Bohr-Rogosinski Inequalities for Bounded Analytic Functions

In this paper we first consider another version of the Rogosinski inequality for analytic functions f(z) = Sigma(infinity)(n=0) anzn in the unit disk vertical bar z vertical bar < 1, in which we replace the coefficients an (n = 0, 1,..., N) of the power series by the derivatives f((n))(z)/n! (n = 0, 1,..., N). Secondly, we obtain improved versions of the classical Bohr inequality and Bohr's inequality for the harmonic mappings of the form f = h+ g, where the analytic part h is bounded by 1 and that vertical bar g '(z)vertical bar <= k vertical bar h' (z)vertical bar in vertical bar z vertical bar < 1 and for some k is an element of [0, 1].