# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Some inequalities of algebraic polynomials having all zeros inside $[-1, 1]$HTML articles powered by AMS MathViewer

by A. K. Varma
Proc. Amer. Math. Soc. 88 (1983), 227-233 Request permission

## 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 $(n = 2m)$ $\int _{ - 1}^1 {{{({P_n}(x))}^2} \geqslant (n/2 + 3/4 + 3/4(n - 1))\int _{ - 1}^1 {P_n^2(x)dx} } ,$ 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 $\int _{ - 1}^1 {({{P’}_n}} (x){)^2}dx \geqslant \frac {n}{4} + \frac {1}{8} + \frac {1}{{8(2n - 1)}},$, where equality holds for ${P_n}(x) = {((1 + x)/2)^n}$.
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