Convex Combinations of Planar Harmonic Mappings Realized Through Convolutions with Half-Strip Mappings

Abstract

Recent investigations into what geometric properties are preserved under the convolution of two planar harmonic mappings on the open unit disk $${\mathbb {D}}$$ have typically involved half-plane and strip mappings. These results rely on having a convolution that is locally univalent and sense-preserving on $${\mathbb {D}}$$ , and thus, much focus has been on trying to satisfy this condition. We introduce a family of right half-strip harmonic mappings, $$\Psi _c : {\mathbb {D}}\rightarrow {\mathbb {C}}$$ , $$c>0$$ , and consider the convolution $$\Psi _c * f$$ for a harmonic mapping $$f = h +\overline{g}: {\mathbb {D}}\rightarrow {\mathbb {C}}$$ . We prove it is sufficient for $$h \pm g$$ to be starlike for $$\Psi _c *f$$ to be locally univalent and sense-preserving. Moreover, $$\Psi _c * f$$ decomposes into a convex combination of two harmonic mappings, one of which is f itself. This decomposition is key in addressing mapping properties of the convolution, and from it, we produce a family of convex octagonal harmonic mappings as well some other families of convex harmonic mappings. Additionally, motivated by the construction of $$\Psi _c$$ , we introduce a generalized harmonic Bernardi integral operator. We demonstrate convolution preserving properties and a weak subordination relationship for this extended operator.