Bloch's Theorem in the Context of Quaternion Analysis

TL;DR

This paper extends Bloch's theorem from complex analysis to quaternion analysis in three-dimensional space, providing explicit bounds for the quaternionic Bloch constant.

Contribution

It generalizes Bloch's theorem to quaternionic functions in 3D and computes explicit lower bounds for the associated Bloch constant.

Findings

Established a quaternionic version of Bloch's theorem.

Derived explicit lower bounds for the quaternionic Bloch constant.

Extended classical complex analysis results to higher-dimensional quaternionic context.

Abstract

The classical theorem of Bloch (1924) asserts that if $f$ is a holomorphic function on a region that contains the closed unit disk $|z|\leq 1$ 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.