The Lowner-Kufarev representations for domains with analytic boundaries

Dmitri Prokhorov

Abstract


We consider the Lowner-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 Lowner-Kufarev chain consisting of univalent quadratic polynomials and compare the Lowner-Kufarev representations of bounded and arbitrary univalent functions.


Keywords


Lowner-Kufarev equation; analytic continuation; univalent polynomials; bounded univalent functions

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References


Kufarev, P. P., On one-parametric families of analytic functions, Mat. Sbornik N. S. 13(55) (1943), 87-118 (Russian).

Lowner, K., Untersuchungen uber schlichte konforme Abbildungen des Einheitskreises. I, Math. Ann. 89 (1923), no. 1-2, 103-121.

Markina, I., Vasil’ev, A., Virasoro algebra and dynamics in the space of univalent functions, Contemp. Math. 525 (2010), 85-116.

Pommerenke, Ch., Uber die Subordination analytischer Funktionen, J. Reine Angew. Math. 218 (1965), 159-173.

Pommerenke, Ch., Univalent Functions, Vandenhoeck & Ruprecht, Gottingen, 1975.

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DOI: http://dx.doi.org/10.2478/v10062-011-0020-7
Date of publication: 2016-07-27 21:54:11
Date of submission: 2016-07-27 13:47:14


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