Abstract
Upper and lower bounds are derived for the average potential energy and Helmholtz free energy of an electron gas with uniform positive background. In the ground-state limit, upper and lower bounds are given for the average kinetic energy, average potential energy, and total ground-state energy. Inequalities are derived for the static form factor $S(k)$ and wave-number-dependent dielectric function $\ensuremath{\epsilon}(k, 0)$, making use of exact sum rules for the Fourier-transformed density-density commutator and of the assumption that $S(k)\ensuremath{\le}1$. Comparison is made with the exact behavior of these quantities for small $k$. The sum rules are used to construct an approximate nonlinear integral equation for the ground-state static form factor of the electron gas.
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