Bernstein's theorem (polynomials)
Bernstein's theorem is an inequality relating the maximum modulus of a complex polynomial function on the unit disk with the maximum modulus of its derivative on the unit disk. It was proven by Sergei Bernstein while he was working on approximation theory.
Let denote the maximum modulus of an arbitrary function on , and let denote its derivative. Then for every polynomial of degree we have
The inequality is best possible with equality holding if and only if
Let be a polynomial of degree , and let be another polynomial of the same degree with no zeros in . We show first that if on , then on .
For an arbitrary polynomial of degree , we obtain Bernstein's Theorem by applying the above result to the polynomials , where is an arbitrary constant exceeding .
In mathematical analysis, Bernstein's 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. Taking the k:th derivative of the theorm,
M. A. Malik showed that if has no zeros in for a given , then .
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