I am trying to understand the proof of a bound for the roots of a polynomial by Kojima, which is roughly sketched by Howard Bell in https://www.jstor.org/stable/2313703?origin=crossref. The proof involves using the companion matrix of a polynomial $f(x)=a_n x^n + \cdots + a_0$ and applying Gershgorins cicle theorem to the columns of the matrix.
The theorem by Kojima states that for positive numbers $\lambda_1,\cdots,\lambda_n$ with $\lambda_1 + \lambda_2 + \cdots + \lambda_n = 1$ the zeros of $f(x)$ satisfy $$ |x|\leq \max\left(\left| \frac{1}{\lambda_j}\frac{a_{n-j}}{a_n} \right|^{\frac{1}{j}} \right). $$
My problem lies in proving the inequality $$ \left| \frac{a_0}{a_n \rho^{n-1}} \right| + \left| \frac{a_1}{a_n \rho^{n-2}} \right| + \cdots + \left| \frac{a_{n-1}}{a_n} \right| \leq (\lambda_1+\lambda_2+\cdots+\lambda_n)\rho = \rho, $$ where we chose $\rho$ to be $$ \rho = \max\left(\left| \frac{1}{\lambda_j}\frac{a_{n-j}}{a_n} \right|^{\frac{1}{j}} \right), \quad j=1,\cdots,n. $$ I have tried using induction starting with a polynomial of degree $n = 1$ but it got really messy and led me nowhere, since I'm not entirely sure how to work with the $\max$ expression in the induction step. Is there another, maybe simpler way?
Any advice on how to prove this inequality would be much appreciated! Thanks!
For $j=1, \ldots, n$ is $$ \left| \frac{1}{\lambda_j}\frac{a_{n-j}}{a_n} \right|^{\frac{1}{j}} \le \rho \implies \left| \frac{a_{n-j}}{a_n} \right| \le \lambda_j\rho^j \implies \left| \frac{a_{n-j}}{a_n \rho^{j-1}} \right| \le \lambda_j\rho $$ and adding these estimates gives the desired $$ \left| \frac{a_{n-1}}{a_n \rho^{0}} \right| + \left| \frac{a_{n-2}}{a_n \rho^{1}} \right| + \cdots + \left| \frac{a_{0}}{a_n \rho^{n-1}} \right| \leq (\lambda_1+\lambda_2+\cdots+\lambda_n)\rho \, . $$