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Michael Brilleslyper, James Rolf (2012)
Anamorphosis, Mapping Problems, and Harmonic Univalent Functions
Liulan Li, S. Ponnusamy (2013)
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Liulan Li, S. Ponnusamy (2012)
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Saminathan Ponnysamy, A. Rasila (2013)
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Zhi-Gang Wang, Zhi-Hong Liu, Yingchun Li (2015)
On convolutions of harmonic univalent mappings convex in the direction of the real axisJournal of Applied Analysis and Computation, 6
Z. Boyd, M. Dorff, M. Nowak, Matthew Romney, Magdalena Woloszkiewicz (2014)
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ON UNIVALENT HARMONIC FUNCTIONS
Yue-Ping Jiang, A. Rasila, Yong Sun (2015)
A note on convexity of convolutions of harmonic mappingsBulletin of The Korean Mathematical Society, 52
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 .
Computational Methods and Function Theory – Springer Journals
Published: Oct 7, 2016
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