Harmonic continuous-time branching moments

We show that the mean inverse populations of nondecreasing, square integrable, continuous-time branching processes decrease to zero like the inverse of their mean population if and only if the initial population k is greater than a first threshold m(1) >= 1. If, furthermore, k is greater than a second threshold m(2) >= m(1), the normalized mean inverse population is at most 1/(k - m(2)). We express m(1) and m(2) as explicit functionals of the reproducing distribution, we discuss some analogues for discrete time branching processes and link these results to the behavior of random products involving i.i.d. nonnegative sums.