On Harmonic Convolutions Involving a Vertical Strip Mapping

Let fβ = hβ + gβ and Fa = Ha +Ga be harmonic mappings obtained by shearing of analytic mappings hβ + gβ = 1/(2isinβ) log ( (1 + ze)/(1 + ze) ) , 0 < β < π and Ha+Ga = z/(1 − z), respectively. Kumar et al. [7] conjectured that if ω(z) = ez(θ ∈ R, n ∈ N) and ωa(z) = (a − z)/(1 − az), a ∈ (−1, 1) are dilatations of fβ and Fa, respectively, then Fa∗̃fβ ∈ S 0 H and is convex in the direction of the real axis, provided a ∈ [(n− 2)/(n+ 2), 1). They claimed to have verified the result for n = 1, 2, 3 and 4 only. In the present paper, we settle the above conjecture, in the affirmative, for β = π/2 and for all n ∈ N.