Monotonicity versions of Epstein's Concavity Theorem and related inequalities

Many trace inequalities can be expressed either as concavity/convexity theorems or monotonicity theorems. A classic example is the joint convexity of quantum relative entropy which equivalent to Data Processing Inequality. The latter says that operations never increase entropy. versions often have many advantages, and direct physical application, in just mentioned. Moreover, results are valid for a larger class maps than, say, (which completely positive). In this paper we prove several new results, first theorem has simple corollary celebrated concavity Epstein. Our starting points Lieb Concavity Convexity Theorems. We also give two proofs these their general forms using interpolation. then our by duality arguments.