  We prove several improved versions of Bohr's inequality for the harmonic
mappings of the form $f=h+\overline{g}$, where $h$ is bounded by 1 and
$|g'(z)|\le|h'(z)|$. The improvements are obtained along the lines of an
earlier work of Kayumov and Ponnusamy, i.e. \cite{KayPon2}, for example a term
related to the area of the image of the disk $D(0,r)$ under the mapping $f$ is
considered. Our results are sharp. In addition, further improvements of the
main results for certain special classes of harmonic mappings are provided.
