Yes. This is a part of my answer to another MO question,

http://mathoverflow.net/questions/134998/zeros-of-polynomials-with-real-positive-coefficients,

and a proof can be found here:
http://www.math.purdue.edu/~eremenko/dvi/saddle10.pdf

@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.