Slice starlike functions over quaternions

TL;DR

This paper extends geometric function theory to quaternion-valued slice regular functions, proving a sharp quaternionic de Branges-type Bieberbach result for the slice-starlike class $\mathcal{S}^*$ and deriving precise growth, distortion, and covering theorems. It introduces coefficient-estimation techniques via the Carathéodory class in the quaternionic setting and employs a convex-combination framework to handle noncommutativity, yielding sharp Fekete-Szegő-type inequalities and radius results. The authors establish quaternionic analogs of Bohr, Rogosinski, one-quarter Koebe covering, and Bloch-Landau theorems, including for functions with convex image, thereby providing a broad suite of sharp tools for higher-dimensional quaternionic analysis. Collectively, the work clarifies the role of slice preservation, extends classical univalent-function theory to quaternions, and lays groundwork for further quaternionic geometric function theory.

Abstract

In this paper, we initiate the study of the geometric function theory for slice starlike functions over quaternions and its subclasses. This allows us to answer negatively some questions about the Bieberbach conjecture, the growth, distortion, and covering theorems for slice regular functions. Precisely, we find that the Bieberbach conjecture holds true for slice starlike functions in contrast to the fact that the Bieberbach conjecture fails for biholomorphic starlike mappings in higher dimensions. We also establish some sharp versions of the growth, distortion, and covering theorems for quaternions.

Theorems & Definitions (62)

• Theorem A
• Theorem 1.1
• Theorem 1.2
• Theorem B
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Theorem 2.3