Bernstein-type inequalities for the derivatives of constrained polynomials
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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 ∈Πnhaving at most k (0 ≤ k ≤ n) zeros in the complex unit disk. Using Lorentz representation and a Markov-type inequality for the derivative of Müntz 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 foranother wide family of classes of constrained polynomials. © 1991 American Mathematical Society.
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