How to show that it holds $|z|<2\max_{0\le k<n}|a_k|^{\frac{1}{n-k}}$ for any root of $X^n+\sum_{k=0}^{n-1}a_kX^k$?

Question

Let $z\in\mathbb{C}$ be a root of the complex polynomial $$f=X^n+\sum_{k=0}^{n-1}a_kX^k$$ I want to show that it holds $$|z|<2\max_{0\le k<n}|a_k|^{\frac{1}{n-k}}$$ Proof: For $s>1$, consider $$f_s:=\frac{1}{s^n}f(sX)=X^n+\sum_{k=0}^{n-1}\frac{a_k}{s^{n-k}}X^k$$ Evaluation of $f_s$ at $z$ yields \begin{equation} \begin{split} f_s(z) &= \sum_{k=0}^{n-1}\frac{a_k}{s^{n-k}}z^k-\sum_{k=0}^{n-1}a_kz^k \\ &=\sum_{k=0}^{n-1}\left(\frac{1}{s^{n-k}}-1\right)a_kz^k \end{split} \end{equation} Since $\sqrt[n-k]{2}\in (1,2]$ and $s>1$ it holds $$\frac{1}{\sqrt[n-k]{2}}<s\;\;\;\Leftrightarrow\;\;\;\frac{1}{2}<s^{n-k}\;\;\;\Leftrightarrow\;\;\;\frac{1}{s^{n-k}}-1<1$$ Thus, \begin{equation} \begin{split} |f_s(z)| &\le \sum_{k=0}^{n-1}\left|\frac{1}{s^{n-k}}-1\right||a_k||z|^k \\ &\le \sum_{k=0}^{n-1}|a_k||z|^k \\ &< \sum_{k=0}^{n-1}|a_k|M^k \end{split} \end{equation} where $$M:=1+\max_{0\le k<n}|a_k|$$ is an upper bound for the length of any root of $f$ by Cauchy's theorem. Now there comes the point where I'm unsure how I should proceed. I would be really thankful if someone could help me out.

Answer 1

By the triangle inequality we have

$$ |f(z)| \geq |z|^n - \sum_{k=0}^{n-1} |a_k| |z|^k, $$

so if $z$ is a root of $f$ then $|z| \leq r$, where $r$ is the unique positive root of the equation

$$ x^n - \sum_{k=0}^{n-1} |a_k| x^k = 0. $$

(Note that Descartes' rule of signs implies that the equation does indeed have a unique positive root.) Let

$$ M := 2 \max_{0 \le k < n} |a_k|^{1/(n-k)}. $$

For all $k$ we have

$$ 2^{n-k} |a_k| \leq M^{n-k}, $$

so

$$ \sum_{k=0}^{n-1} |a_k| M^k \leq M^n \sum_{k=0}^{n-1} 2^{k-n} = M^n (1-2^{-n}) < M^n. $$

Thus $M > r$, from which the result follows.