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Inequalities for derivatives of polynomials with restricted zeros

Author: Attila Máté
Journal: Proc. Amer. Math. Soc. 82 (1981), 221-225
MSC: Primary 26C05; Secondary 26D05, 30C10
MathSciNet review: 609655
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Abstract: Let $ f$ be a polynomial of degree $ n$ that has real roots all of which lie outside the interval $ ( - 1,1)$. P. Erdös proved that if $ \left\vert f \right\vert \leqslant 1$ on this interval then $ \left\vert {f'} \right\vert < ne/2$ on $ [ - 1,1]$; this is much stronger than the results the inequalities of A. Markov and S. N. Bernstein would give. We will show that if only $ n - k$ roots of $ f$ are restricted as above, then $ \left\vert {f'} \right\vert < {c_k}n$ holds on $ [ - 1,1]$ with appropriate $ {c_k}$. An upper estimate for the best $ {c_k}$ is given. Results for higher derivatives and $ {L^p}$ spaces are also obtained.

References [Enhancements On Off] (What's this?)

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Keywords: Bernstein's inequality, Markov's inequality, polynomials with real zeros
Article copyright: © Copyright 1981 American Mathematical Society

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