# Quasiconformal extensions, Loewner chains, and the $$\lambda$$ λ -Lemma

Quasiconformal extensions, Loewner chains, and the $$\lambda$$ λ -Lemma Becker (J Reine Angew Math 255:23–43, 1972) discovered a sufficient condition for quasiconformal extendibility of Loewner chains. Many known conditions for quasiconformal extendibility of holomorphic functions in the unit disk can be deduced from his result. We give a new proof of (a generalization of) Becker’s result based on Slodkowski’s Extended $$\lambda$$ λ -Lemma. Moreover, we characterize all quasiconformal extensions produced by Becker’s (classical) construction and use that to obtain examples in which Becker’s extension is extremal (i.e. optimal in the sense of maximal dilatation) or, on the contrary, fails to be extremal. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# Quasiconformal extensions, Loewner chains, and the $$\lambda$$ λ -Lemma

, Volume 8 (4) – Sep 6, 2018
15 pages

/lp/springer-journals/quasiconformal-extensions-loewner-chains-and-the-lambda-lemma-p0HHcLwEf7
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-018-0247-3
Publisher site
See Article on Publisher Site

### Abstract

Becker (J Reine Angew Math 255:23–43, 1972) discovered a sufficient condition for quasiconformal extendibility of Loewner chains. Many known conditions for quasiconformal extendibility of holomorphic functions in the unit disk can be deduced from his result. We give a new proof of (a generalization of) Becker’s result based on Slodkowski’s Extended $$\lambda$$ λ -Lemma. Moreover, we characterize all quasiconformal extensions produced by Becker’s (classical) construction and use that to obtain examples in which Becker’s extension is extremal (i.e. optimal in the sense of maximal dilatation) or, on the contrary, fails to be extremal.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Sep 6, 2018