The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3775–3806 | Cite as

Slice Starlike Functions Over Quaternions

  • Zhenghua XuEmail author
  • Guangbin Ren


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.


Starlike function Quaternion Bieberbach conjecture Fekete–Szegö inequality Growth and distortion theorems Bloch–Landau theorem 

Mathematics Subject Classification

Primary 30G35 Secondary 30C50 



This work was supported by the NNSF of China (11371337).


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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.School of MathematicsHeFei University of TechnologyHefeiChina
  2. 2.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiChina

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