Abstract:
In this article, we study the geometric properties of Vn(f), the nth De la Vallée Poussin means for univalent starlike harmonic mappings f. In particular, we provide a necessary and sufficient condition for Vn(f) to be univalent and starlike in the unit disk D, when f∈S∗H, the class of all normalized univalent starlike harmonic mappings in D. We determine the radius of fully starlikeness (respectively, fully convexity) of V2(f), when f∈S0H and the result is sharp. Then, we determine the radius rn∈(0,1) so that Vn(f) is univalent and fully starlike in |z|<rn, whenever f is univalent and fully starlike harmonic mapping in D. We also discuss about the geometry preserving nature of Vn(f), when f belongs to some well known geometric subclasses of SH.
