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Convexity of harmonic densities. (English) Zbl 1253.31004
Summary: The convexity of the densities of harmonic measures is proven for subsets of a circle or of the real line. As a consequence, we get the convexity of the densities of equilibrium measures for compact sets lying on circles or the real axis.

MSC:
31A15Potentials and capacity, harmonic measure, extremal length (two-dimensional)
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Full Text: DOI
References:
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[4] Garnett, J. B. and Marshall, D. E.: Harmonic measure. New Mathematical Monographs 2, Cambridge University Press, Cambridge, 2005. · Zbl 1077.31001
[5] Landkof, N. S.: Foundations of modern potential theory. Grundlehren der mathe- matischen Wissenschaften 180, Springer-Verlag, New York-Heidelberg, 1972.
[6] Ransford, T.: Potential theory in the complex plane. Cambridge University Press, Cambridge, 1995. · Zbl 0828.31001
[7] Simon, B.: Orthogonal polynomials on the unit circle, Part 2, Spectral theory. Amer- ican Mathematical Society Colloquium Publications 54, part 2, American Mathemat- ical Society, Providence, RI, 2005. · Zbl 1082.42021