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

The relative operator entropy is defined as S (A|B) =A¹/2 (log A^-1/2BA^-1/2) A¹/2 and the Tsallis relative operator entropy as Tₓ (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 Sⁿ (A|B) and Tⁿₓ (A|B) based on the Taylor expansion. Furthermore, we generalize those entropies to the n-th residual relative operator entropy Rⁿₓ, 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.