Inmathematics, 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]
Let P be a polynomial of degree ≤ n. Then for all nonnegative integers 
Equality is attained for Chebyshev polynomials of the first kind.
Markov's inequality is used to obtain lower bounds in computational complexity theory via the so-called "Polynomial Method".
{{cite journal}}: Cite journal requires |journal= (help) Appeared in German with a foreword by Sergei BernsteinasMarkov, V.A. (1916). "Über Polynome, die in einem gegebenen Intervalle möglichst wenig von Null abweichen". Math. Ann. 77 (2): 213–258. doi:10.1007/bf01456902. S2CID 122406663.
