The generalization of the virial theorem is discussed. The case where the potential energy is a sum of homogeneous functions of various degree is investigated. If the potential energy U is composed of a gravitational (or Coulomb) energy and an energy of the short-range repulsion of particles, then virial inequalities of the form 2-K + Ū < 0 are valid, where K is the kinetic energy. For classical systems of this type, but with a Hamiltonian relativistic in the momenta, the inequality 3Nθ < |Ū| holds, where N is the number of particles in the system, θ = kT, T is the temperature, and k is Boltzmann's constant. © 1979 Plenum Publishing Corporation.