Slice starlike functions over quaternions

TL;DR

This paper explores the geometric function theory of slice starlike functions over quaternions, establishing new results that contrast with higher-dimensional complex cases, including the validity of the Bieberbach conjecture.

Contribution

It introduces the study of slice starlike functions over quaternions and proves the Bieberbach conjecture holds for these functions, unlike in higher-dimensional complex analysis.

Findings

Bieberbach conjecture holds for slice starlike functions

Established sharp growth, distortion, and covering theorems for quaternionic functions

Contrasts with failure of the conjecture in higher-dimensional holomorphic mappings

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.