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Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials

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We prove that max |p′(x)|, where p runs over the set of all algebraic polynomials of degree not higher than n ≥ 3 bounded in modulus by 1 on [−1, 1], is not lower than \( {{\left( {n - 1} \right)} \mathord{\left/{\vphantom {{\left( {n - 1} \right)} {\sqrt {1 - {x^2}} }}} \right.} {\sqrt {1 - {x^2}} }} \) for all x ∈ (−1, 1) such that \( \left| x \right| \in \bigcup\nolimits_{k = 0}^{\left[ {{n \mathord{\left/{\vphantom {n 2}} \right.} 2}} \right]} {\left[ {\cos \frac{{2k + 1}}{{2\left( {n - 1} \right)}}\pi, \cos \frac{{2k + 1}}{{2n}}\pi } \right]} \).

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Correspondence to A. I. Podvysotskaya.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 5, pp. 711–715, May, 2009.

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Podvysotskaya, A.I. Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials. Ukr Math J 61, 847 (2009).

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  • Approximation Theory
  • Extreme Problem
  • Classical Analysis
  • Chebyshev Polynomial
  • Integral Functional