A note on Bernstein and Markov type inequalities

Let D be the unit disk in the complex plane C . We prove that for any polynomial p of degree at most nmaxz∈∂Dp(z)-p(z¯)z-z¯⩽nmax0⩽j⩽npeijπ/n+pe-ijπ/n2,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.