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In mathematical analysis, Bernstein's inequality is named after Sergei Natanovich Bernstein. The inequality states that on the complex plane, within the disk of radius 1, the degree of a polynomial times the maximum value of a polynomial is an upper bound for the similar maximum of its derivative.
Theorem

Let P be a polynomial of degree n on complex numbers with derivative P′. Then

\max_{|z| \le 1}( |P'(z)| ) \le n\cdot\max_{|z| \le 1}( |P(z)| )

The inequality finds uses in the field of approximation theory.

Using the Bernstein's inequality we have for the k:th derivative,

\max_{|z| \le 1}( |P^{(k)}(z)| ) \le \frac{n!}{(n-k)!} \cdot\max_{|z| \le 1}( |P(z)| ).

Markov brothers' inequality
Remez inequality

References

Frappier, Clément (2004). "Note on Bernstein's inequality for the third derivative of a polynomial" (PDF). J. Inequal. Pure Appl. Math. 5 (1). Paper No. 7. ISSN 1443-5756. Zbl 1060.30003.
Natanson, I.P. (1964). Constructive function theory. Volume I: Uniform approximation. Translated by Alexis N. Obolensky. New York: Frederick Ungar. MR 0196340. Zbl 0133.31101.
Rahman, Q. I.; Schmeisser, G. (2002). Analytic theory of polynomials. London Mathematical Society Monographs. New Series 26. Oxford: Oxford University Press. ISBN 0-19-853493-0. Zbl 1072.30006.

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