CONVEX MAPPINGS ASSOCIATED WITH THE ROPER-SUFFRIDGE EXTENSION OPERATOR

doi:10.1007/s10473-019-0612-9

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (6): 1619-1627.doi: 10.1007/s10473-019-0612-9

CONVEX MAPPINGS ASSOCIATED WITH THE ROPER-SUFFRIDGE EXTENSION OPERATOR

张丹莉¹, 徐辉明¹, 王建飞²

1. Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China;
2. School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

收稿日期:2018-05-21 修回日期:2019-05-16 出版日期:2019-12-25 发布日期:2019-12-30
通讯作者: Jianfei WANG,E-mail:wangjf@mail.ustc.edu.cn E-mail:wangjf@mail.ustc.edu.cn
作者简介:Danli ZHANG,E-mail:shirley9519@163.com;Huiming XU,E-mail:xhm@zjnu.cn
基金资助:
The project was partially supported by the National Natural Science Foundation of China (11671362, 11571105), Beijing Municipal Natural Science Foundation (1182008) and the Scientific Research Funds of Huaqiao University.

CONVEX MAPPINGS ASSOCIATED WITH THE ROPER-SUFFRIDGE EXTENSION OPERATOR

Danli ZHANG¹, Huiming XU¹, Jianfei WANG²

1. Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China;
2. School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

Received:2018-05-21 Revised:2019-05-16 Online:2019-12-25 Published:2019-12-30
Contact: Jianfei WANG,E-mail:wangjf@mail.ustc.edu.cn E-mail:wangjf@mail.ustc.edu.cn
Supported by:
The project was partially supported by the National Natural Science Foundation of China (11671362, 11571105), Beijing Municipal Natural Science Foundation (1182008) and the Scientific Research Funds of Huaqiao University.

摘要/Abstract

摘要： Let λ_G(z)|dz|be the hyperbolic metric on a simply connected proper domain G ⊂ C containing the origin, and let||·||_j be the Banach norms of C^n_j for j=1, 2, …, k.This note is to prove that if f is a normalized biholomorphic convex function on G, then
Φ_{N,1/p1,…,1/pk}(f)(z)=F_{1/p1,…,1/pk}(z)=f(z₁), (f'(z₁))^1/p1z, …, (f'(z₁))^1/pkw)
is a normalized biholomorphic convex mapping on
Ω_N={(z₁, z, …, w) ∈ C×C^n₁×…×C^n_k:||z||₁^p1 + … +||w||_k^pk<1/λ_G (z₁)},
where N=1 + n₁ + … + n_k and the branch is chosen such that (f'(z₁))^1/p_j|_z₁=0=1, j=1, …, k. Applying to the Roper-Suffridge extension operator, we obtain a new convex mappings construction of an unbounded domain and a refinement of convex mappings construction on a Reinhardt domain, respectively.

关键词: Roper-Suffridge operator, convex mapping, hyperbolic metric

Abstract: Let λ_G(z)|dz|be the hyperbolic metric on a simply connected proper domain G ⊂ C containing the origin, and let||·||_j be the Banach norms of C^n_j for j=1, 2, …, k.This note is to prove that if f is a normalized biholomorphic convex function on G, then
Φ_{N,1/p1,…,1/pk}(f)(z)=F_{1/p1,…,1/pk}(z)=f(z₁), (f'(z₁))^1/p1z, …, (f'(z₁))^1/pkw)
is a normalized biholomorphic convex mapping on
Ω_N={(z₁, z, …, w) ∈ C×C^n₁×…×C^n_k:||z||₁^p1 + … +||w||_k^pk<1/λ_G (z₁)},
where N=1 + n₁ + … + n_k and the branch is chosen such that (f'(z₁))^1/p_j|_z₁=0=1, j=1, …, k. Applying to the Roper-Suffridge extension operator, we obtain a new convex mappings construction of an unbounded domain and a refinement of convex mappings construction on a Reinhardt domain, respectively.

Key words: Roper-Suffridge operator, convex mapping, hyperbolic metric

中图分类号:

32H02

张丹莉, 徐辉明, 王建飞. CONVEX MAPPINGS ASSOCIATED WITH THE ROPER-SUFFRIDGE EXTENSION OPERATOR[J]. 数学物理学报(英文版), 2019, 39(6): 1619-1627.

Danli ZHANG, Huiming XU, Jianfei WANG. CONVEX MAPPINGS ASSOCIATED WITH THE ROPER-SUFFRIDGE EXTENSION OPERATOR[J]. Acta mathematica scientia,Series B, 2019, 39(6): 1619-1627.

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CONVEX MAPPINGS ASSOCIATED WITH THE ROPER-SUFFRIDGE EXTENSION OPERATOR

CONVEX MAPPINGS ASSOCIATED WITH THE ROPER-SUFFRIDGE EXTENSION OPERATOR

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