  The inequality $m^3\geq \frac{27\pi}{4}
|\mathcal{J}_{2}||\mathcal{J}_{1}-\mathcal{J}_{2}|$ relating total mass and
angular momenta, is established for (possibly dynamical) spacetimes admitting
black holes of ring ($S^1\times S^2$) topology. This inequality is shown to be
sharp in the sense that it is saturated precisely for the extreme
Pomeransky-Sen'kov black ring solutions. The physical significance of this
inequality and its relation to new evidence of black ring instability, as well
as the standard picture of gravitational collapse, are discussed.
