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

Abstract

A class F consisting of analytic functions f ( z ) = ∑ n = 0 ∞ a n z n in the unit disc D = { z ∈ C : | z | < 1 } satisfies a Bohr phenomenon if there exists an 0 $ ]]> r f > 0 such that ∑ n = 1 ∞ | a n | r n ≤ d ( f ( 0 ) , ∂f ( D ) ) for every function f ∈ F , and | z | = r ≤ r f . The largest radius r f is the Bohr radius and the inequality ∑ n = 1 ∞ | a n | r n ≤ d ( f ( 0 ) , ∂f ( D ) ) is Bohr inequality for the class 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.