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 $m_1\ge 1$. If, furthermore, $k$ is greater than a second threshold $m_2\ge m_1$, the normalized mean inverse population is at most $1\mathrm{/}(k-m_2)$. 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.