  Integral representations of quantum relative entropy, and of the directional
second and higher order derivatives of von Neumann entropy, are established,
and used to give simple proofs of fundamental, known data processing
inequalities: the Holevo bound on the quantity of information transmitted by a
quantum communication channel, and, much more generally, the monotonicity of
quantum relative entropy under trace-preserving positive linear maps --
complete positivity of the map need not be assumed. The latter result was first
proved by M\"uller-Hermes and Reeb, based on work of Beigi. For a simple
application of such monotonicities, we consider any `divergence' that is
non-increasing under quantum measurements, such as the concavity of von Neumann
entropy, or various known quantum divergences. An elegant argument due to Hiai,
Ohya, and Tsukada is used to show that the infimum of such a `divergence' on
pairs of quantum states with prescribed trace distance is the same as the
corresponding infimum on pairs of binary classical states. Applications of the
new integral formulae to the general probabilistic model of information theory,
and a related integral formula for the classical R\'enyi divergence, are also
discussed.
