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Generalizing a number of earlier results, P. Borwein established a sharp Markov-type inequality on [-1, 1] for the derivatives of polynomials p E 7rn having at most k (O < k < n) zeros in the complex unit disk. Using Lorentz representation and a Markov-type inequality for the derivative of Muntz polynomials due to D. Newman, we give a surprisingly short proof of Borwein's Theorem. The new result of this paper is to obtain a sharp Bernsteintype analogue of Borwein's Theorem. By the same method we prove a sharp Bernstein-type inequality for another wide family of classes of constrained polynomials.

Bernstein-type inequalities for the derivatives of constrained polynomials