Bernstein's inequality (mathematical analysis)
- 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 with derivative "P′". Then
:
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,
:
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]
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