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Bounds for the derivatives of polynomials: the theorems of Bernstein and Markov

Published online by Cambridge University Press:  15 October 2025

G.J.O. Jameson
G.J.O. Jameson*
Affiliation:
13 Sandown Road, Lancaster LA1 4LN e-mail: pgjameson2@gmail.com
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Extract

This Article is an exposition of two very old theorems. One was first proved by A. A. Markov in 1889 [1], the other by S. N. Bernstein in 1912 [2]. Both Markov (1856–1922) and Bernstein (1880–1968) were authors of numerous pioneering works: there are many other results designated Markov’s theorem or Bernstein’s theorem (or Bernstein’s inequality) in their respective subject areas. “Markov chains” have become a standard concept in probability theory, and “Bernstein polynomials” a standard one in approximation theory.

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Articles
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The Mathematical Gazette , Volume 109 , Issue 576 , November 2025 , pp. 462 - 470
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© The Authors, 2025 Published by Cambridge University Press on behalf of The Mathematical Association

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References

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