In
mathematics, the Markov brothers' inequality is an
inequality proved in the 1890s by brothers
Andrey Markov
Andrey Andreyevich Markov, first name also spelled "Andrei", in older works also spelled Markoff) (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research lat ...
and
Vladimir Markov, two Russian mathematicians. This inequality bounds the maximum of the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s 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.
[ Appeared in German with a foreword by Sergei Bernstein as ]
The statement
Let ''P'' be a polynomial of degree ≤ ''n''. Then for all nonnegative integers
:
Equality is attained for
Chebyshev polynomials
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebys ...
of the first kind.
Related inequalities
*
Bernstein's inequality (mathematical analysis) Bernstein's theorem is an inequality relating the maximum modulus of a complex polynomial function on the unit disk with the maximum modulus of its derivative on the unit disk. It was proven by Sergei Bernstein while he was working on approximation ...
*
Remez inequality
Applications
Markov's inequality is used to obtain lower bounds in
computational complexity theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved ...
via the so-calle
"Polynomial Method"
References
{{Reflist
Theorems in analysis
Inequalities