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The universal means spectrum of conformal mappings has been studied extensively in recent years. In some situations, sharp results are available, in others, only upper and lower estimates have been obtained so far. We review some of the classical results before discussing the recent work of...
We consider the problem of finding a measure from the given values of its logarithmic potential on the support. It is well known that a solution to this problem is given by the generalized Laplacian. The case of our main interest is when the support is contained in a rectifiable curve, and the...
The Hayman-Wu Theorem states that the preimage of a line or circle L under a conformal mapping from the unit disc D to a simply-connected domain Ω has total Euclidean length bounded by an absolute constant. The best possible constant is known to lie in the interval [π2, 4π), thanks to work of...
Let μ be a measure of compact support in the complex plane. In 1991 J. E. Thomson described completely the structure of H
p(μ), the closed subspace of L
p(μ) spanned by the polynomials, when 1 ≤ p < ∞. Here we discuss the extent to which Rp(μ), the closed subspace of L
p(μ) spanned by the...
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