The set ∩m (Rα,m) includes ( - λ, λ ). If we add any value for α outside the interval ( - λ, λ ) the conditions to apply Rouché’s theorem are not satisfied.
So, we need other criteria to find bounds of the roots of Rr,m for fixed m and r is outside the interval ( - λ, λ ), and this new criterion is strongly based on the coefficients of both Q(x) and P(x).
Rouché’s theorem gives an implication, not an equivalence. If the inequality is true then Rα,m has as much zeros in the unit disk as we want so if α is in the (-λ,λ) then surely α is in Am but Am may be larger than that, it need not be symmetric, and I do not know if it is an interval containing (-λ,λ) which is what I ask to be proved.