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A Schwarz lemma at the boundary on complex Hilbert balls and applications to starlike mappings

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Abstract

In this paper, we prove a Schwarz lemma at the boundary for holomorphic mappings f between Hilbert balls, and obtain related consequences. Especially, we obtain estimations of ∥Df(z0)∥ on the holomorphic tangent space for holomorphic mappings f or for homogeneous polynomial mappings f between Hilbert balls. Next, we prove the boundary rigidity theorem for holomorphic self-mappings of a Hilbert ball which have an interior fixed point. We obtain two distortion theorems for various subclasses of starlike mappings on the Euclidean unit ball \({\mathbb{B}^n}\) in ℂn, as applications of the boundary Schwarz lemma for holomorphic mappings between the Euclidean unit balls. Finally, certain coefficient bounds for subclasses of starlike mappings on the unit ball of a complex Hilbert space are also obtained as an application of the estimation of ∥Df(z0)∥ on the holomorphic tangent space for homogeneous polynomial mappings f between Hilbert balls.

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Acknowledgments

The authors would like to thank the referee for useful suggestions which improved the paper.

Some of the research for this paper was carried out in May and August, 2017, when Gabriela Kohr visited the Department of Mathematics of the University of Toronto. She expresses her gratitude to the members of this department for their hospitality during that visit.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada

    Ian Graham

  2. Faculty of Science and Engineering, Kyushu Sangyo University, 3-1 Matsukadai, 2-Chome, Higashi-ku, Fukuoka, 813-8503, Japan

    Hidetaka Hamada

  3. Faculty of Mathematics and Computer Science, Babes-Bolyai University, 1M. Kogalniceanu Str., Cluj-Napoca, 400084, Romania

    Gabriela Kohr

Authors
  1. Ian Graham
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  2. Hidetaka Hamada
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  3. Gabriela Kohr
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Corresponding author

Correspondence to Hidetaka Hamada.

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Note added in proof

After our paper was accepted, we found a paper: Z. Chen, Y. Liu and Y. Pan, A Schwarz lemma at the boundary of Hilbert balls, Chin. Ann. Math. Ser. B 39 (2018), 695–704. In the above paper, the authors obtained a similar result as in Corollary 1.6 in our paper, but in the stronger assumption that f is of class C1+α at \({z_0} \in \partial {\mathbb{B}_1}\), where f is a holomorphic mapping between the unit balls \(\mathbb{B}_1\) and \(\mathbb{B}_2\) of separable complex Hilbert spaces H1 and H2, respectively.

I. Graham was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A9221.

H. Hamada was partially supported by JSPS KAKENHI Grant Number JP16K05217.

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Graham, I., Hamada, H. & Kohr, G. A Schwarz lemma at the boundary on complex Hilbert balls and applications to starlike mappings. JAMA 140, 31–53 (2020). https://doi.org/10.1007/s11854-020-0080-0

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  • DOI: https://doi.org/10.1007/s11854-020-0080-0

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