The sharp refined Bohr–Rogosinski inequalities for certain classes of harmonic mappings

Abstract

A class $\mathcal{F}$ consisting of analytic functions $f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ in the unit disc $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ satisfies a Bohr phenomenon if there exists an $r_f>0$ such that $\sum _{n=1}^{\mathrm{\infty }}|a_n|r^n\le d(f(0),\mathrm{\partial f}(\mathbb{D}))$ for every function $f\in \mathcal{F}$, and $|z|=r\le r_f$. The largest radius $r_f$ is the Bohr radius and the inequality $\sum _{n=1}^{\mathrm{\infty }}|a_n|r^n\le d(f(0),\mathrm{\partial f}(\mathbb{D}))$ is Bohr inequality for the class $\mathcal{F}$, where ‘d’ is the Euclidean distance. In this paper, we prove sharp refinement of the Bohr–Rogosinski inequality for certain classes of harmonic mappings.

Keywords:

• Harmonic functions
• close-to-convex functions
• coefficient estimates
• growth theorem
• Bohr radius
• Bohr-Rogosisnki radius

AMS Subject Classifications:

• Primary 30C45
• 30C50
• 30C80

Acknowledgments

The authors wish to thank the anonymous referees for their helpful suggestions and comments to enhance the clarity and presentation of the paper.

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The author is supported by ‘JU Research Grant’ no.: S-$3/10/22$, dated: $\frac{15}{17}$$\mathrm{.03.2022}$, Jadavpur University, West Bengal, India.