Bernstein's inequality (mathematical analysis)

In the mathematical theory of mathematical analysis, Bernstein's inequality, named after Sergei Natanovich Bernstein, is defined as follows.

Let P be a polynomial of degree $n$ with derivative P′. Then

$\max(P') \le n\cdot\max(P)$

where we define the maximum of a polynomial to be the maximum value attained within a unit disk:

$\max(X) = \max_{|z| \leq 1} \big|X(z)\big|.$

The inequality is named after Sergei Natanovich Bernstein and finds uses in the field of approximation theory.

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

$\max(P^{(k)}) \le \frac{n!}{(n-k)!} \cdot\max(P).$

[edit] See also

C. Frappier, Note on Bernstein's inequality for the third derivative of a polynomial, Journal of Inequalities in Pure and Applied Mathematics, Vol. 5, Issue 1, Article 7, 6 pp., 2004. [1]

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