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
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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