CONTENTSIntroduction..........................................................................................................................................................................5Preliminaries. Complex harmonic functions..........................................................................................................................7I. Spectral values and eigenvalues of a Jordan curve........................................................................................................19 1.1....
Given a quasisymmetric automorphism γ of the unit circle T we define and study a modification Pγ of the classical Poisson integral operator in the case of the unit disk D. The modification is done by means of the generalized Fourier coefficients of γ. For a Lebesgue's integrable complex-valued function f on T, Pγ[f] is a complex-valued harmonic function in D and it coincides with the classical Poisson integral of f provided γ is the identity mapping on T. Our considerations are motivated by the...
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 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...
We give a distortion theorem for quasiconformal automorphisms of the unit disk and its application to improving some results due to Douady and Earle.
Given a quasisymmetric automorphism of the unit circle we define and study a modification of the classical Poisson integral operator in the case of the unit disk . The modification is done by means of the generalized Fourier coefficients of . For a Lebesgue’s integrable complexvalued function on , is a complex-valued harmonic function in and it coincides with the classical Poisson integral of provided is the identity mapping on . Our considerations are motivated by the problem...
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 provided F...
Different aspects of the boundary value problem for quasiconformal mappings and Teichmüller spaces are expressed in a unified form by the use of the trace and extension operators. Moreover, some new results on harmonic and quasiconformal extensions are included.
In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping in the unit disk , if is a convex domain, then the inequality holds for all distinct points . Here and are holomorphic mappings in determined by , up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain in and improve it provided is additionally a quasiconformal mapping in .
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