All roots of the equation $a_0z^n+a_1z^{n-1}+.....+a_{n-1}z+a_n=n$,lie outside the circle with center at the origin and radius $\frac{n-1}{n}$.

Question

Show that all roots of the equation $a_0z^n+a_1z^{n-1}+.....+a_{n-1}z+a_n=n$,where $|a_i|\leq1,i=0,1,2,3...n$ lie outside the circle with center at the origin and radius $\frac{n-1}{n}$.

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It seems that we need to prove $|z|\geq 1-\frac{1}{n}$
But i am stuck and i have no idea how to start with.Please help.

Answer 1

If $|z|\le\dfrac{n-1}n$

$|\sum_{r=0}^na_rz^{n-r}|\le\sum_{r=0}^n|a_rz^{n-r}|\le\sum_{r=0}^n|z^{n-r}|$ as $|a_r|\le1$

Now, $\sum_{r=0}^n|z^{n-r}|=\sum_{u=0}^n|z^u|=\dfrac{1-|z|^{n+1}}{1-|z|}\leq n\left[1-\left(\dfrac{n-1}n\right)^{n+1}\right]<n$