On generalized Csiszar-Kullback inequalities

Classical Csiszar-Kullback inequalities bound the L-1-distance of two probability densities in terms of their relative (convex) entropies. Here we generalize such inequalities to not functions. Also, we analyse the optimality of the necessarily normalized and possibly non-positive L-1 derived Csiszar-Kullback type inequalities and show that they are in many important cases significantly sharper than the classical ones tin terms of the functional dependence of the L-1 bound on the relative entropy). Moreover our construction of these bounds is rather elementary. 2000 Mathematics: Subject Classification: 28A33, 28D20, 52A10, 52A40, 94A17, 46E30.