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Singular values, quasiconformal maps and the Schottky upper bound

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Abstract

Some properties and asymptotically sharp bounds are obtained for singdar values of Ramanujan’s generalized modular equation. from which infinite-product representations of the Hersch-Pfluger ϕdimtortion function ϕ K (r) and the Agard η-distortion function η K (t) follow. By these results, the explicit quasiconformal Schwan lemma is improved, several properties are obtained for the Schottky upper bound, and a conjecture on the linear distortion function λ (K) is proved to be true.

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References

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

    MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  3. He. C-Q., Distortion estimates of quasiconformal mappings,Sci. Sinica, Ser. A, 1984, 27: 225.

    MathSciNet  Google Scholar 

  4. Li. Z., Cui. G. Z., A note on Mori’s thmrem of quasiconformal mappings,Acta Math. Sinica, New Ser., 1993, 1: 55.

    MathSciNet  Google Scholar 

  5. Berndt. B. C.,Ramanujan’s Notebooks, Part II, Berlin-Heidelberg-New York: Springer-Verlag, 1989.

    MATH  Google Scholar 

  6. Hille, E.,Analytic Function Theory, Vol II, New York: Chelsea Publ. Co., 1962.

    MATH  Google Scholar 

  7. Zhang, S. Y., On explicit bounds in Schottky’s theorem,Complex Variubles, 1990, 14: 15.

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Martin. G. J., The distortion theorems for quasiconformal mappings, Schottky’s theorem and holomorphic motions.Proc. Amer. Math. Soc., 1997, 125: 1095.

    Article  MATH  MathSciNet  Google Scholar 

  10. Qiu, S-L., Agard’s η-distortion function and Schottky’s theorem,Sicnce in China, Ser. A, 1997, 40(1): 1.

    Article  MATH  Google Scholar 

  11. Qiu, S-L., Vuorinen, M., Quasimultiplicative properties for the ηK-distortion function,Complex Variables, 1996, 30: 77.

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  13. Anderson G. D., Barnard, R. W., Richards, K. C. et al., Inequalities for zero-balanced hypergeometric functions,Trans. Amer. Muth. Soc., 1995, 347: 1713.

    Article  MATH  MathSciNet  Google Scholar 

  14. Anderson. G. D., Vamanamurthy, M. K., Vuorinen. M., Functional inequalities for hypergeometric functions and complete elliptic integrals,SIAM J. Math. Anal., 1992, 23(2): 512.

    Article  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. Hangzhou Institute of Electronics Engineering, 310037, Hangzhou, China

    Songliang Qiu

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  1. Songliang Qiu
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Qiu, S. Singular values, quasiconformal maps and the Schottky upper bound. Sci. China Ser. A-Math. 41, 1241–1247 (1998). https://doi.org/10.1007/BF02882264

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  • DOI: https://doi.org/10.1007/BF02882264

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