Convolutions of Normalized Harmonic Mappings

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 f1,f2:D→C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_1, f_2: {\mathbb D}\rightarrow {\mathbb C}$$\end{document} from the families above so that f1∗f2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_1 * f_2$$\end{document} 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 f1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_1$$\end{document} and f2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_2$$\end{document} 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.