The generalized Tsallis relative operator entropy via solidarity (Q2790773)

Let \(B(H)\) be the \(C^{*}\)-algebra of all bounded linear operators on a Hilbert space \(H\). The authors obtain \(\Phi(AsB)\leq \Phi(A)s\Phi(B)\) for positive \(A,B\in B(H)\), a positive linear map \(\Phi\) on \(B(H)\) and a solidarity \(s\). Moreover, the authors obtain a reverse inequality of this inequality. Since every operator mean is a solidarity, the first and second inequalities are generalizations of results due to \textit{T. Ando} [Linear Algebra Appl. 26, 203--241 (1979; Zbl 0495.15018)] and \textit{J. Mićić} et al. [Linear Algebra Appl. 318, No. 1--3, 87--107 (2000; Zbl 0971.47014)], respectively. As applications of them, the authors discuss concrete inequalities related to the generalized Tsallis relative operator entropy.