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Some entropy comparison results are presented concerning compound distributions on nonnegative integers. The main result shows that, under a log-concavity assumption, two compound distributions are ordered in terms of Shannon entropy if both the ldquonumbers of claimsrdquo and the ldquoclaim sizesrdquo are ordered accordingly in the convex order. Several maximum/minimum entropy theorems follow as a consequence. Most importantly, two recent results of Johnson (2008) on maximum entropy characterizations of compound Poisson and compound binomial distributions are proved under fewer assumptions and with simpler arguments.