Displaying similar documents to “Removable singularities for Hölder continuous quasiregular mappings in the plane.”

Change of variables formula under minimal assumptions

Piotr Hajłasz (1993)

Colloquium Mathematicae

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In the previous papers concerning the change of variables formula (in the form involving the Banach indicatrix) various assumptions were made about the corresponding transformation (see e.g. [BI], [GR], [F], [RR]). The full treatment of the case of continuous transformation is given in [RR]. In [BI] the transformation was assumed to be continuous, a.e. differentiable and with locally integrable Jacobian. In this paper we show that none of these assumptions is necessary (Theorem 2). We...

On a result by Clunie and Sheil-Small

Dariusz Partyka, Ken-ichi Sakan (2012)

Annales UMCS, Mathematica

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In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping F in the unit disk D, if F(D) is a convex domain, then the inequality |G(z2)− G(z1)| < |H(z2) − H(z1)| holds for all distinct points z1, z2∈ D. Here H and G are holomorphic mappings in D determined by F = H + Ḡ, up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain Ω in ℂ and improve it...

Lebesgue measure and mappings of the Sobolev class W 1 , n

O. Martio (1995)

Banach Center Publications

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We present a survey of the Lusin condition (N) for W 1 , n -Sobolev mappings f : G → ℝ n defined in a domain G of ℝ n . Applications to the boundary behavior of conformal mappings are discussed.

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

Dariusz Partyka (1995)

Banach Center Publications

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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...