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ArticleOriginal scientific text

Title

Submultiplicative properties of the φK-distortion function

Authors 1, 2

Affiliations

  1. Department of Basic Sciences, Hangzhou Institute of Electronics Engineering (HIEE), Hangzhou 310037, P.R. China
  2. Mathematics Department, FIN-00014 University of Helsinki, Helsinki, Finland

Abstract

Some inequalities related to the submultiplicative properties of the distortion function φK(r) are derived.

Bibliography

  1. [AQV] G. D. Anderson, S.-L. Qiu and M. K. Vamanamurthy, Sharp inequalities for elliptic integrals, Report Series No. 302, Univ. of Auckland, 1994.
  2. [AVV1] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Distortion functions for plane quasiconformal mappings, Israel J. Math. 62 (1988), 1-16.
  3. [AVV2] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Special functions of quasiconformal theory, Exposition. Math. 7 (1989), 97-136.
  4. [AVV3] 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.
  5. [AVV4] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Functional inequalities for hypergeometric functions and complete elliptic integrals, ibid. 23 (1992), 512-524.
  6. [AVV5] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Inequalities for plane quasiconformal mappings, in: The Mathematical Heritage of Wilhelm Magnus-Groups, Geometry & Special Functions, W. Abikoff, J. S. Birman and K. Kuiken (eds.), Contemp. Math. 169, Amer. Math. Soc., Providence, R.I., 1994, 1-27.
  7. [B] B. C. Berndt, Ramanujan Notebooks. Part III, Springer, New York, 1991.
  8. [BB] J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, New York, 1987.
  9. [Bo] F. Bowman, Introduction to Elliptic Integrals with Applications, Dover, New York, 1961.
  10. [BF] P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer, Berlin, 1954.
  11. [C] B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, 1977.
  12. [Fr] C.-E. Fröberg, Complete Elliptic Integrals, CWK Gleerup, Lund, 1957.
  13. [He] C.-Q. He, Distortion estimates of quasiconformal mappings, Sci. Sinica Ser. A 27 (1984), 225-232.
  14. [K] R. Kühnau, Eine Methode, die Positivität einer Funktion zu prüfen, Z. Angew. Math. Mech. 74 (1994), 140-142.
  15. [LV] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Grundlehren Math. Wiss. 126, 2nd ed., Springer, New York, 1973.
  16. [M] G. J. Martin, The distortion theorem for quasiconformal mappings, Schottky's theorem and holomorphic motions, Math. Research Report No. MRR 045-94, The Australian National Univ., 1994.
  17. [P1] D. Partyka, An alternative proof of a result due to Douady and Earle, Ann. Univ. Mariae Curie-Skłodowska 42 (1988), 59-68.
  18. [P2] D. Partyka, Approximation of the Hersch-Pfluger distortion function, Ann. Acad. Sci. Fenn. AI 18 (1993), 343-354.
  19. [Q1] S.-L. Qiu, Distortion properties of K-qc. maps and a better estimate of Mori's constant, Acta Math. Sinica 35 (1992), 492-504.
  20. [Q2] S.-L. Qiu, Proof of a conjecture on the first elliptic integrals, J. Hangzhou Inst. Electronics Engrng. 3 (1993), 29-36.
  21. [QV] S.-L. Qiu and M. K. Vamanamurthy, Sharp estimates for complete elliptic integrals, SIAM J. Math. Anal., to appear.
  22. [QVV1] S.-L. Qiu, M. K. Vamanamurthy and M. Vuorinen, Inequalities for distortion functions and quasiconformal mappings, manuscript, 1994.
  23. [QVV2] S.-L. Qiu, M. K. Vamanamurthy and M. Vuorinen, Sharp inequalities for Hersch-Pfluger distortion function, Preprint 93, Nov. 1995, Dept. Math., University of Helsinki.
  24. [VV] M. K. Vamanamurthy and M. Vuorinen, Functional inequalities, Jacobi products, and quasiconformal maps, Illinois J. Math. 38 (1994), 394-419.
  25. [Vu1] M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math. 1319, Springer, 1988.
  26. [Vu2] M. Vuorinen, Singular values, Ramanujan modular equations, and Landen transformations, manuscript 1994.
  27. [W] C.-F. Wang, On the precision of Mori's theorem in Q-mapping, Science Record 4 (1960), 329-333.
  28. [WW] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge Univ. Press, 1958.
  29. [Z] J. Zając, The distortion function ΦK and quasihomographies, in: Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa (eds.), World Sci., Singapore, 1992, 403-428.
Studia Mathematica
1995-1996/117/3
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Pages:
225-242
Main language of publication
English
Received
1995-03-20
Accepted
1995-08-16
Published
1996
Exact and natural sciences