Disk of convexity of sections of univalent harmonic functions

One of the classical results from Szegö shows that if h(z)=z+∑n=2∞anzn is analytic and univalent in the unit disk D:={z∈C:|z|<1}, then the section sn(h)(z)=∑k=1nakzk of hh is univalent in |z|<1/4|z|<1/4. The exact (largest) radius of the univalence rnrn of sn(h)sn(h) remains an open problem. On the other hand, not much is known in the case of harmonic univalent functions. It is then natural to consider the class PH0 of normalized harmonic mappings f=h+g¯ in the unit disk DD satisfying the condition Reh′(z)>|g′(z)| for z∈Dz∈D, where g′(0)=0g′(0)=0. Functions in PH0 are known to be univalent and close-to-convex in DD. In this paper, we first show that each f∈PH0 is convex in the disk |z|<2−1, and then determine the value of rr so that the partial sums of f∈PH0 are convex in |z|