$p(x) = x^3 + ax +b$ show that $2b \le x(r_1^2+ r_2^2+r_3^2)$ if $x > \max (r_1, r_2,r_3)$

Question

Let $p(x) = x^3 + ax +b$ be a polynomial with real coefficients, and $r_1, r_2$ and $r_3$ its roots. Show that $2b \le x(r_1^2+ r_2^2+r_3^2)$ if $x > \max (r_1, r_2,r_3)$.

WLOG Let's assume $ r_1< r_2 <r_3$

We can immediately deduce $r_1+ r_2+r_3= 0$.

Also,$ r_1^2+ r_2^2+r_3^2 \ge 3 \sqrt [3] {r_1^2 r_2^2r_3^2} $

I thought of using more AM-GM but we would need the roots to be positive which it's not the case.

Can you help me out please, thanks.

Answer 1

Since, $b=-r_1r_2r_3$, $r_1=-r_2-r_3$ and $r_3\geq0$ by your work, it's enough to prove that: $$r_3(r_3^2+r_2^2+(r_3+r_2)^2)\geq 2r_3r_2(r_3+r_2)$$ or $$r_3^3\geq0,$$ which is obvious.