Bernstein's inequality (mathematical analysis)

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)| ).

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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