Hashin-Shtrikman bounds for conductivity ($a_1+\frac{3a_1(a_2-a_1)p_2}{3a_1+p_1(a_2-a_1)}\le a^{\ast }\le a_2+\frac{3a_2(a_1-a_2)p_1}{3a_2+p_2(a_1-a_2)}$)

These are the optimal volume-fraction bounds for three-dimensional statistically isotropic two-phase conductivity with $0<a_1<a_2$. Constant phase polarization trial fields reduce the variational principle to a scalar quadratic optimization. The upper trial functional is minimized; the lower one is maximized.