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

 

Some inequalities of algebraic polynomials having real zeros


Author: A. K. Varma
Journal: Proc. Amer. Math. Soc. 75 (1979), 243-250
MSC: Primary 26D05; Secondary 26D10
MathSciNet review: 532144
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Abstract: Let $ {P_n}(x)$ be an algebraic polynomial of degree n having all real zeros. We set

$\displaystyle {I_n} = \frac{{{{\left\Vert {{{P'}_n}(x)\omega (x)} \right\Vert}_{{L_2}[a,b]}}}}{{{{\left\Vert {{P_n}(x)\omega (x)} \right\Vert}_{{L_2}[a,b]}}}}.$

In this work the lower and upper bounds of $ {I_n}$ are investigated under the assumptions that all the zeros of $ {P_n}(x)$ are inside $ [a,b]$ and outside $ [a,b]$, respectively. We restrict ourselves here with two cases, (1) $ \omega (x) = {(1 - {x^2})^{1/2}},[a,b] = [ - 1,1]$; (2) $ \omega (x) = {e^{ - x/2}},[a,b] = [0,\infty )$. Results are shown to be best possible.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0532144-7
PII: S 0002-9939(1979)0532144-7
Article copyright: © Copyright 1979 American Mathematical Society