Bohr inequalities via proper combinations for a certain class of close-to-convex harmonic mappings

Abstract

Let $\mathcal{H}(\Omega )$ be the class of complex-valued functions harmonic in $\Omega \subset \mathbb{C}$ and each $f=h+\overline{g}\in \mathcal{H}(\Omega )$, where $h$ and $g$ are analytic. In the study of Bohr phenomenon for certain class of harmonic mappings, it is to find a constant $r_f\in (0,1)$ such that the inequality \begin{align*} M_f(r):=r+\sum_{n=2}^{\infty}\left(|a_n|+|b_n|\right)r^n\leq d\left(f(0), \partialΩ\right) \;\mbox{for}\;|z|=r\leq r_f, \end{align*} where $d\left(f(0),\partial \Omega \right)$ is the Euclidean distance between $f(0)$ and the boundary of $\Omega :=f(\mathbb{D})$. The largest such radius $r_f$ is called the Bohr radius and the inequality $M_f(r)\le d\left(f(0),\partial \Omega \right)$ is called the Bohr inequality for the class $\mathcal{H}(\Omega )$. In this paper, we study Bohr phenomenon for the class of close-to-convex harmonic mappings establishing several inequalities. All the results are proved to be sharp.

26 pages, 9 figures