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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Inequalities for polynomials on the unit interval
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by Q. I. Rahman and G. Schmeisser
Trans. Amer. Math. Soc. 231 (1977), 93-100
DOI: https://doi.org/10.1090/S0002-9947-1977-0463406-2

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 \[ \begin {array}{*{20}{c}} {|{a_n}| \leqslant {2^{n - 1}}{{\left ( {\cos \frac {\pi }{{4n}}} \right )}^{2n}}\max \limits _{ - 1 \leqslant x \leqslant 1} |{p_n}(x)|,} \\ {\max \limits _{ - 1 \leqslant x \leqslant 1} |{{p’}_n}(x)| \leqslant {{\left ( {n\cos \frac {\pi }{{4n}}} \right )}^2}\max \limits _{ - 1 \leqslant x \leqslant 1} |{p_n}(x)|.} \\ \end {array} \]
References
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Bibliographic Information
  • © Copyright 1977 American Mathematical Society
  • 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