On De la Vallée Poussin means for harmonic mappings

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.