We consider the Löwner-Kufarev differential equations generating univalent maps of the unit disk onto domains bounded by analytic Jordan curves. A solution to the problem of the maximal lifetime shows how long a representation of such functions admits using infinitesimal generators analytically extendable outside the unit disk. We construct a Löwner-Kufarev chain consisting of univalent quadratic polynomials and compare the Löwner-Kufarev representations of bounded and arbitrary univalent functions.
@article{bwmeta1.element.doi-10_2478_v10062-011-0020-7,
author = {Dmitri Prokhorov},
title = {The L\"owner-Kufarev representations for domains with analytic boundaries},
journal = {Annales UMCS, Mathematica},
volume = {65},
year = {2011},
pages = {139-148},
zbl = {1267.30026},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10062-011-0020-7}
}
Dmitri Prokhorov. The Löwner-Kufarev representations for domains with analytic boundaries. Annales UMCS, Mathematica, Tome 65 (2011) pp. 139-148. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10062-011-0020-7/
Kufarev, P. P., On one-parametric families of analytic functions, Mat. Sbornik N. S. 13(55) (1943), 87-118 (Russian).
Löwner, K., Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I, Math. Ann. 89 (1923), no. 1-2, 103-121. | Zbl 49.0714.01
Markina, I., Vasil'ev, A., Virasoro algebra and dynamics in the space of univalent functions, Contemp. Math. 525 (2010), 85-116. | Zbl 1234.81127
Pommerenke, Ch., Über die Subordination analytischer Funktionen, J. Reine Angew. Math. 218 (1965), 159-173. | Zbl 0184.30601
Pommerenke, Ch., Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.
Roth, O., Schippers, E., The Loewner and Hadamard variations, Illinois J. Math. 52 (2008), no. 4, 1399-1415. | Zbl 1187.30005