ESTIMATE ON THE BLOCH CONSTANT FOR CERTAIN HARMONIC MAPPINGS UNDER A DIFFERENTIAL OPERATOR*

doi:10.1007/s10473-024-0116-0

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (1): 295-310.doi: 10.1007/s10473-024-0116-0

ESTIMATE ON THE BLOCH CONSTANT FOR CERTAIN HARMONIC MAPPINGS UNDER A DIFFERENTIAL OPERATOR^*

Jieling Chen, Mingsheng Liu^†

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

收稿日期:2022-08-04 修回日期:2023-07-13 出版日期:2024-02-25 发布日期:2024-02-27
通讯作者: † Mingsheng Liu, E-mail: liumsh@scnu.edu.cn
作者简介:Jieling Chen ,E-mail: 1304889502@qq.com
基金资助:
Natural Science Foundation of Guangdong Province (2021A1515010058).

ESTIMATE ON THE BLOCH CONSTANT FOR CERTAIN HARMONIC MAPPINGS UNDER A DIFFERENTIAL OPERATOR^*

Jieling Chen, Mingsheng Liu^†

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received:2022-08-04 Revised:2023-07-13 Online:2024-02-25 Published:2024-02-27
Contact: † Mingsheng Liu, E-mail: liumsh@scnu.edu.cn
About author:Jieling Chen ,E-mail: 1304889502@qq.com
Supported by:
Natural Science Foundation of Guangdong Province (2021A1515010058).

摘要/Abstract

摘要： In this paper, we first obtain the precise values of the univalent radius and the Bloch constant for harmonic mappings of the form $L(f)=z f_z-\bar{z} f_{\bar{z}}$, where $f$ represents normalized harmonic mappings with bounded dilation. Then, using these results, we present better estimations for the Bloch constants of certain harmonic mappings $L(f)$, where $f$ is a $K$-quasiregular harmonic or open harmonic. Finally, we establish three versions of Bloch-Landau type theorem for biharmonic mappings of the form $L(f)$. These results are sharp in some given cases and improve the related results of earlier authors.

关键词: Bloch-Landau type theorem, Bloch constant, linear complex operator, harmonic mapping, biharmonic mapping, univalent

Abstract: In this paper, we first obtain the precise values of the univalent radius and the Bloch constant for harmonic mappings of the form $L(f)=z f_z-\bar{z} f_{\bar{z}}$, where $f$ represents normalized harmonic mappings with bounded dilation. Then, using these results, we present better estimations for the Bloch constants of certain harmonic mappings $L(f)$, where $f$ is a $K$-quasiregular harmonic or open harmonic. Finally, we establish three versions of Bloch-Landau type theorem for biharmonic mappings of the form $L(f)$. These results are sharp in some given cases and improve the related results of earlier authors.

Key words: Bloch-Landau type theorem, Bloch constant, linear complex operator, harmonic mapping, biharmonic mapping, univalent

中图分类号:

30C99

Jieling Chen, Mingsheng Liu. ESTIMATE ON THE BLOCH CONSTANT FOR CERTAIN HARMONIC MAPPINGS UNDER A DIFFERENTIAL OPERATOR^*[J]. 数学物理学报(英文版), 2024, 44(1): 295-310.

Jieling Chen, Mingsheng Liu. ESTIMATE ON THE BLOCH CONSTANT FOR CERTAIN HARMONIC MAPPINGS UNDER A DIFFERENTIAL OPERATOR^*[J]. Acta mathematica scientia,Series B, 2024, 44(1): 295-310.

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ESTIMATE ON THE BLOCH CONSTANT FOR CERTAIN HARMONIC MAPPINGS UNDER A DIFFERENTIAL OPERATOR^*

ESTIMATE ON THE BLOCH CONSTANT FOR CERTAIN HARMONIC MAPPINGS UNDER A DIFFERENTIAL OPERATOR^*

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