  There are a number of articles which deal with Bohr's phenomenon whereas only
a few papers appeared in the literature on Rogosinski's radii for analytic
functions defined on the unit disk $|z|<1$. In this article, we introduce and
investigate Bohr-Rogosinski's radii for analytic functions defined for $|z|<1$.
Also, we prove several different improved versions of the classical Bohr's
inequality. Finally, we also discuss the Bohr-Rogosinski's radius for a class
of subordinations. All the results are proved to be sharp.
