39 research outputs found
Convolutions of slanted half-plane harmonic mappings
Let denote the class of all univalent, harmonic,
sense-preserving and normalized mappings of the unit disk \ID onto the
slanted half-plane with an
additional condition . Functions in this class can be
constructed by the shear construction due to Clunie and Sheil-Small which
allows by examining their conformal counterpart. Unlike the conformal case,
convolution of two univalent harmonic convex mappings in \ID is not
necessarily even univalent in \ID. In this paper, we fix and show that the convolutions of and some slanted
half-plane harmonic mapping are still convex in a particular direction. The
results of the paper enhance the interest among harmonic mappings and, in
particular, solves an open problem of Dorff, et. al. \cite{DN} in a more
general setting. Finally, we present some basic examples of functions and their
corresponding convolution functions with specified dilatations, and illustrate
them graphically with the help of MATHEMATICA software. These examples explain
the behaviour of the image domains.Comment: 15 pages, preprint of December 2011 (submitted to a journal for
publication
Note on the convolution of harmonic mappings
Dorff et al. \cite{DN} formulated a question concerning the convolution of
two right half-plane mappings, where the normalization of the functions was
considered incorrectly. In this paper, we have reformulated the open problem in
correct form and provided a solution to it in a more general form. In addition,
we also obtain two new theorems which correct and improve some other results.Comment: 11 pages; An extended version of this article was in a couple of
conferences, and also in later workshops in Chennai during 2017 in India.
This version will appear in Bulletin of the Australian Mathematical Societ