On Bohr's inequality for special subclasses of stable starlike harmonic mappings

The focus of this article is to explore the Bohr inequality for a specific subset of harmonic starlike mappings introduced by Ghosh and Vasudevarao (Some basic properties of certain subclass of harmonic univalent functions, Complex Var. Elliptic Equ. 63 (2018), no. 12, 1687–1703.). This set is denoted as ℬ H 0 ( M ) ≔ { f = h + g ¯ ∈ ℋ 0 : ∣ z h ″ ( z ) ∣ ≤ M − ∣ z g ″ ( z ) ∣ } {{\mathcal{ {\mathcal B} }}}_{H}^{0}\left(M):= \{f=h+\overline{g}\in {{\mathcal{ {\mathcal H} }}}_{0}:| z{h}^{^{\prime\prime} }\left(z)| \le M-| z{g}^{^{\prime\prime} }\left(z)| \} for z ∈ D z\in {\mathbb{D}} , where 0 < M ≤ 1 0\lt M\le 1 . It is worth mentioning that the functions belonging to the class ℬ H 0 ( M ) {{\mathcal{ {\mathcal B} }}}_{H}^{0}\left(M) are recognized for their stability as starlike harmonic mappings. With this in mind, this research has a twofold goal: first, to determine the optimal Bohr radius for this specific subclass of harmonic mappings, and second, to extend the Bohr-Rogosinski phenomenon to the same subclass.