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If $|a_k|<3,n\geq k \geq 1$, then prove that all complex numbers $z$ satisfying equation $1+a_1z+a_2z^2+....+a_nz^n=0$ lie outside the circle $|z|=1/4$. This is from my multiple choice test of complex numbers is there a a way to do this in minutes

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    I can't solve this without putting random values of z and n and try to match the option. – 2017-02-13

2 Answers 2

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Hint: It's actually nothing but an application of Rouche's theorem (see this question Finding number of roots of a polynomial in the unit disk ) for the function $g(z)=1$.

The bound is obtained via comparison with the geometric series and the answer obtained is that the claim holds.

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You can also just use simple arithmetic to get, for $|z|<1$, $$ |p(z)|\ge1-\max|a_k|(|z|+|z|^2+…+|z|^n)=1-\max|a_k|\frac{|z|(1-|z|^n)}{1-|z|} \\ \ge\frac{1-(1+\max|a_k|)|z|}{1-|z|} $$ So that inside the circle $$ |z|\le\frac1{1+\max_{k=1,..,n}|a_k|} $$ there can be no roots.

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    Of course.. ... studying Rouche's theorem and its applications was a waste of time after all – 2017-02-15
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    Often Rouche is the scalpel for individualized fine work where the hammer of the more general estimates fails. – 2017-02-15