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## The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle

### Volume 31 / 1995

Banach Center Publications 31 (1995), 303-310 DOI: 10.4064/-31-1-303-310

#### Abstract

This paper provides sufficient conditions on a quasisymmetric automorphism γ of the unit circle which guarantee the existence of the smallest positive eigenvalue of γ. They are expressed by means of a regular quasiconformal Teichmüller self-mapping φ of the unit disc Δ. In particular, the norm of the generalized harmonic conjugation operator $A_γ:ℍ → ℍ$ is determined by the maximal dilatation of φ. A characterization of all eigenvalues of a quasisymmetric automorphism γ in terms of the smallest positive eigenvalue of some other quasisymmetric automorphism σ is given.

#### Authors

• Dariusz Partyka

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