Estimate of roots of polynomial with positive, decreasing coefficients

I am looking for guidance about the size of roots of a polynomial $\sum a_kx^k$ where the coefficients are positive and decreasing, $0<a_{k+1}<a_k$ for each $k$. My hope is that the roots (real or complex) of such a polynomial are always bigger than one in modulus. Could someone suggest a proof (or counterexample) of this? (I'm aware of the bound on the size of the roots $1+\max a_k$ for general polynomials, but am hoping for something tighter based on the positivity or decreasing nature of the coefficients.)

More generally, I'd be interested in a reference for results about bounds of roots of polynomials based solely on their coefficients.

1 Answer

You may wish to consider the following review and the references cited therein for the general question.

Reviewed Works:

Analytic Theory of Polynomials by Qazi Ibadur Rahman, Gerhard Schmeisser;

Complex Polynomials by Terry Sheil-Small

Review by: Kenneth B. Stolarsky The American Mathematical Monthly Vol. 112, No. 7 (Aug. - Sep., 2005), pp. 664-671

• I checked out the references. It looks like the Enestrom-Kakeya theorem,mentioned in all three that you listed, is the way to go. Many thanks for the pointers. – Rus May Sep 14 '15 at 16:29