In mathematics, the Markov brothers' inequality is an inequality proved in the 1890s by brothers Andrey Markov and Vladimir Markov, two Russian mathematicians. This inequality bounds the maximum of the derivatives of a polynomial on an interval in terms of the maximum of the polynomial.^[1] For k = 1 it was proved by Andrey Markov,^[2] and for k = 2,3,... by his brother Vladimir Markov.^[3]

The statement

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

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

Equality is attained for Chebyshev polynomials of the first kind.

Related inequalities

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

Applications

Markov's inequality is used to obtain lower bounds in computational complexity theory via the so-called "Polynomial Method".

References