Convolutions of harmonic convex mappings

Details Convolutions of harmonic convex mappings Convolutions of harmonic convex mappings

2009-03-09

arxiv.org/abs/0903.1595v1

Mathematics

Complex Variables

15 pages, 4 figure files

Scientific paper

The first author proved that the harmonic convolution of a normalized right half-plane mapping with either another normalized right half-plane mapping or a normalized vertical strip mapping is convex in the direction of the real axis. provided that it is locally univalent. In this paper, we prove that in general the assumption of local univalency cannot be omitted. However, we are able to show that in some cases these harmonic convolutions are locally univalent. Using this we obtain interesting examples of univalent harmonic maps one of which is a map onto the plane with two parallel slits.

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