Generalizations of operator Shannon inequality based on Tsallis and Rényi relative entropies

Recently, we obtained relations among relative operator entropies of sequences. For example, for relative operator entropy S(A|B), Rényi relative operator entropy It(A|B) and Tsallis relative operator entropy Tt(A|B) of sequences of strictly positive operators A=(A1,…,An) and B=(B1,…,Bn) such that ∑i=1nAi=∑i=1nBi=I,S(A|B)⩽It(A|B)⩽Tt(A|B)⩽0 holds for 0