  Using a population dynamics inspired by an ensemble of growing cells, a set
of fluctuation theorems linking observables measured at the lineage and
population levels are derived. One of these relations implies specific
inequalities comparing the population doubling time with the mean generation
time at the lineage or population levels. While these inequalities have been
derived before for age controlled models with negligible mother-daughter
correlations, we show that they also hold for a broad class of size-controlled
models. We discuss the implications of this result for the interpretation of a
recent experiment in which the growth of bacteria strains has been probed at
the single cell level.
