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Enestrom-Kakeya Theorem
Suppose that $0<a_0\leq a_1\leq\cdots\leq a_n$, then prove that the complex polynomial $$P_n(z)=a_0z^n+a_1z^{n-1}+\cdots+a_{n-1}z+a_n$$ cannot have a root in the unit disc, i.e., such that $|z|<1$.
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This question was asked and answered fairly recently here but I don't have a specific link that I can point to. – Dilip Sarwate Sep 05 '12 at 11:32
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See also http://math.stackexchange.com/questions/188039/showing-that-the-roots-of-a-polynomial-with-descending-positive-coefficients-lie?lq=1 – Hans Lundmark Sep 05 '12 at 11:37