Illinois Journal of Mathematics

The Loewner and Hadamard variations

Oliver Roth and Eric Schippers

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 18 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258554369

Digital Object Identifier
doi:10.1215/ijm/1258554369

Mathematical Reviews number (MathSciNet)
MR2595774

Zentralblatt MATH identifier
1187.30005

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

Citation

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


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