Abstract
Loewner’s theorem provides a simple characterization of \({\mathcal M}_\infty (a,b)\) but it is not so simple to describe which functions are in a general \({\mathcal M}_n(a,b)\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
L. Bers, The action of the universal modular group on certain boundary points, Contributions to Algebra (collection of papers dedicated to Ellis Kolchin), pp. 25–36, Academic Press, New York, 1977.
F. Burstall, F. Pedit, and U. Pinkall, Schwarzian derivatives and flows of surfaces, Differential Geometry and Integrable Systems (Tokyo, 2000), pp. 39–61, Contemp. Math., 308, American Mathematical Society, Providence, RI, 2002.
M. Chuaqui, P. Duren, and B. Osgood, Curvature properties of planar harmonic mappings, Comput. Methods Funct. Theory 4 (2004), 127–142.
O. Dobsch, Matrixfunktionen beschränkter Schwankung, Math. Z. 43 (1937), 353–388.
W. F. Donoghue, Monotone Matrix Functions and Analytic Continuation, Springer, New York-Heidelberg-Berlin, 1974.
R. C. Gunning, Special coordinate coverings of Riemann surfaces, Math. Ann. 170 (1967), 67–86.
F. Hansen and J. Tomiyama, Differential analysis of matrix convex functions, Linear Algebra Appl. 420 (2007), 102–116.
F. Hansen and J. Tomiyama, Differential analysis of matrix convex functions, II, J. Inequal. Pure Appl. Math. 10 (2009), Article 32, 5 pp.
O. Heinävaara, private communication,
O. Kozlovski, Structural stability in 1D dynamics, Topics in Dynamics and Ergodic Theory, pp. 107–124, London Math. Soc. Lecture Note Ser., 310, Cambridge University Press, Cambridge, 2003.
O. Lehto, Univalent Functions and Teichmüller Spaces, Graduate Texts in Mathematics, 109. Springer-Verlag, New York, 1987.
K. Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177–216.
Z. Nehari, The Schwarzian derivative and Schlicht functions, Bull. A.M.S. 55 (1949), 545–551.
Z. Nehari, Conformal Mapping, McGraw-Hill, New York-Toronto-London, 1952; reprinted by Dover, New York, 1975.
H. Oliveira and J. Sousa Ramos, On iterated positive Schwarzian derivative maps, Dynamical Systems and Functional Equations (Murcia, 2000), Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), 1673–1681.
B. Osgood, Old and new on the Schwarzian derivative, Quasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), pp. 275–308, Springer, New York, 1998.
T. Sasaki and M. Yoshida, Schwarzian derivatives and uniformization, The Kowalevski Property (Leeds, 2000), pp. 271–286, CRM Proc. Lecture Notes, 32, American Mathematical Society, Providence, RI, 2002.
H. A. Schwarz, Über einige Abbildungsaufgaben, J. Reine Angew. Math. 70 (1869), 65–83.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Simon, B. (2019). 2 × 2 Matrices: The Donoghue and Hansen–Tomiyama Theorems. In: Loewner's Theorem on Monotone Matrix Functions. Grundlehren der mathematischen Wissenschaften, vol 354. Springer, Cham. https://doi.org/10.1007/978-3-030-22422-6_14
Download citation
DOI: https://doi.org/10.1007/978-3-030-22422-6_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-22421-9
Online ISBN: 978-3-030-22422-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)