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 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)| ).

See also[edit]