Some extensions of the Markov inequality for polynomials

Let D denote the unit disc of the complex plane and Pn the class of polynomials of degree at most n with complex coefficients. We prove that max z ∈ ∂D ∣∣∣∣pk(z) − pk(z̄)z − z̄ ∣∣∣ ∣ ≤ n1+k max0≤j≤n ∣∣∣∣p(eijpi/n) + p(e−ijpi/n)2 where p0: = p belongs to Pn and for k ≥ 0, pk+1(z): = zp′k(z). We also show how this result contains or sharpens certain classical inequalities for polynomials due to Bernstein, Markov, Duffin and Schaeffer and others