  In this paper we first consider another version of the Rogosinski inequality
for analytic functions $f(z)=\sum_{n=0}^\infty a_nz^n$ in the unit disk $|z| <
1$, in which we replace the coefficients $a_n$ $(n= 0,1,\ldots ,N)$ of the
power series by the derivatives $f^{(n)}(z)/n!$ $(n= 0,1,\ldots ,N)$. Secondly,
we obtain improved versions of the classical Bohr inequality and Bohr's
inequality for the harmonic mappings of the form $f = h + \overline{g}$, where
the analytic part $h$ is bounded by $1$ and that $|g'(z)| \le k|h'(z)|$ in $|z|
< 1$ and for some $k \in [0,1]$.
