Abstract
In this paper, we establish some results concerning the convolutions of harmonic mappings convex in the horizontal direction with harmonic vertical strip mappings. Furthermore, we provide examples illustrated graphically with the help of Maple to illuminate the results.
Highlights
For real-valued harmonic functions u and v in the open unit disk E = fz ∈ C : jzj < 1g, the complex-valued continuous function f = u + iv is said to be harmonic and can be expressed as f = h + g, where h and g are analytic in E
The Jacobian of f = h + g is given by J f = jh′j2 − jg′j2
We denote by SH the class of all harmonic, sense-preserving, and univalent mappings f = h + g in E, which are normalized by the condition hð0Þ = gð0Þ = 0 and h′ð0Þ = 1
Summary
We investigate the conditions under which the convolutions of harmonic mappings Pδ, f γ, and Fa with prescribed dilatations are univalent and CHD provided that the convolutions are locally univalent and sense-preserving. Let f γ = hγ + gγ ∈ S0CHD be given by (6) with dilatation ω = gγ′/hγ′ and Fa = Ha + Ga be a mapping defined by (2) and (3).
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