"Injectivity of sections of convex harmonic mappings and convolution th" by Liulan Li and Saminathan Ponnusamy
 

Injectivity of sections of convex harmonic mappings and convolution theorems

Article Type

Research Article

Publication Title

Czechoslovak Mathematical Journal

Abstract

We consider the class H0of 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 H0and show that if f ∈ PH0(α) and F ∈ GH0(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) (Formula presented.), where f = h + g ∈ 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 ∈ H0is 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 ∈ CH0is not convex in the disk |z| < 1/4 but it is convex in a smaller disk.

First Page

331

Last Page

350

DOI

10.1007/s10587-016-0259-9

Publication Date

6-1-2016

Comments

Open Access; Green Open Access

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