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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. For k = 1 it was proved by Andrey Markov, and for k = 2,3,... by his brother Vladimir Markov.

Contents

The statement

Let P be a polynomial of degree ≤ n. Then

max − 1 ≤ x ≤ 1 | P ( k ) ( x ) | ≤ n 2 ( n 2 − 1 2 ) ( n 2 − 2 2 ) ⋯ ( n 2 − ( k − 1 ) 2 ) 1 ⋅ 3 ⋅ 5 ⋯ ( 2 k − 1 ) max − 1 ≤ x ≤ 1 | P ( x ) | .

Equality is attained for Chebyshev polynomials of the first kind.

  • 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

    Markov brothers' inequality Wikipedia
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