Lower bound for roots of a polynomial with positive coefficients.

Question

Suppose $\large{P(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_{n-1} x^{n-1} + x^n}$

where all the $\large{a_i}$ are strictly positive.

Find a sharp (ish?) lower bound for the smallest magnitude of all the roots in terms of the coefficients.

Can this be done? Or is it impossible, apart from special cases?

The case I am after is for strictly decreasing coefficients. $$a_0 > a_1 > a_2 > \cdots > a_{n-2} > a_{n-1}$$

Answer 1

In 1912, Kakeya has obtained following result:

The roots of the polynomial $$p(z) = a_0 + a_1 z + \cdots + a_{n-1} z^{n-1} + a_n z^n$$ with real and positive coefficients lie in the annulus $R_1 \le |z| \le R_2$ where $$R_1 = \min_{0\le j \le n-1} \frac{a_j}{a_{j+1}}\quad\text{and}\quad R_2 = \max_{0\le j \le n-1} \frac{a_j}{a_{j+1}}$$

For the problem at hand, one can take $a_n = 1$. This leads to following lower bound for the smallest magnitude of the roots: $$|z| \ge \min\left\{ \frac{a_0}{a_1}, \frac{a_1}{a_2}, \cdots, \frac{a_{n-2}}{a_{n-1}}, a_{n-1} \right\}$$ Other types of bounds are available, a good search key is the keyword "Eneström-Kakeya Theorem".

Update

I just notice there is an error in Kakeya's 1912 paper$\color{blue}{{}^{[1]}}$. It wrongly assert the inequalities are strict (i.e $R_1 < |z| < R_2$ instead of the correct version $R_1 \le |z| \le R_2$). When $R_1 < R_2$, it is possible for some roots lie on one (but not both) of the circles $|z| = R_1$ and $|z| = R_2$. For a sufficient and necessary condition for this to happen, please refer to Anderson's paper$\color{blue}{{}^{[2]}}$ below.

References

• $\color{blue}{[1]}$ Kakeya, S., On the Limits of the Roots of an Algebraic Equation with Positive Coefficients, Tôhoku Mathematical Journal (First Series),2, 140–142 (1912–13).
  ( an online copy can be found here)

• $\color{blue}{[2]}$ N. Anderson, E. B. Saff, and R. S. Varga, On the Eneström-Kakeya theorem and its sharpness, Linear Algebra Appl. 28 (1979), 5-16.