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

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 ${P_n}(x) = {((1 + x)/2)^n}$.