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

doi:10.1007/s10473-024-0116-0

Abstract

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

CLC Number:

30C99

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^*

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