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Question Statement-

If $z$ is a complex number and $x_0, x_1,..., x_{2n}$ are real then prove that all the roots of the equation of $$(sinx_0)z^{2n}+(cosx_1)z^{2n-1}+(sinx_2)z^{2n-2}+...+(cosx_{2n-1})z+sinx_{2n}=b$$ lie in the region $$|z|>1-\frac{1}{|b|}$$ My working- I tried using the conventional methods of vieta's relations and taking conjugates and somehow arrive on some conclusions. Then I tried proving that maybe all the roots have to be real or all the root's complex conjugate must also be a root, as only possible complex coefficient is $b$. I have notice that the inequality essentially means we have to prove $|z|>1$ and also all the sin and cos functions essentially imply that all coefficients are less than or equal to 1 in magnitude. But I haven't gotten much out of it. Would love some hints!

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2 Answers 2

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Hint: assume for the same of contradiction that $z$ is a root of the polynomial equation and $|z| \le 1-1/|b|$. Find an upper bound for the left-hand side using the triangle inequality and the sum of a geometric series.

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  • $\begingroup$ Thanks! I tried using proof by contradiction by same claim but couldn't see how, thanks again for the help. $\endgroup$
    – bm27
    Feb 21 at 7:38
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The $\cos$ and $\sin$ here are not relevant, we can simplify to: $$ \sum_{n=0}^N a_n \ z^n = b, \ \mbox{with all} \ |a_n| \leq 1 $$ Then use that for $|z|\geq 1$ the claim it is trivially true and that for $|z|<1$ we can write: $$ |b| = \left|\sum_{i=0}^{2n} a_i \ z^i\right| \leq \left|\sum_{i=0}^{2n} |a_i| \cdot |z|^i\right| \leq \left|\sum_{i=0}^{2n} 1 \cdot |z|^i\right| < \left|\sum_{i=0}^\infty |z|^i\right| \leq \frac{1}{1-|z|}, $$ which gives the required result since there is one $<$ sign in the sequence.

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    $\begingroup$ This is basically what Greg hinted me to do, anyways thanks for the answer $\endgroup$
    – bm27
    Feb 21 at 9:27
  • $\begingroup$ Yes Greg's answer showed up too late on my side... $\endgroup$ Feb 21 at 9:36
  • $\begingroup$ yeah I think we were composing our answers simultaneously :) $\endgroup$ Feb 21 at 17:17

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