Skip to main content
SpringerLink
Log in

Conformal invariants, quasiconformal maps, and special functions

  • Conference paper
  • First Online:
Quasiconformal Space Mappings

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1508))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 29.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 39.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Abramowitz and I. A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1965.

    Google Scholar 

  2. S. Agard and F. W. Gehring, Angles and quasiconformal mappings, Proc. London Math. Soc. (3) 14A (1965), 1–21.

    Article  MathSciNet  MATH  Google Scholar 

  3. L. Ahlfors and A. Beurling, Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950), 101–129.

    Article  MathSciNet  MATH  Google Scholar 

  4. G. D. Anderson, Extremal rings in n-space for fixed and varying n, Ann. Acad. Sci. Fenn. A I 575 (1974), 1–21.

    MATH  Google Scholar 

  5. G. D. Anderson and J. S. Frame, Numerical estimates for a Grötzsch ring constant, Constr. Approx. 4 (1988), 223–242.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. D. Anderson and M. K. Vamanamurthy, Hölder continuity of quasiconformal mappings of the unit ball, Proc. Amer. Math. Soc. 104 (1988), 227–230.

    MathSciNet  MATH  Google Scholar 

  7. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Dimension-free quasiconformal distortion in n-space, Trans. Amer. Math. Soc. 297 (1986), 687–706.

    MathSciNet  MATH  Google Scholar 

  8. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Sharp distortion theorems for quasiconformal mappings, Trans. Amer. Math. Soc. 305 (1988), 95–111.

    Article  MathSciNet  MATH  Google Scholar 

  9. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for the extremal distortion function, Proc. of the 13th Rolf Nevanlinna Colloquium, Lecture Notes in Math. Vol. 1351, 1–11, Springer-Verlag, 1988.

    Google Scholar 

  10. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen: Distortion functions for plane quasiconformal mappings, Israel J. Math. 62 (1988), 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Special functions of quasiconformal theory, Exposition. Math. 7 (1989), 97–138.

    MathSciNet  MATH  Google Scholar 

  12. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Functional inequalities for complete elliptic integrals and their ratios, SIAM J. Math. Anal. 21 (1990), 536–549.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Functional inequalities for hypergeometric functions and complete elliptic integrals, SIAM J. Math. Anal. 23 (1992) (to appear).

    Google Scholar 

  14. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for quasiconformal mappings in the plane and in space (in preparation).

    Google Scholar 

  15. E. F. Beckenbach and R. Bellman, Inequalities, Ergebnisse der Mathematik, Heft 30, Springer-Verlag, 1961.

    Google Scholar 

  16. J. Becker and Ch. Pommerenke, Hölder continuity of conformal maps with quasiconformal extension, Complex Variables Theory Appl. 10 (1988), 267–272.

    Article  MathSciNet  MATH  Google Scholar 

  17. B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasicon-formal mappings in R n, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), 257–324.

    Article  MathSciNet  MATH  Google Scholar 

  18. E. D. Callender, Hölder continuity of n-dimensional quasiconformal mappings, Pacific J. Math. 10 (1960), 499–515.

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Caraman, n-Dimensional Quasiconformal (QCf) Mappings, Abacus Press, Tunbridge Wells, Kent, England, 1974.

    MATH  Google Scholar 

  20. R. Fehlmann and M. Vuorinen, Mori’s theorem for n-dimensional quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I 13 (1988), 111–124.

    MathSciNet  MATH  Google Scholar 

  21. J. Ferrand, G. Martin, and M. Vuorinen, Lipschitz conditions in conformally invariant metrics, J. Analyse Math. 56 (1991), 187–210.

    Article  MathSciNet  MATH  Google Scholar 

  22. B. Fuglede, Extremal length and functional completion, Acta Math. 98 (1957), 171–219.

    Article  MathSciNet  MATH  Google Scholar 

  23. F. W. Gehring, Symmetrization of rings in space, Trans. Amer. Math. Soc. 101 (1961), 499–519.

    Article  MathSciNet  MATH  Google Scholar 

  24. F. W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353–393.

    Article  MathSciNet  MATH  Google Scholar 

  25. F. W. Gehring, Topics in quasiconformal mappings, Proc. Internat. Congr. Math. (Berkeley, California 1986), Vol. 1, 62–80, AMS, 1987.

    MathSciNet  MATH  Google Scholar 

  26. F. W. Gehring and J. Väisälä, The coefficients of quasiconformality of domains in space, Acta Math. 114 (1965), 1–70.

    Article  MathSciNet  MATH  Google Scholar 

  27. H. Grötzsch, Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhängende Erweiterung des Picardschen Satzes, Ber. Verh. Sächs. Akad. Wiss. Leipzig 80 (1928), 503–507.

    MATH  Google Scholar 

  28. J. Hersch, Longeurs extrémales et théorie des fonctions, Comment. Math. Helv. 29 (1955), 301–337.

    Article  MathSciNet  MATH  Google Scholar 

  29. T. Iwaniec, Some aspects of partial differential equations and quasiregular mappings, Proc. Internat. Congr. Math. (Warsaw, 1983), Vol. 2, 1193–1208, PWN, Warsaw, 1984.

    MATH  Google Scholar 

  30. J. A. Jenkins, The method of the extremal metric, Proc. Congr. on the Occasion of the solution of the Bieberbach conjecture, Amer. Math. Soc. 1986, 95–104.

    Google Scholar 

  31. J. A. Kelingos, Characterization of quasiconformal mappings in terms of harmonic and hyperbolic measure, Ann. Acad. Sci. Fenn. Ser. A I, No. 368 (1965), 1–16.

    Google Scholar 

  32. M. Kreines, Sur une classe de fonctions de plusieurs variables, Mat. Sbornik 9 (1941), 713–719.

    MathSciNet  MATH  Google Scholar 

  33. R. Kühnau, Eine Methode, die Positivität einer Funktion zu prüfen, Zeitschrift f. angew. Math. u. Mech. (to appear).

    Google Scholar 

  34. M. Lavrentiev, Sur un critère différentiel des transformations homéomorphes des domaines à trois dimensions, Dokl. Akad. Nauk SSSR 22 (1938), 241–242.

    MATH  Google Scholar 

  35. O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, 2nd ed., Die Grundlehren der math. Wissenschaften, Band 126, Springer-Verlag, New-York-Heidelberg-Berlin, 1973.

    Book  MATH  Google Scholar 

  36. J. Lelong-Ferrand, Invariants conformes globaux sur les varietes riemanniennes, J. Differential Geom. 8 (1973), 487–510.

    MathSciNet  MATH  Google Scholar 

  37. C. Loewner, On the conformal capacity in space, J. Math. Mech. 8 (1959), 411–414.

    MathSciNet  MATH  Google Scholar 

  38. A. Markushevich, Sur certaines classes de transformations continues, Dokl. Akad. Nauk SSSR 28 (1940), 301–304.

    MathSciNet  Google Scholar 

  39. O. Martio, S. Rickman, and J. Väisälä, Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I 465 (1970), 1–13.

    MathSciNet  MATH  Google Scholar 

  40. A. P. Prudnikov, Yu. A. Brychkov, and O. Marichev, Integrals and Series, Vol. 3: More Special Functions, transl. from the Russian by G. G. Gould, Gordon and Breach Science Publishers, New York, 1988.

    Google Scholar 

  41. Yu. G. Reshetnyak, A sufficient condition for Höler continuity of a mapping, Soviet Math. Doklady 1 (1960), 76–78.

    MATH  Google Scholar 

  42. Yu. G. Reshetnyak, Bounds on moduli of continuity for certain mappings, Siberian Math. J. 7 (1966), 879–886.

    Article  MATH  Google Scholar 

  43. Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, Translations of Mathematical Monographs Vol. 73, Amer. Math. Soc. Providence, R.I., 1989.

    Google Scholar 

  44. S. Rickman, Quasiregular Mappings (to appear).

    Google Scholar 

  45. M. Schiffer, On the modulus of doubly-connected domains, Quart. J. Math. Oxford, Ser. 17 (1946), 197–213.

    Article  MathSciNet  MATH  Google Scholar 

  46. B. V. Shabat, On the theory of quasiconformal mappings in space, Soviet Math. 1 (1960), 730–733.

    MathSciNet  MATH  Google Scholar 

  47. O. Teichmüller, Untersuchungen über konforme und quasikonforme Abbildung, Deutsche Math. 3 (1938), 621–678.

    MATH  Google Scholar 

  48. J. Väisälä, On quasiconformal mappings in space, Ann. Acad. Sci. Fenn. Ser. A I 298 (1961), 1–36.

    MathSciNet  MATH  Google Scholar 

  49. J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Math. Vol. 229, Springer-Verlag, Berlin-Heidelberg-New York, 1971.

    MATH  Google Scholar 

  50. J. Väisälä, Free quasiconformality in Banach spaces I, Ann. Acad. Sci. Fenn. Ser. A I 15 (1990), 355–379.

    MathSciNet  MATH  Google Scholar 

  51. M. Vuorinen, Conformal invariants and quasiregular mappings, J. Analyse Math. 45 (1985), 69–115.

    Article  MathSciNet  MATH  Google Scholar 

  52. M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math. Vol. 1319, Springer-Verlag, Berlin-Heidelberg-New York, 1988.

    MATH  Google Scholar 

  53. M. Vuorinen, Quadruples and spatial quasiconformal mappings, Math. Z. 205 (1990), 617–628.

    Article  MathSciNet  MATH  Google Scholar 

  54. M. Vuorinen, Conformally invariant extremal problems and quasiconformal mappings, Quart. J. Math. Oxford (to appear).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Michigan State University, 48824, East Lansing, MI, USA

    G. D. Anderson, M. K. Vamanamurthy & M. Vuorinen

  2. University of Auckland, Auckland, New Zealand

    G. D. Anderson, M. K. Vamanamurthy & M. Vuorinen

  3. University of Helsinki, SF-00100, Helsinki, Finland

    G. D. Anderson, M. K. Vamanamurthy & M. Vuorinen

Authors
  1. G. D. Anderson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. K. Vamanamurthy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Vuorinen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Matti Vuorinen

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag

About this paper

Cite this paper

Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M. (1992). Conformal invariants, quasiconformal maps, and special functions. In: Vuorinen, M. (eds) Quasiconformal Space Mappings. Lecture Notes in Mathematics, vol 1508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094235

Download citation

  • DOI: https://doi.org/10.1007/BFb0094235

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55418-9

  • Online ISBN: 978-3-540-47061-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation