  In this paper we investigate a notion 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 $\mathbf{A}=(A_1,...,A_n)$ and
$\mathbf{B}=(B_1,...,B_n)$ 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 $$
S_q^f(\mathbf{A}|\mathbf{B}):=\sum_{j=1}^nA_j^{1/2}(A_j^{-1/2}B_jA_j^{-1/2})^qf(A_j^{-1/2}B_jA_j^{-1/2})A_j^{1/2}\,,
$$ 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.
