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The estimates of all homogeneous expansions for a subclass of biholomorphic mappings which have parametric representation in several complex variables

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Abstract

In this paper, we obtain the estimates of all homogeneous expansions for a subclass of biholomorphic mappings which have parametric representation on the unit ball of complex Banach spaces. Meanwhile, we also establish the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in Cn. Especially, the above estimates are only sharp for biholomorphic starlike mappings and starlike mappings of order α under restricted conditions. Our derived results generalize many known results.

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Acknowledgements

We thank the referees for their useful comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, 524048, P. R. China

    Xiao Song Liu

  2. Department of Mathematics, Huzhou Teachers College, Huzhou, 313000, P. R. China

    Tai Shun Liu

Authors
  1. Xiao Song Liu
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  2. Tai Shun Liu
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Correspondence to Xiao Song Liu.

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Supported by National Natural Science Foundation of China (Grant No. 11471111) and Guangdong Natural Science Foundation (Grant No. 2014A030307016)

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Liu, X.S., Liu, T.S. The estimates of all homogeneous expansions for a subclass of biholomorphic mappings which have parametric representation in several complex variables. Acta. Math. Sin.-English Ser. 33, 287–300 (2017). https://doi.org/10.1007/s10114-016-5226-8

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  • DOI: https://doi.org/10.1007/s10114-016-5226-8

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