On generalized Csiszár-Kullback inequalities

Classical Csiszár-Kullback inequalities bound the L¹-distance of two probability densities in terms ot their relative (convex) entropies. Here we generalise such inequalities to not necessarily normalized and possibly non-positive L¹ functions. Also, we analyse the optimality of the derived Csiszár-Kullback type inequalities and show that they are in many important cases significantly sharper than the classical ones (in terms of the functional dependence of the L¹ bound on the relative entropy). Moreover our construction of these bounds is rather elementary.