Chapter 1 - History and introduction: classical Markov–Bernstein inequalities

Abstract

Inequalities of Markov and Bernstein type have been fundamental for the proofs of many inverse theorems in polynomial approximation theory. The first chapter provides an account on history and a short introduction to the classical Markov–Bernstein inequalities, as well as their basic generalizations and refinements, and divided into four sections. A brief history and introduction of Markov's inequality along with some of its generalizations, including inequalities for higher derivatives, is given in Section 1.1. Section 1.2 deals with the Markov–Duffin–Schaeffer inequalities, which show that Markov-type inequalities are true under the lighter hypothesis than ones in the original Markov-type inequalities. Markov type inequalities for the so-called curved majorants are also treated in this section. An introduction to Bernstein's inequality along with some of its generalizations and refinements are treated in Section 1.3. Finally, Bernstein's type inequalities on several intervals, as well as the so-called Szegő-variants of Bernstein's inequalities are shortly presented in Section 1.4.