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

Let \({\textbf{A}}=\{A_{1},...,A_{n}\}\) $\mathbf{A}=\{A_1,...,A_n\}$ and \({\textbf{B}}=\{B_{1},...,B_{n}\}\) $\mathbf{B}=\{B_1,...,B_n\}$ be two finite sequences of strictly positive operators on a Hilbert space \( {\mathcal {H}}\) $\mathcal{H}$ and f,  \(h:{\mathbb {I}}\rightarrow {\mathbb {R}}\) $h:\mathbb{I}\to \mathbb{R}$ continuous functions with \(h>0\) $h>0$ .. We consider the generalized Csiszár f-divergence operator mapping defined by \(\begin{aligned} {\textbf{I}}_{f\Delta h}({\textbf{A}},{\textbf{B}})=\sum _{i=1}^{n}P_{f\Delta h}(A_{i},B_{i}), \end{aligned}\) $$\begin{array}{c}\hfill {\mathbf{I}}_{f\mathrm{\Delta }h}(\mathbf{A},\mathbf{B})=\sum _{i=1}^n{P}_{f\mathrm{\Delta }h}(A_i,B_i),\end{array}$$ where \(\begin{aligned} P_{f\Delta h}(A,B):=h(A)^{1/2}f(h(A)^{-1/2}Bh(A)^{-1/2})h(A)^{1/2} \end{aligned}\) $$\begin{array}{c}\hfill P_{f\mathrm{\Delta }h}(A,B):=h{(A)}^{1/2}f(h{(A)}^{-1/2}Bh{(A)}^{-1/2})h{(A)}^{1/2}\end{array}$$ is introduced for every strictly positive operator A and every self-adjoint operator B, where the spectrum of the operators \(\begin{aligned} A, A^{-1/2}BA^{-1/2}\text { and }h(A)^{-1/2}Bh(A)^{-1/2} \end{aligned}\) $$\begin{array}{c}\hfill A,A^{-1/2}B{A}^{-1/2}\text{and}h{(A)}^{-1/2}Bh{(A)}^{-1/2}\end{array}$$ are contained in the closed interval \({\mathbb {I}}\) $\mathbb{I}$ . In this paper we obtain some lower and upper bounds for \({\textbf{I}}_{f\Delta h}({\textbf{A}},{\textbf{B}})\) ${\mathbf{I}}_{f\mathrm{\Delta }h}(\mathbf{A},\mathbf{B})$ with applications to the geometric operator mean and the relative operator entropy. We verify the information monotonicity for the Csisz ár f-divergence operator mapping and the generalized Csiszár f-divergence operator mapping.