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The hereditary property of convexity and starlikeness for conformal mappingsdoes not generalize to univalent harmonic mappings. This failure leads us tothe notion of fully starlike and convex mappings of order \alpha, (0\leq\alpha<1). A bound for the radius of fully starlikeness and fully convexity oforder \alpha is determined for certain families of univalent harmonic mappings.Convexity is not preserved under the convolution of univalent harmonic convexmappings, unlike in the analytic case. Given two univalent harmonic convexmappings f and g, the problem of finding the radius r_{0} such that f*g is aunivalent harmonic convex mapping in |z|<r_{0}, is being considered.

Fully Starlike and Convex Harmonic Mappings of order \alpha