and then . The second term on the right hand side of Bogoliubov inequality can be calculated as

where . Thus, for system of N spins on a lattice of coordination number z, is given by

where the factor of 1/2 is to take care of double counting of pairs over the whole lattice. Then by Bogoliubov inequality one has

Taking derivative of the right hand side with respect to λ and setting it to zero we could determine λ as,

The derivative to become zero, should be zero. Therefore, and hence the magnetization is given by

the same mean field equation as obtained by Weiss. Note that, tanh is due to Ising spins (two states problem) and nothing to do with mean-field approximation.

If , is the mean field. The mean field free energy is then given by