2013/07/24 by Raj Kumar, Kumar, Raj, Sushma Gupta +5
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #math.CV
paper · pdf · doi:10.48550/arxiv.1307.6292
18 Pages
arxiv created 2013/07/24 · arxiv updated 2013/07/25
Let fβ= hβ+gβand Fa = Ha +Ga be harmonic mappings obtained by shearing of analytic mappings hβ+gβ= 1/(2isinβ)log((1 + zeiβ)/(1 + ze-iβ)), 0<β<πand Ha+Ga = z/(1-z), respectively. Kumar et al. [5] conjectured that if ω(z)=eiθzn (θ∈ R, n∈ N) and ωa(z)=(a-z)/(1-az), a∈(-1,1) are dilatations of fβand Fa, respectively, then Fa∗ fβ∈ SH0 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.