Lower and Upper Bounds for the Generalized Csiszar f-divergence Operator Mapping

Let A = {A(1),...,A(n)} and B = {B-1,...,B-n} be two finite sequences of strictly positive operators on a Hilbert space H and f, h : I -> R continuous functions with h > 0.. We consider the generalized Csiszar f-divergence operator mapping defined by I-f Delta h(A,B)=Sigma P-n(i=1)f Delta h(A(i),B-i), where P-f Delta h(A,B) := h(A)(1/2)f(h(A)(-1/2)Bh(A()-1/2))h(A)(1/2) is introduced for every strictly positive operator A and every self-adjoint operator B, where the spectrum of the operators A,A(-1/2)BA(-1/2) and h(A)(-1/2)Bh(A)(-1/2) are contained in the closed interval I. In this paper we obtain some lower and upper bounds for If Delta h(A,B) with applications to the geometric operator mean and the relative operator entropy. We verify the information monotonicity for the Csisz ar f-divergence operator mapping and the generalized Csiszar f-divergence operator mapping.