  We develop connections between Stein's approximation method, logarithmic
Sobolev and transport inequalities by introducing a new class of functional
inequalities involving the relative entropy, the Stein kernel, the relative
Fisher information and the Wasserstein distance with respect to a given
reference distribution on $\mathbb{R}^d$. For the Gaussian model, the results
improve upon the classical logarithmic Sobolev inequality and the Talagrand
quadratic transportation cost inequality. Further examples of illustrations
include multidimensional gamma distributions, beta distributions, as well as
families of log-concave densities. As a by-product, the new inequalities are
shown to be relevant towards convergence to equilibrium, concentration
inequalities and entropic convergence expressed in terms of the Stein kernel.
The tools rely on semigroup interpolation and bounds, in particular by means of
the iterated gradients of the Markov generator with invariant measure the
distribution under consideration. In a second part, motivated by the recent
investigation by Nourdin, Peccati and Swan on Wiener chaoses, we address the
issue of entropic bounds on multidimensional functionals $F$ with the Stein
kernel via a set of data on $F$ and its gradients rather than on the Fisher
information of the density. A natural framework for this investigation is given
by the Markov Triple structure $(E, \mu, \Gamma)$ in which abstract
Malliavin-type arguments may be developed and extend the Wiener chaos setting.
