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A refinement about estimation of expansion coefficients for normalized biholomorphic mappings

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Abstract

A refining estimation of homogeneous expansion forf is discussed, wheref belongs to a subclass of all normalized biholomorphic mappings defined on the unit polydisk in -n or the unit ball in complex Banach spaces, andx = 0 is a zero of orderk + 1 off(x) - x. Moreover, an estimation of homogeneous expansion for subordinate mappings defined on the unit ball in complex Banach spaces is also given.

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References

  1. Gong, S., The Bieberbach Conjecture (ed. Providence, R. I.), Amer. Math. Soc., International Press, 1999, 126–176.

    MATH  Google Scholar 

  2. Cartan, H., Sur la possibilité d’étendre aux fonctions de plusieurs variables complexes la theorie des fonctions univalents, in Lecons sur les Fonctions Univalents ou Mutivalents (eds. Montel, P., Gauthier-Villar), 1933, 129–155.

  3. Zhang, W. J., Dong, D. Z., Wang, Y. Z., The growth theorem for convex maps on the Banach space, Chin. Quart. J. of Math. (in Chinese), 1991, 7(2): 84–87.

    Google Scholar 

  4. Kohr, G., On some best bounds for coefficients of several subclasses of biholomorphic mappings in ℂn, Complex Variables, 1998, 36: 261–284.

    MATH  MathSciNet  Google Scholar 

  5. Roper, K. A., Suffridge, T. J., Convexity properties of holomorphic mappings in ℂn, Trans.Amer.Math.Soc., 1999, 351: 1803–1833.

    Article  MATH  MathSciNet  Google Scholar 

  6. Suffridge, T. J., Starlike and convex maps in Banach spaces, Pacific Jour. of Math., 1973, 46: 575–589.

    MATH  MathSciNet  Google Scholar 

  7. Hamada, H., Kohr, G., Liczberski, P., Starlike mappings of order α on the unit ball in complex Banach spaces, Glas. Mat. Ser. III, 2001, 36: 39–48.

    MATH  MathSciNet  Google Scholar 

  8. Gong, S., Liu, T. S., Criterion for the family of ε starlike mappings, J. Math. Anal. and Appl., 2002, 274: 696–704.

    Article  MATH  MathSciNet  Google Scholar 

  9. Zhang, W. J., Liu, T. S., On growth and covering theorems for quasi-convex mappings in the unit ball of a complex Banach space, Science in China, Ser. A, 2002, 45(12): 1538–1547.

    MATH  Google Scholar 

  10. Lin, Y. Y., Hong, Y., Some properties of holomorphic maps in Banach spaces, Acta Math. Sinica (in Chinese), 1995, 38(2): 234–241.

    MATH  MathSciNet  Google Scholar 

  11. Boyd, A. V., Coefficient estimates for starlike functions of order α, Proc. Amer. Math. Soc., 1966, 17: 1016–1018.

    Article  MATH  MathSciNet  Google Scholar 

  12. Duren, P. L., Univalent Functions, New York: Springer-Verlag, 1983, 195–196.

    MATH  Google Scholar 

  13. Graham, I., Kohr, G., Geometric Function Theory in One and Higher Dimensions, New York: Marcel Dekker, 2003, 34–35.

    MATH  Google Scholar 

  14. Hamada, H., Kohr, G., Growth and distortion results for convex mappings in infinite dimensional spaces, Complex Variables, 2002, 47: 291–301.

    Article  MATH  MathSciNet  Google Scholar 

  15. Zhang, W. J., Dong, D. Z., The growth and 1/4-theorems for starlike mappings in a Banach space, Chin. Ann. of Math., Ser. A (in Chinese), 1992, 13(4): 417–423.

    MATH  MathSciNet  Google Scholar 

  16. Roper, K. A., Suffridge, T. J., Convex mappings on the unit ball of ℂn, J. Anal. Math., 1995, 65: 333–347.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of China, 230026, Hefei, China

    Liu Taishun & Liu Xiaosong

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  1. Liu Taishun
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  2. Liu Xiaosong
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Correspondence to Liu Taishun.

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Taishun, L., Xiaosong, L. A refinement about estimation of expansion coefficients for normalized biholomorphic mappings. Sci. China Ser. A-Math. 48, 865–879 (2005). https://doi.org/10.1007/BF02879070

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  • DOI: https://doi.org/10.1007/BF02879070

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