Using the set of trial spinors$$\left\{ {N_i \left( {_{\hat \Omega _i u_i }^{u_i } } \right), i = 1, \ldots , N} \right\}$$ and the Dirac-Coulomb Hamiltonian (H_DC) we discuss the role of the minimax theorem in relativistic Hartree-Fock calculations. In principle, the minimax theorem guarantees the occurrence of an upper bound. We also consider a scaling of the functionsu_i and discuss the condition to derive the relativistic hypervirial theorem; the variational procedure represented by the condition serves as an example of the minimax technique. Single zeta calculations onH₂⁺,H₂ and He are analysed. The effect of enlarging the basis is investigated for the He atom. The “upper bound” obtained by usingcoherent basis spinors differs from the result of the (random) linear variation using the kinetically balanced basis set by an amount which is at most of orderc⁻⁴. Use of thecoherent basis set is advocated.