True or False #1

Let us assume that some root(s) of the given equation lie within a distance of $1$ unit from the origin. For, if there is(are) no such root(s) the given statement will be false.

So, check the given statement for $|z| < 1$

$a_{1}z^3+a_{2}z^2+a_{3}z+a_{4}=3$

$|a_{1}z^3+a_{2}z^2+a_{3}z+a_{4}|=|3|$

Using the property :

$|z_{1}|+|z_{2}|+|z_{3}|+\cdots+|z_{n}| \ge |z_{1}+z_{2}+z_{3}+\cdots+z_{n}|$

We get:

$|a_{1}||z^3|+|a_{2}||z^2|+|a_{3}||z|+|a_{4}| \ge 3$

Let $|z|=r$

Using $|a_{i}| \le 1, i=1,2,3,4$ and $|z|^n=|z^n|$

$3 \le 1+r+r^2+r^3 < 1+r+r^2+r^3+r^4+...$

$3 < \dfrac{1}{1-r}$

$3-3r <1 \implies r > \dfrac{2}{3}$

Hence , the given statement is false.

I want to construct a counterexample. If we want z=1 to be a root, we can set all the as to 3/4 and it works.