  We consider general N-particle wave functions that have the form of a product
of the Laughlin state with filling factor $1/\ell$ and an analytic function of
the N variables. This is the most general form of a wave function that can
arise through a perturbation of the Laughlin state by external potentials or
impurities, while staying in the lowest Landau level and maintaining the strong
correlations of the original state. We show that the perturbation can only
shift or lower the 1-particle density but nowhere increase it above a maximum
value. Consequences of this bound for the response of the Laughlin state to
external fields are discussed.
