Sum of square of coefficients in a polynomial when the modulus of roots are less than one

Question

Let $P(x) = x^p + a_1x^{p-1}+ a_2x^{p-2}+\dots + a_{p-1}x+ a_p$ with roots having a modulus less than 1.

Can we prove that \begin{equation} a_1^2 +a_2^2 + \dots +a_p^2 < 1? \end{equation}

Any hints to prove or disprove this claim.

Answer 1

This is obviously false. Take $P(x)=(x-0.8)^2=x^2-1.6x+0.64$, then $a_1^2+a_2^2>1$ because $|a_1|=1.6>1$.