Bloch's Theorem in the Context of Quaternion Analysis

Abstract

The classical theorem of Bloch (1924) asserts that if $f$ is a holomorphic function on a region that contains the closed unit disk $|z|\le 1$ such that $f(0)=0$ and $|f^{\prime }(0)|=1$, 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.

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

Cited by in corpus (1)

• A Bloch-Landau Theorem for slice regular functions