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

:$max(P\text{'})\; le\; ncdotmax(P)$

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

:

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,

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

ee also

* Markov brothers' inequality
* Remez inequality

References

* 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. [http://www.emis.de/journals/JIPAM/images/154_03_JIPAM/154_03.pdf|online]