  Our aim is to provide a short and self contained synthesis which generalise
and unify various related and unrelated works involving what we call
Phi-Sobolev functional inequalities. Such inequalities related to Phi-entropies
can be seen in particular as an inclusive interpolation between Poincare and
Gross logarithmic Sobolev inequalities. In addition to the known material,
extensions are provided and improvements are given for some aspects. Stability
by tensor products, convolution, and bounded perturbations are addressed. We
show that under simple convexity assumptions on Phi, such inequalities hold in
a lot of situations, including hyper-contractive diffusions, uniformly strictly
log-concave measures, Wiener measure (paths space of Brownian Motion on
Riemannian Manifolds) and generic Poisson space (includes paths space of some
pure jumps Levy processes and related infinitely divisible laws). Proofs are
simple and relies essentially on convexity. We end up by a short parallel
inspired by the analogy with Boltzmann-Shannon entropy appearing in Kinetic
Gases and Information Theories.
