MAPPING PROPERTIES OF A FAMILY OF PLANAR HARMONIC FUNCTIONS

MAPPING PROPERTIES OF A FAMILY OF PLANAR HARMONIC FUNCTIONS

A family of complex-valued close-to-convex harmonic mappings, Ψ_c,𝛽 : 𝔻 → ℂ (c > 0, 𝛽 ≥ 0), defined on the open unit disc 𝔻, is constructed and the convolution Ψ_c,𝛽 * f is studied for a harmonic mapping f = h + ḡ. It is proved that this convolution is locally univalent and sense-preserving whenever h ± 𝜖g are starlike, for any fixed 𝜖 with |𝜖| = 1. Some conditions on univalent harmonic mapping f = h + ḡ are also determined so that the convolution, Ψ_c,𝛽 * f, is close-to-convex or is convex in certain direction for every 𝛽 ≥ 1. Apart from proving an Alexander-type result for the mapping, Ψ_c,𝛽 * f, it is also established that this convolution decomposes into a convex combination of two harmonic mappings. Finally, weak subordination properties of the mapping Ψ_c,𝛽 are examined in the case when 𝛽 is a positive integer.