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.
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Communicated by See Keong Lee.
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Muir, S. Convex Combinations of Planar Harmonic Mappings Realized Through Convolutions with Half-Strip Mappings. Bull. Malays. Math. Sci. Soc. 40, 857–880 (2017). https://doi.org/10.1007/s40840-016-0336-0
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DOI: https://doi.org/10.1007/s40840-016-0336-0