Markov brothers' inequality

Markov brothers' inequality bounds the maximum magnitude of a polynomial’s derivative on an interval using the polynomial’s degree and maximum magnitude there. It also has higher-derivative forms with degree-dependent bounds.

Markov inequality: For a polynomial of degree at most n on [−1,1], the derivative’s maximum magnitude is at most n² times the polynomial’s maximum magnitude. This is the first-derivative form at the heart of the brothers' inequality.
Andrey Markov: A Russian mathematician known for work in probability, analysis, and approximation theory. He established the foundational polynomial derivative bound in the late nineteenth century.
Polynomial interpolation: The construction of a polynomial taking prescribed values at a finite set of distinct points. Derivative bounds help analyze how interpolation polynomials vary between data points.
Bernstein–Walsh inequality: A bound on a polynomial outside a compact set in terms of its supremum on that set and a conformal growth factor. It extends polynomial growth control beyond the interval used in Markov’s theorem.
Bernstein inequality for polynomials: A polynomial derivative bound on [−1,1] weighted by the square root of one minus the variable squared. It complements Markov’s uniform endpoint-sensitive bound with a sharper interior estimate.
Vladimir Markov: A Russian mathematician who extended his brother Andrey’s polynomial derivative inequality to higher derivatives. His work gives the higher-derivative bounds behind the plural name.
Runge's phenomenon: The divergence of certain polynomial interpolation sequences near interval endpoints as the number of equally spaced nodes increases. Markov-type growth bounds help explain how polynomial derivatives can become large.
Remez inequality: A bound on a polynomial’s maximum over an interval using its maximum on a measurable subset. It controls polynomial size from partial-set information rather than derivative size.
Chebyshev polynomial: A family of orthogonal polynomials characterized by cosine identities and extremal properties on [−1,1]. Chebyshev polynomials attain equality in Markov’s first-derivative bound.
Approximation theory: The study of approximating functions by simpler objects, especially polynomials and rational functions. The inequality arose within classical questions about polynomial approximation and extremal behavior.

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