On convex combinations of harmonic mappings.

Let ψ μ , ν (z) = (1 − 2 cos ν e i μ z + e 2 i μ z 2) − 1 , μ , ν ∈ [ 0 , 2 π) 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 ∫ 0 z ψ μ , ν (ξ) p (ξ) d ξ 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. [ABSTRACT FROM AUTHOR]