Abstract: In this paper we first consider another version of the Rogosinski inequality for analytic functions (Formula presented.) in the unit disk (Formula presented.), in which we replace the coefficients (Formula presented.) of the power series by the derivatives (Formula presented.). Secondly, we obtain improved versions of the classical Bohr inequality and Bohr’s inequality for the harmonic mappings of the form (Formula presented.), where the analytic part (Formula presented.) is bounded by (Formula presented.) and that (Formula presented.) in |z|<1 and for some (Formula presented.). © 2020, Pleiades Publishing, Ltd.