  Based on differences of generalized R\'enyi entropies nontrivial constraints
on the shape of the distribution function of broadly distributed observables
are derived introducing a new parameter in order to quantify the deviation from
lognormality. As a test example the properties of the two--measure random
Cantor set are calculated exactly and finally using the results of numerical
simulations the distribution of the eigenvector components calculated in the
critical region of the lowest Landau--band is analyzed.
