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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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On a problem of Turán about polynomials with curved majorants
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by Q. I. Rahman
Trans. Amer. Math. Soc. 163 (1972), 447-455
DOI: https://doi.org/10.1090/S0002-9947-1972-0294586-5

Addendum: Trans. Amer. Math. Soc. 168 (1972), 517-518.

Abstract:

Let $\phi (x) \geqq 0$ for $- 1 \leqq x \leqq 1$. For a fixed ${x_0}$ in $[ - 1,1]$ what can be said for $\max |{p’_n}({x_0})|$ if ${p_n}(x)$ belongs to the class ${P_\phi }$ of all polynomials of degree $n$ satisfying the inequality $|{p_n}(x)| \leqq \phi (x)$ for $- 1 \leqq x \leqq 1$? The case $\phi (x) = 1$ was considered by A. A. Markov and S. N. Bernstein. We investigate the problem when $\phi (x) = {(1 - {x^2})^{1/2}}$. We also study the case $\phi (x) = |x|$ and the subclass consisting of polynomials typically real in $|z| < 1$.
References
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Bibliographic Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 163 (1972), 447-455
  • MSC: Primary 26A75; Secondary 30A06
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0294586-5
  • MathSciNet review: 0294586