On convolution, convex, and starlike mappings
DOI:
https://doi.org/10.24193/subbmath.2022.2.17Keywords:
Convolution, convex mapping, starlike mapping, convex polygon, slit mapping, P´olya-Schoenberg conjecture.Abstract
Let C and S∗ stand for the classes of convex and starlike mapping in D, and let co(C), co(S∗) denote the closures of the respective convex hulls. We derive characterizations for when the convolution of mappings in co(C) is convex, as well as when the convolution of mappings in co(S∗) is starlike. Several characterizations in terms of convolution are given for convexity within co(C) and for starlikeness within co(S∗). 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.
Mathematics Subject Classification (2010): 30C45, 30C30.
Received 19 January 2022; Accepted 20 January.
References
Brickman, MacGregor, T.H., Wilken, D.R., Convex hulls of some classical families of univalent functions, Trans. Amer. Math. Soc. 156 (1971), 91-107
Brickman, Hallenbeck, D.J., MacGregor, T.H., Wilken, D.R., Convex hulls and extreme points of families of convex and starlike mappings, Trans. Amer. Math. Soc., 185(1973), 413-428.
Chuaqui, M., Duren, P., Osgood, B., Schwarzian derivatives of convex mappings, Ann. Acad. Scie. Fenn. Math., 36(2011), 1-12.
Chuaqui, M., Duren, P., Osgood, B., Concave conformal mappings and pre-vertices of Schwarz-Christoffel mappings, Proc. AMS, 140(2012), 3495-3505.
Chuaqui, M., Pommerenke, C., On Schwarz-Christoffel mappings, Pacific J. Math., 270(2014), 319-334.
Duren, P., Univalent Functions, Grundlehreen der Mathematischen Wissenschaften 259, Springer-Verlag, New York, 1983.
Hallenbeck, D.J., Applications of extreme point of families of starlike and close-to-convex mappings, Pacific. J. Math., 57(1975), 167-176.
Nehari, Z., A property of convex conformal maps, J. Analyse Math., 30(1976), 390-393. [9] Ruscheweyh, S., Sheil-Small, T., Hadamard products of schlicht functions and the P´olya-Schoenberg conjecture, Comment. Math. Helv., 48(1973), 119-135.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
