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 𝒫ℋ0(α) (0 ≤ α< 1) of univalent close-to-convex harmonic mappings f = h + g̅ defined on the open unit disk 𝔻 ⊂ ℂ. By incorporating non-negative monotone increasing functions associated with the planar area integral Sr/π of the image domain f(𝔻r), we establish new sharp Bohr-type inequalities expressed in terms of the Euclidean distance d(f(0), ∂ f(𝔻)). Furthermore, we establish sharp Bohr--Rogosinski-type inequalities involving powers of the modulus of the mapping |f(z)|p (p ≥ 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.
Related
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BibTeX
@misc{ahamed2026sharp,
title = {{Sharp Bohr and Bohr-Rogosinski inequalities involving area measure for close-to-convex harmonic mappings}},
author = {Molla Basir Ahamed and Partha Pratim Roy},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08600},
eprint = {2609.08600},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08600},
}
RIS
TY - UNPB
TI - Sharp Bohr and Bohr-Rogosinski inequalities involving area measure for close-to-convex harmonic mappings
AU - Molla Basir Ahamed
AU - Partha Pratim Roy
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08600
UR - https://arxiv.org/abs/2609.08600
AB - In this article, we investigate refined and generalized versions of the Bohr and Bohr--Rogosinski inequalities for a normalized subclass 𝒫_ℋ^0(α) (0 ≤ α< 1) of univalent close-to-convex harmonic mappings f = h + g̅ defined on the open unit disk 𝔻 ⊂ ℂ. By incorporating non-negative monotone increasing functions associated with the planar area integral S_r/π of the image domain f(𝔻_r), we establish new sharp Bohr-type inequalities expressed in terms of the Euclidean distance d(f(0), ∂ f(𝔻)). Furthermore, we establish sharp Bohr--Rogosinski-type inequalities involving powers of the modulus of the mapping |f(z)|^p (p ≥ 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.
ER -