Let $f(x)=x^5-x^4-x^3-x^2-x-c$, where $c>2$ is a real number. It is easy to prove that there exists a positive real root $\alpha>2$ of $f(x)$ and all the other roots are non real.

Also, by using Kakeya-Enestrom theorem, it is possible to prove that if $\beta$ is a non real root of $f(x)$, then $|\beta|\leq \alpha$.

However, I would like to prove that this inequality is strict, but I am not being able to prove this. Someone has some suggestion?

The same problem happens by defining $f_c(x)=x^k-x^{k-1}-\cdots - x- c$, where $c>2$. So, $f_c(x)$ has only one positive root, say $\alpha$, which must be dominant, i.e., if $\beta$ is another root, then $|\beta|<\alpha$.

Thanks in advance!