Illinois Journal of Mathematics

The Loewner and Hadamard variations

Oliver Roth and Eric Schippers

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We give an explicit formula relating the infinitesimal generators of the Loewner differential equation and the Hadamard variation. This is applied to establish an extension of the Hadamard variation to the case of arbitrary simply-connected domains and to prove the existence of Loewner chains with arbitrary smooth initial generator starting at an arbitrary univalent function which is sufficiently smooth up to the boundary. As another application of this method, we show that every subordination chain $f_t$ is differentiable almost everywhere and satisfies a Loewner equation, without assuming that $f_t'(0)$ is continuous.

Article information

Illinois J. Math., Volume 52, Number 4 (2008), 1399-1415.

First available in Project Euclid: 18 November 2009

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Zentralblatt MATH identifier

Primary: 30C35: General theory of conformal mappings 30C55: General theory of univalent and multivalent functions


Roth, Oliver; Schippers, Eric. The Loewner and Hadamard variations. Illinois J. Math. 52 (2008), no. 4, 1399--1415. doi:10.1215/ijm/1258554369.

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