2014/01/01 by Raj Kumar, Kumar, Raj, Sushma Gupta +3
Mathematics · #30C45 #Advanced Banach Space Theory #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1401.0259
openalex publication_date 2014/01/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Let ∗ and \widetilde ∗ denote the convolution of two analytic maps and that of an analytic map and a harmonic map respectively. Pokhrel [1] proved that if f = h+g is a harmonic map convex in the direction of eiγ and ϕ is an analytic map in the class DCP, then f\widetilde∗ ϕ= h\widetilde∗ϕ+ g\widetilde∗ϕ is also convex in the direction of eiγ, provided f\widetilde∗ϕ is locally univalent and sense-preserving. In the present paper we obtain a general condition under which f\widetilde∗ ϕ is locally univalent and sense-preserving. Some interesting applications of the general result are also presented.