Advanced versions of the inverse function theorem
DOI:
https://doi.org/10.24193/subbmath.2022.2.04Abstract
This short opus is dedicated to the bright memory of the distinguished mathematician Gabriela Kohr and her mathematical heritage. Gabriela Kohr ’s contribution to analysis of one and several complex variables brought new knowledge into the modern theory as well as new colors to the subject. During our meetings with Gabriela at various conferences she always proposed some interesting and often nonstandard questions related to classical issues as well as new directions. It is worth to be mentioned her excellent book [50] together with Ian Graham on classical and modern problems in Geometric Function Theory in complex spaces (see also, [46], [56], [27], [22], [24], [49] and [48]).
Mathematics Subject Classification (2010): 30C45.
Received 28 February 2022; Accepted 10 March 2022.
References
Abate, M., The infinitesimal generators of semigroups of holomorphic maps, Ann. Mat. Pura Appl., 161(1992), 167-180.
Aharonov, D., Elin, M., Reich, S., Shoikhet, D., Parametric representations of semi-complete vector fields on the unit balls in Cn and in Hilbert space, Rend. Mat. Acc. Lincei Analisi Matematicas, 10(1999), 229-253.
Aharonov, D., Reich, S., Shoikhet, D., Flow invariance conditions for holomorphic mappings in Banach spaces, Math. Proc. R. Ir. Acad. 99A(1999), 93-104.
Aizenberg, L., Reich, S., Shoikhet, D., One-Sided Estimates for the existence of null points of holomorphic mappings in Banach spaces, J. Math. Anal. Appl., 203(1996), 38-54.
Alpay, D., Dijksma, A., Langer, H., Reich, S., Shoikhet, D., Boundary interpolation and rigidity for generalized Nevanlinna functions, Math. Nachr., 283(2010), 335-364.
Alpay, D., Reich, S., Shoikhet, D., Rigidity theorems, boundary interpolation and re- producing kernels for generalized Schur functions, Comput. Methods Funct. Theory, 9(2009), 347-364.
Arosio, L., Bracci, F., Hamada, H., Kohr, G., An abstract approach to Loewner chains, arXiv:1002.4262 [pdf, ps, other] math.CV math. DS, 2011.
Becker, J., Uber die Losungstruktur der einer Differentialgleichung in der konformen Abbildung, J. Reine Angew. Math., 285(1976), 6-74.
Berkson, E., Porta, H., Semigroups of analytic functions and composition operators, Michigan Math. J., 25(1978), 101-115.
Betker, T., Lowner chains and quasiconformal extensions, Complex Var., 20(1992), 107- 111.
Bolotnikov, V., Elin, M., Shoikhet, D., Inequalities for angular derivatives and boundary interpolation, Anal. Math. Phys., 3(2013).
Bracci, F., Contreras, M.D., Diaz-Madrigal, S., Evolution families and the Loewner equation I: the unit disc, J. Reine Angew. Math., 672(2012), 1-37.
Bracci, F., Contreras, M.D., Diaz-Madrigal, S., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, 2020.
Bracci, F., Contreras, M.D., Diaz-Madrigal, S., Elin, M., Shoikhet, D., Filtrations of infinitesimal generators, Funct. Approx. Comment. Math., 59(2018), no. 1.
Bracci, F., Contreras, M.D., Diaz-Madrigal, S., Gaussier, H., A characterization of orthogonal convergence in simply connected domains, Math. Ann. (to appear), arXiv:1806.06582.
Bracci, F., Elin, M., Shoikhet, D., Growth estimates for pseudo-dissipative holomorphic maps in Banach spaces, J. Nonlinear Convex Anal., 15(2014), 191-198.
Bracci, F., Graham, I., Hamada, H., Kohr, G., Variation of Loewner chains, extreme and support points in the class S0 in higher dimensions, arXiv:1402.5538.
Bracci, F., Kosinski, L., Zwonek, W., Slice rigidity property of holomorphic maps Kobayashi-isometrically preserving complex geodesics, arXiv:2012.13701.
Bracci, F., Kozitsky, Y., Shoikhet, D., Abel averages and holomorphically pseudo- contractive maps in Banach spaces, J. Math. Anal. Appl., 423(2015), 1580-1593.
Bracci, F., Kraus, D., Roth, O., A new Schwarz-Pick Lemma at the boundary and rigidity of holomorphic maps, arXiv:2003.0(2019), 63-96.
Brickman, L., Hallenbeck, D.J., MacGregor, T.H., Wilken, D.R., Convex hulls and extreme points of families of starlike and convex mappings, Trans. Amer. Math. Soc., 185(1973), 413-428.
Burckel, R.B., An Introduction to Classical Complex Analysis, Vol. 1, Academic Press, New York and London, 1979.
Burns, D.M., Krantz, S.G., Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary, J. Amer. Math. Soc., 7(1994), 661-676.
Chae, S.B., Holomorphy and Calculus in Normed Spaces, Marcel Dekker, New York, 1985.
Contreras, M.D., Dıaz-Madrigal, S., Pommerenke, C., On boundary critical points for semigroups of analytic functions, Math. Scand., 98(2006), 125-142.
Contreras, M.D., Dıaz-Madrigal, S., Pommerenke, C., Second angular derivatives and parabolic iteration in the unit disk, Trans. Amer. Math. Soc., 362(2010), 357-388.
Cowen, C., MacCluer, B., Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995.
Duren, P.L., Univalent Functions, Springer-Verlag, New York, 1983.
Earle, C.J., Hamilton, R.S., A fixed point theorem for holomorphic mappings, Proc. Symp. Pure Math., Amer. Math. Soc., Providence, RI, 16(1970), 61-65.
Elin, M., Jacobzon, F., Analyticity of semigroups on the right half-plane, J. Math. Anal. Appl., 448(2017), 750-766.
Elin, M., Jacobzon, F., Some geometric feachers of non-linear resolvents, Preprint, https://arxiv.org/pdf/2104.00758.pdf.
Elin, M., Jacobzon, F., Levenshtein, M., Shoikhet, D., The Schwarz Lemma. Rigidity and Dynamics, In: Harmonic and Complex Analysis and Applications, Birkhauser Basel, 2014, 135-230.
Elin, M., Levenshtein, M., Reich, S., Shoikhet, D., Commuting semigroups of holomorphic mappings, Math. Scand., 103(2008), 295-319.
Elin, M., Reich, S., Shoikhet, D., Holomorphically accretive mappings and spiral-shaped functions of proper contractions, Nonlinear Anal. Forum, 5(2000), 149-161.
Elin, M., Reich, S., Shoikhet, D., Numerical Range of Holomorphic Mappings and Applications, Birkha¨user, Basel, 2010.
Elin, M., Reich, S., Shoikhet, D., Yacobzon, F., Asymptotic behavior of one-parameter semigroups and rigidity of holomorphic generators, Compl. Anal. Oper. Theory, 2(2008), 55-86.
Elin, M., Shoikhet, D., Linearization Models for Complex Dynamical Systems. Topics in univalent functions, functional equations and semigroup theory, Birkha¨user, Basel, 2010.
Elin, M., Shoikhet, D., Boundary behavior and rigidity of semigroups of holomorphic mappings, Analysis Math. Physics, 1(2011), 241-258.
Elin, M., Shoikhet, D., Sugawa, T., Filtration of Semi-Complete Vector Fields Revisited, In: Complex Analysis and Dynamical Systems, New Trends and Open Problems, Birhauser, Trends in Mathematics, 2017.
Elin, M., Shoikhet, D., Sugawa, T., Geometric properties of the nonlinear resolvent of holomorphic generators, Journal of Mathematical Analysis and Applications, 483(2020), no. 2, 1-18.
Elin, M., Shoikhet, D., Tarkhanov, N., Analytic semigroups of holomorphic mappings and composition operators, Comput. Methods Funct. Theory, 18(2018), 269-294.
Elin, M., Shoikhet, D., Tuneski, N., Parametric embedding of starlike functions, Compl. Anal. Oper. Theory, 11(2017), 1543-1556.
Elin, M., Shoikhet, D., Zalcman, L., A flower structure of backward flow invariant do- mains for semigroups, Ann. Acad. Sci. Fenn. Math., 33(2008), 3-34.
Franzoni, T., Vesentini, E., Holomorphic Maps and Invariant Distances, North-Holand, Amsterdam, 1980.
Goebel, K., Reich, S., Uniform Convexity, Hyperbolic Geometry and Nonexpensive Mappings, Marcel Dekker, New York and Basel, 1982.
Goluzin, G.M., Geometric Function Theory of Complex Variable, Moscow-Leningrad, Tech. Literatura, 1952.
Goodman, A.W., Univalent Functions, Vol. I., Mariner Publishing Co., Inc., Tampa, 1983.
Graham, I., Hamada, H., Honda, T., Kohr, G., Shon, H.H., Distortion and coefficient bounds for Carath´eodory families in Cn and complex Banach spaces, J. Math. Anal. Appl., 416(2014), 449-469.
Graham, I., Hamada, H., Kohr, G., Radius problems for holomorphic mappings on the unit ball in Cn, Math. Nachr., 279(2006), 1474-1490.
Graham, I., Kohr, G., Geometric Function Theory in One and Higher Dimensions, Marcel Dekker, Inc, NY-Basel, 2006.
Gurganus, K.R., Φ-like holomorphic functions in Cn and Banach space, Trans. Amer. Math. Soc., 205(1975), 389-400.
Harris, L.A., The numerical range of holomorphic functions in Banach spaces, Amer. J. Math., 93(1971), 1005-1019.
Harris, L.A., On the size of balls covered by analytic transformations, Monatshefte Math., 83(1977), 9-23.
Harris, L.A., Fixed point theorems for infinite dimensional holomorphic functions, J. Korean Math. Soc., 41(2004), 175-192.
Harris, L.A., Reich, S., Shoikhet, D., Dissipative holomorphic functions, Bloch radii and the Shwarz lemma, J. Anal. Math., 82(2000), 221-232.
Hille, E., Phillips, R., Functional Analysis and Semi-Groups (revised eddition), Amer. Soc. Colloq. Publ., 31, Providence, R.I., 808 pp.
Jack, I.S., Functions starlike and convex of order α, J. London Math. Soc., 3(1971), 469-474.
Khatskevich, V., Reich, S., Shoikhet, D., Global implicit function and fixed point theorems for holomorphic mappings and semigroups, Complex Variables, 28(1996), 347-356.
Khatskevich, V., Shoikhet, D., Differential Operatos and Nonlinear Equations, Operator Theory: Advances and Applications, 66, Birkha¨user, Basel, 1994.
Kohr, G., On some conditions of spiralikeness for mappings of C1 class, Complex Variables, 32(1997), 79-98.
Kohr, G., Some sufficient conditions of spiralikeness in Cn, Complex Variables, 36(1998), 1-9.
Krasnoselskii, M.A., Zabreiko, P.P., Geometric Methods of Nonlinear Anlysis, Nauka, 1975.
Kresin, G., Some estimates for the implicit function, Personal Communications, 2017.
Kresin, G., Maz’ya, V.G., Sharp Real-Part Theorems. A Unified Approach, Lecture Notes in Mathematics, Springer, Berlin, 2007.
Kresin, G., Shoikhet, D., Some estimates for Bloch radii, Personal Communications, 2010.
Kresin, G., Shoikhet, D., An inverse function theorem for holomorphic mappings, Preprint, 2015.
Kriete, T.L., MacCluer, B.D., A rigidity theorem for composition operators on certain Bergman spaces, Michigan Math. J., 42(1995), 379-386.
Levenshtein, M., Shoikhet, D., A Burns-Krantz type theorem for pseudo-contractive map- pings, Advances in Complex Analysis and Operator Theory, (2017), 237-246.
Ma, W., Minda, D., Hyperbolically convex functions II, Ann. Polon. Math., 71(1999), 273-285.
Mej´ıa, D., Pommerenke, C., On hyperbolically convex functions, J. Geom. Anal., 10(2000), 365-378.
Mejıa, D., Pommerenke, C., The analytic point function in the disk, Comput. Methods Funct. Theory, 5(2005), 275-299.
Mejıa, D., Pommerenke, C., Vasil’ev, A., Distortion theorems for hyperbolically convex functions, Complex Variables Theory Appl., 44(2001), no. 2, 117-130.
Miller, S.S., Mocanu, P.T., Differential Subordinations: Theory and Applications, Marcel Dekker, New York, NY, 2000.
Osserman, R., A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc., 128(2000), 3513-3517.
Pinchuk, B., On starlike and convex functions of order α, Duke Math. J., 35(1968), 721-734.
Pommerenke, C., Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.
Poreda, T., On generalized differential equations in Banach spaces, Dissertationes Math. (Rozprawy Mat.), 310(1991).
Prokhorov, D.V., Bounded Univalent Functions, In: Handbook of Complex Analysis, Geometric Function Theory I, Elsiever, 2002, 207-228.
Reich, S., Shoikhet, D., Generation theory for semigroups of holomorphic mappings in Banach spaces, Abstr. Appl. Anal., 1(1996), 1-44.
Reich, S., Shoikhet, D., Semigroups and generators on convex domains with the hyperbolic metric, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei, 9(1997), 231-250.
Reich, S., Shoikhet, D., Metric domains, holomorphic mappings and nonlinear semi- groups, Abstr. Appl. Anal., 3(1998), 203-228.
Reich, S., Shoikhet, D., Nonlinear Semigroups, Fixed Points, and the Geometry of Do- mains in Banach Spaces, World Scientific Publisher, Imperial College Press, London, 2005.
Reich, S., Shoikhet, D., Zem´anek, J., Ergodicity, numerical range, and fixed points of holomorphic mappings, J. Anal. Math., 119(2013), 275-303.
Robertson, M.S., Radii of star-likeness and close-to-convexity, Proc. Amer. Math. Soc., 16(1966), 847-852.
Shih, M.H., Bolzano’s theorem in several complex variables, Proc. Amer. Math. Soc., 79(1980), 32-34.
Shoikhet, D., Semigroups in Geometrical Function Theory, Kluwer Academic Publishers, Dordrecht, 2001.
Shoikhet, D., Representations of holomorphic generators and distortion theorems for starlike functions with respect to a boundary point, Int. J. Pure Appl. Math., 5(2003), 335-361.
Shoikhet, D., Another look at the Burns-Krantz theorem, J. Anal. Math., 105(2008), 19-42.
Shoikhet, D., Rigidity and parametric embedding of semi-complete vector fields on the unit disk, Milan J. Math., 77(2010), 127-150.
Shoikhet, D., Parametric Embedding and rigidity of semi-complete vector fields, Milano Journal of Mathematics, 84(2016), 159-202.
Shoikhet, D., Remarks and review on the inverse function theorem, Preprint, 2022.
Siskakis, A., Composition semigroups and the Ces‘aro operator on Hp, J. Lond. Math. Soc., 36(1987), 153-164.
Solynin, A.Yu, Hyperbolic convexity and the analytic fixed point function, Proc. Amer. Math. Soc., 135(2007), 1181-1186.
Suffridge, T.J., Starlike and convex maps in Banach space, Pacific J. Math., 46(1973), 575-589.
Sugawa, T., Personal Communications, 2019.
Tauraso, R., Vlacci, F., Rigidity at the boundary for holomorphic self-maps of the unit disk, Complex Variables Theory Appl., 45(2001), 151-165.
Tuneski, N., On certain sufficient conditions for starlikeness, Internat. J. Math. & Math. Sci., 23(2000), 521-527.
Tuneski, N., Some simple sufficient conditions for starlikeness and convexity, Applied Mathematics Letters, 22(2009), 693-697.
Xiu-Shuang, M., Ponnusamy, S., Sugawa, T., Harmonic spirallike functions and harmonic strongly starlike function, arXiv:2108.11622.
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.
