Disk of convexity of sections of univalent harmonic functions

One of the classical results from Szegö shows that if h ( z ) = z + ∑ n = 2 ∞ a n z n is analytic and univalent in the unit disk D : = { z ∈ C : | z | < 1 } , then the section s n ( h ) ( z ) = ∑ k = 1 n a k z k of h is univalent in | z | < 1 / 4 . The exact (largest) radius of the univalence r n of s n ( 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 P H 0 of normalized harmonic mappings f = h + g ¯ in the unit disk D satisfying the condition Re h ′ ( z ) > | g ′ ( z ) | for z ∈ D , where g ′ ( 0 ) = 0 . Functions in P H 0 are known to be univalent and close-to-convex in D . In this paper, we first show that each f ∈ P H 0 is convex in the disk | z | < 2 − 1 , and then determine the value of r so that the partial sums of f ∈ P H 0 are convex in | z | < r .