In this paper we investigate a notion, related to $f$-divergence functional, of relative operator entropy, which develops the theory started by J.I. Fujii and E. Kamei [Math. Japonica 34 (1989), 341--348]. For two finite sequences $\\\\mathbf{A}=(A_1,\\\\cdots,A_n)$ and $\\\\mathbf{B}=(B_1,\\\\cdots,B_n)$ of positive operators acting on a Hilbert space, a real number $q$ and an operator monotone function $f$ we extend the concept of entropy by $$ S_q^f(\\\\mathbf{A}|\\\\mathbf{B}):=\\\\sum_{j=1}^nA_j^{\\\\frac{1}{2}}\\\\left(A_j^{-\\\\frac{1}{2}}B_jA_j^{-\\\\frac{1}{2}}\\\\right)^qf\\\\left(A_j^{-\\\\frac{1}{2}}B_jA_j^{-\\\\frac{1}{2}}\\\\right)A_j^{\\\\frac{1}{2}}\\\\,, $$ and then give upper and lower bounds for $S_q^f(\\\\mathbf{A}|\\\\mathbf{B})$ as an extension of an inequality due to T. Furuta [Linear Algebra Appl. 381 (2004), 219--235] under certain conditions. Afterwards, some inequalities concerning the classical Shannon entropy are drawn from it.

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