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INVERSE BERNSTEIN INEQUALITIES AND MIN–MAX–MIN PROBLEMS ON THE UNIT CIRCLE

  • Tamás Erdélyi (a1), Douglas P. Hardin (a2) and Edward B. Saff (a3)

Abstract

We give a short and elementary proof of an inverse Bernstein-type inequality found by S. Khrushchev for the derivative of a polynomial having all its zeros on the unit circle. The inequality is used to show that equally-spaced points solve a min–max–min problem for the logarithmic potential of such polynomials. Using techniques recently developed for polarization (Chebyshev-type) problems, we show that this optimality also holds for a large class of potentials, including the Riesz potentials $1/r^{s}$ with $s>0.$

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1.Ambrus, G., Analytic and Probabilistic Problems in Discrete Geometry. PhD Thesis, University College London, 2009.
2.Ambrus, G., Ball, K. and Erdélyi, T., Chebyshev constants for the unit circle. Bull. Lond. Math. Soc. 45(2) 2013, 236248.
3.Bernstein, S. N., Leçons sur les Propriétés Extrémales et la Meilleure Approximation des Fonctions Analytiques d’une Variable Réelle. Gauthier-Villars (Paris, 1926).
4.Borwein, P. and Erdélyi, T., Polynomials and Polynomial Inequalities. Springer (New York, NY, 1995).
5.Brauchart, J. S., Hardin, D. P. and Saff, E. B., The Riesz energy of the Nth roots of unity: an asymptotic expansion for large N. Bull. Lond. Math. Soc. 41(4) 2009, 621633.
6.DeVore, R. A. and Lorentz, G. G., Constructive Approximation. Springer (Berlin, Heidelberg, 1993).
7.Erdélyi, T. and Saff, E. B., Riesz polarization in higher dimensions. J. Approx. Theory 171 2013, 128147.
8.Hardin, D. P., Kendall, A. P. and Saff, E. B., Polarization optimality of equally spaced points on the circle for discrete potentials. Discrete Comput. Geom. 50(2) 2013, 236243.
9.Khrushchev, S., Rational compacts and exposed quadratic irrationalities. J. Approx. Theory 159(2) 2009, 243289.
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INVERSE BERNSTEIN INEQUALITIES AND MIN–MAX–MIN PROBLEMS ON THE UNIT CIRCLE

  • Tamás Erdélyi (a1), Douglas P. Hardin (a2) and Edward B. Saff (a3)

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