Relative operator entropies and operator inequalities based on Young's inequality

The relative operator entropy is defined as S(A|B)=A^1/2(log A^-1/2BA^-1/2)A^1/2 and the Tsallis relative operator entropy as T_x(A|B)=A ♮_x B-Ax for strictly positive operators A and B on a Hilbert space. We extend these relative operator entropies to the n-th relative operator entropies S^[n](A|B) and T^[n]_x(A|B) based on the Taylor expansion. Furthermore, we generalize those entropies to the n-th residual relative operator entropy R^[n]_x,y(A|B). By using them, we introduce operator valued divergences which are extensions of the α-divergence. We construct new inequalities among those relative operator entropies and operator valued divergences. Those inequalities contain a refinement of Young's inequality as a special case.