Science in China Series A: Mathematics

, Volume 40, Issue 1, pp 1–9 | Cite as

Agard’s η-distortion function and Schottky’s theorem

  • Songliang Qiu
Science in China (Series A)


The monotoneity properties of certain functions defined in terms of the η-distortion function ηκ(t) in quasiconformal theory are studied and asymptotically sharp bounds are obtained for ηκ(t), thus proving some properties of the upper bound functionK(t, r) in Schottky’s theorem on analytic functions and improving the known explicit bounds forK (t, r).


quasiconformal theory special function Schottky’s theorem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Borwein, J. M., Borwein, P.B.,Pi and the AGM, New York: John Wiley and Sons, 1987.MATHGoogle Scholar
  2. 2.
    Lehto, O., Virtanen, K. I.,Quasiconformal Mappings in the Plane, 2nd ed., New York-Heidelberg-Berlin: Springer-Verlag, 1973.MATHGoogle Scholar
  3. 3.
    Berndt, B.C.,Ramanujan Notebooks, Part III, Berlin, New York; Springer-Verlag, 1991.MATHGoogle Scholar
  4. 4.
    Beurling, A., Ahlfors, L. V., The boundary correspondence under quasi-conformal mappings,Acta Math., 1956, 96: 125.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Agard, S., Distortion theorems for quasiconformal mappings,Ann. Acad. Sci. Fenn., Ser AI, 1968, 413: 1.MathSciNetGoogle Scholar
  6. 6.
    Li, Z., Cui. G., A note on Mori’s theorem of K-quasiconformal mappings,ActaMath. Sinica, New Ser., 1993, 1: 55.MathSciNetGoogle Scholar
  7. 7.
    Martin,G.J., The distortion theorem for quasiconformal mappings, Schottky’s theorem and holomorphic motions,Math. Research Report, No. 045–94, The Australian National Univ., 1994.Google Scholar
  8. 8.
    Hayman, W.K.,Subharmonic Functions, Vol.2, San Diego: Academic Press, 1989.MATHGoogle Scholar
  9. 9.
    Hayman, W.K., Some remarks on Schottky’s theorem,Proc. Cambr. Phil. Soc., 1947, 43: 442.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jenkins, J., On explicit bounds for Schottky’s theorem,Canad. J. Math., 1955, 7: 76.MATHMathSciNetGoogle Scholar
  11. 11.
    Hempel. J., Precise bounds in the theorems of Schottky and Picard,J. London Math. Soc., 1980, 21(2): 279.CrossRefMathSciNetGoogle Scholar
  12. 12.
    Zhang, S., On explicit bounds in Schottky’s theorem,Complex Variables, 1990, 14: 15.MATHGoogle Scholar
  13. 13.
    Anderson, G. D., Vamanamurthy, M. K., Vuorinen, M., Functional inequalities for complete elliptic integrals and their ratios,SIAM J. Math. Anal., 1990, 21(2): 536.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Qiu, S.L., Vamanamurthy, M. K., Elliptic integrals and the modulus of Grötzsch ring,Pan. Amer. Math. J., 1995, 5(2): 41.MATHMathSciNetGoogle Scholar
  15. 15.
    Anderson, G. D., Vamanamurthy, M. K., Vuorinen, M., Inequalities for quasiconformal mappings in space,Pacific J. Math., 1993, 160: 1.MATHMathSciNetGoogle Scholar

Copyright information

© Science in China Press 1997

Authors and Affiliations

  • Songliang Qiu
    • 1
  1. 1.School of Science and ArtsHangzhou Institute of Electronics EngineeringHangzhouChina

Personalised recommendations