数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1491-1502.doi: 10.1007/s10473-023-0402-2

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DISTORTION THEOREMS FOR CLASSES OF g-PARAMETRIC STARLIKE MAPPINGS OF REAL ORDER IN $\mathbb{C}^n*$

Hongyan Liu1, Zhenhan Tu1, Liangpeng XIONG2,†   

  1. 1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;
    2. School of Mathematics and Computer Science, Jiangxi Science and Technology Normal University, Nanchang 330038, China
  • 收稿日期:2022-04-15 修回日期:2022-09-26 发布日期:2023-08-08
  • 通讯作者: †Liangpeng XIONG, E-mail: lpxiong2016@whu.edu.cn
  • 作者简介:Hongyan Liu, E-mail: hongyanliu@whu.edu.cn; Zhenhan Tu, E-mail: zhhtu.math@whu.edu.cn
  • 基金资助:
    * National Natural Science Foundation of China (12071354); XIONG was supported by the National Natural Science Foundation of China (12061035), the Jiangxi Provincial Natural Science Foundation (20212BAB201012), the Research Foundation of Jiangxi Provincial Department of Education (GJJ201104) and the Research Foundation of Jiangxi Science and Technology Normal University (2021QNBJRC003).

DISTORTION THEOREMS FOR CLASSES OF g-PARAMETRIC STARLIKE MAPPINGS OF REAL ORDER IN $\mathbb{C}^n*$

Hongyan Liu1, Zhenhan Tu1, Liangpeng XIONG2,†   

  1. 1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;
    2. School of Mathematics and Computer Science, Jiangxi Science and Technology Normal University, Nanchang 330038, China
  • Received:2022-04-15 Revised:2022-09-26 Published:2023-08-08
  • Contact: †Liangpeng XIONG, E-mail: lpxiong2016@whu.edu.cn
  • About author:Hongyan Liu, E-mail: hongyanliu@whu.edu.cn; Zhenhan Tu, E-mail: zhhtu.math@whu.edu.cn
  • Supported by:
    * National Natural Science Foundation of China (12071354); XIONG was supported by the National Natural Science Foundation of China (12061035), the Jiangxi Provincial Natural Science Foundation (20212BAB201012), the Research Foundation of Jiangxi Provincial Department of Education (GJJ201104) and the Research Foundation of Jiangxi Science and Technology Normal University (2021QNBJRC003).

摘要: In this paper, we define the class $\widehat{\mathcal {S}}^{\gamma}_g(\mathbb{B}_{\mathbb{X}})$ of $g$-parametric starlike mappings of real order $\gamma$ on the unit ball $\mathbb{B}_{\mathbb{X}}$ in a complex Banach space $\mathbb{X}$, where $g$ is analytic and satisfies certain conditions. By establishing the distortion theorem of the Fr\'{e}chet-derivative type of $\widehat{\mathcal {S}}^{\gamma}_g(\mathbb{B}_{\mathbb{X}})$ with a weak restrictive condition, we further obtain the distortion results of the Jacobi-determinant type and the Fr\'{e}chet-derivative type for the corresponding classes (compared with $\widehat{\mathcal {S}}^{\gamma}_g(\mathbb{B}_{\mathbb{X}})$) defined on the unit polydisc (resp. unit ball with the arbitrary norm) in the space of $n$-dimensional complex variables, $n\geqslant2$. Our results extend the classic distortion theorem of holomorphic functions from the case in one-dimensional complex space to the case in the higher dimensional complex space. The main theorems also generalize and improve some recent works.

关键词: Banach space, distortion theorem of Jacobi-determinanttype, distortion theorems of the Fréchet-derivative type, $g$-parametric starlike mappings

Abstract: In this paper, we define the class $\widehat{\mathcal {S}}^{\gamma}_g(\mathbb{B}_{\mathbb{X}})$ of $g$-parametric starlike mappings of real order $\gamma$ on the unit ball $\mathbb{B}_{\mathbb{X}}$ in a complex Banach space $\mathbb{X}$, where $g$ is analytic and satisfies certain conditions. By establishing the distortion theorem of the Fr\'{e}chet-derivative type of $\widehat{\mathcal {S}}^{\gamma}_g(\mathbb{B}_{\mathbb{X}})$ with a weak restrictive condition, we further obtain the distortion results of the Jacobi-determinant type and the Fr\'{e}chet-derivative type for the corresponding classes (compared with $\widehat{\mathcal {S}}^{\gamma}_g(\mathbb{B}_{\mathbb{X}})$) defined on the unit polydisc (resp. unit ball with the arbitrary norm) in the space of $n$-dimensional complex variables, $n\geqslant2$. Our results extend the classic distortion theorem of holomorphic functions from the case in one-dimensional complex space to the case in the higher dimensional complex space. The main theorems also generalize and improve some recent works.

Key words: Banach space, distortion theorem of Jacobi-determinanttype, distortion theorems of the Fréchet-derivative type, $g$-parametric starlike mappings