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Based on differences of generalized R\'enyi entropies nontrivial constraintson the shape of the distribution function of broadly distributed observablesare derived introducing a new parameter in order to quantify the deviation fromlognormality. As a test example the properties of the two--measure randomCantor set are calculated exactly and finally using the results of numericalsimulations the distribution of the eigenvector components calculated in thecritical region of the lowest Landau--band is analyzed.Comment: LaTeX 4 pages, 3 EPS included, to appear in Europhysics Letter

Multifractality beyond the parabolic approximation: Deviations from the log-normal distribution at criticality in quantum Hall systems