Bernstein's theorem (polynomials)

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

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

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

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] More generally, if ${\displaystyle P(z)}$ has no zeros in ${\displaystyle |z|<k,}$ for ${\displaystyle k\geq 1}$, then ${\displaystyle \max _{|z|=1}|P'(z)|\leq {\frac {n}{1+k}}\max _{|z|=1}|P(z)|}$.^[4]

• Markov brothers' inequality
• Remez inequality