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