Markov brothers' inequality

From Wikipedia, the free encyclopedia

You have new messages (last change).

In mathematics, the Markov brothers' inequality is an inequality proved by Andrey Markov for k = 1 and by his brother Vladimir Markov for k = 2,3,\dots

Proposition: Let P be a polynomial of degree \leq n. Then

\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.

[edit] See also

[edit] References

  • N.I.Achiezer (Akhiezer), Theory of approximation, Translated from the Russian and with a preface by Charles J.~Hyman, Dover Publications, Inc., New York, 1992. x+307 pp.
  • A.A.Markov, On a question by D.I.Mendeleev, Zap. Imp. Akad. Nauk SPb. 62 (1890), 1-24
  • V.A.Markov, O funktsiyakh, naimeneye uklonyayushchikhsya ot nulya v dannom promezhutke (1892). Appeared in German with a foreword by Sergei Bernstein as Über Polynome, die in einem gegebenen Intervalle möglichst wenig von Null abweichen, Math. Ann. 77 (1916), 213-258
Retrieved from "http://en.wikipedia.org../../../m/a/r/Markov_brothers%27_inequality.html"
In other languages