Bernstein-type inequalities for the derivatives of constrained polynomials Academic Article

Generalizing a number of earlier results, P. Borwein established a sharp Markov-type inequality on [ 1 , 1 ] [ - 1,1] for the derivatives of polynomials p n p in {pi _n} having at most k ( 0 k n ) k(0 leq k leq n) zeros in the complex unit disk. Using Lorentz representation and a Markov-type inequality for the derivative of Mntz polynomials due to D. Newman, we give a surprisingly short proof of Borweins Theorem. The new result of this paper is to obtain a sharp Bernstein-type analogue of Borweins Theorem. By the same method we prove a sharp Bernstein-type inequality for another wide family of classes of constrained polynomials.