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Sharp distortion theorems for a subclass of biholomorphic mappings which have a parametric representation in several complex variables

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Abstract

In this paper, the sharp distortion theorems of the Fréchet-derivative type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball of complex Banach spaces are established, and the corresponding results of the above generalized mappings on the unit polydisk in Cn are also given. Meanwhile, the sharp distortion theorems of the Jacobi determinant type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball with an arbitrary norm in Cn are obtained, and the corresponding results of the above generalized mappings on the unit polydisk in Cn are got as well. Thus, some known results in prior literatures are generalized.

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References

  1. Poreda, T., On the univalent holomorphic maps of the unit polydisc of Cn which have the parametric representation, I—the geometric properties, Ann. Univ. Mariae Curie Sklodowsda Sect. A, 41, 1987, 105–113.

    MathSciNet  MATH  Google Scholar 

  2. Kohr, G., Using the method of Löwner chains to introduce some subcalsses of biholomorphic mappings in Cn, Rev. Roum. Math. Pures Appl., 46, 2001, 743–760.

    MathSciNet  MATH  Google Scholar 

  3. Graham, I., Hamada, H. and Kohr, G., Parametric representation of univalent mappings in several complex variables, Canadian J. Math., 54(2), 2002, 324–351.

    Article  MathSciNet  MATH  Google Scholar 

  4. Kubicka, E. and Poreda, T., On the parametric representation of starlike maps of the unit ball in Cn into Cn, Demonstration Math., 21, 1988, 345–355.

    MathSciNet  MATH  Google Scholar 

  5. Kohr, G. and Liczberski, P., Univalent Mappings of Several Complex Variables, Cluj University Press, Cluj-Napoca, Romania, 1998.

    MATH  Google Scholar 

  6. Hamada, H., Honda, T. and Kohr, G., Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation, J. Math. Anal. Appl., 317(1), 2006, 302–319.

    Article  MathSciNet  MATH  Google Scholar 

  7. Hamada, H. and Honda, T., Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables, Chin. Ann. Math. Ser. B, 29(4), 2008, 353–368.

    Article  MathSciNet  MATH  Google Scholar 

  8. Xu, Q. H. and Liu, T. S., Sharp growth and distortion theorems for a subclass of biholomorphic mappings, Computer Math. Appl., 59(12), 2010, 3778–3784.

    Article  MathSciNet  MATH  Google Scholar 

  9. Xu, Q. H., Liu, T. S. and Xu, H. M., Growth and distortion theorems for a subclass of holomorphic mappings in several complex variables, Chin. Ann. Math. Ser. A, 35(5), 2014, 565–574 (in Chinese).

    MathSciNet  MATH  Google Scholar 

  10. Barnard, R. W., FitzGerald C. H. and Gong S., A distortion theorem of biholomorphic convex mappings in C2, Trans. Amer. Math. Soc., 344(2), 1994, 907–924.

    MathSciNet  MATH  Google Scholar 

  11. Liu, T. S. and Zhang, W. J., A distortion theorem of biholomorphic convex mappings in Cn, Chin. Ann. Math. Ser. A, 20(4), 1999, 505–512 (in Chinese).

    Google Scholar 

  12. Gong, S., Wang, S. K. and Yu, Q. H., Biholomorphic convex mappings of ball in Cn, Pacif. J. Math., 161(2), 1993, 287–306.

    Article  MathSciNet  MATH  Google Scholar 

  13. Gong, S. and Liu, T. S., Distortion theorems for biholomorphic convex mappings on bounded convex circular domains, Chin. Ann. Math. Ser. B, 20(3), 1999, 297–304.

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, T. S. and Zhang, W. J., A distortion theorem of biholomorphic convex mappings in a Banach space, Acta Math. Sin., 46(6), 2003, 1041–1046 (in Chinese).

    MathSciNet  MATH  Google Scholar 

  15. Zhu, Y. C. and Liu, M. S., Distortion theorems for biholomorphic convex mappings in Banach spaces, Complex Variables, 50(1), 2005, 57–68.

    Article  MathSciNet  MATH  Google Scholar 

  16. Chu, C. H., Hamada, H., Honda T. and Kohr G., Distortion theorems for convex mappings on homogeneous balls, J. Math. Anal. Appl., 369(2), 2010, 437–442.

    Article  MathSciNet  MATH  Google Scholar 

  17. Hamada, H. and Kohr, G., Growth and distortion results for convex mappings in infinite dimensional spaces, Complex Variables, 47(4), 2002, 291–301.

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu, T. S., Wang, J. F. and Lu, J., Distortion theorems for starlike mappings in several complex variables, Taiwanese J. Math., 15(6), 2011, 2601–2608.

    MathSciNet  MATH  Google Scholar 

  19. Liu, X. S. and Liu, T. S., On the sharp distortion theorems for a subclass of starlike mappings in several complex variables, Taiwanese J. Math., 19(2), 2015, 363–379.

    MathSciNet  Google Scholar 

  20. Hamada, H., Kohr, G. and Liczberski, P., Starlike mappings of order a on the unit ball in complex Banach spaces, Glas. Mat. Ser. III, 36(1), 2001, 39–48.

    MathSciNet  MATH  Google Scholar 

  21. Xu, Q. H. and Liu, T. S., On coefficient estimates for a class of holomorphic mappings, Sci. China Ser. A-Math. 52(4), 2009, 677–686.

    Article  MathSciNet  MATH  Google Scholar 

  22. Kohr, G. and Liczberski, P., On strongly starlikeness of order a in several complex variables, Glas. Mat., 33(53), 1998, 185–198.

    MathSciNet  MATH  Google Scholar 

  23. Feng, S. X. and Lu, K. P., The growth theorem for almost starlike mappings of order a on bounded starlike circular domains, Chin. Quart. J. Math., 15(2), 2000, 50–56.

    MathSciNet  MATH  Google Scholar 

  24. Chuaqui, M., Applications of subordination chains to starlike mappings in Cn, Pacif. J. Math., 168(1), 1995, 33–48.

    Article  MathSciNet  MATH  Google Scholar 

  25. Honda, T., The growth theorem for k-fold symmetric convex mappings, Bull. London Math. Soc., 34(6), 2002, 717–724.

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Computation Science, Lingnan Normal University, Zhanjiang, Guangdong, 524048, China

    Xiaosong Liu

  2. Department of Mathematics, Huzhou University, Huzhou, Zhejiang, 313000, China

    Taishun Liu

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

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11031008, 11471111) and Guangdong Natural Science Foundation (No. 2014A030307016).

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Liu, X., Liu, T. Sharp distortion theorems for a subclass of biholomorphic mappings which have a parametric representation in several complex variables. Chin. Ann. Math. Ser. B 37, 553–570 (2016). https://doi.org/10.1007/s11401-016-1019-8

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  • DOI: https://doi.org/10.1007/s11401-016-1019-8

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