Bernstein's theorem (polynomials)

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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.[1]

Statement[edit]

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

.

[2]

Proof[edit]

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 .

By Rouché's theorem, with has all its zeros in . By virtue of the Gauss–Lucas theorem, has all its zeros in as well. It follows that on , otherwise we could choose an with such that has a zero in .

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 .

Bernstein's inequality[edit]

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,

Similar results[edit]

Paul Erdős conjectured that if has no zeros in , then . This was proved by Peter Lax.[3]

M. A. Malik showed that if has no zeros in for a given , then .[4]

See also[edit]

References[edit]

  1. ^ R. P. Boas, Jr., Inequalities for the derivatives of polynomials, Math. Mag. 42 (1969), 165–174.
  2. ^ M. A. Malik, M. C. Vong, Inequalities concerning the derivative of polynomials, Rend. Circ. Mat. Palermo (2) 34 (1985), 422–426.
  3. ^ P. D. Lax, Proof of a conjecture of P. Erdös on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509–513.
  4. ^ M. A. Malik, On the derivative of a polynomial J. London Math. Soc (2) 1 (1969), 57–60.

Further reading[edit]

  • Frappier, Clément (2004). "Note on Bernstein's inequality for the third derivative of a polynomial" (PDF). J. Inequal. Pure Appl. Math. 5 (1). Paper No. 7. ISSN 1443-5756. Zbl 1060.30003.
  • Natanson, I.P. (1964). Constructive function theory. Volume I: Uniform approximation. Translated by Alexis N. Obolensky. New York: Frederick Ungar. MR 0196340. Zbl 0133.31101.
  • Rahman, Q. I.; Schmeisser, G. (2002). Analytic theory of polynomials. London Mathematical Society Monographs. New Series. 26. Oxford: Oxford University Press. ISBN 0-19-853493-0. Zbl 1072.30006.