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Bohr Radius for Some Classes of Harmonic Mappings

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Abstract

In this note, we introduce a general class of sense-preserving harmonic mappings defined as follows:

$$\begin{aligned} \mathcal {B}_{\mathcal {H}}^{0}(M):= \{f=h+\bar{g}: \sum _{m=2}^{\infty }(\gamma _m|a_m|+\delta _m|b_m|)\le M, \; M>0 \}, \end{aligned}$$

where \(h(z)=z+\sum _{m=2}^{\infty }a_mz^m\), \(g(z)=\sum _{m=2}^{\infty }b_m z^m\) are analytic functions in \(\mathbb {D}:=\{z\in \mathbb {C}: |z|\le 1 \}\) and

$$\begin{aligned} \gamma _m,\; \delta _m \ge \alpha _2:=\min \{\gamma _2, \delta _2\}>0, \end{aligned}$$

for all \(m\ge 2\). We obtain Growth Theorem, Covering Theorem, and derive the Bohr radius for the class \(\mathcal {B}_{\mathcal {H}}^{0}(M)\). As an application of our results, we obtain the Bohr radius for many classes of harmonic univalent functions and some classes of univalent functions.

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Acknowledgement

We are thankful to the Editor and the Reviewers for their valuable suggestions to improve the previous version of this manuscript.

Funding

The work of Kamajeet Gangania is supported by University Grant Commission, New-Delhi, India under UGC-Ref. No.:1051/(CSIR-UGC NET JUNE 2017).

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Correspondence to K. Gangania.

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The work of Kamajeet Gangania is supported by University Grant Commission, New-Delhi, India under UGC-Ref. No.:1051/(CSIR-UGC NET JUNE 2017)

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Gangania, K., Kumar, S.S. Bohr Radius for Some Classes of Harmonic Mappings. Iran J Sci Technol Trans Sci 46, 883–890 (2022). https://doi.org/10.1007/s40995-022-01304-7

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