  We study the harmonic moments of Galton-Watson processes, possibly non
homogeneous, with positive values. Good estimates of these are needed to
compute unbiased estimators for non canonical branching
  Markov processes, which occur, for instance, in the modeling of the
polymerase chain reaction. By convexity, the ratio of the harmonic mean to the
mean is at most 1. We prove that, for every square integrable branching
mechanisms, this ratio lies between 1-A/k and 1-B/k for every initial
population of size k greater than A. The positive constants A and B, such that
B is at most A, are explicit and depend only on the generation-by-generation
branching mechanisms. In particular, we do not use the distribution of the
limit of the classical martingale associated to the Galton-Watson process.
Thus, emphasis is put on non asymptotic bounds and on the dependence of the
harmonic mean upon the size of the initial population. In the Bernoulli case,
which is relevant for the modeling of the polymerase chain reaction, we prove
essentially optimal bounds that are valid for every initial population.
Finally, in the general case and for large enough initial populations, similar
techniques yield sharp estimates of the harmonic moments of higher degrees.
