Sharp Bohr and Bohr-Rogosinski inequalities involving area measure for close-to-convex harmonic mappings

Abstract

In this article, we investigate refined and generalized versions of the Bohr and Bohr--Rogosinski inequalities for a normalized subclass ${\mathcal{P}}_{\mathcal{H}}^0(\alpha )$ ($0\le \alpha <1$) of univalent close-to-convex harmonic mappings $f=h+\overline{g}$ defined on the open unit disk $\mathbb{D}\subset \mathbb{C}$. By incorporating non-negative monotone increasing functions associated with the planar area integral $S_r/\pi$ of the image domain $f({\mathbb{D}}_r)$, we establish new sharp Bohr-type inequalities expressed in terms of the Euclidean distance $d(f(0),\partial f(\mathbb{D}))$. Furthermore, we establish sharp Bohr--Rogosinski-type inequalities involving powers of the modulus of the mapping $|f(z){|}^p$ ($p\ge 1$). All associated radii are proven to be sharp, and extremal functions realizing the equality cases are explicitly identified. As applications, our results generalize and unify several well-known classical and recent theorems in geometric function theory.

13 pages, 2 figures