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Univalency of Convolutions of Univalent Harmonic Right Half-Plane Mappings

Univalency of Convolutions of Univalent Harmonic Right Half-Plane Mappings We consider the convolution of half-plane harmonic mappings with respective dilatations $$(z+a)/(1+az)$$ ( z + a ) / ( 1 + a z ) and $$e^{i\theta }z^{n}$$ e i θ z n , where $$-1<a<1$$ - 1 < a < 1 and $$\theta \in \mathbb {R},n\in \mathbb {N}$$ θ ∈ R , n ∈ N . We prove that such convolutions are locally univalent for $$n=1$$ n = 1 , which solves an open problem of Dorff et al. (see J Anal 18:69–81 [3, Problem 3.26]). Moreover, we provide some numerical computations to illustrate that such convolutions are not univalent for $$n\ge 2$$ n ≥ 2 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Univalency of Convolutions of Univalent Harmonic Right Half-Plane Mappings

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Publisher
Springer Journals
Copyright
Copyright © 2016 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-016-0180-0
Publisher site
See Article on Publisher Site

Abstract

We consider the convolution of half-plane harmonic mappings with respective dilatations $$(z+a)/(1+az)$$ ( z + a ) / ( 1 + a z ) and $$e^{i\theta }z^{n}$$ e i θ z n , where $$-1<a<1$$ - 1 < a < 1 and $$\theta \in \mathbb {R},n\in \mathbb {N}$$ θ ∈ R , n ∈ N . We prove that such convolutions are locally univalent for $$n=1$$ n = 1 , which solves an open problem of Dorff et al. (see J Anal 18:69–81 [3, Problem 3.26]). Moreover, we provide some numerical computations to illustrate that such convolutions are not univalent for $$n\ge 2$$ n ≥ 2 .

Journal

Computational Methods and Function TheorySpringer Journals

Published: Oct 7, 2016

References