Roots less than 1 if at least one coefficient is greater than one

Question

I have this doubt. If you have this equation with $\alpha_i \in \mathbb R$ $$P(z)=1-\alpha_{1}z-\alpha_{2}z^{2}- \cdots - \alpha_{p}z^{p}=0$$ I believe that if there exist an $\alpha$ greater or equal to one, there exist a root less or equal to one in module. I know it is true if p is 1 or p is 2 but i don't know how to prove it in general, if it is true, of course, In fact, i know the conditions if p=2 for not root inside the unit circle are $$-1<\alpha_{2}<1 $$ $$\alpha_{2}-\alpha_{1}<1 $$ $$ \alpha_{2}+\alpha_{1}<1 $$

Answer 1

Your conjecture (if I understand it correctly) seems to be wrong, even for $p=2$. As an example, $$ P(z) = \frac{(z+ 1.5)^2}{2.25} = 1 + \frac{3}{2.25}z + \frac{1}{2.25}z^2 $$ has no root in the unit disk, but $\alpha_1 > 1$.

Remark: This answer applies to the initial version of the question where the polynomial was defined as $P(z)=1+\alpha_{1}z+\alpha_{2}z^{2}+ \cdots + \alpha_{p}z^{p}$.