Convolutions of planar harmonic strip mappings

We consider classes of harmonic univalent functions f(k) = h(k) + (g) over bar (k), (k = 1, 2) that are shears of the analytic map h(k) - g(k) = 1/2 log[(1 + z) / (1 - z)] with dilatation omega(k) = e(i theta k) z(k). We prove that if the convolution f(1) * f(2) is locally one-to-one and sense-preserving, then f(1) * f(2) is convex in the direction of the real axis.