  In 1984, Clunie and Sheil-Small proved that a sense-preserving harmonic
function whose analytic part is convex, is univalent and close-to-convex. In
this paper, certain cases are discussed under which the conclusion of this
result can be strengthened and extended to fully starlike and fully convex
harmonic mappings. In addition, we investgate the properties of functions in
the class $\mathcal{M}(\alpha)$ $(|\alpha|\leq 1)$ consisting of harmonic
functions $f=h+\overline{g}$ with $g'(z)=\alpha zh'(z)$, $\RE
(1+{zh''(z)}/{h'(z)})>-{1}/{2} $ $ \mbox{for} |z|<1 $. The coefficient
estimates, growth results, area theorem and bounds for the radius of
starlikeness and convexity of the class $\mathcal{M}(\alpha)$ are determined.
In particular, the bound for the radius of convexity is sharp for the class
$\mathcal{M}(1)$.
