On convolution, convex, and starlike mappings
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2022
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Abstract
Let C and S* stand for the classes of convex and starlike mapping in D, and let <(co(C))over bar>, <(co(S*))over bar> denote the closures of the respective convex hulls. We derive characterizations for when the convolution of mappings in <(co(C))over bar> is convex, as well as when the convolution of mappings in <(co(S*))over bar> is starlike. Several characterizations in terms of convolution are given for convexity within <(co(C))over bar> and for starlikeness within <(co(S*))over bar>. We also obtain a correspondence via convolution between C and S*, as well as correspondences between the subclasses of convex and starlike mappings that have n-fold symmetry.
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Convolution, convex mapping, starlike mapping, convex polygon, slit mapping, POlya-Schoenberg conjecture