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Mathematical Notes

, Volume 80, Issue 1–2, pp 31–35 | Cite as

Lemniscates and inequalities for the logarithmic capacities of continua

  • V. N. Dubinin
Article

Abstract

It is shown that if P(z) = z n + ⋯ is a polynomial with connected lemniscate E(P) = {z: ¦P(z)¦ ≤ 1} and m critical points, then, for any n− m+1 points on the lemniscate E(P), there exists a continuum γ ⊂ E(P) of logarithmic capacity cap γ ≤ 2−1/n which contains these points and all zeros and critical points of the polynomial. As corollaries, estimates for continua of minimum capacity containing given points are obtained.

Key words

lemniscate of a polynomial logarithmic capacity continuum of minimal capacity Riemann surface conformal map Chebyshev polynomial 

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • V. N. Dubinin
    • 1
  1. 1.Institute of Applied Mathematics, Far-Eastern DivisionRussian Academy of SciencesRussia

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