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Let $0<a_0<a_1<...<a_n$, $a_i\in \mathbb R$. I need to show that if $$a_0+a_1z+...+a_nz^n=0$$ then $|z|<1$. Any hints? I don’t know how to begin.

I can’t use Rouche’s theorem as it wasn’t introduced in the classes yet, and the answer to the duplicated question shown only that $|z|\leq1$, but I don’t know how to rule out the $|z|=1$ case.

Maja Blumenstein
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    You want to use Rouche's Theorem of course, but it barely doesn't work. The trick is to slightly modify your function. The details can be found in the linked duplicate. – Brevan Ellefsen Oct 19 '19 at 21:12
  • @BrevanEllefsen Rouche’s theorem has not been introduced in the classes yet, so I hoped for a more elementary approach – Maja Blumenstein Oct 19 '19 at 22:03
  • then please edit the post with more details, any attempts you've made, etc and appeal to have it re-opened. – Brevan Ellefsen Oct 20 '19 at 03:12
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    To rule out $|z|=1$ note that equality happens in the triangle inequality only if the respective complex numbers have all the same argument ($|a_1+a_2+... a_k| = |a_1|+..|a_k|$ means $a_k=r_ke^{i\alpha}$) for same $\alpha$, so checking the linked answer you will note that you need a root $z$ on the unit circle to have same argument as $a_0$, so it would have to be $1$ but plainly $1$ is not a root of the original equation – Conrad Oct 20 '19 at 13:02

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