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
Let P be a polynomial of degree ≤ n. Then for all nonnegative integers [math]\displaystyle{ k }[/math]
- [math]\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)|. }[/math]
Equality is attained for Chebyshev polynomials of the first kind.
Related inequalities
Applications
Markov's inequality is used to obtain lower bounds in computational complexity theory via the so-called "Polynomial Method".
References
- ↑ Achiezer, N.I. (1992). Theory of approximation. New York: Dover Publications, Inc..
- ↑ Markov, A.A. (1890). "On a question by D. I. Mendeleev". Zap. Imp. Akad. Nauk. St. Petersburg 62: 1–24.
- ↑ 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 (2): 213–258. doi:10.1007/bf01456902. https://zenodo.org/record/1428284.
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