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The aim of this paper is to study the effective transport coefficients λ_e of macroscopically isotropic two-phase composites for the case, where dielectric coefficients λ₁ and λ₂ of components are real. As an input we take: (i) N coefficients of the power expansion of λ_e(x) at x = 0, where x = (λ₂/λ₁) - 1; (ii) the analytical property of λ_e(x), namely λ_e(-1) ≥ 0; (iii) the Schulgasser inequality λ_e(x) λ_e(y) = (λ₁)², y = -x / (x + 1). By starting from (i), (ii) and (iii), an infinite set of bounds on λ_e(x) has been established and compared with the corresponding ones reported in literature. As an example of illustration of the obtained results, the regular arrays of spheres has been investigated numerically.