Bounds for kinetic and exchange energies of fermion systems

We show that the physical quantities of an A-fermion system expressed by means of integrals of powers of the single-particle density ρ(r) are bounded by any kth moment around the origin 〈$r^k$〉 of ρ and A in a simple manner. As applications, the kinetic and semiclassical exchange energies of a neutral atom (A=Z) turn out to be bounded from below by 0.2184$Z^{5/3}$〈$r^{\mathrm{-}1}$ ${〉}^2$ and from above by 0.3046$Z^{4/3}$〈r〉, respectively. Comparison of these bounds with the microscopic values of the corresponding quantities is made. Similar bounds for the same quantities in nuclei are also given and analyzed.