Abstract
In this paper, we discover new connections between topics in classical geometric function theory and complex dynamics. In particular, we study some classes of starlike functions and their embeddings in the classes of semigroup generators. In addition, we find explicit formulas for the radii of starlikeness for those classes and uniform rates of convergence of semigroups to their Denjoy–Wolff points.
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References
Berkson, E., Porta, H.: Semigroups of analytic functions and composition operators. Mich. Math. J. 25(1), 101–115 (1978)
Bracci, F., Contreras, M.D., Díaz-Madrigal, S., Elin, M., Shoikhet, D.: Filtrations of semigroup generators. Funct. Approx. 59, 99–115 (2018)
Brickman, L., Hallenbeck, D.J., MacGregor, T.H., Wilken, D.R.: Convex hulls and extreme points of families of starlike and convex mappings. Trans. Am. Math. Soc. 185, 413–428 (1973)
Elin, M., Shoikhet, D.: Linearization Models for Complex Dynamical Systems. Birkhäuser, Basel (2010)
Elin, M., Shoikhet, D., Sugawa, T.: Filtration of semi-complete vector fields revisited, Complex analysis and dynamical systems. New trends and open problems, pp. 93–102, Trends Math., Birkhäuser/Springer, Cham (2018)
Elin, M., Shoikhet, D., Tarkhanov, N.: Analytic semigroups of holomorphic mappings and composition operators. Comput. Methods Funct. Theory (2017). https://doi.org/10.1007/s40315-017-0227-x
Goodman, A.W.: Univalent Functions, vol. I. Mariner, Tampa (1983)
Hamada, H., Kohr, G.: Simple criterions for strongly starlikeness and starlikeness of certain order. Math. Nachr. 254(255), 165–171 (2003)
Kanas, S., Sugawa, T.: Strong starlikeness for a class of convex functions. J. Math. Anal. Appl. 336, 1005–1017 (2007)
Kim, Y.C., Sugawa, T.: Correspondence between spirallike functions and starlike functions. Math. Nachr. 285, 322–331 (2012)
Krzyż, J.G.: A counter example concerning univalent functions. Folia Soc. Sci. Lub. 2, 57–58 (1962)
Lyzzaik, A.: On a conjecture of M. S. Robertson. Proc. Am. Math. Soc. 91, 108–110 (1984)
MacGregor, T.H.: The radius of univalence of certain analytic functions. Proc. Am. Math. Soc. 14, 514–520 (1963)
Miller, S.S., Mocanu, P.T.: Differential Subordinations: Theory and Applications. Marcel Dekker, New York (2000)
Mocanu, P.T.: Some simple criteria for starlikeness and convexity. Lib. Math. 13, 27–40 (1993)
Pinchuk, B.: On starlike and convex functions of order \(\alpha \). Duke Math. J. 35, 721–734 (1968)
Robertson, M.S.: Univalent functions starlike with respect to a boundary point. J. Math. Anal. Appl. 81, 327–345 (1981)
Shoikhet, D.: Semigroups in Geometrical Function Theory. Kluwer, Dordrecht (2001)
Shoikhet, D.: Rigidity and parametric embedding of semi-complete vector fields on the unit disk. Milan J. Math. 84(1), 159–202 (2016)
Sokół, J., Nunokawa, M., Choc, N.E., Tang, H.: On some applications of Noshiro–Warschawski’s theorem. Filomat 31, 107–112 (2017). https://doi.org/10.2298/FIL1701107S
Tuan, P.D., Anh, V.V.: Radii of starlikeness and convexity for certain classes of analytic functions. J. Math. Anal. Appl. 64(1), 146–158 (1978)
Tuneski, N.: On some simple sufficient conditions for univalence. Math. Bohem. 126(1), 229–236 (2001)
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The authors are very grateful to the anonym referee whose recommendations essentially improved the content of the paper.
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Communicated by Dmitry Khavinson.
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Elin, M., Shoikhet, D. & Tuneski, N. Radii Problems for Starlike Functions and Semigroup Generators. Comput. Methods Funct. Theory 20, 297–318 (2020). https://doi.org/10.1007/s40315-020-00311-2
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DOI: https://doi.org/10.1007/s40315-020-00311-2