Abstract
This work is devoted to the so-called filtration theory of semigroup generators in the unit disk. It should be noted that numerous filtrations studied to nowadays have been introduced for different purposes and considered from different points of view. So, our aim is to summarize the known facts, to present common and distinct properties of filtrations as well as to study some new filtrations with an emphasis on their connection with geometric function theory and the dynamic features of semigroups generated by elements of different filtration families. Among the dynamic properties, we mention the uniform convergence on the unit disk and the sectorial analytical extension of semigroups with respect to their parameter. We also solve the Fekete–Szegö problem over various filtration classes, as well as over non-linear resolvents.
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References
Abate, M. 1989. Iteration theory of holomorphic maps on taut manifolds. Rende: Mediterranean Press.
Abate, M. 1992. The infinitesimal generators of semigroups of holomorphic maps. Annali di Matematica Pura ed Applicata 161: 167–180.
Aharonov, D., M. Elin, S. Reich, and D. Shoikhet. 1999. Parametric representations of semi-complete vector fields on the unit balls in \(\mathbb{C} ^{n}\) and in Hilbert space. Atti dell’Accademia Nazionale dei Lincei 10: 229–253.
Aharonov, D., S. Reich, and D. Shoikhet. 1999. Flow invariance conditions for holomorphic mappings in Banach spaces. Mathematical Proceedings of the Royal Irish Academy 99A: 93–104.
Aksentiev, L.A. 1958. Sufficient conditions for univalence of regular functions. Izvestiya Vysshikh Uchebnykh Zavedenii Matematika 3: 3–7 ((Russian)).
Avicou, C., I. Chalendar, and J.R. Partington. 2016. Analyticity and compactness of semigroups of composition operators. Journal of Mathematical Analysis and Applications 437: 545–560.
Balasubramanian, R., S. Ponnusamy, and M. Vuorinen. 2002. On hypergeometric functions and function spaces. Journal of Computational and Applied Mathematics 139 (2): 299–322.
Berkson, E., and H. Porta. 1978. Semigroups of analytic functions and composition operators. The Michigan Mathematical Journal 25: 101–115.
Bracci, F., M.D. Contreras, and S. Díaz-Madrigal. 2020. Continuous semigroups of holomorphic self-maps of the unit disc. Springer Monographs in Mathematics. New York: Springer.
Bracci, F., M.D. Contreras, S. Díaz-Madrigal, M. Elin, and D. Shoikhet. 2018. Filtrations of infinitesimal generators. Functiones et Approximatio Commentarii Mathematici 59: 99–115.
Bracci, F., and P. Gumenyuk. 2016. Contact points and fractional singularities for semigroups of holomorphic self-maps in the unit disc. Journal d’Analyse Mathematique 130: 185–217.
Brickman, L., D.J. Hallenbeck, T.H. MacGregor, and D.R. Wilken. 1973. Convex hulls and extreme points of families of starlike and convex mappings. Transactions of the American Mathematical Society 185: 413–428.
Duren, P. 1983. Univalent Functions. New York: Springer.
Contreras, M.D., S. Díaz-Madrigal, and Ch. Pommerenke. 2006. On boundary critical points for semigroups of analytic functions. Mathematica Scandinavica 98: 125–142.
Efraimidis, I., and D. Vukotić. 2018. Applications of Livingston-type inequalities to the generalized Zalcman functional. Mathematische Nachrichten 291: 1502–1513. https://doi.org/10.1002/mana.201700022.
Elin, M., and F. Jacobzon. 2017. Analyticity of semigroups in the right half-plane. Journal of Mathematical Analysis and Applications 448: 750–766.
Elin, M., and F. Jacobzon. 2020. Coefficient body for nonlinear resolvents. Annales UMCS 2: 41–53.
M. Elin and F. Jacobzon. 2022. The Fekete-Szegö problem for spirallike mappings and non-linear resolvents in Banach spaces. Studia Universitatis Babeş-Bolyai Mathematica 67, 329–344. https://doi.org/10.24193/subbmath.2022.2.09
Elin, M., and F. Jacobzon. 2022. Families of inverse functions: coefficient bodies and the Fekete-Szegö problem. Mediterranean Journal of Mathematics. https://doi.org/10.1007/s00009-022-02017-2.
Elin, M., and F. Jacobzon. 2023. Estimates on some functionals over non-linear resolvents. Filomat 37: 797–808.
Elin, M., F. Jacobzon, and D. Shoikhet. 2024. Filtration families of semigroup generators. The Fekete-Szegö problem. Applied Set-Valued Analysis and Optimization 6: 13–29.
M. Elin, F. Jacobzon and N. Tuneski. 2022. The Fekete–Szego functional and filtration of generators. Rendiconti del Circolo Matematico di Palermo Series II. https://doi.org/10.1007/s12215-022-00824-w
Elin, M., S. Reich, and D. Shoikhet. 2002. Asymptotic behavior of semigroups of \(\rho \)-non-expansive and holomorphic mappings on the Hilbert ball. Annali di Matematica Pura ed Applicata 181: 501–526.
M. Elin, S. Reich and D. Shoikhet. 2004. Complex Dynamical Systems and the Geometry of Domains in Banach Spaces, Dissertationes Math. (Rozprawy Mat.) 427, 62 pp.
Elin, M., S. Reich, and D. Shoikhet. 2019. Numerical range of holomorphic mappings and applications. Cham: Birkhäuser.
M. Elin and D. Shoikhet. 2010. Linearization Models for Complex Dynamical Systems. Topics in univalent functions, functional equations and semigroup theory, Birkhäuser Basel.
M. Elin, D. Shoikhet, and T. Sugawa. 2018. Filtration of semi-complete vector fields revisited. In Complex analysis and dynamical systems. New trends and open problems, 93–102. Birkhäuser/Springer, Cham: Trends in Math.
Elin, M., D. Shoikhet, and N. Tarkhanov. 2018. Analytic extension of semigroups of holomorphic mappings and composition operators. Computer Methods Function Theory 18: 269–294.
Elin, M., D. Shoikhet, and N. Tuneski. 2017. Parametric embedding of starlike functions. Complex Analysis and Operator Theory 11: 1543–1556.
Elin, M., D. Shoikhet, and N. Tuneski. 2020. Radii problems for starlike functions and semigroup generators. Computational Methods and Function Theory 20: 297–318.
Elin, M., D. Shoikhet, and L. Zalcman. 2008. A flower structure of backward flow invariant domains for semigroups. Annales Academiæ Scientiarum Fennicæ Mathematica 33: 3–34.
Fukui, S. 1997. On \(\alpha \)-convex functions of order \(\beta \). International Journal of Mathematics and Mathematical Sciences 20: 769–772.
S. Giri and S. S. Kumar. Radius and convolution problems of analytic functions involving semigroup generators, available in: arXiv:2205.10777.
G. M. Goluzin. 1950. Some estimates for bounded functions. Mat. Sb. (N.S.), 26(68):1, 7–18.
A. W. Goodman. 1983. Univalent functions, Vol 1, Mariner publishing company, Inc.
I. Graham, H. Hamada and G. Kohr. 2020. Loewner chains and nonlinear resolvents of the Carathéodory family on the unit ball in \(\mathbb{C}^n\). Journal of Mathematical Analysis and Applications (2020). https://doi.org/10.1016/j.jmaa.2020.124289
Graham, I., and G. Kohr. 2003. Geometric function theory in one and higher dimensions. New York and Basel: Marcel Dekker.
Gurganus, K.R. 1975. \(\Phi \)-like holomorphic functions in \(\mathbb{C}^n\) and Banach spaces. Transactions of the American Mathematical Society 205: 389–406.
Hallenbeck, D.J. 1974. Convex hulls and extreme points of some families of univalent functions. Transactions of the American Mathematical Society 192: 285–292.
Hallenbeck, D.J., and S. Ruscheweyh. 1971. Subordination by convex functions. Proceedings of the American Mathematical Society 52: 191–195.
Hamada, H., G. Kohr, M. Kohr. 2021. The Fekete–Szegö problem for starlike mappings and nonlinear resolvents of the Carathéodory family on the unit balls of complex Banach spaces. Anal. Math. Phys. 11 (2021). https://doi.org/10.1007/s13324-021-00557-6.
Jacobzon, F., M. Levenshtein, and S. Reich. 2011. Convergence characteristics of one-parameter continuous semigroups. Analysis and Mathematical Physics 1: 311–335.
Jahangiri, M., H. Silverman, and E.M. Silvia. 1991. Classes of functions defined by subordination. In New Trends in Geometric Function theory and applications, ed. R. Parvatham and S. Ponnusamy, 34–41. Word Scientific: Singapore.
Janowski, W. 1973. Some extremal problems for certain families of analytic functions. Annales Polonici Mathematici 28: 297–326.
Keogh, F.R., and E.P. Merkes. 1969. A coefficient inequality for certain classes of analytic functions. Proceedings of the American Mathematical Society 20: 8–12.
Kim, Y.C., and R. Rønning. 2001. Integral transforms of certain subclasses of analytic functions. Journal of Mathematical Analysis and Applications 258: 466–486.
Komatu, Y. 1961. On starlike and convex mappings of a circle. Kodai Mathematical Sem. Rep. 13: 123–126.
Li, L., S. Ponnusamy, and K.-J. Wirths. 2022. Relations of the class \(U_{\lambda }\) to other families of functions. Bulletin of the Malaysian Mathematical Sciences Society 45 (3): 955–972.
Ma , W. C., and D. Minda. 1992. A unifed treatment of some special classes of univalent functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157–169, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA.
Martin, G.J., and B.G. Osgood. 1986. The quasihyperbolic metric and associated estimates on the hyperbolic metric. Journal of Analyse Mathematics 47: 37–53.
Marx, A. 1932. Untersuchungen über schlichte Abbildungen. Mathematische Annalen 107: 40–67.
Miller, S.S., and P.T. Mocanu. 1985. Univalent solutions of Briot-Bouquet differential equations. Journal of Differ Equations 56: 297–309.
Miller, S.S., and P.T. Mocanu. 2000. Differential Subordinations. Theory and Applications: Marcel Dekker Inc., New York.
Miller, S.S., P.T. Mocanu, and M.O. Reade. 1972. All alpha-convex functions are starlike. Revue Roumaine de Mathematiques Pures et Appliquees 17: 1395–1397.
Mocanu, P.T. 1969. Une proprié de convexité généralisée dans la théoric de la représentation conforme. Mathemattca (Cluj) 11: 127–133.
Obradović, M., S. Ponnusamy, and K.J. Wirths. 2016. Geometric Studies on the Class \(U(\lambda )\). Bulletin of the Malaysian Mathematical Sciences Society 39: 1259–1284. https://doi.org/10.1007/s40840-015-0263-5.
Ponnusamy, S. 1993. Convolution properties of some classes of meromorphic univalent functions. Proceedings of the Indian Academy of Sciences 103 (1): 73–89.
Ponnusamy, S. 1994. Differential subordination concerning starlike functions. Proceedings of the Indian Academy of Sciences 104 (2): 397–411.
Ponnusamy, S., and V. Singh. 2002. Criteria for univalent, starlike and convex functions. Bulletin of the Belgian Mathematical Society 9 (4): 511–531.
Poreda, T.. 1991. On generalized differential equations in Banach space, Dissert. Math. 310.
Reich, S., and D. Shoikhet. 1996. Generation theory for semigroups of holomorphic mappings in Banach spaces. Abstract and Applied Analysis 1: 1–44.
Reich, S., and D. Shoikhet. 1997. Semigroups and generators on convex domains with the hyperbolic metric, Atti Acad. Naz. Lincei Cl. Sci. Fiz. Mat. Natur. Rend. Lincei (9) Mat. Appl. 8, 231–250.
Reich, S., and D. Shoikhet. 1997. The Denjoy-Wolff theorem. Ann. Univ. Mariae Curie-Sklodowska 51: 219–240.
Reich, S., and D. Shoikhet. 2005. Nonlinear Semigroups, Fixed Points, and the Geometry of Domains in Banach Spaces. Imperial College Press, London: World Scientific Publisher.
Ruscheweyh, St. 1975. New criteria for univalent functions. Proceedings of the American Mathematical Society 49: 109–115.
Ruscheweyh, St. 1977. Linear operators between classes of prestarlike functions. Commentarii Mathematici Helvetici 52: 497–509.
Shoikhet, D. 2003. Representations of holomorphic generators and distortion theorems for spirallike functions with respect to a boundary point. International Journal of Pure and Applied Mathematics 5: 335–361.
Shoikhet, D. 2001. Semigroups in Geometrical Function Theory. Dordrecht: Kluwer Academic Publishers.
Shoikhet, D. 2016. Rigidity and parametric embedding of semi-complete vector fields on the unit disk. Milan Journal of Mathematics 84: 159–202. https://doi.org/10.1007/s00032-016-0254-5.
Simon, B. 2005. Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, Colloquium Publications, Amer. Math. Society.
Strohhäcker, E. 1933. Beiträge zür Theorie der schhlichten Functionen. Mathematische Zeitschrift 37: 356–380.
Suffridge, T. J. 1976. Starlike functions as limits of polynomials, Advances in complex function theory, in: Lecture Notes in Maths., 505, Springer, Berlin–Heidelberg–New York, 164–202.
Sugawa, T., and L.-M. Wang. 2017. Spirallikeness of shifted hypergeometric functions. Annales Academiæ Scientiarum Fennicæ Mathematica 42: 963–977.
Tuneski, N. 2009. Some simple sufficient conditions for starlikeness and convexity. Appl. Math. Letters 22: 693–697.
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Elin, M., Jacobzon, F. Survey on filtrations (parametric embeddings) of infinitesimal generators. J Anal (2024). https://doi.org/10.1007/s41478-024-00754-z
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DOI: https://doi.org/10.1007/s41478-024-00754-z