English

Bohr radius for some classes of Harmonic mappings

Complex Variables 2020-10-06 v1

Abstract

We introduce a general class of sense-preserving harmonic mappings defined as follows: \begin{equation*} \mathcal{S}^0_{h+\bar{g}}(M):= \{f=h+\bar{g}: \sum_{m=2}^{\infty}(\gamma_m|a_m|+\delta_m|b_m|)\leq M, \; M>0 \}, \end{equation*} where h(z)=z+m=2amzmh(z)=z+\sum_{m=2}^{\infty}a_mz^m, g(z)=m=2bmzmg(z)=\sum_{m=2}^{\infty}b_m z^m are analytic functions in D:={zC:z1}\mathbb{D}:=\{z\in\mathbb{C}: |z|\leq1 \} and \begin{equation*} \gamma_m,\; \delta_m \geq \alpha_2:=\min \{\gamma_2, \delta_2\}>0, \end{equation*} for all m2m\geq2. We obtain Growth Theorem, Covering Theorem and derive the Bohr radius for the class Sh+gˉ0(M)\mathcal{S}^0_{h+\bar{g}}(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.

Keywords

Cite

@article{arxiv.2010.01304,
  title  = {Bohr radius for some classes of Harmonic mappings},
  author = {S. Sivaprasad Kumar and Kamaljeet Gangania},
  journal= {arXiv preprint arXiv:2010.01304},
  year   = {2020}
}
R2 v1 2026-06-23T18:59:42.763Z