Revisiting Bohr Inequalities with Analytic and Harmonic Mappings on unit disk

Abstract

In this paper, we study some improved and refined versions of the classical Bohr inequality applicable to the class $\mathcal{B}$, which consists of self-analytic mappings defined on the unit disk $\mathbb{D}$. First, we improve the Bohr inequality for the class $\mathcal{B}$ of analytic self-maps, incorporating the area measurements of sub-disks ${\mathbb{D}}_r$ of $\mathbb{D}$. Secondly, we establish a sharp inequality with suitable setting as an improved version of the classic Bohr inequality. Then we obtain a sharp refined Bohr inequality in which the coefficients $|a_k|$ $(k=0,1,2,3)$ in the majorant series $M_f(r)$ of $f$ are replaced by $|f^{(k)}(z)|/k!$. Finally, for a certain class ${\mathcal{P}}_{\mathcal{H}}^0(M)$ of harmonic mappings of the form $f=h+\overline{g}$, we generalize the Bohr inequality incorporating a sequence $\{{\phi }_n(r){\}}_{n=0}^{\infty }$ of continuous functions of $r$ in $[0,1)$ and give some applications.

25 pages, 0 figures