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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Markov's inequality for polynomials with real zeros


Author: Peter Borwein
Journal: Proc. Amer. Math. Soc. 93 (1985), 43-47
MSC: Primary 41A17
DOI: https://doi.org/10.1090/S0002-9939-1985-0766524-4
MathSciNet review: 766524
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Abstract: Markov's inequality asserts that $ \vert\vert{p'_n}\vert\vert \leqslant {n^2}\vert\vert{p_n}\vert\vert$ for any polynomial $ {p_n}$ of degree $ n$. (We denote the supremum norm on $ [ - 1,1]$ by $ \vert\vert.\vert\vert$.) In the case that $ {p_n}$ has all real roots, none of which lie in $ [ - 1,1]$, Erdös has shown that $ \vert\vert{p'_n}\vert\vert \leqslant en\vert\vert{p_n}\vert\vert/2$. We show that if $ {p_n}$ has $ n - k$ real roots, none of which lie in $ [ - 1,1]$, then $ \vert\vert{p'_n} \leqslant cn(k + 1)\vert\vert{p_n}\vert\vert$, where $ c$ is independent of $ n$ and $ k$. This extension of Markov's and Erdös' inequalities was conjectured by Szabados.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0766524-4
Article copyright: © Copyright 1985 American Mathematical Society