Bernstein's theorem (polynomials)

In mathematics, 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]

Let ${\displaystyle \max _{|z|=1}|f(z)|}${\displaystyle \max _{|z|=1}|f(z)|} denote the maximum modulus of an arbitrary function ${\displaystyle f(z)}${\displaystyle f(z)} on ${\displaystyle |z|=1}${\displaystyle |z|=1}, and let ${\displaystyle f'(z)}${\displaystyle f'(z)} denote its derivative. Then for every polynomial ${\displaystyle P(z)}${\displaystyle P(z)} of degree ${\displaystyle n}${\displaystyle n} we have

$${\displaystyle \max _{|z|=1}|P'(z)|\leq n\max _{|z|=1}|P(z)|}$${\displaystyle \max _{|z|=1}|P'(z)|\leq n\max _{|z|=1}|P(z)|}

and equality holds if and only if ${\displaystyle P(z)=\alpha z^{n}}${\displaystyle P(z)=\alpha z^{n}}.^[2]

Paul Erdős conjectured that if ${\displaystyle P(z)}${\displaystyle P(z)} has no zeros in ${\displaystyle |z|<1}${\displaystyle |z|<1}, then ${\displaystyle \max _{|z|=1}|P'(z)|\leq {\frac {n}{2}}\max _{|z|=1}|P(z)|}${\displaystyle \max _{|z|=1}|P'(z)|\leq {\frac {n}{2}}\max _{|z|=1}|P(z)|}. This was proved by Peter Lax.^[3] More generally, if ${\displaystyle P(z)}${\displaystyle P(z)} has no zeros in ${\displaystyle |z|<k,}${\displaystyle |z|<k,} for ${\displaystyle k\geq 1}${\displaystyle k\geq 1}, then ${\displaystyle \max _{|z|=1}|P'(z)|\leq {\frac {n}{1+k}}\max _{|z|=1}|P(z)|}${\displaystyle \max _{|z|=1}|P'(z)|\leq {\frac {n}{1+k}}\max _{|z|=1}|P(z)|}.^[4]

• Markov brothers' inequality
• Remez inequality

• 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. OCLC 179746249. Zbl 0133.31101.
• Rahman, Q.I.; Schmeisser, G. (2002). Analytic theory of polynomials. London Mathematical Society Monographs. New Series. Vol.26. Oxford: Oxford University Press. doi:10.1093/oso/9780198534938.001.0001. ISBN 0-19-853493-0. Zbl 1072.30006.