Bohr--Rogosinski inequalities for bounded analytic functions

In this paper we first consider another version of the Rogosinski inequality for analytic functions $f(z)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ in the unit disk $|z| < 1$,inwhichwereplacethecoefficients$a_n$ $(n=0,1,\dots ,N)$ of the power series by the derivatives $f^{(n)}(z)\mathrm{/}n!$ $(n=0,1,\dots ,N)$. Secondly, we obtain improved versions of the classical Bohr inequality and Bohr’s inequality for the harmonic mappings of the form $f=h+\bar{g}$, where the analytic part $h$ is bounded by $1$ and that $\mathrm{\mid }{g}^{\mathrm{\prime }}(z)\mathrm{\mid }\le k\mathrm{\mid }{h}^{\mathrm{\prime }}(z)\mathrm{\mid }$ in $|z| < 1$andforsome$k \in [0,1]$.