ON THE UNIVALENCE OF CERTAIN POLYHARMONIC MAPPINGS WITH BOUNDED LENGTH DISTORTIONS

doi:10.1007/s10473-024-0605-1

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (6): 2125-2138.doi: 10.1007/s10473-024-0605-1

ON THE UNIVALENCE OF CERTAIN POLYHARMONIC MAPPINGS WITH BOUNDED LENGTH DISTORTIONS

Xin WANG¹, Saminathan PONNUSAMY², Mingsheng LIU^3,†

1. Department of Mathematics, Shenzhen Polytechnic University, Shenzhen 518055, China;
2. Department of Mathematics, Indian Institute of Technology Madras, Chennai-600036, India;
3. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

收稿日期:2023-10-08 修回日期:2024-05-05 发布日期:2024-12-06
通讯作者: † Mingsheng LIU, E-mail: liumsh@scnu.edu.cn
作者简介:Xin WANG, E-mail: 574149386@qq.com; Saminathan PONNUSAMY, E-mail: samy@iitm.ac.in
基金资助:
Liu's research was supported by the Natural Science Foundation of Guangdong Province (2021A1515010058). Wang's research was supported by the Youth Innovation Foundation of Shenzhen Polytechnic University (6024310023K).

ON THE UNIVALENCE OF CERTAIN POLYHARMONIC MAPPINGS WITH BOUNDED LENGTH DISTORTIONS

Xin WANG¹, Saminathan PONNUSAMY², Mingsheng LIU^3,†

1. Department of Mathematics, Shenzhen Polytechnic University, Shenzhen 518055, China;
2. Department of Mathematics, Indian Institute of Technology Madras, Chennai-600036, India;
3. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received:2023-10-08 Revised:2024-05-05 Published:2024-12-06
Contact: † Mingsheng LIU, E-mail: liumsh@scnu.edu.cn
About author:Xin WANG, E-mail: 574149386@qq.com; Saminathan PONNUSAMY, E-mail: samy@iitm.ac.in
Supported by:
Liu's research was supported by the Natural Science Foundation of Guangdong Province (2021A1515010058). Wang's research was supported by the Youth Innovation Foundation of Shenzhen Polytechnic University (6024310023K).

摘要/Abstract

摘要： The main purpose of this paper is to investigate the univalence of normalized polyharmonic mappings with bounded length distortions in the unit disk. We first establish the coefficient estimates for polyharmonic mappings with bounded length distortions. Then, using these results, we establish five Landau-type theorems for subclasses of polyharmonic mappings $F$ and $L(F)$, where $F$ has bounded length distortion and $L$ is a differential operator.

关键词: harmonic mappings, biharmonic mappings, polyharmonic mappings, coefficients estimates, Landau-type theorems

Abstract: The main purpose of this paper is to investigate the univalence of normalized polyharmonic mappings with bounded length distortions in the unit disk. We first establish the coefficient estimates for polyharmonic mappings with bounded length distortions. Then, using these results, we establish five Landau-type theorems for subclasses of polyharmonic mappings $F$ and $L(F)$, where $F$ has bounded length distortion and $L$ is a differential operator.

Key words: harmonic mappings, biharmonic mappings, polyharmonic mappings, coefficients estimates, Landau-type theorems

中图分类号:

30C99

Xin WANG, Saminathan PONNUSAMY, Mingsheng LIU. ON THE UNIVALENCE OF CERTAIN POLYHARMONIC MAPPINGS WITH BOUNDED LENGTH DISTORTIONS[J]. 数学物理学报(英文版), 2024, 44(6): 2125-2138.

Xin WANG, Saminathan PONNUSAMY, Mingsheng LIU. ON THE UNIVALENCE OF CERTAIN POLYHARMONIC MAPPINGS WITH BOUNDED LENGTH DISTORTIONS[J]. Acta mathematica scientia,Series B, 2024, 44(6): 2125-2138.

参考文献

[1] Abdulhadi Z, Abu Muhanna Y, Khuri S.On some properties of solutions of the biharmonic equation. Appl Math Comput, 2006, 117: 346-351
[2] Bai X X, Liu M S.Landau-type theorems of polyharmonic mappings and log-$p$-harmonic mappings. Complex Anal Oper Theory, 2019, 13: 321-340
[3] Bhowmik B, Sen S.Landau and Bloch constants for meromorphic functions. Monatsh Math, 2023, 201(2): 359-373
[4] Chen H H, Gautheir P M, Hengartner W.Bloch constants for planar harmonic mappings. Proc Amer Math Soc, 2000, 128(11): 3231-3240
[5] Chen J L, Liu M S.Estimate on the Bloch constant for certain harmonic mappings under differential operator. Acta Math Sci, 2024, 44B(1): 295-310
[6] Chen J, Rasila A, Wang X.Landau's theorem for polyharmonic mappings. J Math Anal Appl, 2014, 409: 934-945
[7] Chen S, Ponnusamy S, Rasila A.Coefficient estimates, Landau's theorem and Lipschitz-type spaces on planar harmonic mappings. J Aust Math Soc, 2014, 96(2): 198-215
[8] Chen S, Ponnusamy S, Wang X.Bloch constant and Landau's theorem for planar $p$-harmonic mappings. J Math Appl Appl, 2011, 373: 102-110
[9] Chen S, Ponnusamy S, Wang X.Coefficient estimates and Landau-Bloch's theorem for planar harmonic mappings. Bull Malaysian Math Sciences Soc, 2011, 34(2): 255-265
[10] Chen S, Ponnusamy S, Wang X.On some properties of solutions of the $p$-harmonic equation. Filomat, 2013, 27(4): 577-591
[11] Chen S, Ponnusamy S, Wang X.Properties of some classes of planar harmonic and planar biharmonic mappings. Complex Anal Oper Theory, 2011, 5: 901-916
[12] Chen S F, Liu M S.Landau-type theorems and bi-lipschitz theorems for bounded biharmonic mappinngs. Monatsh Math, 2020, 193: 783-806
[13] Duren P.Harmonic Mappings in the Plane. Cambridge: Cambridge University Press, 2004
[14] Grigoryan A.Landau and Bloch theorems for harmonic mappings. Complex Variable Theory Appl, 2006, 51(1): 81-87
[15] Huang X Z.Sharp estimate on univalent radius for planar harmonic mappings with bounded Fréchet derivative
(in Chinese). Sci Sin Math, 2014, 44(6): 685-692
[16] Liu M S.Landau's theorems for biharmonic mappings. Complex Var Elliptic Eq, 2008, 53(9): 843-855
[17] Liu M S, Chen H H.The Landau-Bloch type theorems for planar harmonic mappings with bounded dilation. J Math Anal Appl, 2018, 468(2): 1066-1081
[18] Liu M S, Liu Z W, Zhu Y C.Landau's theorems for certain biharmonic mappings. Acta Math Sinica (Chinese Series), 2011, 54(1): 69-80
[19] Liu M S, Liu Z X.Landau-type theorems for $p$-harmonic mappings or $\log$-$p$-harmonic mappings. Applicable Analysis, 2014, 93(11): 2462-2477
[20] Liu M S, Luo L F.Precise values of the Bloch constants of certain log-$p$-harmonic mappings. Acta Math Sci, 2021, 41B(1): 297-310
[21] Liu M S, Ponnusamy S.Landau-type theorems for certain bounded bi-analytic functions and biharmonic mappings. Canadian Mathematical Bulletin, 2024, 67(1): 152-165
[22] Liu M S, Wang X, Kou K I.Estimates on Bloch constants for certain log-$p$-harmonic mappings. Monatsh Math, 2024, 203(1): 175-198
[23] Luo X, Liu M S.Landau-Bloch type theorems of certain subclasses of polyharmonic mappings. Comput. Methods and Function Theory, 2023, 23(2): 303-325
[24] Luo X, Liu M S, Li T.Landau-Bloch type theorems of certain subclasses of biharmonic mappings. Filomat, 2023, 37(2): 403-416
[25] Mocanu P T.Starlikeness and convexity for nonanalytic functions in the unit disc. Mathematica (Cluj), 1980, 22(1): 77-83
[26] Rajala K.A lower bound for the Bloch radius of $K$-quasiregular mappings. Proc Amer Math Soc, 2004, 132(9): 2593-2601
[27] Zhu Y C, Liu M S.Landau-type theorems for certain planar harmonic mappings or biharmonic mappings. Complex Var Elliptic Eq, 2013, 58(12): 1667-1676

ON THE UNIVALENCE OF CERTAIN POLYHARMONIC MAPPINGS WITH BOUNDED LENGTH DISTORTIONS

ON THE UNIVALENCE OF CERTAIN POLYHARMONIC MAPPINGS WITH BOUNDED LENGTH DISTORTIONS

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 2

Metrics

本文评价

推荐阅读 10

[1]	郭常予, 向长林. REGULARITY OF P-HARMONIC MAPPINGS INTO NPC SPACES[J]. 数学物理学报(英文版), 2021, 41(2): 633-645.
[2]	刘名生, 骆莉芳. PRECISE VALUES OF THE BLOCH CONSTANTS OF CERTAIN LOG-p-HARMONIC MAPPINGS[J]. 数学物理学报(英文版), 2021, 41(1): 297-310.