On harmonic convolutions involving a vertical strip mapping

TL;DR

This paper proves a conjecture about the harmonic convolution of specific harmonic mappings involving vertical strip and shear mappings, confirming convexity in the direction of the real axis for all natural numbers n when eta=rac{ ext{pi}}{2}.

Contribution

It confirms the conjecture for eta=rac{ ext{pi}}{2} and all natural numbers n, extending previous partial results to a complete proof.

Findings

Convolution is convex in the direction of the real axis for the specified mappings.

The conjecture holds for all natural numbers n when eta=rac{ ext{pi}}{2}.

The result generalizes previous verified cases for n=1,2,3,4.

Abstract

Let f_\beta = h_\beta+\bar{g}_\beta and F_a = H_a +\bar{G}_a be harmonic mappings obtained by shearing of analytic mappings h_\beta +g_\beta = 1/(2i\sin\beta)log((1 + ze^{i\beta})/(1 + ze^{-i\beta})), 0<\beta<\pi and H_a+G_a = z/(1-z), respectively. Kumar et al. [5] conjectured that if \omega(z)=e^{i\theta}z^n (\theta\in R, n\in N) and \omega_a(z)=(a-z)/(1-az), a\in(-1,1) are dilatations of f_\beta and F_a, respectively, then F_a\ast f_\beta \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 \beta =\pi/2 and for all n\in N.