Bernstein's inequality (mathematical analysis)

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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.


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

The inequality finds uses in the field of approximation theory.

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

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