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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 A = (A₁,...,Aₙ) and B = (B₁,...,Bₙ) 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 setting $S_q^f\left(A\right|B):={\sum }_{j=1}^n{A}_j^{1/2}{\left(A_j^{-1/2}{B}_j{A}_j^{-1/2}\right)}^{q}f\left(A_j^{-1/2}{B}_j{A}_j^{-1/2}\right)A_j^{1/2}$, and then give upper and lower bounds for $S_q^f\left(A\right|B)$ as an extension of an inequality due to T. Furuta [Linear Algebra Appl. 381 (2004), 219-235] under...