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.)