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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An analogue of some inequalities of P. Turán concerning algebraic polynomials having all zeros inside $ [-1,+{}1]$. II


Author: A. K. Varma
Journal: Proc. Amer. Math. Soc. 69 (1978), 25-33
MSC: Primary 26A82; Secondary 26A75
MathSciNet review: 0473124
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Abstract: Let $ {P_n}(x)$ be an algebraic polynomial of degree $ \leqslant n$ having all zeros inside $ [ - 1, + 1]$; then we have

$\displaystyle \int_{ - 1}^1 {P'}_n^2 (x)dx > \left( {\frac{n}{2} + \frac{3}{4} + \frac{3}{{4n}}} \right)\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

$\displaystyle \int_{ - 1}^1 {P'}_n^2 (x)dx \geqslant \left( {\frac{n}{2} + \frac{3}{4} + \frac{3}{{4(n - 1)}}} \right)\int_{ - 1}^1 {P_n^2(x)dx,} $

equality for $ {P_n}(x) = {(1 - {x^2})^m},n = 2m$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0473124-9
PII: S 0002-9939(1978)0473124-9
Article copyright: © Copyright 1978 American Mathematical Society