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Bohr’s phenomenon for the classes of Quasi-subordination and K-quasiregular harmonic mappings

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Abstract

In this paper, we investigate the Bohr radius for K-quasiregular sense-preserving harmonic mappings \(f=h+{\overline{g}}\) in the unit disk \({\mathbb {D}}\) such that the translated analytic part \(h(z)-h(0)\) is quasi-subordinate to some analytic function. The main aim of this article is to extend and to establish sharp versions of four recent theorems by Liu and Ponnusamy (Bull Malays Math Sci Soc 42:2151–2168, 2019) and, in particular, we settle affirmatively the two conjectures proposed by them. Furthermore, we establish two refined versions of Bohr’s inequalities and determine the Bohr radius for the derivatives of analytic functions associated with quasi-subordination.

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Acknowledgements

We thank the referee for his/her careful reading of our paper and invaluable comments. The research of the first author was supported by Guangdong Natural Science Foundation of China (No. 2018A030313508). The work of the second author was supported by Mathematical Research Impact Centric Support (MATRICS) of the Department of Science and Technology (DST), India (MTR/2017/000367). The third author was supported by the Natural Science Foundation of China (No. 11771090). The first author would also thank the Laboratory of Mathematics of Nonlinear Sciences, Fudan University (LMNS) for its support during his visit to Fudan University.

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Correspondence to Saminathan Ponnusamy.

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Liu, MS., Ponnusamy, S. & Wang, J. Bohr’s phenomenon for the classes of Quasi-subordination and K-quasiregular harmonic mappings. RACSAM 114, 115 (2020). https://doi.org/10.1007/s13398-020-00844-0

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