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

Part of: General convexity Approximations and expansions Polynomials and matrices Geometric function theory

Published online by Cambridge University Press:  13 August 2014

Tamás Erdélyi ,
Douglas P. Hardin  and
Edward B. Saff
Tamás Erdélyi
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843, U.S.A. email terdelyi@math.tamu.edu
Douglas P. Hardin
Affiliation:
Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, TN 37240, U.S.A. email doug.hardin@vanderbilt.edu
Edward B. Saff
Affiliation:
Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, TN 37240, U.S.A. email edward.b.saff@vanderbilt.edu
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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 $s>0.$

MSC classification

Primary: 11C08: Polynomials 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) 41A17: Inequalities in approximation (Bernstein, Jackson, Nikol'skiĭ-type inequalities) 52A40: Inequalities and extremum problems
Type
Research Article
Information
Mathematika , Volume 61 , Issue 3 , September 2015 , pp. 581 - 590
Copyright
Copyright © University College London 2014 

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References

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