Markov-Bernstein Type Inequalities for Classes of Polynomials with Restricted Zeros

We prove that there exists an absolute constant c > 0 such that |p′(y)| ≤ c min { n(k + 1), ( n(k + 1) 1 − y2 )1/2} max −1≤x≤1 |p(x)|, −1 ≤ y ≤ 1 for every real algebraic polynomial of degree at most n having at most k zeros in the open unit disk {z ∈ C : |z| < 1}. This inequality, which has been conjectured for at least a decade, improves and generalizes several earlier results. Up to the multiplicative absolute constant c, it is a sharp generalization of both Markov’s and Bernstein’s inequalities.