A Unified Study of Bohr's Inequality for analytic and harmonic mappings on the Unit Disk
Authors
Molla Basir Ahamed
Partha Pratim Roy
Sujoy Majumder
Abstract
We investigate improved forms of the Bohr inequality, using the quantity $S_r/π$, 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 $\mathcal{W}^0_{\mathcal{H}}(α)$ 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.
