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Some inequalities of algebraic polynomials having all zeros inside $ [-1,\,1]$


Author: A. K. Varma
Journal: Proc. Amer. Math. Soc. 88 (1983), 227-233
MSC: Primary 41A17; Secondary 26C05
DOI: https://doi.org/10.1090/S0002-9939-1983-0695248-5
MathSciNet review: 695248
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Abstract: Let $ {H_n}$ be the set of all algebraic polynomials whose degree is $ n$ and whose zero are all real and lie inside $ [ - 1,1]$. Then for $ n$ even we have $ (n = 2m)$

$\displaystyle \int_{ - 1}^1 {{{({P_n}(x))}^2} \geqslant (n/2 + 3/4 + 3/4(n - 1))\int_{ - 1}^1 {P_n^2(x)dx} } ,$

where equality holds iff $ {P_n}(x) = {(1 - {x^2})^m}$. If $ n$ is an odd positive integer, a similar inequality is valid (see (1.6) below). In the case $ {P_n} \in {H_n}$ and subject to the condition $ {P_n}(1) = 1$, then

$\displaystyle \int_{ - 1}^1 {({{P'}_n}} (x){)^2}dx \geqslant \frac{n}{4} + \frac{1}{8} + \frac{1}{{8(2n - 1)}},$

, where equality holds for $ {P_n}(x) = {((1 + x)/2)^n}$.

References [Enhancements On Off] (What's this?)

  • [1] J. Eröd, Bizonyos polinomok maximumar ial, Mat. Fiz. Lapok 46 (1939), 58.
  • [2] J. Szabadös and A. K. Varma, Approximation theory. III (E. W. Cheney, ed.), Academic Press, New York, 1980, pp. 881-888.
  • [3] P. Turán, Über die Ableitund von Polynomen, Compositio Math. 7 (1939), 89-95. MR 0000228 (1:37b)
  • [4] A. K. Varma, An analogue of some inequalities of P. Turán concerning algebraic polynomials having all zeros inside $ [ - 1,1]$, Proc. Amer. Math. Soc. 69 (1978), 25-33. MR 0473124 (57:12802)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0695248-5
Article copyright: © Copyright 1983 American Mathematical Society

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