  Discrete convex Sobolev inequalities and Beckner inequalities are derived for
time-continuous Markov chains on finite state spaces. Beckner inequalities
interpolate between the modified logarithmic Sobolev inequality and the
Poincar\'e inequality. Their proof is based on the Bakry-Emery approach and on
discrete Bochner-type inequalities established by Caputo, Dai Pra, and Posta
and recently extended by Fathi and Maas for logarithmic entropies. The abstract
result for convex entropies is applied to several Markov chains, including
birth-death processes, zero-range processes, Bernoulli-Laplace models, and
random transposition models, and to a finite-volume discretization of a
one-dimensional Fokker-Planck equation, applying results by Mielke.
