A Markov-type inequality for the derivatives of constrained polynomials
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Markov's inequality asserts that max -1≤x≤1|p′(x)|≤n2 max -1≤x≤1|p(x)| (1) for every polynomial of degree at most n. The magnitude of sup p∈S max-1≤x≤1|p′(x)| max-1≤x≤1|p(x)| (2) was examined by several authors for certain subclasses S of Πn. In this paper we introduce S = Snm(r) (0 ≤ m ≤ n, 0 < r ≤ 1), the set of those polynomials from Πn which have all but at most m zeros outside the circle with center 0 and radius r, and establish the exact order of the above expression up to a multiplicative constant depending only on m. © 1990.
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