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The fundamental inequalities for two-point Padé approximants corresponding to two asymptotic expansions of the effective transport coefficients λ_e(x)/λ₁ , x =λ₂/λ₁ - 1 have been derived, where λ₁ and λ₂ denote the transport moduli of the composite components. The inequalities achieved constitute the new bounds on the values of λ_e(x)/λ₁ - the best with respect to the given number of coefficients of the asymptotic expansions of λ_e(x)/λ₁ at x = 0 and x = ∞. For the particular cases, our two-point Padé bounds reduce to the classical estimations of λ_e(x)/λ₁ available in literature [7, 9, 17, 24].