On harmonic convolutions involving a vertical strip mapping

Abstract

Let f_β= h_β+\bar{g}_βand F_a = H_a +\bar{G}_a be harmonic mappings obtained by shearing of analytic mappings h_β+g_β= 1/(2i\sinβ)log((1 + ze^{iβ})/(1 + ze^{-iβ})), 0<β<πand H_a+G_a = z/(1-z), respectively. Kumar et al. [5] conjectured that if ω(z)=e^{iθ}z^n (θ\in R, n\in N) and ω_a(z)=(a-z)/(1-az), a\in(-1,1) are dilatations of f_βand F_a, respectively, then F_a\ast f_β\in S_H^0 and is convex in the direction of the real axis provided a\in[(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\in N.

18 Pages