N-point Padé approximants to real-valued Stieltjes series with nonzero radii of convergence

By employing special continued fractions to two Stieltjes series with nonzero radii of convergence we extend the inequalities for one-point Padé approximants reported by Baker (1975, Corollory 17.1) to the case of two-point Padé approximants. We prove that some convergents of the continued fractions form a monotone sequences of upper and lower bounds converging uniformly to Stieltjes function x1(x) on compact subsets of (−R, ∞), where R is a radius of convergence of an expansion of x f1(x) at x = 0. For an illustration of theoretical results we provide nontrivial numerical examples. As an application to real physical problems second order Padé approximantsʹ bounds on the effective conductivity of a square array of cylinders are evaluated.