mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Markov brothers' inequality is an

inequality Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...

proved in the 1890s by brothers Andrey Markov and

Vladimir Markov Vladimir Markov may refer to: * Vladimir Markov (mathematician) (1871–1897), Russian mathematician * Vladimir Markov (footballer, born 1889) (1889–1942), Russian football defender * Vladimir Markov (politician) (1859–1919), Finnish general an ...

, 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. F ...

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 Sergei Natanovich Bernstein (russian: Серге́й Ната́нович Бернште́йн, sometimes Romanized as ; 5 March 1880 – 26 October 1968) was a Ukrainian and Russian mathematician of Jewish origin known for contributions to parti ...

The statement

Let ''P'' be a polynomial of degree ≤ ''n''. Then for all nonnegative integers

k

\max_ , P^(x),  \leq \frac \max_ , P(x), .

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 Chebyshe ...

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 In mathematics, the Remez inequality, discovered by the Soviet mathematician Evgeny Yakovlevich Remez , gives a bound on the sup norms of certain polynomials, the bound being attained by the Chebyshev polynomials. The inequality Let ''σ'' be an ...

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 by ...

via the so-calle
"Polynomial Method"

References

{{Reflist Theorems in analysis Inequalities