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 ($\mathcal{P}^0{\mathcal{H}}(M)$ and $\mathcal{W}^0_{\mathcal{H}}(\alpha)$ of the form $f = h + \overline{g}$) by introducing a sequence ${\varphi_n(r)}_{n=0}^\infty$ 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.