Examine the polynomial $$ x^{\tau+1}-\left(1+m\right)x^{\tau}+mx^{\tau-1}+\left(1-m\right)\alpha=0\,. $$ with positive parameters $\tau,\alpha$ and $m<1$, and denote $\left|x_{\max}\left(\tau,\alpha,m\right)\right|$ as the maximal magnitude of a root of this equation. For $\tau>1$, I observed numerically that this root lies on the unit circle, i.e. $$ \left|x_{\max}\left(\tau,\alpha,m\right)\right|=1 $$ if and only if $$ \frac{1}{\alpha}= a(m)(\tau-1)+1 $$ for some positive function $a(m)$ which increases with $m$. This numerical relation could be seen here
I wonder if this linear relationship could be proved (at least in some limit), and perhaps even if one can find a simple form for $a(m)$.
Comment: a simpler special case of this question was asked and answered here, but I couldn't extend the method used there to the general problem here (I get a more complicated expression which I do not know how to solve).
Thanks in advance!
