  Let $f=h+\overline{g}$ be a harmonic univalent map in the unit disk
$\mathbb{D}$, where $h $ and $g$ are analytic. We obtain an improved estimate
for the second coefficient of $h$. This indeed is the first qualitative
improvement after the appearance of the papers by Clunie and Sheil-Small in
1984, and by Sheil-Small in 1990. Also, when the sup-norm of the dilatation is
less than $1$, it is shown that the spherical area of the covering surface of
$h$ is dominated by the spherical area of the covering surface of $f.$
