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Computational Methods and Function Theory

, Volume 12, Issue 2, pp 541–558 | Cite as

Bloch’s Theorem in the Context of Quaternion Analysis

  • João Pedro MoraisEmail author
  • Klaus Gürlebeck
Article

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.

Keywords

Quaternion analysis Riesz system monogenic functions Bloch’s Theorem Bloch constant 

2000 MSC

30G35 32A05 

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Copyright information

© Heldermann  Verlag 2012

Authors and Affiliations

  1. 1.Institute of Applied AnalysisFreiberg University of Mining and TechnologyFreibergGermany
  2. 2.Institut für Mathematik/PhysikBauhaus-Universität WeimarWeimarGermany

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