Mapping properties of a family of planar harmonic functions
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A family of complex-valued close-to-convex harmonic mappings, $\Psi_{c,\beta}:\mathbb{D}\to \mathbb{C}~(c>0,\beta\ge 0),$ defined on the open unit disc $\mathbb{D},$ is constructed and the convolution $\Psi_{c,\beta}*f$ is studied for a harmonic mapping $f=h+\overline{g}.$ It is proved that this convolution is locally univalent and sense-preserving whenever $h\pm \epsilon g$ are starlike, for any fixed $\epsilon$ with $|\epsilon|=1.$ Some conditions on univalent harmonic mapping $f=h+\overline{g}$ are also determined so that the convolution, $\Psi_{c,\beta}*f,$ is close-to-convex or is convex in certain direction for every $\beta\geq 1.$ Apart from proving an Alexander-type result for the mapping, $\Psi_{c,\beta}*f,$ it is also established that this convolution decomposes into a convex combination of two harmonic mappings. Finally, weak subordination properties of the mapping $\Psi_{c,\beta}$ are examined in the case when $\beta$ is a positive integer.
Abstract
A family of complex-valued close-to-convex harmonic mappings, $\Psi_{c,\beta}:\mathbb{D}\to \mathbb{C}~(c>0,\beta\ge 0),$ defined on the open unit disc $\mathbb{D},$ is constructed and the convolution $\Psi_{c,\beta}*f$ is studied for a harmonic mapping $f=h+\overline{g}.$ It is proved that this convolution is locally univalent and sense-preserving whenever $h\pm \epsilon g$ are starlike, for any fixed $\epsilon$ with $|\epsilon|=1.$ Some conditions on univalent harmonic mapping $f=h+\overline{g}$ are also determined so that the convolution, $\Psi_{c,\beta}*f,$ is close-to-convex or is convex in certain direction for every $\beta\geq 1.$ Apart from proving an Alexander-type result for the mapping, $\Psi_{c,\beta}*f,$ it is also established that this convolution decomposes into a convex combination of two harmonic mappings. Finally, weak subordination properties of the mapping $\Psi_{c,\beta}$ are examined in the case when $\beta$ is a positive integer.