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Bloch’s Theorem in the Context of Quaternion Analysis

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Abstract

The classical Theorem of Bloch (1924) asserts that if f is a holomorphic function on a region that contains the closed unit disk ¦z¦ ≤ 1 such that \(f(0)=0\ {\rm and}\mid f'(0)\mid=1\), then the image domain contains discs of radius

$${3\over 2}-{\sqrt 2}>{1\over 12}$$

The optimal value is known as Bloch’s constant and 1/12 is not the best possible. In this paper we give a direct generalization of Bloch’s Theorem to the three-dimensional Euclidean space in the framework of quaternion analysis. We compute explicitly a lower bound for the Bloch constant.

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Authors and Affiliations

  1. Institute of Applied Analysis, Freiberg University of Mining and Technology, 9596, Freiberg, Germany

    João Pedro Morais

  2. Institut für Mathematik/Physik, Bauhaus-Universität Weimar, Coudraystr. 13B, 99421, Weimar, Germany

    Klaus Gürlebeck

Authors
  1. João Pedro Morais
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  2. Klaus Gürlebeck
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Correspondence to João Pedro Morais.

Additional information

The first author’s research is supported by Foundation for Science and Technology (FCT) via the post-doctoral grant SFRH/BPD/66342/2009.

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Morais, J.P., Gürlebeck, K. Bloch’s Theorem in the Context of Quaternion Analysis. Comput. Methods Funct. Theory 12, 541–558 (2012). https://doi.org/10.1007/BF03321843

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