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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Inequalities for polynomials on the unit interval


Authors: Q. I. Rahman and G. Schmeisser
Journal: Trans. Amer. Math. Soc. 231 (1977), 93-100
MSC: Primary 30A06
DOI: https://doi.org/10.1090/S0002-9947-1977-0463406-2
MathSciNet review: 0463406
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Abstract: Let $ {p_n}(z) = \sum\nolimits_{k = 0}^n {{a_k}{z^k}} $ be a polynomial of degree at most n with real coefficients. Generalizing certain results of I. Schur related to the well-known inequalities of Chebyshev and Markov we prove that if $ {p_n}(z)$ has at most $ n - 1$ distinct zeros in $ ( - 1,1)$, then

\begin{displaymath}\begin{array}{*{20}{c}} {\vert{a_n}\vert \leqslant {2^{n - 1}... ...1 \leqslant x \leqslant 1} \vert{p_n}(x)\vert.} \\ \end{array} \end{displaymath}


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DOI: https://doi.org/10.1090/S0002-9947-1977-0463406-2
Keywords: Extremal problems, inequalities for polynomials, Chebyshev's inequality, Markov's inequality
Article copyright: © Copyright 1977 American Mathematical Society

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