Area bound for an axisymmetric Kerr merger

For initially well-separated Kerr black holes with no orbital angular momentum, exact axisymmetry gives $J_f=J_1+J_2$. Put $X=(A_1+A_2)/(8\pi )$. The black-hole area theorem and the irreducible mass identity implyIndeed, $X\ge |J_1|+|J_2|\ge |J_f|$, and $\frac{1}{2}(Y+J_f^2/Y)$ is nondecreasing for $Y\ge X$. Opposite spin signs lower $J_f^2$ at fixed initial areas and hence raise the allowed radiated-energy bound; this need not be achievable in an actual merger.