Isospin bounds for energy partition in ee and NN annihilation

In reactions such as $\text{e}$e → anything, $\text{N}$N → anything, the final state (neglecting secondary isospin violating decays) is customarily supposed to have isospin zero or one. We show that for such states the average fraction (X) of the energy carried away by neutral pions is bounded by $\text{X<}\text{(9+√41)}\text{20}\text{≈0.77}$, when I₃ = 0 (e.g., for ee, pp, nn) and by $\text{X>}\text{(11?√41)}\text{40}\text{≈0.115}$, when I₃=±1 (e.g., for pn or np).