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Cauchy transforms of self-similar measures: Starlikeness and univalence


Authors: Xin-Han Dong, Ka-Sing Lau and Hai-Hua Wu
Journal: Trans. Amer. Math. Soc. 369 (2017), 4817-4842
MSC (2010): Primary 28A80; Secondary 30C55, 30E20
DOI: https://doi.org/10.1090/tran/6819
Published electronically: December 7, 2016
MathSciNet review: 3632551
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Abstract: For the contractive iterated function system $ S_kz=e^{2\pi ik/m}+{\rho (z-e^{2\pi ik/m})}$ with $ 0<\rho <1, k=0,\cdots , m-1$, we let $ K\subset \mathbb{C}$ be the attractor, and let $ \mu $ be a self-similar measure defined by $ \mu =\frac 1m\sum _{k=0}^{m-1}\mu \circ S_k^{-1}$. We consider the Cauchy transform $ F$ of $ \mu $. It is known that the image of $ F$ at a small neighborhood of the boundary of $ K$ has very rich fractal structure, which is coined the Cantor boundary behavior. In this paper, we investigate the behavior of $ F$ away from $ K$; it has nice geometry and analytic properties, such as univalence, starlikeness and convexity. We give a detailed investigation for those properties in the general situation as well as certain classical cases of self-similar measures.


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Additional Information

Xin-Han Dong
Affiliation: College of Mathematics and Econometrics, Hunan University, Changsha, 410082, People’s Republic of China – and – Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China
Email: xhdong@hunnu.edu.cn

Ka-Sing Lau
Affiliation: College of Mathematics and Econometrics, Hunan University, Changsha, 410082, People’s Republic of China – and – Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
Email: kslau@math.cuhk.edu.hk

Hai-Hua Wu
Affiliation: College of Mathematics and Econometrics, Hunan University, Changsha, 410082, People’s Republic of China – and – Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China
Email: hunaniwa@163.com

DOI: https://doi.org/10.1090/tran/6819
Keywords: Self-similar measure, Cauchy transform, radius of starlikeness, convexity, univalence
Received by editor(s): December 12, 2014
Received by editor(s) in revised form: July 16, 2015
Published electronically: December 7, 2016
Additional Notes: This research was supported in part by the NNSF of China (No. 11571099), an HKRGC grant, SRFDP of Higher Education (No. 20134306110003), and Scientific Research Fund of Hunan Provincial Education Department (No. 14K057). The first author is the corresponding author
Article copyright: © Copyright 2016 American Mathematical Society