Hierarchies of Stieltjes functions and their application to the calculation of bounds for the dielectric constant of a two-component composite medium

The bulk effective dielectric constant of a two-component composite medium with a fixed microstructure, is a Stieltjes function with special analytical properties. It is shown how other related Stieltjes functions can be constructed by exploiting partial information that is available about the values of dielectric constant and about its series expansion. These functions are used to construct continued fraction representations that can provide exact bounds that incorporate that information. Explicit algorithmic constructions are developed for obtaining these bounds in the real as well as in the complex domain.