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

doi:10.1007/s10473-023-0402-2

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

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

Hongyan Liu¹, Zhenhan Tu¹, Liangpeng XIONG^2,†

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 XIONG^2,†

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/Abstract

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

Hongyan Liu, Zhenhan Tu, Liangpeng XIONG. DISTORTION THEOREMS FOR CLASSES OF g-PARAMETRIC STARLIKE MAPPINGS OF REAL ORDER IN $\mathbb{C}^n*$[J]. 数学物理学报(英文版), 2023, 43(4): 1491-1502.

Hongyan Liu, Zhenhan Tu, Liangpeng XIONG. DISTORTION THEOREMS FOR CLASSES OF g-PARAMETRIC STARLIKE MAPPINGS OF REAL ORDER IN $\mathbb{C}^n*$[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1491-1502.

参考文献

[1] Barnard R, FitzGerald C, Gong S. A distortion theorem for biholomorphic mappings in $\mathbb{C}^2$. Trans Am Math Soc, 1994, 344: 907-924
[2] Cartan H.Sur la possibilité détendre aux fonctions de plusieurs variables complexes la théorie des fonctions univalentes// Montel P. Lecons sur les Fonctions Univalentes ou Multivalentes. Paris: Gauthier-Villars, 1933
[3] Chu C H, Hamada H, Honda T, Kohr G. Distortion theorems for convex mappings on homogeneous balls. J Math Anal Appl, 2010, 369: 437-442
[4] Chu C H, Hamada H, Honda T, Kohr G. Distortion for locally biholomorphic Bloch mappings on bounded symmetric domains. J Math Anal Appl, 2016, 441: 830-843
[5] Duren P L. Univalent Functions.New York: Springer-Verlag, 1983
[6] Gong S, Wang S K, Yu Q H. Biholomorphic convex mappings of ball in $\mathbb{C}^n$. Pacif J Math, 1993, 161: 287-306
[7] Gong S, Liu T S. Distortion theorems for biholomorphic convex mappings on bounded convex circular domains. Chin Ann Math Ser B, 1999, 20: 297-304
[8] Graham I, Hamada H, Kohr G. Parametric representation of univalent mappings in several complex variables. Canadian J Math, 2002, 54: 324-351
[9] Graham I, Hamada H, Kohr G. Loewner chains, Bloch mappings and Pfaltzgraff-Suffridge extension operators on bounded symmetric domains. Complex Var Elliptic Equ, 2020, 65: 57-73
[10] Hamada H, Honda T, Kohr G.Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation. J Math Anal Appl, 2006, 317: 302-319
[11] Kohr G, Liczberski P.Univalent Mappings of Several Complex Variables. Cluj-Napoca: Cluj University Press, 1998
[12] Kohr G. Using the method of Löwner chains to introduce some subcalsses of biholomorphic mappings in $\mathbb{C}^n$. Rev Roum Math Pures Appl,2001, 46: 743-760
[13] Liu T S, Wang J F, Lu J. Distortion theorems for starlike mappings in several complex variables. Taiwanese J Math, 2011, 15: 2601-2608
[14] Liu X S, Liu T S. Sharp distortion theorems for a subclass of biholomorphic mappings which have a parametric representation in several complex variables. Chin Ann Math, 2016, 37B: 553-570
[15] Liu X S. Sharp distortion theorems for a class of biholomorphic mappings in several complex variables. Acta Mathematica Scientia, 2022, 42B(2): 454-466
[16] Nasr M A, Aouf M K. Radius of convexity for the class of starlike functions of complex order. Bull Fac Sci Assiut Univ Sect A, 1983, 12: 153-159
[17] Poreda T. On the univalent holomorphic maps of the unit polydisc of $\mathbb{C}^n$ which have the parametric representation, I-the geometric properties. Ann Univ Mariae Curie Sklodowsda Sect A, 1987, 41: 105-113
[18] Srivastava H M, Altιntç O, Serenbay S K. Coefficient bounds for certain subclasses of starlike functions of complex order. Appl Math Lett,2011, 24: 1359-1363
[19] Tu Z H, Xiong L P. Growth and distortion results for a class of biholomorphic mapping and extremal problem with parametric representation in $\mathbb{C}^n$. Complex Anal Oper Theory, 2019, 13: 2747-2769
[20] Wang J F. Distortion theorem for locally biholomorphic Bloch mappings on the unit ball $\mathbb{B}^n$. Bull malays Math Sci Soc, 2015, 38: 1657-1667
[21] Xu Q H, Liu T S. On the growth and covering theorem for normalized biholomorphic mappings. Chin J Cont Math, 2009, 30(2): 167-174
[22] Xu Q H, Liu T S. Sharp growth and distortion theorems for a subclass of biholomorphic mappings. Computer Math Appl, 2010, 59: 3778-3784
[23] Xiong L P. Distortion results for a certain subclass of biholomorphic mappings in $\mathbb{C}^n$. Complex Var Elliptic Equ, 2022, 67(4): 887-897
[24] Zhu Y C, Liu M S. Distortion theorems for biholomorphic convex mappings in Banach spaces. Complex Variables, 2005, 50: 57-68

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

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

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	Meng WEN, Haiyang LI, Changsong HU, Jigen PENG. ITERATIVE METHODS FOR OBTAINING AN INFINITE FAMILY OF STRICT PSEUDO-CONTRACTIONS IN BANACH SPACES[J]. 数学物理学报(英文版), 2022, 42(5): 1765-1778.
[2]	Ravinder Kumar SHARMA, Sumit CHANDOK. THE GENERALIZED HYPERSTABILITY OF GENERAL LINEAR EQUATION IN QUASI-2-BANACH SPACE[J]. 数学物理学报(英文版), 2022, 42(4): 1357-1372.
[3]	孙玉奇, 张文. COARSE ISOMETRIES BETWEEN FINITE DIMENSIONAL BANACH SPACES[J]. 数学物理学报(英文版), 2021, 41(5): 1493-1502.
[4]	Yekini SHEHU. SINGLE PROJECTION ALGORITHM FOR VARIATIONAL INEQUALITIES IN BANACH SPACES WITH APPLICATION TO CONTACT PROBLEM[J]. 数学物理学报(英文版), 2020, 40(4): 1045-1063.
[5]	沈钦锐. A VIEWPOINT TO MEASURE OF NON-COMPACTNESS OF OPERATORS IN BANACH SPACES[J]. 数学物理学报(英文版), 2020, 40(3): 603-613.
[6]	张子厚, Vicente MONTESINOS, 刘春燕. SOME METRIC AND TOPOLOGICAL PROPERTIES OF NEARLY STRONGLY AND NEARLY VERY CONVEX SPACES[J]. 数学物理学报(英文版), 2020, 40(2): 369-378.
[7]	Ioannis K. ARGYROS, Yeol Je CHO, Santhosh GEORGE, 肖义彬. LOCAL CONVERGENCE OF INEXACT NEWTON-LIKE METHOD UNDER WEAK LIPSCHITZ CONDITIONS[J]. 数学物理学报(英文版), 2020, 40(1): 199-210.
[8]	戴端旭, 郑本拓. STABILITY OF A PAIR OF BANACH SPACES FOR ε-ISOMETRIES[J]. 数学物理学报(英文版), 2019, 39(4): 1163-1172.
[9]	Muaadh ALMAHALEBI, Abdellatif CHAHBI. APPROXIMATE SOLUTION OF P-RADICAL FUNCTIONAL EQUATION IN 2-BANACH SPACES[J]. 数学物理学报(英文版), 2019, 39(2): 551-566.
[10]	于林, 王汝慧, 赵守江. CARLESON MEASURES AND THE GENERALIZED CAMPANATO SPACES OF VECTOR-VALUED MARTINGALES[J]. 数学物理学报(英文版), 2018, 38(6): 1779-1788.
[11]	Jasbir Singh MANHAS, Ruhan ZHAO. PRODUCTS OF WEIGHTED COMPOSITION AND DIFFERENTIATION OPERATORS INTO WEIGHTED ZYGMUND AND BLOCH SPACES[J]. 数学物理学报(英文版), 2018, 38(4): 1105-1120.
[12]	程立新, 罗思捷. YET ON LINEAR STRUCTURES OF NORM-ATTAINING FUNCTIONALS ON ASPLUND SPACES[J]. 数学物理学报(英文版), 2018, 38(1): 151-156.
[13]	Santhosh GEORGE, C. D. SREEDEEP. LAVRENTIEV'S REGULARIZATION METHOD FOR NONLINEAR ILL-POSED EQUATIONS IN BANACH SPACES[J]. 数学物理学报(英文版), 2018, 38(1): 303-314.
[14]	王紫, 王玉文. A NEW PERTURBATION THEOREM FOR MOORE-PENROSE METRIC GENERALIZED INVERSE OF BOUNDED LINEAR OPERATORS IN BANACH SPACES[J]. 数学物理学报(英文版), 2017, 37(6): 1619-1631.
[15]	邱德华, ?陈平炎, Volodin ANDREI. COMPLETE MOMENT CONVERGENCE FOR L^P-MIXINGALES[J]. 数学物理学报(英文版), 2017, 37(5): 1319-1330.