2015/06/01 by Liulan Li, Li, Liulan, Saminathan Ponnusamy +1
Mathematics · #30C45 #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics #Holomorphic and Operator Theory #math.CV #msc:30C45
paper · pdf · doi:10.48550/arxiv.1506.00542
16 pages, 3 figures; To appear in Czechoslovak Mathematical Journal
arxiv created 2015/06/01 · openalex publication_date 2015/06/01 · arxiv updated 2015/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the article the authors consider the class \mathcal H0 of sense-preserving harmonic functions f=h+g defined in the unit disk |z|<1 and normalized so that h(0)=0=h'(0)-1 and g(0)=0=g'(0), where h and g are analytic in the unit disk. In the first part of the article we present two classes PH0(α) and GH0(β) of functions from \mathcal H0 and show that if f∈ PH0(α) and F\inGH0(β), then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters α and β are satisfied. In the second part we study the harmonic sections (partial sums) sn, n(f)(z)=sn(h)(z)+sn(g)(z), where f=h+g∈ \mathcal H0, sn(h) and sn(g) denote the n-th partial sums of h and g, respectively. We prove, among others, that if f=h+g∈\mathcal H0 is a univalent harmonic convex mapping, then sn, n(f) is univalent and close-to-convex in the disk |z|< 1/4 for n≥ 2, and sn, n(f) is also convex in the disk |z|< 1/4 for n≥2 and n≠ 3. Moreover, we show that the section s3,3(f) of f∈ \mathcal CH0 is not convex in the disk |z|<1/4 but is shown to be convex in a smaller disk.