  This paper is focused on derivations of data-processing and majorization
inequalities for $f$-divergences, and their applications in information theory
and statistics. For the accessibility of the material, the main results are
first introduced without proofs, followed by exemplifications of the theorems
with further related analytical results, interpretations, and
information-theoretic applications. One application 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, 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. Almost all the analysis is relegated to the
appendices, which form a major part of this manuscript.
