  We prove that in the context of general Markov semigroups Beckner
inequalities with constants separated from zero as $p\to 1^+$ are equivalent to
the modified log Sobolev inequality (previously only one implication was known
to hold in this generality). Further, by adapting an argument by Boucheron et
al. we derive Sobolev type moment estimates which hold under these functional
inequalities.
  We illustrate our results with applications to concentration of measure
estimates (also of higher order, beyond the case of Lipschitz functions) for
various stochastic models, including random permutations, zero-range processes,
strong Rayleigh measures, exponential random graphs, and geometric functionals
on the Poisson path space.
