Markov brothers' inequality

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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[edit]

Let P be a polynomial of degree ≤ n. Then

Equality is attained for Chebyshev polynomials of the first kind.

Related inequalities[edit]


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


  1. ^ Achiezer, N.I. (1992). Theory of approximation. New York: Dover Publications, Inc.
  2. ^ Markov, A.A. (1890). "On a question by D. I. Mendeleev". Zap. Imp. Akad. Nauk. St. Petersburg. 62: 1–24.
  3. ^ Markov, V.A. (1892). "О функциях, наименее уклоняющихся от нуля в данном промежутке (On Functions of Least Deviation from Zero in a Given Interval)". Appeared in German with a foreword by Sergei Bernstein as Markov, V.A. (1916). "Über Polynome, die in einem gegebenen Intervalle möglichst wenig von Null abweichen". Math. Ann. 77: 213–258. doi:10.1007/bf01456902.