Stable geometric properties of analytic and harmonic functions

Given any sense preserving harmonic mapping f=h+g in the unit disk, we prove that for all |λ|=1 the functions fλ=h+λ are univalent (resp. close-to-convex, starlike, or convex) if and only if the analytic functions Fλ=h+λg are univalent (resp. close-to-convex, starlike, or convex) for all such λ. We also obtain certain necessary geometric conditions on h in order that the functions fλ belong to the families mentioned above. In particular, we see that if fλ are univalent for all λ on the unit circle, then h is univalent.