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 ε ₑ ( ε ₁ ,ε ₂ ) of a two-component composite medium with a fixed microstructure, considered as function of the ratio ε ₁ /ε ₂, 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 ε ₑ ( ε ₁ ,ε ₂ ) and about its series expansion. These functions are used to construct continued fraction representations for ε ₑ ( ε ₁ ,ε ₂ ) that can provide exact bounds for ε ₑ that incorporate that information. Explicit algorithmic constructions are developed for obtaining these bounds in the real as well as in the complex domain.