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]

Statement

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

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

The inequality cannot be improved and equality holds if and only if ${\displaystyle P(z)=\alpha z^{n}}$. ^[2]

Bernstein's inequality

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. Applying the theorem k times yields

${\displaystyle \max _{|z|\leq 1}|P^{(k)}(z)|\leq {\frac {n!}{(n-k)!}}\cdot \max _{|z|\leq 1}|P(z)|.}$

Similar results

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

M. A. Malik showed that if ${\displaystyle P(z)}$ has no zeros in ${\displaystyle |z|<k}$ for a given ${\displaystyle k\geq 1}$, then ${\displaystyle \max _{|z|=1}|P'(z)|\leq {\frac {n}{1+k}}\max _{|z|=1}|P(z)|}$.^[4]

See also

• Markov brothers' inequality
• Remez inequality