Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
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 all zeros inside $ [-1,\,1]$


Author: A. K. Varma
Journal: Proc. Amer. Math. Soc. 88 (1983), 227-233
MSC: Primary 41A17; Secondary 26C05
MathSciNet review: 695248
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 41A17, 26C05

Retrieve articles in all journals with MSC: 41A17, 26C05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0695248-5
PII: S 0002-9939(1983)0695248-5
Article copyright: © Copyright 1983 American Mathematical Society