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We give a quasiconformal version of the proof for the classical Lindelöf theorem: Let f map the unit disk D conformally onto the inner domain of a Jordan curve C. Then C is smooth if and only if arh f'(z) has a continuous extension to D. Our proof does not use the Poisson integral representation of harmonic functions in the unit disk.
It is shown that the -fine topology defined via a Wiener criterion is the coarsest topology making all supersolutions to the -Laplace equation
continuous. Fine limits of quasiregular and BLD mappings are also studied.
The rate of growth of the energy integral of a quasiregular mapping is estimated in terms of a special isoperimetric condition on . The estimate leads to new Phragmén-Lindelöf type theorems.
Let be a mapping from a metric space X to a metric space Y, and let α be a positive real number. Write dim (E) and H(E) for the Hausdorff dimension and the s-dimensional Hausdorff measure of a set E. We give sufficient conditions that the equality dim (f(E)) = αdim (E) holds for each E ⊆ X. The problem is studied also for the Cantor ternary function G. It is shown that there is a subset M of the Cantor ternary set such that H(M) = 1, with s = log2/log3 and dim(G(E)) = (log3/log2) dim (E), for every...
We give a quasiconformal version of the proof for the classical Lindelof theorem: Let map the unit disk conformally onto the inner domain of a Jordan curve : Then is smooth if and only if arg has a continuous extension to . Our proof does not use the Poisson integral representation of harmonic functions in the unit disk.
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