  This work provides data-processing and majorization inequalities for
$f$-divergences, and it considers some of their applications to coding
problems. This work also provides tight bounds on the R\'{e}nyi entropy of a
function of a discrete random variable with a finite number of possible values,
where the considered function is not one-to-one, and their derivation is based
on majorization and the Schur-concavity of the R\'{e}nyi entropy. One
application of the $f$-divergence inequalities refers to the performance
analysis of list decoding with either fixed or variable list sizes; some
earlier bounds on the list decoding error probability are reproduced in a
unified way, and new bounds are obtained and exemplified numerically. Another
application is related to a study of the quality of approximating a probability
mass function, which is induced by the leaves of a Tunstall tree, by an
equiprobable distribution. The compression rates of finite-length Tunstall
codes are further analyzed for asserting their closeness to the Shannon entropy
of a memoryless and stationary discrete source. In view of the tight bounds for
the R\'{e}nyi entropy and the work by Campbell, non-asymptotic bounds are
derived for lossless data compression of discrete memoryless sources.
