TITLE: Stability of real polynomials with positive coefficients
QUESTION [20 upvotes]: Say that a polynomial in an indeterminate $x$ with real coefficients of degree $d$ has positive coefficients if each of the coefficients of $x^d,\ldots,x^1,x^0$ is (strictly) positive. 
For $f$ a monic univariate polynomial with real coefficients, if there exists a positive integer $m_0$ such that its $m_0$th power $f^{m_0}$ has positive coefficients, then does there exist a positive integer $m_1$ such that for each positive integer $m$ at least $m_1$, the $m$th power $f^m$ has positive coefficients?

REPLY [9 votes]: Yes. This is a part of my answer to another MO question,
Zeros of polynomials with real positive coefficients,
@David Handelman: thanks for the reference! I will insert it to my paper.
EDIT. In fact, Theorem 1 in the preprint cited above is not new. This was found as a
result of David's answer. The revised preprint is posted on the arxiv and here
http://www.math.purdue.edu/~eremenko/dvi/saddle13.pdf