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.