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Abstract and Applied Analysis
Volume 2011, Article ID 671765, 8 pages
http://dx.doi.org/10.1155/2011/671765
Research Article

Monotonicity, Convexity, and Inequalities Involving the Agard Distortion Function

1Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
3Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received 22 June 2011; Accepted 10 November 2011

Academic Editor: Martin D. Schechter

Copyright © 2011 Yu-Ming Chu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. F. Bowman, Introduction to Elliptic Functions with Applications, Dover Publications, New York, NY, USA, 1961.
  2. O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer, New York, NY, USA, 2nd edition, 1973.
  3. B. C. Berndt, Ramanujan's Notebooks. Part III, Springer, New York, NY, USA, 1991.
  4. A. Beurling and L. Ahlfors, “The boundary correspondence under quasiconformal mappings,” Acta Mathematica, vol. 96, pp. 125–142, 1956. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. S. B. Agard and F. W. Gehring, “Angles and quasiconformal mappings,” Proceedings of the London Mathematical Society. Third Series, vol. 14a, pp. 1–21, 1965. View at Google Scholar · View at Zentralblatt MATH
  6. S. L. Qiu, “Some distortion properties of K-quasiconformal mappings and an improved estimate of Mori's constant,” Acta Mathematica Sinica, vol. 35, no. 4, pp. 492–504, 1992 (Chinese). View at Google Scholar
  7. G. D. Anderson and M. K. Vamanamurthy, “Some properties of quasiconformal distortion functions,” New Zealand Journal of Mathematics, vol. 24, no. 1, pp. 1–15, 1995. View at Google Scholar · View at Zentralblatt MATH
  8. G. J. Martin, “The distortion theorem for quasiconformal mappings, Schottky's theorem and holomorphic motions,” Proceedings of the American Mathematical Society, vol. 125, no. 4, pp. 1095–1103, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, “Distortion functions for plane quasiconformal mappings,” Israel Journal of Mathematics, vol. 62, no. 1, pp. 1–16, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. S.-L. Qiu and M. Vuorinen, “Quasimultiplicative properties for the η-distortion function,” Complex Variables. Theory and Application, vol. 30, no. 1, pp. 77–96, 1996. View at Google Scholar
  11. G. D. Anderson, S. Qiu, and M. K. Vuorinen, “Bounds for the Hersch-Pfluger and Belinskii distortion functions,” in Computational Methods and Function Theory 1997 (Nicosia), vol. 11 of Ser. Approx. Decompos., pp. 9–22, World Scientific, River Edge, NJ, USA, 1999. View at Google Scholar
  12. S. Qiu, “Agard's η-distortion function and Schottky's theorem,” Science in China. Series A, vol. 40, no. 1, pp. 1–9, 1997. View at Publisher · View at Google Scholar
  13. G. D. Anderson, S.-L. Qiu, M. K. Vamanamurthy, and M. Vuorinen, “Generalized elliptic integrals and modular equations,” Pacific Journal of Mathematics, vol. 192, no. 1, pp. 1–37, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. G. D. Anderson, M. K. Vamanamurthy, and M. K. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, NY, USA, 1997.
  15. G. D. Anderson, S.-L. Qiu, and M. Vuorinen, “Modular equations and distortion functions,” Ramanujan Journal, vol. 18, no. 2, pp. 147–169, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH