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

  • Science in China (Series A)
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Abstract

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).

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References

  1. Borwein, J. M., Borwein, P.B.,Pi and the AGM, New York: John Wiley and Sons, 1987.

    MATH  Google Scholar 

  2. Lehto, O., Virtanen, K. I.,Quasiconformal Mappings in the Plane, 2nd ed., New York-Heidelberg-Berlin: Springer-Verlag, 1973.

    MATH  Google Scholar 

  3. Berndt, B.C.,Ramanujan Notebooks, Part III, Berlin, New York; Springer-Verlag, 1991.

    MATH  Google Scholar 

  4. Beurling, A., Ahlfors, L. V., The boundary correspondence under quasi-conformal mappings,Acta Math., 1956, 96: 125.

    Article  MATH  MathSciNet  Google Scholar 

  5. Agard, S., Distortion theorems for quasiconformal mappings,Ann. Acad. Sci. Fenn., Ser AI, 1968, 413: 1.

    MathSciNet  Google Scholar 

  6. Li, Z., Cui. G., A note on Mori’s theorem of K-quasiconformal mappings,ActaMath. Sinica, New Ser., 1993, 1: 55.

    MathSciNet  Google Scholar 

  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.

  8. Hayman, W.K.,Subharmonic Functions, Vol.2, San Diego: Academic Press, 1989.

    MATH  Google Scholar 

  9. Hayman, W.K., Some remarks on Schottky’s theorem,Proc. Cambr. Phil. Soc., 1947, 43: 442.

    Article  MATH  MathSciNet  Google Scholar 

  10. Jenkins, J., On explicit bounds for Schottky’s theorem,Canad. J. Math., 1955, 7: 76.

    MATH  MathSciNet  Google Scholar 

  11. Hempel. J., Precise bounds in the theorems of Schottky and Picard,J. London Math. Soc., 1980, 21(2): 279.

    Article  MathSciNet  Google Scholar 

  12. Zhang, S., On explicit bounds in Schottky’s theorem,Complex Variables, 1990, 14: 15.

    MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  14. Qiu, S.L., Vamanamurthy, M. K., Elliptic integrals and the modulus of Grötzsch ring,Pan. Amer. Math. J., 1995, 5(2): 41.

    MATH  MathSciNet  Google Scholar 

  15. Anderson, G. D., Vamanamurthy, M. K., Vuorinen, M., Inequalities for quasiconformal mappings in space,Pacific J. Math., 1993, 160: 1.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Science and Arts, Hangzhou Institute of Electronics Engineering, 310037, Hangzhou, China

    Songliang Qiu

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  1. Songliang Qiu
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Qiu, S. Agard’s η-distortion function and Schottky’s theorem. Sci. China Ser. A-Math. 40, 1–9 (1997). https://doi.org/10.1007/BF03182864

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  • DOI: https://doi.org/10.1007/BF03182864

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