On the Maximum Entropy Properties of the Binomial Distribution

It is shown that the Binomial(n,p) distribution maximizes the entropy in the class of ultra-log-concave distributions of order n with fixed mean np. This result, which extends a theorem of Shepp and Olkin (1981), is analogous to that of Johnson (2007), who considers the Poisson case. The proof constructs a Markov chain whose limiting distribution is Binomial(n,p) and shows that the entropy never decreases along the iterations of this Markov chain.