To solve a problem almost identical to the one in this question I would like to locate the zeros of the function $$ Q(z) = \sum_{k=0}^{m-1}(m-k)z^k = \frac{m - (m+1)z + z^{m+1}}{{(z-1)}^2} $$ where $m$ is a positive integer. In particular, I want to show that $Q(z)=0 \implies |z|>R$ for some $R \geq 1$.
- Based on computer solutions it seems that the statement holds at least for $R=\frac{m+1}{m}$, perhaps even $R=\frac{m+2}{m}$, so I have mostly worked with discs of such radii while trying to prove the statement.
- I have tried using Rouché's theorem, but so far it has only given me bounds with $R<1$.
- Inspired by plots of the roots I considered the equation $Q(z-c)=0$ shifting the roots by some real number $c$ but it didn't seem to help.
- A possibly related observation that I have not proven is that the roots of $Q(z)$ sum up to $-2$.
The following theorem is due to Eneström (1893):
THEOREM. Let $p_n(z) = \sum_{i=0}^n a_iz^i$ be a polynomial with positive real coefficients ($a_i>0$ for all $0\le i \le n$). Then all the zeros of $p_n(z)$ are contained in the annulus $$\alpha\le |z|\le \beta, $$ where $$ \alpha=\min_{0\le i<n}\frac{a_i}{a_{i+1}};\quad \beta=\max_{0\le i<n}\frac{a_i}{a_{i+1}}. $$
Apply this to your polynomial in its original form.