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Buchdahl, by imposing a few physical assumptions on the matter, i.e., itsdensity is a nonincreasing function of the radius and the fluid is a perfectfluid, and on the configuration, such as the exterior is the Schwarzschildsolution, found that the radius $r_0$ to mass $m$ ratio of a star would obeythe Buchdahl bound $r_0/m\geq9/4$. He noted that the bound was saturated by theSchwarzschild interior solution, the solution with $\rho_{\rm m}(r)= {\rmconstant}$, where $\rho_{\rm m}(r)$ is the energy density of the matter at $r$,when the central central pressure blows to infinity. Generalizations of thisbound have been studied. One generalization was given by Andr\'easson byincluding electrically charged matter and imposing that $p+2p_T \leq\rho_{\rmm}$, where $p$ is the radial pressure and $p_T$ the tangential pressure. Hisbound is given by $r_0/m\geq9/\left(1+\sqrt{1+3\,q^2/r_0^2}\right)^{2}$, theBuchdahl-Andr\'easson bound, with $q$ being the star's total electric charge.Following Andr\'easson's proof, the configuration that saturates the Buchdahlbound is an uncharged shell, rather than the Schwarzschild interior solution.By extension, the configurations that saturate the Buchdahl-Andr\'easson boundare charged shells. One expects then, in turn, that there should exist anelectrically charged equivalent to the interior Schwarzschild limit. We findhere that this equivalent is provided by the equation $\rho_{\rm m}(r) +{Q^2(r)}/ {\left(8\pi\,r^4\right)}= {\rm constant}$, where $Q(r)$ is theelectric charge at $r$. This equation was put forward by Cooperstock and de laCruz, and Florides, and realized in Guilfoyle's stars. When the centralpressure goes to infinity Guilfoyle's stars are configurations that saturatethe Buchdahl-Andr\'easson bound. It remains to find a proof in Buchdahl'smanner such that these configurations are the limiting configurations of thebound.

Sharp bounds on the radius of relativistic charged spheres: Guilfoyle's stars saturate the Buchdahl-Andr\'easson bound