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

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.