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The Enestrom-Kakeya theorem states that all roots of the polynomial:

$p(z):=\sum_{k=0}^n a_kz^k$

lie outside the open unit disk if the sequence $(a_k)$ is positive and decreasing.

A proof can be found in Marden- Geometry of Polynomials.

I was wondering if the following idea of mine constitutes a valid proof as well:

$(1-z)p(z)=a_0+\sum_{k=1}^n (a_k-a_{k-1})z^k-a_nz^{n+1}$

so

$|(1-w)p(w)|\geq a_0-\sum_{k=1}^n (a_{k-1}-a_k)|w|^k -a_n > a_0-\sum_{k=1}^n (a_{k-1}-a_k) -a_n=0$

for all $|w|<1.$

(Using the reverse triangle inequality. The step at which strict inequality is introduced is only valid if some $a_{k-1}-a_k$ is nonzero. If this is not the case, our polynomial trivially has all roots on the unit circle.)

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    As a corollary of a more general result. Thank you, this does indeed seem to be the most straightforward proof.2012-08-23

0 Answers 0