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Transactions of the American Mathematical Society

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On a problem of Turán about polynomials with curved majorants


Author: Q. I. Rahman
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
Addendum: Trans. Amer. Math. Soc. 168 (1972), 517-518.
MathSciNet review: 0294586
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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 \vert{p'_n}({x_0})\vert$ if $ {p_n}(x)$ belongs to the class $ {P_\phi }$ of all polynomials of degree $ n$ satisfying the inequality $ \vert{p_n}(x)\vert \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) = \vert x\vert$ and the subclass consisting of polynomials typically real in $ \vert z\vert < 1$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0294586-5
Keywords: Polynomials with curved majorants, Chebyshev polynomial of the first kind, Chebyshev polynomial of the second kind, polynomials typically real in $ \vert z\vert < 1$, GaussLucas theorem
Article copyright: © Copyright 1972 American Mathematical Society

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