Stable geometric properties of analytic and harmonic functions

Source abstract

Abstract Given any sense preserving harmonic mapping f=h+ḡ in the unit disk, we prove that for all |λ|=1 the functions f λ =h+λḡ are univalent ( resp . close-to-convex, starlike, or convex) if and only if the analytic functions F λ =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 f λ belong to the families mentioned above. In particular, we see that if f λ are univalent for all λ on the unit circle, then h is univalent.