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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]$


Author: A. K. Varma
Journal: Proc. Amer. Math. Soc. 55 (1976), 305-309
DOI: https://doi.org/10.1090/S0002-9939-1976-0396878-7
MathSciNet review: 0396878
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Abstract | References | Additional Information

Abstract: Let $ {P_n}(x)$ be an algebraic polynomial of degree $ \leqslant n$ having all its zeros inside $ [ - 1, + 1]$; then we have

$\displaystyle \int_{ - 1}^1 {P_n^{'2}(x)dx > (n/2)\int_{ - 1}^1 {P_n^2(x)dx.} } $

The result is essentially best possible. Other related results are also proved.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0396878-7
Article copyright: © Copyright 1976 American Mathematical Society