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Homogeneous expansions of normalized biholomorphic convex mappings overB P

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Abstract

The power series expansions of normalized biholomorphic convex mappings on the Reinhardt domain\(B^p = \left\{ z \right. \in \mathbb{C}^n :\left\| { z } \right\| _p = [\mathop \sum \limits_{j = 1} \left| {Z_j } \right| ^p ]^{1/p}< 1\} (p > 2 )\) are studied. It is proved that the first (k+1) terms of the expansions of the jth componentf j of such a mapf depend only onz j , for 1 ⩽j⩽n, wherek is the natural number that satisfiesk < ρ ⩽k +I. Whenp→ ∞, this gives the result on the unit polydisc obtained by Suffridge in 1970.

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Authors and Affiliations

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

    Taishun Liu

  2. Department of Mathematics, Shenzhen University, 518060, Shenzhen, China

    Wenjun Zhang

Authors
  1. Taishun Liu
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  2. Wenjun Zhang
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Project supported in part by the National Natural Science Foundation of China.

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Liu, T., Zhang, W. Homogeneous expansions of normalized biholomorphic convex mappings overB P . Sci. China Ser. A-Math. 40, 799–806 (1997). https://doi.org/10.1007/BF02878918

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  • DOI: https://doi.org/10.1007/BF02878918

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