Abstract
Recent results on the convolution of two planar harmonic mappings is built upon the theory that when the convolution of functions from certain families of mappings, such as half-plane or strip mappings, is locally univalent, then the convolution will possess certain direction-convexity properties. Thus, much of the latest work on harmonic convolutions centers around establishing conditions on the dilatations of $$f_1, f_2: {\mathbb D}\rightarrow {\mathbb C}$$ from the families above so that $$f_1 * f_2$$ is locally univalent. Recently, it was noted that normalizations for these families were not treated properly when some dilatations considered did not fix zero. In this paper, we account for a variety of dilatations that do not fix zero by broadening the family from which $$f_1$$ and $$f_2$$ are chosen. Additionally, we show that when removing a hypothesis from one of our results it is possible to have a locally univalent convolution that fails to be univalent. This demonstrates that some of the previous work on convolutions cannot simply be modified by a re-normalization while affirming the necessity of the hypothesis.
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