Entropy inequalities for discrete channels

The sharp lower boundf(x)on the per-symbol output entropy for a given per-symbol input entropyxis determined for stationary discrete memoryless channels; it is the lower convex envelope of the boundg(x)for a single channel use. The bounds agree for all noiseless channels and all binary channels. However, for nonbinary channels,gis not generally convex so that the bounds differ. Such is the case for the Hamming channels that generalize the binary symmetric channels. The bounds are of interest in connection with multiple-user communication, as exemplified by Wyner's applications of "Mrs. Gerber's lemma" (the bound for binary symmetric channels first obtained by Wyner and Ziv). These applications extend from the binary symmetric case to the. Hamming case. Doubly stochastic channels are characterized by the property of never decreasing entropy.