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This paper is focused on derivations of data-processing and majorizationinequalities for $f$-divergences, and their applications in information theoryand statistics. For the accessibility of the material, the main results arefirst introduced without proofs, followed by exemplifications of the theoremswith further related analytical results, interpretations, andinformation-theoretic applications. One application refers to the performanceanalysis of list decoding with either fixed or variable list sizes; someearlier bounds on the list decoding error probability are reproduced in aunified way, and new bounds are obtained and exemplified numerically. Anotherapplication is related to a study of the quality of approximating a probabilitymass function, induced by the leaves of a Tunstall tree, by an equiprobabledistribution. The compression rates of finite-length Tunstall codes are furtheranalyzed for asserting their closeness to the Shannon entropy of a memorylessand stationary discrete source. Almost all the analysis is relegated to theappendices, which form a major part of this manuscript.

On Data-Processing and Majorization Inequalities for $f$-Divergences with Applications