MARKOV AND BERNSTEIN TYPE INEQUALITIES IN L(P) FOR CLASSES OF POLYNOMIALS WITH CONSTRAINTS
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The Markov-type inequality is proved for all real algebraic polynomials f of degree at most n having at most k, with 0 k n, zeros (counting multiplicities) in the open unit disk of the complex plane, and for all p > 0, where c(p) = cp + 1(l + p2) with some absolute constant c > 0. This inequality has been conjectured since 1983 when the L case of the above result was proved. It improves and generalizes many earlier results. Up to the multiplicative constant c(p)> 0 the above inequality is sharp. A sharp Bernstein-type analogue for real trigonometric polynomials is also established, which is interesting on its own, and plays a key role in the proof of the Markov-type inequality. 1995 The London Mathematical Society.
