Operator monotones, the reduction criterion and the relative entropy
  We introduce the theory of operator monotone functions and employ it to derive a new inequality relating the quantum relative entropy and the quantum conditional entropy. We present applications of this new inequality and in particular we prove a new lower bound on the relative entropy of entanglement and other properties of entanglement measures. 
M.B. Plenio, S. Virmani, P. Papadopoulos