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

Full Text:

PDF

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.

Roth, O., Schippers, E., The Loewner and Hadamard variations, Illinois J. Math. 52 (2008), no. 4, 1399-1415.




DOI: http://dx.doi.org/10.17951/a.2011.65.2.139-148
Data publikacji: 2016-07-27 21:54:11
Data złożenia artykułu: 2016-07-27 13:47:14

Refbacks

  • There are currently no refbacks.


Copyright (c) 2011 Dmitri Prokhorov