A Unified Study of Bohr's Inequality for analytic and harmonic mappings on the Unit Disk

Abstract

We investigate improved forms of the Bohr inequality, using the quantity $S_r/\pi$, for analytic selfmaps in class $\mathcal{B}$ of $\mathbb{D}$, where $S_r$ is the area measure of ${\mathbb{D}}_r$. We then generalize the inequality for harmonic mappings (${\mathcal{P}}_{\mathcal{H}}^0(M)$ and ${\mathcal{W}}_{\mathcal{H}}^0(\alpha )$ of the form $f=h+\overline{g}$) by introducing a sequence $\{{\phi }_n(r){\}}_{n=0}^{\infty }$ of differentiable, increasing functions on $[0,1)$. The Hurwitz Lerch Zeta function is utilized for some consequences, and all results are shown to be sharp.

28 pages. arXiv admin note: substantial text overlap with arXiv:2402.15689, arXiv:2212.05710