Generalized Shannon inequalities based on Tsallis relative operator entropy

Tsallis relative operator entropy is defined and then its properties are given. Shannon inequality and its reverse one in Hilbert space operators derived by Furuta [Linear Algebra Apll. 381 (2004) 219] are extended in terms of the parameter of the Tsallis relative operator entropy. Moreover the generalized Tsallis relative operator entropy is introduced and then several operator inequalities are derived.