The spherical metric and univalent harmonic mappings

Let $f=h+\bar{g}$ 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 $f\mathrm{.}$