ON THE UNIVALENCE OF CERTAIN POLYHARMONIC MAPPINGS WITH BOUNDED LENGTH DISTORTIONS

doi:10.1007/s10473-024-0605-1

Acta mathematica scientia,Series B ›› 2024, Vol. 44 ›› Issue (6): 2125-2138.doi: 10.1007/s10473-024-0605-1

Previous Articles Next Articles

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).

Abstract

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

CLC Number:

30C99

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.

Trendmd

References

[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

Abstract

Cite this article

share this article

References

Related Articles 2

Metrics

Comments

Recommended 10

[1]	Changyu GUO, Changlin XIANG. REGULARITY OF P-HARMONIC MAPPINGS INTO NPC SPACES [J]. Acta mathematica scientia,Series B, 2021, 41(2): 633-645.
[2]	Mingsheng LIU, Lifang LUO. PRECISE VALUES OF THE BLOCH CONSTANTS OF CERTAIN LOG-p-HARMONIC MAPPINGS [J]. Acta mathematica scientia,Series B, 2021, 41(1): 297-310.