Shannon like inequalities for f-connections of positive linear maps and positive operators

In this paper, we investigate a map (T,B)→TfT−(B), called an f-connection, induced by an operator convex (concave) function f, where T− denotes a reflexive generalized inverse of a positive linear map T between unital C⁎-algebras A and B of Hilbert space operators, and B∈B. Some special cases of f-connection with invertible T are related to geometric operator mean, relative operator entropy and Csiszár operator f-divergence. We formulate conditions under which the inequality f(1)I≤∑k=1nTkfTk−(Bk) holds, where Tk:A→B are positive linear maps and Bk∈RanTk. In particular, the following Shannon like inequality 0≤∑k=1nTkfTk−(Bk) is shown for an operator convex function f with f(1)=0. The obtained results are specified for Csiszár operator f-divergence. Some recent results by Isa et al. (2013) [15] are recovered.