Abstract
In this paper we survey various results concerning extremal problems related to Loewner chains, the Loewner differential equation, and Herglotz vector fields on the Euclidean unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\). First, we survey recent results related to extremal problems for the Carathéodory families \({\mathcal M}\) and \({\mathcal N}_A\) on the Euclidean unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\), where \(A\in L(\mathbb {C}^n)\) with m(A) > 0. In the second part of this paper, we present recent results related to extremal problems for the family \(S_A^0(\mathbb {B}^n)\) of normalized univalent mappings with A-parametric representation on the Euclidean unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\), where \(A\in L(\mathbb {C}^n)\) with k +(A) < 2m(A). In the last section we survey certain results related to extreme points and support points for a special compact subset of \(S_A^0(\mathbb {B}^n)\) consisting of bounded mappings on \(\mathbb {B}^n\). Particular cases, open problems, and questions will be also mentioned.
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References
Arosio, L.: Resonances in Loewner equations. Adv. Math. 227, 1413–1435 (2011)
Arosio, L., Bracci, F., Hamada, H., Kohr, G.: An abstract approach to Loewner chains. J. Anal. Math. 119, 89–114 (2013)
Arosio, L., Bracci, F., Wold, F.E.: Solving the Loewner PDE in complete hyperbolic starlike domains of \(\mathbb {C}^n\). Adv. Math. 242, 209–216 (2013)
Bracci, F.: Shearing process and an example of a bounded support function in \(S^0(\mathbb {B}^2)\). Comput. Methods Funct. Theory 15, 151–157 (2015)
Bracci, F., Roth, O.: Support points and the Bieberbach conjecture in higher dimension. Preprint (2016); arXiv: 1603.01532
Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: Evolution families and the Loewner equation II: complex hyperbolic manifolds. Math. Ann. 344, 947–962 (2009)
Bracci, F., Elin, M., Shoikhet, D.: Growth estimates for pseudo-dissipative holomorphic maps in Banach spaces. J. Nonlinear Convex Anal. 15, 191–198 (2014)
Bracci, F., Graham, I., Hamada, H., Kohr, G.: Variation of Loewner chains, extreme and support points in the class S 0 in higher dimensions. Constr. Approx. 43, 231–251 (2016)
Duren, P.: Univalent Functions. Springer, New York (1983)
Duren, P., Graham, I., Hamada, H., Kohr, G.: Solutions for the generalized Loewner differential equation in several complex variables. Math. Ann. 347, 411–435 (2010)
Goodman, G.S.: Univalent functions and optimal control. Ph.D. Thesis, Stanford University (1968)
Graham, I., Kohr, G.: Geometric Function Theory in One and Higher Dimensions. Marcel Dekker, New York (2003)
Graham, I., Hamada, H., Kohr, G.: Parametric representation of univalent mappings in several complex variables. Can. J. Math. 54, 324–351 (2002)
Graham, I., Kohr, G., Pfaltzgraff, J.A.: The general solution of the Loewner differential equation on the unit ball of \(\mathbb {C}^n\). In: Complex Analysis and Dynamical Systems II. Contemporary Mathematics, vol. 382, pp. 191–203. American Mathematical Society, Providence (2005)
Graham, I., Hamada, H., Kohr, G., Kohr, M.: Asymptotically spirallike mappings in several complex variables. J. Anal. Math. 105, 267–302 (2008)
Graham, I., Hamada, H., Kohr, G., Kohr, M.: Extreme points, support points and the Loewner variation in several complex variables. Sci. China Math. 55, 1353–1366 (2012)
Graham, I., Hamada, H., Kohr, G., Kohr, M.: Extremal properties associated with univalent subordination chains in \(\mathbb {C}^n\). Math. Ann. 359, 61–99 (2014)
Graham, I., Hamada, H., Honda, T., Kohr, G., Shon, K.H.: Growth, distortion and coefficient bounds for Carathéodory families in \(\mathbb {C}^n\) and complex Banach spaces. J. Math. Anal. Appl. 416, 449–469 (2014)
Graham, I., Hamada, H., Kohr, G., Kohr, M.: Support points and extreme points for mappings with A-parametric representation in \(\mathbb {C}^n\). J. Geom. Anal. 26, 1560–1595 (2016)
Gurganus, K.: Φ-like holomorphic functions in \(\mathbb {C}^n\) and Banach spaces. Trans. Am. Math. Soc. 205, 389–406 (1975)
Gustafson, K.E., Rao, D.K.M.: Numerical Range. The Field of Values of Linear Operators and Matrices. Springer, New York (1997)
Hamada, H.: Polynomially bounded solutions to the Loewner differential equation in several complex variables. J. Math. Anal. Appl. 381, 179–186 (2011)
Hamada, H.: Approximation properties on spirallike domains of \(\mathbb {C}^n\). Adv. Math. 268, 467–477 (2015)
Harris, L.: The numerical range of holomorphic functions in Banach spaces. Am. J. Math. 93, 1005–1019 (1971)
Iancu, M.: A density result for parametric representations in several complex variables. Comput. Methods Funct. Theory 15, 247–262 (2015)
Kirwan, W.E.: Extremal properties of slit conformal mappings. In: Brannan, D., Clunie, J. (eds.) Aspects of Contemporary Complex Analysis, pp. 439–449. Academic Press, London (1980)
Muir, J.: A Herglotz-type representation for vector-valued holomorphic mappings on the unit ball of \(\mathbb {C}^n\). J. Math. Anal. Appl. 440, 127–144 (2016)
Pell, R.: Support point functions and the Loewner variation. Pac. J. Math. 86, 561–564 (1980)
Pfaltzgraff, J.A.: Subordination chains and univalence of holomorphic mappings in \(\mathbb {C}^n\). Math. Ann. 210, 55–68 (1974)
Pommerenke, C.: Univalent Functions. Vandenhoeck and Ruprecht, Göttingen (1975)
Poreda, T.: On the univalent holomorphic maps of the unit polydisc in \(\mathbb {C}^n\) which have the parametric representation, I-the geometrical properties. Ann. Univ. Mariae Curie Skl. Sect. A. 41, 105–113 (1987)
Poreda, T.: On the univalent holomorphic maps of the unit polydisc in \(\mathbb {C}^n\) which have the parametric representation, II-the necessary conditions and the sufficient conditions. Ann. Univ. Mariae Curie Skl. Sect. A. 41, 115–121 (1987)
Poreda, T.: On generalized differential equations in Banach spaces. Diss. Math. 310, 1–50 (1991)
Prokhorov, D.V.: Bounded univalent functions. In: Kühnau R. (ed.) Handbook of Complex Analysis: Geometric Function Theory, vol. I, pp. 207–228. Elsevier, New York (2002)
Reich, S., Shoikhet, D.: Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces. Imperial College Press, London (2005)
Roth, O.: Control theory in \({\mathcal H}(\mathbb {D})\). Dissertation, Bayerischen University Würzburg (1998)
Roth, O.: Pontryagin’s maximum principle for the Loewner equation in higher dimensions. Can. J. Math. 67, 942–960 (2015)
Schleissinger, S.: On support points of the class \(S^0(\mathbb {B}^n)\). Proc. Am. Math. Soc. 142, 3881–3887 (2014)
Suffridge, T.J.: Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions. In: Lecture Notes in Mathematics, vol. 599, pp. 146–159. Springer, Berlin (1977)
Voda, M.: Loewner theory in several complex variables and related problems. PhD. Thesis, University of Toronto (2011)
Voda, M.: Solution of a Loewner chain equation in several complex variables. J. Math. Anal. Appl. 375, 58–74 (2011)
Acknowledgements
I. Graham was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A9221. H. Hamada was partially supported by JSPS KAKENHI Grant Number JP16K05217.
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Graham, I., Hamada, H., Kohr, G., Kohr, M. (2017). Loewner Chains and Extremal Problems for Mappings with A-Parametric Representation in ℂn . In: Bracci, F. (eds) Geometric Function Theory in Higher Dimension. Springer INdAM Series, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-73126-1_13
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