TITLE: How prove this polynomials inequality
QUESTION [5 upvotes]: Let $f=a_0x^n+a_1x^{n-1}+\ldots+a_{n-1}x+a_n\in \mathbb{R}[X]$ a polynomial which has the roots contained in $(-1,1)$. Prove that:
$$\left|\frac{a_1+a_3+a_5+\ldots}{a_0+a_2+a_4+\ldots}\right|<1$$
Thank you .

REPLY [10 votes]: Without loss of generality I will assume $a_0 > 0$.  Let $p = a_1 + a_3 + \dotsc$ and $q = a_0 + a_2 +\dotsc$.  If $n$ is even then
$$0 < f(-1) = q - p$$
$$0 < f(1) = q + p$$
and combining these equations we get $q > 0$ and $\left|\tfrac{p}{q}\right|<1$.  If $n$ is odd then
$$0 < -f(-1) = q - p$$
$$0 < f(1) = q + p$$
and again $q > 0$ and $\left|\tfrac{p}{q}\right| < 1$.