Let ψ μ,ν (z) = (1-2 cos νe iμ z + e 2iμ z 2)-1 , μ, ν ∈ [0, 2π) and p be an analytic mapping with Re p > 0 on the open unit disk. We consider the sense-preserving planar harmonic mappings f = h + g, which are shears of the mapping z 0 ψ μ,ν (ξ)p(ξ) dξ in the direction μ. These mappings include the harmonic right half-plan mappings, vertical strip mappings, and their rotations. For various choices of dilatations g /h of f , sufficient conditions are found for the convex combinations of these mappings to be univalent and convex in the direction μ.

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We investigate the univalency and the directional convexity of the convolution φ * f = φ * h + φ * g of the harmonic mapping f = h + ḡ with a mapping φ whose convolution with the mapping z + ∞ k=2 k n z k is starlike (and such a mapping φ is called n-starlike). In addition, we investigate the directional convexity of (i) the convolution of an analytic convex mapping with the slanted half-plane mapping, and (ii) the partial sums of the convolution of a 6-starlike mapping with the harmonic Koebe mapping and the harmonic half-plane mapping.

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