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Bohr-Type Inequalities for Harmonic Mappings with a Multiple Zero at the Origin

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Abstract

In this paper, we first determine Bohr’s inequality for the class of harmonic mappings \(f=h+\overline{g}\) in the unit disk \(\mathbb {D}\), where either both \(h(z)=\sum _{n=0}^{\infty }a_{pn+m}z^{pn+m}\) and \(g(z)=\sum _{n=0}^{\infty }b_{pn+m}z^{pn+m}\) are analytic and bounded in \(\mathbb {D}\), or satisfies the condition \(|g'(z)|\le d|h'(z)|\) in \(\mathbb {D}\backslash \{0\}\) for some \(d\in [0,1]\) and h is bounded. In particular, we obtain Bohr’s inequality for the class of harmonic p-symmetric mappings. Also, we investigate the Bohr-type inequalities of harmonic mappings with a multiple zero at the origin and that most of results are proved to be sharp.

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Data Availability Statement

The authors declare that this research is purely theoretical and does not associate with any data.

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Acknowledgements

This research of the first two authors is partly supported by Guangdong Natural Science Foundations (Grant No. 2021A030313326). The work of the third author was supported by Mathematical Research Impact Centric Support (MATRICS) of the Department of Science and Technology (DST), India (MTR/2017/000367).

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, Guangdong, China

    Yong Huang & Ming-Sheng Liu

  2. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600 036, India

    Saminathan Ponnusamy

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  1. Yong Huang
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  2. Ming-Sheng Liu
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  3. Saminathan Ponnusamy
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Correspondence to Ming-Sheng Liu.

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Huang, Y., Liu, MS. & Ponnusamy, S. Bohr-Type Inequalities for Harmonic Mappings with a Multiple Zero at the Origin. Mediterr. J. Math. 18, 75 (2021). https://doi.org/10.1007/s00009-021-01726-4

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  • DOI: https://doi.org/10.1007/s00009-021-01726-4

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