Bloch's Theorem in the Context of Quaternion Analysis

Details Bloch's Theorem in the Context of Quaternion Analysis Bloch's Theorem in the Context of Quaternion Analysis

2012-01-02

arxiv.org/abs/1201.0530v1

Mathematics

Complex Variables

The present article is a preliminary version, submitted to Computational Methods and Function Theory

Scientific paper

The classical theorem of Bloch (1924) asserts that if $f$ is a holomorphic function on a region that contains the closed unit disk $|z|\leq 1$ such that $f(0) = 0$ and $|f'(0)| = 1$, then the image domain contains discs of radius $3/2-\sqrt{2} > 1/12$. 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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