Explicit solution of the degree-eight extremum equation

Derive an explicit solution of the degree-eight polynomial equation obtained when computing the supremum of the lower-bound function for the weighted shift operator T(1,sq,q^2,tq^3,q^4,sq^5,q^6,tq^7,\ldots), for arbitrary parameters s,t>0 and 0<q<1.

Background

The paper studies bounds for the numerical radius of the weighted shift operator with weights (1,sq,q^2,tq^3,q^4,sq^5,q^6,tq^7,\ldots), where s,t>0 and 0<q<1. The lower bound involves maximizing the function ((1-z)\sqrt{z}/(1-q^4z^4))((1+sqz)+q^2z^2(1+tqz)) over 0<z<1.

The authors state that finding the maximizing point requires solving a polynomial equation of degree eight. They explicitly identify the lack of an explicit solution for arbitrary s and t as an open question, although the paper subsequently derives the lower bound without solving that equation in closed form.