A note on Bernstein and Markov type inequalities

Abstract

Let D be the unit disk in the complex plane C . We prove that for any polynomial p of degree at most n max z ∈ ∂ D p ( z ) - p ( z ¯ ) z - z ¯ ⩽ n max 0 ⩽ j ⩽ n p e ij π / n + p e - ij π / n 2 , where ∂ D denotes the boundary of D . We show how this result is related to classical inequalities of Bernstein and Markov and to more recent results due to Duffin and Schaeffer.