Markov brothers' inequality

Let P be a polynomial of degree ≤ n. Then for all nonnegative integers ${\displaystyle k}$

${\displaystyle \max _{-1\leq x\leq 1}{\big |}P^{(k)}(x){\big |}\leq {\frac {n^{2}(n^{2}-1^{2})(n^{2}-2^{2})\cdots (n^{2}-(k-1)^{2})}{1\cdot 3\cdot 5\cdots (2k-1)}}\max _{-1\leq x\leq 1}|P(x)|.}$

This inequality is tight, as equality is attained for Chebyshev polynomials of the first kind.

• Bernstein's inequality (mathematical analysis)
• Remez inequality

Markov's inequality is used to obtain lower bounds in computational complexity theory via the so-called "polynomial method".^[4]