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Some inequalities for the generalized linear distortion function

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Abstract

In this paper, we establish several inequalities for the the generalized linear distortion function λ(a,K) by using the monotonicity and convexity of certain combinations λ(a,K)

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References

  1. M Abramowitz, I A Stegun (Eds.). Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1965.

    Google Scholar 

  2. H Alzer, S L Qiu. Monotonicity theorems and inequalities for the complete elliptic integrals, J Comput Appl Math, 2004, 172: 289–312.

    Article  MathSciNet  MATH  Google Scholar 

  3. G D Anderson, S L Qiu, M K Vamanamurthy, M Vuorinen. Generalized elliptic integrals and modular equations, Pacific J Math, 2000, 192: 1–37.

    Article  MathSciNet  Google Scholar 

  4. G D Anderson, S L Qiu, M Vuorinen. Modular equations and distortion functions, Ramanujan J, 2009, 18: 147–169.

    Article  MathSciNet  MATH  Google Scholar 

  5. G D Anderson, M K Vamanamurthy, M Vuorinen. Distortion functions for plane quasiconformal mappings, Israel J Math, 1988, 62: 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  6. G D Anderson, M K Vamanamurthy, M Vuorinen. Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997.

    MATH  Google Scholar 

  7. S András, Á Baricz. Bounds for complete elliptic integral of the first kind, Expo Math, 2010, 28: 357–364.

    Article  MathSciNet  MATH  Google Scholar 

  8. B C Berndt, S Bhargave, F G Garvan. Ramanujan’s theories of elliptic functions to alternative bases, Trans Amer Math Soc, 1995, 347: 4163–4244.

    Article  MathSciNet  MATH  Google Scholar 

  9. B N Guo, F Qi. Some bounds for the complete elliptic integrals of the first and second kinds, Math Inequal Appl, 2011, 14: 323–334.

    MathSciNet  MATH  Google Scholar 

  10. C Q He. Distortion estimates of quasiconformal mappings, Sci Sinica Ser A, 1984, 27: 225–232.

    MathSciNet  MATH  Google Scholar 

  11. O Lehto, K I Virtanen. Quasiconformal Mappings in the Plane, Springer-Verlag, New York-Heidelberg, 1973.

    MATH  Google Scholar 

  12. X Y Ma, S L Qiu. Properties of the generalized elliptic integrals, J Zhejiang Sci-Tech Univ, 2007, 24: 200–205.

    Google Scholar 

  13. S L Qiu, M K Vamanamurthy, M Vuorinen. Bounds for quasiconformal distortion functions, J Math Anal Appl, 1997, 205: 43–64.

    Article  MathSciNet  MATH  Google Scholar 

  14. S L Qiu, M Vuorinen. Duplication inequalities for the ratios of hypergeometric functions, Forum Math, 2000, 12: 109–133.

    MathSciNet  MATH  Google Scholar 

  15. S L Qiu, M Vuorinen. Special functions in geometric function theory, In: Handbook of Complex Analysis: Geometric Function Theory, Vol. 2, Elsevier Sci B V, Amsterdam, 2005, 621-659.

    Google Scholar 

  16. E D Rainville. Special Functions, MacMillan, New York, 1960.

    MATH  Google Scholar 

  17. M K Vamanamurthy, M Vuorinen. Functions inequalities, Jacobi products, and quasiconformal maps, Illinois J Math, 1994, 38: 394–419.

    MathSciNet  MATH  Google Scholar 

  18. M Vuorinen. Conformal Geometry and Quasiregular Mappings, Springer-Verlag, Berlin, 1988.

    MATH  Google Scholar 

  19. G D Wang, X H Zhang, Y M Chu. A Hölder mean inequality for the Hersch-Pfluger distortion function, Sci Sin Math, 2010, 40: 783–786.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Xiao-yan Ma, Song-liang Qiu & Gen-hong Zhong

  2. Department of Mathematics and Computing Science, Hunan City University, Yiyang, 413000, China

    Yu-ming Chu

Authors
  1. Xiao-yan Ma
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  2. Song-liang Qiu
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  3. Gen-hong Zhong
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  4. Yu-ming Chu
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Corresponding author

Correspondence to Yu-ming Chu.

Additional information

Supported by the National Natural Science Foundation of China (11071069, 11171307) and the Natural Science Foundation of Hunan Province (09JJ6003).

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Ma, Xy., Qiu, Sl., Zhong, Gh. et al. Some inequalities for the generalized linear distortion function. Appl. Math. J. Chin. Univ. 27, 87–93 (2012). https://doi.org/10.1007/s11766-012-2896-6

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  • DOI: https://doi.org/10.1007/s11766-012-2896-6

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