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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