Abstract
In this paper, we mainly consider the convolutions of slanted half-plane mappings and strip mappings of the unit disk D. If f1 is a slanted half-plane mapping and f2 is a slanted half-plane mapping or a strip mapping, then we prove that f1 * f2 is convex in some direction if f1 * f2 is locally univalent in D. We also obtain two sufficient conditions for f1 * f2 to be locally univalent in D. Our results extend many of the recent results in this direction. Moreover, we consider a class of harmonic mappings including slanted half-plane mappings and strip mappings, and as a consequence, we prove that the any convex combination of such locally univalent and sense-preserving mappings is also convex.
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