Math 2.0

Approximations and expansions Rates of approximation: direct and inverse theorems Theorem

Markov brothers' inequality

Builds on Polynomial functionDegree of a polynomialHigher derivative

Statement

Let n≥1n \ge 1 and let PP be a real polynomial function of degree at most nn. Write ∥Q∥=max⁡−1≤x≤1∣Q(x)∣\norm{Q} = \max_{-1 \le x \le 1} \abs{Q(x)} for a function QQ continuous on [−1,1][-1, 1].

  1. ∥P′∥≤n2∥P∥\norm{P'} \le n^2 \norm{P}.
  2. More generally, for 1≤k≤n1 \le k \le n, the kk-th derivative satisfies ∥P(k)∥≤n2(n2−12)(n2−22)⋯(n2−(k−1)2)1⋅3⋅5⋯(2k−1) ∥P∥.\norm{P^{(k)}} \le \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)} \, \norm{P}.

Remarks

Both bounds are attained by the Chebyshev polynomial TnT_n, defined by Tn(cos⁡θ)=cos⁡(nθ)T_n(\cos \theta) = \cos(n \theta). The constant in the second bound equals Tn(k)(1)T_n^{(k)}(1); for k=1k = 1 it is n2n^2.

Example: P(x)=T2(x)=2x2−1P(x) = T_2(x) = 2x^2 - 1 has ∥P∥=1\norm{P} = 1 and P′(x)=4xP'(x) = 4x, so ∥P′∥=4=22∥P∥\norm{P'} = 4 = 2^2 \norm{P}.

In the interior, Bernstein's inequality for polynomials gives the stronger pointwise bound ∣P′(x)∣≤n1−x2∥P∥\abs{P'(x)} \le \frac{n}{\sqrt{1 - x^2}} \norm{P} for −1<x<1-1 < x < 1. The factor n2n^2 in the first bound is caused by the endpoints ±1\pm 1, where this pointwise bound degenerates.

Going deeper

For a student

Markov-type inequalities bound the derivative of a polynomial by its size, with a constant depending only on the degree. They are used in approximation theory to prove inverse theorems: if a function is well approximated by polynomials, then it is smooth. The first inequality is due to A. A. Markov and the general one to his brother V. A. Markov.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionRing, definition, no evidence levelRingdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionPolynomial function, definition, no evidence levelPolynomial functiondefinitionDerivative function, definition, no evidence levelDerivative functiondefinitionHigher derivative, definition, no evidence levelHigher derivativedefinitionMarkov brothers' inequality, statement, no evidence levelMarkov brothers' ineq…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Group concept
  2. Limit point of a set definition
  3. Ring definition
  4. Limit of a function definition
  5. Polynomial ring definition
  6. Continuity at a point definition
  7. Derivative definition
  8. Degree of a polynomial definition
  9. Continuous function definition
  10. Differentiable function definition
  11. Polynomial function definition
  12. Derivative function definition
  13. Higher derivative definition
  14. Markov brothers' inequality statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Markov_brothers%27_inequality
About this page
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Kindstatement, theorem
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Changed2 November 2026

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