enclosing roots of polynomial equations and their applications to iterative processes

We introduce a special class of real recurrentpolynomials f_n (n ≥ 1) of degree n,with unique positive roots s_n, which are decreasingas n increases. The first root s₁, as well asthe last one denoted by s_∞ are expressed in closedform, and enclose all s_n (n > 1).
This technique is also used to find weaker than before[Kantorovich and Akilov, 1982] sufficient convergence conditions for some popular iterative processes converging to solutions of equations.