An analogue of some inequalities of P. Turán concerning algebraic polynomials having all zeros inside $[-1,+1]$
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- by A. K. Varma PDF
- Proc. Amer. Math. Soc. 55 (1976), 305-309 Request permission
Abstract:
Let ${P_n}(x)$ be an algebraic polynomial of degree $\leqslant n$ having all its zeros inside $[ - 1, + 1]$; then we have \[ \int _{ - 1}^1 {P_n^{’2}(x)dx > (n/2)\int _{ - 1}^1 {P_n^2(x)dx.} } \] The result is essentially best possible. Other related results are also proved.References
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S. N. Bernstein, Sur l’order de la meilleure approximation des fonctions continues par des polynômes de dégré donné, Mém. Acad. Belgique, 1912.
János Eröd, Bizonyos polinomok maximumának, Mat. Fiz. Lapok. 46(1939), 58-82 [see Zentralblatt 21(1940), p. 395].
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Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 55 (1976), 305-309
- DOI: https://doi.org/10.1090/S0002-9939-1976-0396878-7
- MathSciNet review: 0396878