A note on operator inequalities of Tsallis relative operator entropy

Abstract

Tsallis relative operator entropy was defined as a parametric extension of relative operator entropy and the generalized Shannon inequalities were shown in the previous paper. After the review of some fundamental properties of Tsallis relative operator entropy, some operator inequalities related to Tsallis relative operator entropy are shown in the present paper. Our inequalities give the upper and lower bounds of Tsallis relative operator entropy. The operator equality on Tsallis relative operator entropy is also shown by considering the tensor product. This relation generalizes the pseudoadditivity for Tsallis entropy. As a corollary of our operator equality derived from the tensor product manipulation, we show several operator inequalities including the superadditivity and the subadditivity for Tsallis relative operator entropy. Our results are generalizations of the superadditivity and the subadditivity for Tsallis entropy.

New sharp inequalities for operator means

“…Direct calculations show that f 2.5 (7) > 0 while f 5 (8) < 0, which means that neither (2.5) nor (2.7) is uniformly better than the other. However, this entails the following refinement of (2.5)…”

“…As an application of Proposition 2.1, we estimate the bounds of Tsallis relative operator entropy defined by [7]:…”

Some New Karamata Type Inequalities and Their Applications to Some Entropies

“…Here S r (X||Y ) ≡ X♮r Y −X r = X 1/2 ln r (X −1/2 Y X −1/2 )X 1/2 defined for r ∈ R and a self-adjoint positive operators X and Y , is called Tsallis relative operator entropy. See [8,9], for example.…”

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“…Extending a work of Fujii [6] for relative operator entropy, Furuichi et al [9,Proposition 2.3] obtained:…”