The sharp refined Bohr–Rogosinski inequalities for certain classes of harmonic mappings.

A class $ \mathcal {F} $ F consisting of analytic functions $ f(z)=\sum _{n=0}^{\infty }a_nz^n $ f (z) = ∑ n = 0 ∞ a n z n in the unit disc $ \mathbb {D}=\{z\in \mathbb {C}:|z| D = { z ∈ C : | z | < 1 } satisfies a Bohr phenomenon if there exists an $ r_f>0 $ r f > 0 such that \[ \sum_{n=1}^{\infty}|a_n|r^n\leq{d}\left(f(0),\partial f(\mathbb{D})\right) \] ∑ n = 1 ∞ | a n | r n ≤ d (f (0) , ∂f (D)) for every function $ f\in \mathcal {F} $ f ∈ F , and $ |z|=r\leq r_f $ | z | = r ≤ r f . The largest radius $ r_f $ r f is the Bohr radius and the inequality $ \sum _{n=1}^{\infty }|a_n|r^n\leq {d}\left (f(0),\partial f(\mathbb {D})\right) $ ∑ n = 1 ∞ | a n | r n ≤ d (f (0) , ∂f (D)) is Bohr inequality for the class $ \mathcal {F} $ F , where 'd' is the Euclidean distance. In this paper, we prove sharp refinement of the Bohr–Rogosinski inequality for certain classes of harmonic mappings. [ABSTRACT FROM AUTHOR]