Stanisław Tokarzewski

Two-point Pade´ approximants for the expansions of Stieltjes functions in real domain

The fundamental inequalities for the sequences of subdiagonal and diagonal one-point Pade approximants to Stieltjes function has been extended to the case of certain two-point Pade approximants. The results can be applied to the theory of inhomogeneous media for calculating the bounds for the effective transport coefficients of two-components heterogeneous materials.

Continued fractions, two-point Padé approximants and errors in the Stieltjes case

A Stieltjes function is expanded in mixed T- and S-continued fraction. The relations between approximants of this continued fraction and two-point Pade approximants are established. The method used by Gilewicz and Magnus (J. Comput. Appl. Math. 49 (1993) 79; Integral Transforms Special Functions 1 (1993) 9) has been adapted to obtain the exact relations between the errors of the contiguous two-point Pade approximants in the whole cut complex plane.

A Contribution to the Bounds on Real Effective Moduli of Two-Phase Composite Materials

Effective transport coefficients of two-phase composite materials λe(x) can be represented by power expansions of four Stieltjes functions: λe(x)/λ1, λ2(x)/λe, λe(y)/λ2, λ1(y)/λe, where x = (λ2/λ1) - 1 and y = -x/(x + 1), while λ1 and λ2 denote the real moduli of a matrix and inclusions respectively.5 By constructing Pade approximants to power expansions of these functions, we derive an infinite set of fundamental inequalities identifying real-valued Miltons bounds20 as a lower and upper estimations of λe(x). From coefficients of a power expansion of λe(x) not exactly known, but only within the limits, the infinite set of new bounds on λe(x) has been derived. Due to Schulgasser inequality21 some improvement of existing bounds20 is proposed. For an illustration of the results achieved, the improved bounds on the effective conductivity λe(x) of a regular array of spheres are evaluated.