrossRefGoogle Scholar
[12]Tukia, P. and Väisälä, J.. Quasiconformal extension from dimension n to n + 1. Ann. of Math. 115 (1982), 331–348.Google Scholar
[13]Tukia, P. and Väisälä, J.. Extension of embeddings close to isometries and similarities. Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 153–175.CrossRefGoogle Scholar
[14]Väisälä, J.. Lectures on n-dimensional Quasiconformal Mappings. Lecture Notes in Math. 229 (Springer-Verlag, 1971).CrossRefGoogle Scholar
[15]Väisälä, J.. Quasisymmetric embeddings in euclidean spaces. Trans. Amer. Math. Soc. 264 (1981), 191–204.Google Scholar
[16]Väisälä, J.. Free quasiconformality in Banach spaces II. Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991), 255–310.CrossRefGoogle Scholar
[17]Vuorinen, M.. Conformal Geometry and Quasiregular Mappings. Lecture Notes in Math. 1319 (Springer-Verlag, 1988).CrossRefGoogle Scholar