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An Analogue of some Inequalities of P. Turán Concerning Algebraic Polynomials having all Zeros Inside $\lbrack -1, + 1 \rbrack$. II

A. K. Varma
Proceedings of the American Mathematical Society
Vol. 69, No. 1 (Apr., 1978), pp. 25-33
DOI: 10.2307/2043182
Stable URL: http://www.jstor.org/stable/2043182
Page Count: 9
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
An Analogue of some Inequalities of P. Turán Concerning Algebraic Polynomials having all Zeros Inside $\lbrack -1, + 1 \rbrack$. II
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Abstract

Let $P_n(x)$ be an algebraic polynomial of degree $\leqslant n$ having all zeros inside $\lbrack -1, + 1 \rbrack$; then we have $$\int^1_{-1} P_n^{'2}(x) dx > \bigg(\frac{n}{2} + \frac{3}{4} + \frac{3}{4n} \bigg) \int^1_{-1} P_n^2(x)dx.$$ This bound is much sharper than found in [2]. Moreover, if $P_n(1) = P_n(-1) = 0$, then under the above conditions we have $$\int^1_{-1} P_n^{'2}(x)dx \geqslant \bigg(\frac{n}{2} + \frac{3}{4} + \frac{3}{4(n - 1)} \bigg) \int^1_{-1} P_n^2(x)dx,$$ equality for $P_n(x) = (1 - x^2)^m, n = 2m$.

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