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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Roper-Suffridge extension operator and its applications to convex mappings in $ {\mathbb{C}}^{2}$


Authors: Jianfei Wang and Taishun Liu
Journal: Trans. Amer. Math. Soc. 370 (2018), 7743-7759
MSC (2010): Primary 32H02; Secondary 30C55
DOI: https://doi.org/10.1090/tran/7221
Published electronically: May 3, 2018
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Abstract: The purpose of this paper is twofold. The first is to investigate the Roper-Suffridge extension operator which maps a biholomorhic function $ f$ on $ D$ to a biholomorphic mapping $ F$ on

$\displaystyle \Omega _{n,p_{2},\cdots ,p_{n}}(D)=\left \{(z_1,z_0)\in D\times {... ...\vert z_{j}\vert^{p_{j}}<\frac {1}{\lambda _{D}(z_1)}\right \},\,\,\,p_j\geq 1,$    

where $ z_0=(z_2,\ldots ,z_n)$ and $ \lambda _{D}$ is the density of the Poincar $ \acute {e}$ metric on a simply connected domain $ D\subset \mathbb{C}$. We prove this Roper-Suffridge extension operator preserves $ \varepsilon $-starlike mapping: if $ f$ is $ \varepsilon $-starlike, then so is $ F$. As a consequence, we solve a problem of Graham and Kohr in a new method. By introducing the scaling method, the second part is to construct some new convex mappings of domain $ \Omega _{2,\,m}=\{(z_1,z_2)\in {\mathbb{C}}^{2}:\vert z_1\vert^2+\vert z_2\vert^m<1\}$ with $ m\geq 2$, which can be applied to discuss the extremal point of convex mappings on the domain. This scaling idea can be viewed as providing an alternative approach to studying convex mappings on $ \Omega _{2,\,m}$. Moreover, we propose some problems.

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Additional Information

Jianfei Wang
Affiliation: School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian, 362021, People’s Republic of China–and–Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, People’s Republic of China
Email: wangjf@mail.ustc.edu.cn

Taishun Liu
Affiliation: Department of Mathematics, Huzhou University, Huzhou, Zhejiang, 313000, People’s Republic of China
Email: lts@ustc.edu.cn

DOI: https://doi.org/10.1090/tran/7221
Received by editor(s): August 9, 2016
Received by editor(s) in revised form: December 18, 2016, and February 5, 2017
Published electronically: May 3, 2018
Additional Notes: The first author’s research was supported by the National Natural Science Foundation of China (Nos. 11671362 and 11001246) and the Natural Science Foundation of Zhejiang Province (No. LY16A010004). The first author is the corresponding author.
The second author’s research was supported by the National Natural Science Foundation of China (Nos. 11471111 and 11571105).
Dedicated: Dedicated to the memory of Professor Sheng Gong
Article copyright: © Copyright 2018 American Mathematical Society

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